REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
JULY 2018
MATHEMATICS
REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION
JULY 2018
MATHEMATICS
1. Joseph gave ¼ of his candies to Joy and Joy gave 1/5 of what she got to
Tim. If Tim received 2 candies, how many candies did Joseph have
originally?
A. 30
B. 20
C. 50
D. 40
SOLUTION:
1
𝐽𝑜𝑠𝑒𝑝ℎ 𝑤𝑎𝑠 𝑔𝑖𝑣𝑒𝑛 𝑡𝑜 𝐽𝑜𝑦
4
1
𝐽𝑜𝑦 𝑤𝑎𝑠 𝑔𝑖𝑣𝑒𝑛 𝑡𝑜 𝑇𝑖𝑚
5
1
𝑇𝑖𝑚 = 𝐽𝑜𝑦
5
1 1
2 = ( 𝐽𝑜𝑠𝑒𝑝ℎ)
5 4
𝐽𝑜𝑠𝑒𝑝ℎ = 40
2. What conic section is described by the equation 4x2-y2+8x+4y=15?
A. parabola
B. hyperbola
C. circle
D. ellipse
3. Find the maximum area of a rectangle which can be inscribed in an ellipse
having the equation x2 + 4y2 = 4
A. 4
B. 3
C. 2
D. 5
SOLUTION:
4. If the general equation of the conic Ax2 + Bxy +Cy2 + Dx + Ey +F = 0. If B2 –
AC>0 the equation describes is _____________.
A. ellipse
B. hyperbola
C. parabola
D. circle
5. Determine the equation that expressed that G is proportional to x and
inversely proportional to C and z. Symbols a, b, and c are constants.
𝑐𝑘
A. G= 𝐺𝐺
𝑎
B. G = 𝑏𝑐
𝒄𝒌
C. G = 𝒛𝑪
𝑏𝑐
D. G = 𝑧𝐾
6. The chord passing through the focus of the parabola and is perpendicular to
its axis is termed as
A. axis
B. latus rectum
C. directrix D.translated axis
7. What’s the equation of the hyperbola with focus at (-3 -3√13 , 1) asymptotes
intersecting at (-3, 1) and one asymptotes passing thru the point (1, 7)?
A. 4x2- 9y2 + 54x + 8y - 247 = 0
C. 9x2- 4y2 + 54x + 8y - 247 = 0
B. 4x2+ 9y2 + 54x - 8y + 284 = 0
D. 9x2 + 9y2 + 54x - 8y + 284 = 0
8. Find the ratio of the sides of triangle if its sides form an arithmetic
progression and one of the angles is 90 degrees.
A. 4 : 5 : 6
B. 1 : 2 : 3
C. 3 : 4 : 5 D. 2 : 3 : 4
SOLUTION:
Let a = first term
d= common difference
(a-d) , a , (a+d)
By Pythagorean Theorem,
(a-d)2 + a2 = (a+d)2
a2- 2ad + d2 + a2 = a2 + 2ad + d2
a2-4ad = 0
a(a-4d) = 0
a= 0
a-4d = 0
a= 4d
(4d-d) , 4d , (4d+d)
3d, 4d, 5d
3 :4:5
9. The area enclosed by the ellipse 4x2 + 9y2 = 36 is revolved about the line x =
3, what is the volume generated?
A. 370.3
B. 360.1
C. 355.3
D. 365.1
SOLUTION:
(4x2 + 9y2 = 36) 1/36
x2/ 9 + y2/ 4 = 1
a= √9 = 3
b= √4 = 2
v = ac
= (𝜋𝑎𝑏)(2𝜋𝑎)
= 2𝜋2a2b
=2𝜋2(3)2(2)
v = 355.3
10. The polynomial x2 + 4x + 4 is the area of a square floor. What is the length
of its side?
A. x + 2
B. x – 2
C. x + 1
D. x – 1
SOLUTION:
A = x2 + 4x + 4
A = (x+2) (x+2) = (x+2)2
Asquare = s2
s = x+2
11. Given a conic section, if B2 – AC = 0, it is called?
A. circle
B. parabola
C. hyperbola
D. ellipse
12. Find the height of a right circular cylinder of maximum volume which can
be inscribed in a sphere of radius 10cm.
A. 11.55 cm
B. 14.55 cm
C. 12.55 cm
D. 18.55 cm
SOLUTION:
ℎ
R2= r2 + ( 2)2
ℎ
r2 = R2- (2)2
ℎ
r2 = 102- (4)2
v= 𝜋 r2 h
ℎ
v= 𝜋(102- (4)2)(h)
=100 𝜋h -
𝜋ℎ3
4
3𝜋ℎ2
dv = 100 𝜋 0 = 100 𝜋 100 𝜋 =
4
4
3𝜋ℎ2
4
3𝜋ℎ2
4
100 (3) = h2
400/ 3 = h2
400
h = √ 3 = 11.55 cm
13. The length of the latus rectum of the parabola y = 4px2 is:
A. 4p
B. 2p
C. p
D. -4p
SOLUTION:
y = 4px2
LR = 4a = 4p
14. The area bounded by the curve y2 = 12x and the line x = 3 is revolved
about the line x = 3. What is the volume generated?
A. 186
B. 179
C. 181
D. 184
SOLUTION:
r= xr – xl
∫ 𝑑𝑣 = ∫ 𝜋𝑟3 dh
𝑦ˆ2
6
(3
−
) 𝑑𝑦
𝑣 = 𝜋∫
12
−6
v=
288𝜋
5
𝑜𝑟 𝟏𝟖𝟏
15. What is the length of the shortest line segment in the first quadrant drawn
tangent to the ellipse b2x2 + a2y2 = a2b2 and meeting to the coordinates axes?
A. a/b
B. a + b
C. ab
D. b/a
16. Find the radius of the circle inscribed in the triangle determined by the
lines y=x+4, y= -x-4 and y = 7x-2.
5
𝟓
A. √2
3
B. 𝟐√𝟐
3
C. √2
D. 2√2
SOLUTION:
Radius of Circle
y=x+4 ; y= -x-4 ; y = 7x-2
 Solve for 1st pt.,
y= x+4 ; x= -4-y
y= (-4-y) + 4
y=0
x= -4-0
x= -4
(-4, 0)
 Solve for 2nd pt.,
y=x+4
y+2
y=7x-2 ; x = 7
y=
y+2
+ 4 ; y=5
7
5+2
x=
7
+4 =1
(1, 5)
 Solve for 3rd pt.,
y= -x-4
y+2
y= 7x-2 ; x= 7
y+2
y= − (
x=
7
) − 4; y= -15 / 4
−15
+2
4
7
; x= -1/4
(-1/4 , -15 / 4 )
−4 1
𝑥1 𝑥2 𝑥3 𝑥1
1
A= 2 (
)=2(
𝑦1 𝑦2 𝑦3 𝑦1
0 5
1
1
= 2 |(−20 −
15
4
5
−1
4
−15
4
− 0) − (0 − 4 + 15)|
−4
)
0
A= 75/4
Find the perimeter:
 Side between (-4, 0) and (1,5)
d= √(−1 + 4)2 + (5 − 0)2 = 5√2
 Side between (-4, 0) and (-1/4 , -15 / 4 )
1
15
d= √(− 4 + 4)2 + (−
4
− 0)2 =
15√2
4
 Side between (1, 5) and (-1/4 , -15 / 4 )
1
15
d= √(− 4 − 1)2 + (−
P=5√2 +
15√2
4
+
4
− 5)2 =
25√2
4
25√2
4
= 15√2
One-half of the perimeters
=
15√2
2
Radius of inscribed circle in a triangle
=
75/4
15√2
2
𝟓
= 𝟐√𝟐
17. Find the moment of inertia of the area bounded by the parabola y2=4x and
the line x=1, with respect to the x-axis.
A. 2.133
B. 1.333
C. 3.333
D. 4.133
SOLUTION:
y2=4x, x=1
y = yR – yL
y = 1- y2 / 4
𝑏
Ix = ∫𝑎 𝑟2dA
2
(1− y2 / 4)dy
=∫−2 𝑦2
𝑑𝐴
Ix = 32 / 15 or 2.133
18.
What is the unit vector which is orthogonal both to 9i + 9j and 9i+9k?
𝑖
𝑗
𝑘
A. √3 + √3 + √3
𝑖
𝑗
𝑘
𝒊
𝒋
𝒌
𝑖
C. √𝟑 - √𝟑 − √𝟑
B. 3 + 3 + 3
𝑗
SOLUTION:
a=9i + 9j ; (i, j, k) ; (9, 9, 0)
b=9i+9k ; (i, j, k) ; (9, 0, 9)
By determinants,
𝑖 𝑗 𝑘
9
9 9 0= i(
0
9 0 9
0
9 0
9 9
)–j(
)+𝑘(
)
9
9 9
9 0
= i( 81 -0 ) – j ( 81- 0) + k (0-81)
= i( 81) – j ( 81) + k (-81)
Solving for modulus,
= √81ˆ2 + (−81)ˆ2 + 81ˆ2
=81 √3
The unit vector is,
1
=81 √3 (81i-81j-81k)
81𝑖
=81 √3 −
19.
81𝑗
81 √3
81𝑘
𝒊
𝒋
𝒌
− 81 √3 = √𝟑 - √𝟑 − √𝟑
Express in polar form: -3 -4i
4
C. √5eˆ-𝑖(𝜋 + tan−1 3)
4
D. √5eˆ𝒊(𝝅 + 𝐭𝐚𝐧−𝟏 𝟑)
A. 5eˆ-𝑖(𝜋 + tan−1 3)
B. 5eˆ𝑖(𝜋 + tan−1 3)
4
SOLUTION:
Mode 2, RAD
5𝑒 𝑖(−2.214)
−3 − 4𝑖 ≫ 𝑟 𝑐𝑖𝑠𝜃
𝑐𝑖𝑠 − 2.214
𝑟𝑒 𝜃𝑖
choose A
𝑘
D. 3 - 3 - 3
𝟒
20. The axis of the hyperbola through its foci is known as:
A. conjugate axis
B. transverse axis C. major axis
D. minor axis
21. Describe the locus represented by l z+2i l + l z-2i l = 6.
A. circle
B. parabola
C. ellipse
D. hyperbola
22. If the radius of the sphere is increased by a factor of 3, by what factor
does the volume of the sphere change?
A. 9
B. 18
C. 27
D. 54
SOLUTION:
V = 4/3 𝜋𝑟3 = k 𝑟3
r2 = 3r1
3
v2 / v1 = r2 / r13 = 33 r13 / 𝑟3 = 27
Evaluate the ∫(7x 3 − 4x 2 )dx.
23.
A.
B.
7x4
4
7x4
4
−
+
4𝑥 2
4𝑥 2
3
3
+𝐶
+𝐶
C.
D.
7x4
4
𝟕𝐱 𝟒
𝟒
+
−
4𝑥 3
3
𝟒𝒙𝟑
𝟑
+𝐶
+𝑪
24. Describe the locus represented by l z-3 l – l z+3 l = 4.
A. ellipse
B. circle
C. hyperbola
D. parabola
25. Melissa is 4 times as old as Jun. Pat is 5 years older than Melissa. If Jim
is y, how old is Pat?
A. 4y + 5
B. y + 5
C. 5y + 4
D. 4 + 5y
SOLUTION:
Melissa – 4y
Jim – y
Pat – 4y + 5
Therefore, Pat = 4y + 5
26. A conic section whose eccentricity, is less than one is known as:
A. a parabola B. an ellipse
C. a circle
D. a hyperbola
27. Two lines passing through the point (2,3) make an angle of 45 degrees
with each other. If the pipe of one of the lines is 2, find the slope of the other.
A. -2
B. -1
C. -3
D. 0
SOLUTION:
(2,3)
𝜃 = 45 m1= 2
Tan 𝜃
= m2 –m1 / 1+ m2 m1
Tan 45 = m2 –2 / 1+ m2 (2)
m2 = -3
28. From the top of a building the angle of depression of the foot of a pole is
48 deg 10 min. From the foot of a building the angle of elevation of the top of
a pole is 18 deg 50min. Both building and pole are on a level ground. If the
height of a pole is 4m, how high is the building?
A. 13.10m
B. 12.10m
C. 10.90m
D. 11.60m
SOLUTION:
Tan 𝜃
=y/x
x= y / Tan 𝜃
= 4 / tan 18°50’
x= 12.13
Tan 𝜃 = x / h
h = x / tan 𝜃
= 12.13 / tan 48° 10’
h = 10.90m
29. The locus of a point which moves so that the sum of its distances between
two fixed points is constant is called
A. ellipse
B. parabola
C. circle
D. hyperbola
30. Totoy is 5 feet 11 inches tall and Nancy is 6 feet 5 inches tall. How much
taller is Nancy than Totoy?
A. 1 foot 7 inches
B. 1 foot
C. 7 inches D. 6 inches
SOLUTION:
h2 = 5’ 11’’ = 5.917
h2 = 6’ 5’’ = 6.417
= h2- h2
= 6.417 - 5.917
12𝑖𝑛
= 0.5ft ( 1𝑓𝑡 )
= 6 inches
31. If log64 x = 3/2, find x.
A. 512
B. 521
SOLUTION:
3
log64 x = 2
𝑙𝑜𝑔𝑥
𝑙𝑜𝑔64
C. 253
D. 258
3
logx = 2 log64 ; x = 64ˆ3/2= 512
3
=2
32. What is the product of -9p3r and 2p-3r?
A. 18p4r + 27p6r2
C. 18p2r + 27p2r3
B. -18p4r + 27p3r2
D. -18p2r + 27p2r3
SOLUTION:
= (-9p3r) (2p-3r)
= 18p4r + 27p3r2
33.
𝑥2
Evaluate ∫ √𝑥 2 +25dx , using trigonometric substitution x = 5 tan 𝜃.
𝟏
𝑨. 𝟑 (𝒙𝟐 + 𝟐𝟓)3/2 – 25(𝒙𝟐 + 𝟐𝟓)1/2 + C
1
𝐵. 3 (𝑥 2 + 25)3/2 + 25(𝑥 2 + 25)1/2 + C
𝐶.
𝐷.
25
3
25
3
(𝑥 2 + 25)3/2 – 25(𝑥 2 + 25)1/2 + C
(𝑥 2 + 25)3/2 + 25(𝑥 2 + 25)1/2 + C
SOLUTION:
= 125∫
(sin ˆ3 𝜃 / cosˆ3 𝜃) ( 1 / cosˆ2 𝜃 )
(1 /cos𝜃 )
(sin ˆ3 𝜃)
= 125∫ (cosˆ4 𝜃 ) d 𝜃
d𝜃
(sin ˆ3 𝜃)
= 125∫ (cosˆ4 𝜃 ) sin 𝜃 d 𝜃
u = cos 𝜃 , du= - sin 𝜃 d 𝜃
= 125∫ −
(1−cos ˆ2 ) (− sin 𝜃 )d 𝜃
(cosˆ4 𝜃 )
−1+𝑢ˆ2
= 125∫
𝑢ˆ4
= 125 (
1
3𝑢ˆ3
-
d𝑢
1
𝑢
1
+ 𝐶)
1
= 125 (3𝑐𝑜𝑠ˆ3θ - 𝑐𝑜𝑠𝜃 + 𝐶)
1
= 125 (3 sec ˆ3 𝜃 − 𝑠𝑒𝑐𝜃 + 𝐶)
1
= 125 (3 (√𝑡𝑎𝑛𝜃 + 1) ˆ3 − √𝑡𝑎𝑛𝜃 + 1 + 𝐶)
1
= 3 5ˆ3 (√𝑡𝑎𝑛𝜃 + 1)ˆ3 − 25 (5) √𝑡𝑎𝑛𝜃 + 1 + 𝐶 )
=
34.
𝟏
𝟑
(𝒙𝟐 + 𝟐𝟓)3/2 – 25(𝒙𝟐 + 𝟐𝟓)1/2 + C
Michael’s favorite cake recipe calls for 0.75 pounds of flour, he has a 5
pound bag. He wants to make several cakes for the school bake sale. How
many cakes can he make?
A. 5
B. 6
C. 7
D. 8
SOLUTION:
Five pounds of flour divided by .75 equals
= 6.6666
Michael can make 6 cakes.
35.
Find the minimum amount of tin sheet that can be made into a closed
cylinder having a volume of 108 cu. Inches in square inches.
A. 125
SOLUTION:
V = 108 cu. in,
B. 137
C. 150
D. 120
V = 𝜋𝑟 2 h
h=r
𝑣 = 𝜋𝑟 3
100 = 𝜋𝑟 3
3
100
r= √
𝜋
= 3.17 𝑖𝑛.
AT = 2𝜋𝑟ℎ + 2𝜋𝑟 2
= 2𝜋𝑟 2 + 2𝜋𝑟 2
= 2𝜋(3.17)2 + 2𝜋(3.17)2
= 126. 28 𝑖𝑛.2 = 125 𝒊𝒏.𝟐
36.
A chord of a circle 10 ft. in diameter is increasing at the rate of 1 ft/s. Find
the rate of change on the smaller arc subtended by the chord when the cord
is 8 ft. long.
A. 5/2 ft/min.
37.
B. 2/5 ft/min.
C. 5/3 ft/min.
D. 3/5 ft/min.
Find the centroid of a semicircular area of radius a.
A. 2a/π
B. 4a/π
C. 2a/3π
SOLUTION:
By second proposition of Pappus
𝑉 = 𝐴 𝑥 2𝜋𝑑
𝐴=
1 2
𝜋𝑟
2
𝑉=
4 3
𝜋𝑟
3
𝑟=𝑎
D. 4a/3π
4 3
1
𝜋𝑎 = 𝜋𝑎2 𝑥 2𝜋𝑑
3
2
𝑑=
38.
4𝑎
3𝜋
An equilateral triangle with side “a” is revolved about its altitude. Find the
volume of the solid generated.
A. 0.32a3
B. 0.23a3
C. 0.41a3
D. 0.14a3
SOLUTION:
ℎ2 = 𝑎 2 −
1 2
𝑎
4
1
√3
ℎ = √𝑎2 − 𝑎2 =
𝑎
4
2
By second proposition of Pappus
𝐴=
1 𝑎
ℎ( )
2 2
𝐴=
√3 2
𝑎
8
𝑉 = 𝐴 𝑥 2𝜋𝑑
𝑉=
1
√3 2
𝑎 𝑥 2𝜋 𝑥 𝑎
8
3
𝑉 = .23𝑎3
39.
If the area bounded by the parabolas y=x2-C2 and y=C2-x2 is 576 square
units, find the value of C.
A. 5
40.
B. 6
C. 7
Solve y”-5y’+4y = sin 3x.
1
A. y= 25 (3 cos 3𝑥 − sin 3𝑥) + 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 4𝑥
1
B. y= 25 (3 sin 3𝑥 − cos 3𝑥) + 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 4𝑥
D. 8
𝟏
C. y= 𝟓𝟎 (𝟑 𝐜𝐨𝐬 𝟑𝒙 − 𝐬𝐢𝐧 𝟑𝒙) + 𝑪𝟏 𝒆𝒙 + 𝑪𝟐 𝒆𝟒𝒙
1
D. y= 50 (3 sin 3𝑥 − cos 3𝑥) + 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 4𝑥
41.
A car is travelling at a rate of 36 m/s towards a statue of height 6m. What
is the rate of change of a distance of the car towards the top of the statue
when it is 8m from the statue?
A. 32.4 m/s
B. 39.6 m/s
C. 26.6 m/s
D. 28.8 m/s
SOLUTION:
S2=s12 + 62
S2= (36t) 2 + 36
S2 = 1296t2 + 36
Differentiate
𝑑𝑠
2s𝑑𝑡 = 2592𝑡
𝑑𝑠 2592𝑡
=
𝑑𝑡
2𝑠
@ S1 = 8m
8 = 36 t
t = 0.222 sec.
@ t = 0.222 sec.
S= √1296t 2 + 36
= √1296 (0.222)2 + 36
S= 9.99 m
Ds/ dt = 2592t / 2s
@ S= 9.99 m
@ t = 0.222 sec
dS/ dt = 2592 (0.222) / 2(9.99)
dS / dt = 28.8 m/s
42.
A fencing is limited to 20 ft. length. What is the maximum rectangular area
that can be fenced in using two perpendicular corner sides of an existing
wall?
A. 120
B. 100
C. 140
SOLUTION:
x+y=20
D. 190
y = 20-x
A = xy
Subs. Y
A = x (20-x)
A = 20x – x2
Differentiate:
𝑑𝐴
𝑑𝑥
= 20 − 2𝑥
0 = 20-2x
X = 10 ft.
y = 20-x
y = 20 - 10
y = 10 ft.
A = (10) (10)
A = 100 ft.2
43.
Evaluate Laplace transform of t cos kt.
A. s2/(s2+k2)2
C. (-s2+k2)/(s2+k2)2
B. k2/(s2+k2)2
D. (s2-k2)/(s2+k2)2
SOLUTION:
ℒ(t cos kt) =
44.
(𝐬𝟐−𝐤𝟐)
(𝐬𝟐+𝐤𝟐)𝟐
Carmela and Marian got summer jobs at the ice cream shop and were
supposed to work 15 hours per week each for 8 weeks. During that time
Marian was ill for one week and Carmela took her shifts. How many hours did
Carmela work during the 8 weeks?
A. 120
B. 135
C. 150
D. 185
SOLUTION:
Total hours in 8 weeks
15 ℎ𝑜𝑢𝑟𝑠
𝑥 8 𝑤𝑒𝑒𝑘𝑠 = 120 ℎ𝑜𝑢𝑟𝑠
𝑤𝑒𝑒𝑘
Total hours Carmela works when Marian was ill for 1 week
120 hours + 15 hours = 135 hours
45.
Manuelita had 75 stuffed animals. Her grandmother gave 15 of them to
her. What percentage of the stuffed animals did her grandmother give her?
A. 20%
B. 15%
C. 25%
D. 10%
SOLUTION:
75
=5
15
100% / 5 = 20 %
46.
Find the coordinates of an object that has been displaced from the point (-
4,9) by the vector 4i-5i.
A. (0,4)
B. (0,-4)
C. (4,0)
D. (-4,0)
SOLUTION:
P( -4, 9)
Vector (4i -5i) = P ( 4, -5)
X = -4 + 4 = 0
X = 9 + (-5) = 4
P (0, 4)
47.
A triangle has two congruent sides and the measured of one angle 40
degrees. Which of the following types of triangle is it?
A. Isosceles
48.
C. right
D. scalene
The parabola defined by the equation 3y2+4x=0 opens ___________.
A. Upward B. downward
49.
B. equilateral
C. to the left
D. to the right
If a place on the earth is 12 degrees south of the equator, find its distance
in nautical miles from the North Pole.
A. 6,021
B. 6,102
C. 6,210
D. 6,120
SOLUTION:
R = 3959 Statute Miles
𝜋
Θ = 102° (180°) =
S = r𝜃 = (3959)(
S = 17047.95 SM (
17𝜋
30
17𝜋
30
5280 𝑓𝑡.
15 𝑀
)
1 𝑁𝑀
)(6080 𝑓𝑡.)
S = 6120 NM
50.
If the standard deviation of a population is 9, the population variance is.
A. 9
B. 3
C. 21
D. 81
SOLUTION:
σ=9
σ = √𝑣
v = σ2 = 9 2
v = 81
51.
Simply the equation 𝑠𝑖𝑛2 𝜃 (1 + 𝑐𝑜𝑠 2 𝜃).
B. sin2 𝜃
A. 1
52.
C. tan2 𝜃
D. cos2 𝜃
What is the complement of a 60 degree angle?
A. 120 degrees
B. 30 degrees
C. 40 degrees
D. 20 degrees
SOLUTION:
Complementary 𝜃 = 90 °
90° = 𝜃1 + 𝜃2
𝜃2 = 90°- 𝜃1 = 90° - 60°
𝜽2 = 30
53.
If 2xy-y2=3, find y”
A. 2/(x-y)4
B. -2/(x-y)4
C. 3/(x-y)3
D. 1/(2-x)
54.
The Rotary Club and the Jaycee Club had a joint party, 120 members of
the Rotary Club and 100 members of the Jaycees Club also attended but 30
of those attended are members of both clubs. How many persons attended
the party?
A. 220
B. 190
C. 150
D. 250
SOLUTION:
120 -x + x + 100 – x = 30
X = 190
55.
Two numbers have a harmonic mean of 9 and a geometric mean of 6.
Determine the arithmetic mean.
A. ¼
B. 4
C. 1/9
D. 9
SOLUTION:
HM = 9
GM = 6
GM2 = (HM)(AM)
AM =
56.
𝐺𝑀2
𝐻𝑀
=
62
9
=4
Find the force on one force of a right triangle of sides 4m and altitude of
3m. The altitude is submerged vertically with the 4m side in the surface.
A. 58.86 kN B. 62.64 kN
C. 53.22 kN
SOLUTION:
W(0) = 4 m
W(3) = 0
0−4
−4
=
3−0
3
4
W (h) = 4 - 3 ℎ
D. 66.67 kN
F = ∫ 𝛾𝐻2𝑂 ℎ 𝑤(ℎ) 𝑑ℎ
3
F = ∫0 (9810)(ℎ) (4 −
3
F = (9810) ∫0 (4ℎ −
57.
4
3
4
3
ℎ ) 𝑑ℎ
𝑥 2 ) 𝑑ℎ
= 58.86 kN
An airplane flying with the wind, took 2 hours to travel 1000 km and
2.5hours in flying back. What was the wind velocity in kph?
A. 40
B. 50
C. 60
D. 70
SOLUTION:
V1 – V2
t2 = 2.5 hours
V1 = Airplane Ve;ocity
V2 = Wind velocity
D = Vt
; V = D/t
@ flying with wind
V1 + V2 = 1000/2 = 500
@ flying bak
V1 – V2 = 1000/ 25
= 400
(V1 + V2) - (V1 - V2) = 500-400
V1 + V 2 - V1 + V 2
= 100
V2 = 50kph
58.
In how many ways can 6 people be lined up to get on a bus, if certain 3
persons insist on following each other?
A. 72
B. 144
C. 480
SOLUTION:
(4 !) (3 !) = 144
D. 120
59.
If 3x3y=27 and 2x+y=5, find x.
A. 3
B. 4
60.
C. 2
D. 1
Find the work done in moving an object along a vector a= 3i + 4i if the
force applied is b = 2i + i.
A. 8
B.9
C. 10
D. 12
SOLUTION:
d = a = 3i + 4i
F = b = 2i + i
W=Fxd
Using dot product
W = (a1)(b1) + (a2) (b2)
= (3)(2) + (4)(1)
W = 10
61.
If the line 3x-ky-8 = 0 passes through the point (-2,4), then k is equal to
A.-7/2
B. -5/2
C. -3/2
D. -1/2
SOLUTION:
3x –ky – 8 = 0
@ (-2,4)
k=?
3 (-2) – k (4) – 8 = 0
𝟕
k = −𝟐
62. What is the allowable error in measuring the edge of the cube that is
intended to
hold 8 cu. M. of the error of the computed volume is not to
exceed 0.03 cu. m?
A. 0.002
B. 0.003
C. 0.0025
D. 0.001
SOLUTION:
3
3
Edge = √𝑣 = √8
=2
dv = 3E2dE
𝑑𝑣
dE = 3𝐸2 =
0.03
(3)(2)2
dE = 0.0025
63. A man can do a job in 8 days. After the man has worked for 3 days, his son
joins him together they complete the job in 3 more days. How long will it take
the son to do job alone?
A. 12 days
B. 10 days
C. 13 days
D. 11 days
SOLUTION:
Let x = For son
Man = 1/8
Son = 1/x
1
1
1
3 (8) + 3 (𝑥 + 8) = 1
3
8x (8 +
3
𝑥
+
3
8
= 1) 8𝑥
3x + 24 + 3x = 8x
6X + 24 + 8X
X = 12 days
64. The probability that a randomly chosen safes prospects will make a
purchase is 0.18. If a salesman calls on 5 prospects, what is the probability
that the salesmen will make exactly 3 sales?
A. 0.0392
SOLUTION:
B. 0.0239
C. 0.0329
D. 0.0293
( 5 C3 ) ( 0.18 )3 (1 – 0.18 )2
X = 0.0392
5
65. If 𝑠𝑒𝑐 2 𝐴 = 2 , 𝑡ℎ𝑒𝑛 1 − 𝑠𝑖𝑛2 𝐴 =
A. 0.20
B. 0.30
C. 0.40
D. 0.50
SOLUTION:
5
𝑠𝑒𝑐 2 𝐴 = 2
------- 1
1 − 𝑠𝑖𝑛2 𝐴 = 𝑐𝑜𝑠 2 𝐴
--------- 2
𝑠𝑒𝑐 2 𝐴 + 𝑐𝑜𝑠 2 𝐴
cos 𝐴 =
=1
1
= 𝑐𝑜𝑠 2 𝐴 =
𝑠𝑒𝑐𝐴
1 − 𝑠𝑖𝑛2 𝐴 =
1
𝑠𝑒𝑐 2 𝐴
=
1
5
2
1
𝑠𝑒𝑐 2 𝐴
= 𝟎. 𝟒𝟎
66. What is the angle between the diagonal of a cube and one of its edges?
A. 44.74°
B. 54.74°
C. 64.74°
SOLUTION:
A (1, 1, 1)
B (0, 0, 1)
Cos𝜃 =
(𝑎)(𝑏)
𝑙𝑎𝑙𝑙𝑏𝑙
= cos −1(
(1,1,1)(0,0,1)
√3
)
𝜽 = 𝟓𝟒. 𝟕 °
67. The line 3x-4y=5 is perpendicular to the line
A. 3x-4y=1
C. 4x+3y=3
B. 4x-3y=1
D. 3x+4y=0
SOLUTION:
D. 74.74°
3x-4y=5
4y= 3x-5
𝑦=
𝑦=
3𝑥 − 5
4
3
20
(𝑥 − )
4
3
@ perpendicular
𝑚2 = −
1
−1
4
=
= −
3
𝑚1
3
4
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)
4
𝑦 − 𝑘 = − (𝑥 − ℎ)
3
3𝑦 − 3ℎ = −4𝑥 + 4ℎ
4𝑥 + 3𝑦 = (3𝑘 + 4ℎ)
𝟒𝒙 + 𝟑𝒚 = 𝟑
68.
If the plane 3x+2y-3x=0 is perpendicular to the plane 9x-3ky+y-t=0
A.2
B. -2
C. 3
D. -3
SOLUTION:
3x + 2y -3z = 0
9x – 3ky + 5zy =0
For parallel
𝐴 𝐹
=
𝐵 𝐺
3
9
=
2 −3𝑘
K= -2
69. A solid has a circular base of radius r. Find the volume of the solid if every
plane section perpendicular to z fixed diameter is in semicircle.
A.1.20r3
B. 2.09r3
C. 2.51r3
D. 4.10r3
70. Find the y-intercept of the line given by the equation 7x+4y=8
B. 2
B. -2
C. 3
D. -3
SOLUTION:
7x+4y=8
𝑦 = 𝑚𝑥 + 𝑏
4𝑦
4
=
8−7𝑥
4
𝑦= −
7𝑥
4
+2
b=2
71.
Find the area inside the cardioid r=1+cos ϴ and outside the circle r=1.
A.2.97
B. 2.79
C. 2.85
D. 2.58
SOLUTION:
𝜃2
∫ (𝑟 2
2 𝜃1
1
− 𝑟2 2 )𝑑𝜃 = 𝐴
𝑟 = 1 + 𝑐𝑜𝑠𝜃 , 𝑟 = 1
1 = 1 + cos 𝜃
𝜃= ±
𝜋
2
𝜋
1 2
∫ (1 + 𝑐𝑜𝑠𝜃)2 − (12 )𝑑𝜃 = 𝐴
2 −𝜋
2
𝐴 = 2.79 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠
72. A person had a rectangular-shaped garden with sides of lengths 16 feet and
9 feet. The garden was changed into square design with the same area as the
original rectangular-shaped garden. How many feet in length are each sides
of the new square-shape garden.
a. 7
B. 9
C. 12
D. 16
SOLUTION:
Δ =(16)(9)
= 144 sq.ft
= √144
= 12
73.
Which of the following rope length is longest?
a. 1 meter
B. 1 yard
C. 32 inches
D. 85 cm
74. Martin , a motel housekeeper, has finished cleaning about 40% of the 32
rooms he's been assigned. About how many more rooms does he have left to
clean?
a. 29
SOLUTION:
B. 25
C. 21
D. 19
Room left to clean = 60% (30)
= 19 Room
75. A horse tied to a post with twenty-foot rope. What is the longest path that
the horse can walk?
a. 20 feet
B. 40 feet
C. 62.83 feet
D.125.66feet
SOLUTION:
𝐶 = 2𝜋𝑟
𝐶 = 2𝜋(20)
𝐶 = 125.66 𝑓𝑡
76. Doming wants to know the height of a telephone pole. He measures his
shadow, which is 3 feet long , and the pole's shadow, whcih 10 feet long .
Domingo's height is 6 feet. How tall is the pole ?
a. 40 ft
B. 30 ft
C. 20 ft
SOLUTION:
By similar triangles
3
6
=
10
ℎ
D. 10 ft
ℎ = 20 𝑓𝑡
77. A weight of 60 pounds rests on the end of an 8-foot lever and is 3 feet from
the fulcrum. What weight must be placed on the other and of the lever to
balance the 60 pound weight.
a. 36pounds
B. 32pounds
C. 40pounds
D. 46pounds
SOLUTION:
5x =60(3) =180
X= 36
78. A number is 1 more than twice another. Their squares differ by 176. What
is the larger number?
a. 9
B. 7
C. 15
D. 16
SOLUTION:
X = larger number
𝑥 = 2𝑦 + 1
𝑥 2 − 𝑦 2 = 176
𝑥−1 2
𝑥 −(
) = 176
2
𝑥 = 15
2
79. The sides of a right triangle is in arithmetic progression whose common
difference is 6cm. Its area is
a. 216sq.cm
B. 270sq.cm
C. 360sq.cm
D. 144sq.cm
SOLUTION:
A.P.
(x)
(x+6)
(x+12) ^2 = x^2 + (x+6)^2
X^2+24x+144 = x^2 + x^2 +12x +36
(x+12) ---- hypo
C^2 = A^2 + B^2
X^2 – 12x – 108 = 0
X^2 – 18x + 6x – 108 = 0
X(x-18) + 6(x-18)=0
(x-18)(x+6)=0
X=-6 or 18
Area = ½ (18)(24) = 216
80. A tank has 100 liters of brine with 40 N dissolved salt. Pure water enters the
tank at the rate of 2 liters per minute abd the resulting mixture leaves the tank
at the same rate. When will the concentration in the tank be 0.20 N/L
a. 24.6min B. 34.7min
C. 44.8min
D. 54.9min
SOLUTION:
𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 =
𝑄
𝑉
. 2(100) = 𝑄
𝑄 = 20
𝑑𝑞
2𝑄
= 0(2) −
𝑑𝑡
100
20
−∫
40
𝑑𝑞
1 𝑡
=
∫ 𝑑𝑡
𝑄
50 0
. 693 =
1
(𝑡 − 0)
50
𝑡 = 34.7 𝑚𝑖𝑛
81. The base of an isosceles triangle is 20.4 and the base angles are 48°20'.
Find the altitude of the triangle.
a. 9.8
B. 10.8
C. 11.6
D.12.7
SOLUTION:
tan ϴ = h / 10.2
h = 10.2 tan 48°20'
h = 11.46
82. A lady gives a party dinner party for six guests. In how many ways may
they be selected from among 10 friends if two of the friends will not attend the
party together?
A.112
B. 128
C. 140
D. 160
83. A rubber ball is dropped from a height of 81m. Each time is strikes the
ground, it rebounds two-thirds of the distance through which it last fell. Find
total distance it travels in coming to rest?
A. 243m
B. 162m
C. 405m
D. 324m
SOLUTION:
Infite geometric Progression
𝑆𝑛 = 𝐴𝑛 +
𝐴𝑛
1−𝑟
h = 81 + 2 ( 81 (2/3) / 1- (2/3))
h = 405m
84. Find the diameter of a pulley which is driven at 360 rpm by a belt moving at
40ft/s.
A.2.12ft
B. 1.11ft
1.24ft
SOLUTION:
v = rw
r = v/w
r = 40 ft/s / (2)(π)(360/60)8 (R/S)
r = 1.06
d = 2r
d = 2(1.06)
C. 2.43ft
D.
d = 2.12ft
85. Find the volume generated by the circle x2+y2=25 if it is revolved about the
line 4x+3y=40.
A 3,498c.u.
B. 3,948c.u.
C. 4,624c.u.
D. 4,426c.u.
SOLUTION:
r = 5, c (0,0)
d = Ax + By + C / √A2 + B2
d = 4(0) + 3(0) - 40 / √42 + 32
d=8
2nd Prop. Pappus
V = 2π A (d)
V = 2π (π 52)(8)
V= 3,948 c.u.
86. A ranch has cattle and horses in a ratio of 9:5. If there are 80 more head of
cattle than horses, how many animals are on the ranch?
A.140
B. 168
C. 238
D. 280
SOLUTION:
x = 80 + y
x = cattle
5y = 9x
y = horses
x = 180
y = 100
total = 280
87. The first term of a geometric sequence is 375 and the fourth term is 192.
Find the common ratio.
A 5/4
B. 4/5
SOLUTION:
an = a1 rn-1
C. 3/2
D. 2/3
92 = 378 (r)4-1
r = 4/5
88. In how many ways can a person choose 1 or more of 4 electrical
appliances?
A.16
B. 15
C. 12
D. 20
SOLUTION:
Choose 1 or more
2n - 1 , 24 - 1 = 15
89. The probability that a certain man will be alive 25 years hence is 3/7, and
the probability that his wife will be alive 25years hence is 4/5. Determine the
probability that 25 years hence, only the man will be alive.
A 12/35
B. 4/35
C. 31/35
D. 3/35
SOLUTION:
P = ( Probability of man alive ) ( Probability of wife not alive )
P = 3/7 ( 1 - 4/5 )
P = 3/35
90. Find the point in the parabola y2=4x at which ratio of change of the ordinate
and abscissa are equal.
A .(1,2)
B. (1,-2)
C. (2,1)
1)
SOLUTION:
y2 = 4x
2ydy = 4dx
,
y=2,
( 2 )2 = 4x
x=1
(1,2)
91.
(1-2i)-1 can be written as
dy = dx
substitute to Eq.1
D. (2,-
A.1/5 + 2/5i
B. 1/5 - 2/5i
C. -1/3 - 2/3i
D. -1/3 + 2/3i
92. Find the y-intercept of the line tangent to the parabola x=2y2 at the point
(2,1).
A .-7
B. 7
C. 3/2
D. 1/2
SOLUTION:
y2 = x / 3
( 2,1 )
y' = 1/4
y - 1 = 1/4 ( x- 2 )
@ y intercept x = 0
y - 1 = 1/4 ( 0 - 2 )
y = -1/2
93.
A growth curve is given by A=10 e2t at what value of t is A=100?
A .5.261
B. 3.070
C. 1.151
D.
0.726
SOLUTION:
10 = e2t
ln 10 = 2t ln e
t = ln10 / 2
t = 1.151
94. If the short leg of a right triangle is 5 units long and the long leg is 7 units
long , find the angle opposite the short leg in degrees.
a. 26.3
B. 28.9
C. 31.2
D. 35.5
SOLUTION:
a=5 ,
b=7
c = √52 + 72 = √74
𝑎2 = 𝑏 2 + ( 𝑐)2 − 2𝑏𝑐𝑐𝑜𝑠𝜃
52 = 72 + (√74 )2 - 2(√74 )(7) cos ϴ
ϴ = 35. 5
95. Express 2 sin2 theta as a function of cos2 theta.
A. cos2ϴ-2 B.cos2ϴ+1 .
C. cos2ϴ+2
D.1-cos2ϴ
SOLUTION:
Using half angle formula
(2)
sin2ϴ = 1/2 ( 1 - cos ϴ )
(2)
sin2ϴ = ( 1 - cos ϴ )
96. The x and y axes are asymptotes of a hyperbola that passes through the
point (2,2). Its equation is
A.x2-y2=0
C. y2-x2=0
B. xy=4
D. x2+y2=4
SOLUTION:
Since x and y are asymptotes , hyperbola is 45°
xy = a2
xy = 4
97.
If the area of the equilateral triangle is 4√3 find the perimeter.
A .16
B. 12
C. 18
SOLUTION:
A = (√3 / 4) ( S2 )
4√3 = (√3 / 4 )( S2 )
S=4
P = 3S
P = (3)(4)
P = 12
98.
Find the area bounded by the curve r=8 cosϴ
A 50.27
B. 12.57
SOLUTION:
r = 8 cosϴ
C. 8
D. 67.02
D. 14
r = a cosϴ
A = π (a/2)2
A = π ( 8/2 )2
A = 50.27
99. What is the length of the transverse axis of the hyperbola whose equation
is 9y2-16x2=144?
A.6
B. 9
C. 8
D. 7
SOLUTION:
9y2-16x2=144
( y2/16 ) - ( x2 / 9 ) = 1
a2 = 16
a=4
Length Transverse
2a = (2)(4) = 8
100. Find the area bounded by x=2y-y2 and the y-axis.
A. 4/3
B.5/3
C. 2/3
SOLUTION:
∫ |2y -y2 | dy ,
upper limit = 2
, lower limit = 0
A = 4/3
D. 1/3
REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
MARCH 2018
MATHEMATICS
1. A tangent to a conic is line
A. which is parallel to the normal C. which passes inside the conic
B. which touches the conic at only one point D. All of the above
2. Simplify 1/(csc x+1) + 1/(csc x -1)
A. 2 sec x tan x
B. 2 csc x cot x
C. 2 sec x
D. 2 csc x
SOLUTION:
1/(csc x+1) + 1/(csc x -1) .csc²-1
csc x -1 + csc x +1 = 2 csc x
3. Find the coordinates of the centroid of the plane are bounded by the
parabola y= 4-x² and the x-axis
A. (0 , 1.5)
B. (0 , 1)
C. (0 , 2)
D. (0 , 1.6)
SOLUTION:
y=4-x² ; x²= -1(y-4)
at y=0 x=+/-2
lower limit =0 upper limit = 2
A=2/3bh = 2/3(4)(4)= 32/3
Ay= 2 ∫ ydx (y/2)
32/3 (y)= 2 ∫ (4-x²)²dx
y=1.6 x=0 therefore centroid is (0, 1.6)
4. Evaluate Γ(-3/2)
A. -2(sqrt of pi)/3
C. -4(sqrt of pi)/3
B. 2(sqrt of pi)/3
D. 4(sqrt of pi)/3
SOLUTION:
Γ-3/2 = Γ(-3/2 +1)/(-3/2) = Γ (-1/2)/(-3/2) = -2/3(-2√π) = 4/3√π
5. Ben is two years away from being twice Ellen’s age. The sum of twice Ben’s
age and Thrice Ellen’s age is 66. Find Ben’s age now.
A. 19
B. 20
C. 18
D.21
SOLUTION:
x + 2 = 2y ; x= 2y-2
2x + 3y = 66
2(2y-2) + 3y = 66
Y=10
X= 2(10) - 2 = 18
6. Find the area bounded by the outside the first curve and inside the second
curve r=5, r=10sin theta
A. 47.83
SOLUTION:
B. 34.68
C. 73.68
D. 54.26
Area = ½(5²) π – area of sector1 – area of sector2
area of sector1 = 1/2(5²)(π/3)= 25 π /6
lower limit = π/3 upper limit = π/2
area of sector2= ½ ∫ (10cos Θ)² = 25 π /6 – 25/4 √3
Area= ½(5²) π - 25 π /6 - 25 π /6 – 25/4 √3 = 47.83
7. In polar coordinate system, the polar angle is negative when
A. measured counterclockwise C.measured at the terminal side of theta
B. measured clockwise
D. none of these
8. A balloon rising vertically 150m from and observer. At exactly 1min, the
angle of elevation is 29 deg 28min. How fast is the balloon rising at that
instant?
A. 104 m/min
B. 102 m/min
C.106 m/min
D.108 m/min
SOLUTION:
y= 150 tan Θ
dy/dt = 150 sec²Θ dΘ/dt
Θ=29deg 28min = 0.5143rad
dΘ/dt = Θ/t = 0.5143/1 = 0.5143 rad/min
dy/dt = 150 sec²(29deg 28min)(0.5143 rad/min)
= 101.77 m/min = 102 m/min
9. When the ellipse is rotated about its longer axis, the ellipsoid is
A. spheroid
B. oblate
C. prolate
D. paraboloid
10. For the formula R= E/C, find the maximum error if C= 20 with possible error
0.1 and E= 120 with a possible error of 0.05
A. 0.0325
B. 0.0275
C. 0.0235
SOLUTION:
dR = 1/C dE – E/C² dC
D. 0.0572
dR = 1/20(0.05) – 120/20² (-0.01) = 0.0325
11. The probability that a married man watches a certain television show is 0.4
and the probability that a married woman watches the show is 0.5. The
probability that a man watches the show, given that his wife does is 0.7.
Find the probability that a wife watches the show given that her husband
does.
A. 0.875
B. 0.745
C. 0.635
D. 0.925
SOLUTION:
Let :
M – the event that the man watch the show
W - the event that the woman watch the show
Given : P(m) = o.4
P(w) = 0.5
P(m/w) = 0.7
Solution : P(m or w) = P(w)*P(m/w) = 0.5 x 0.7 = 0.35
P(w/m) = P(w or m)/P(m) = P(m or w)/P(m) = 0.35/0.4 = 0.875
12. Four friends took the EE Board exam, each with a probability 0.6 passing
the said exam. Find the probability that at least one of them will pass the
exam.
A. 0.7494
B. 0.7449
C. 0.9744
D. 0.9474
SOLUTION:
Let: x – probability of passing the said exam
Y – probability that at least one of them will pass the exam.
Z – probability that fail the exam
Given: x = 0.6
z = 1 – x = 1 – 0.6 = 0.4
Y = 1 – 0.44 = 0.9744
13. Evaluate lim ( sin-19x )/2x , when x = 0.
A. 9/2
B.π
C. ∞
SOLUTION:
Let: x = 0.0000001
D. - ∞
( sin-19x )/2x = ( sin-19(.0000001) )/2(0.0000001) = 4.5 or 9/2
Note : Set in radian mode
14. A sequence of numbers where the succeeding term is greater than the
preceding term is called.
A. Dissonant series
C. Isometric series
B. Convergent series
D. Divergent series
15. Find the initial point of v = <-3,1,2> if the terminal point is <5,0,-1>
A. <8,1,-3>
B. <8,-1,-3>
C. <-8,1,3>
D. <-8,-1,3>
SOLUTION:
Given : <-3,1,2> , <5,0,-1>
( 5 – (-3), 0 – (1), -1 –(2)) = ( 8,-1,-3 )
16. What do you call the integral divided by difference of the abscissa?
A. Average value
C. Abscissa value
B. Mean value
D. Integral value
17. Solve (D2-3D+2)y=4x
A. c1ex + c2e2x
C. c1ex + c2e2x + 3
B. c1ex + c2e2x + 2
D. c1ex + c2e2x + 2x + 3
SOLUTION:
(D2-3D+2)y=4x
(D – 1)(D – 2),
Therefore D1 = 1, D2 = 2
Yc = c1eD1x + c2eD2x = c1ex + c2e2x
Yp = Ax + B
Yp’ = A
Yp’’ = 0
Subst. to equation,
O – 3(A) +2(Ax + B) = 4x
@ x : 2A = 4
A=2
@ k : -3A + 2B = 0
B=3
Yp = 2x + 3
Y = Yc + Yp = c1ex + c2e2x + 2x + 3
18. Find the second derivative of the function y=5x3 +2x + 1
A. 2x
B. x
C. 30x
D. 24x
SOLUTION:
Given : y = 5x3 +2x + 1
y’ = 15x2 +2
y’’ = 30x
19. Three circle of radai 3, 4, and 5 inches respectively, are tangent to each
other extremely. Find the largest angle of a triangle found by joining the
center of the circles.
A. 72.6 deg
B. 75.1 deg
C. 73.4 deg
SOLUTION:
Given: r1 = 3, r2 = 4, r3 = 5
sides of a triangle are 7, 8, 9
S = ( 7 + 8 + 9 )/2 = 12
A = √(s(s-7)(s-8)(s-9)) = 26.83 sq. unit
Angle 1:
26.83 = (1/2)(7)(8)sinƟ
Ɵ = 73.4 deg
Angle 2:
26.83 = (1/2)(7)(9)sinƟ
Ɵ = 58.4 deg
26.83 = (1/2)(9)(8)sinƟ
Ɵ = 48.18 deg
Therefore: Ɵ = 73.4 deg is the highest
Angle 3:
D. 73.5 deg
20. A reflecting telescopes has a parabolic mirror for witch the distance from
the vertex to the focus is 30 ft. If the distance across the top of the mirror
is 64 in, how deep is the mirror of the center?
A. 32/45 in
B. 30/43 in
C. 32/47 in
D. 35/46 in
SOLUTION:
Given: x = 64/2 = 32 , p = 30x12 = 360
at origin at the center
X2 = 4py
y = x2/4p = 322/4(360) = 32/45 in
21. An observer wishes to determine the height of a tower. He takes sights
at the top of the tower from A and B, which are 50 ft apart at the same
elevation on a direct line with the tower. The vertical angle at point A is
30 degrees and at point B is 40 degrees. What is the height of the tower?
A. 85.60 ft
C. 110.29 ft
B. 143.97 ft
D. 95.24 ft
SOLUTION:
Tan40=h/x
X=h/tan40 - eq 1
Tan30= h/50+x
X=h/tan30
- eq 2
Equate 1 and 2
h/tan40 = h/tan 30
h=95.24ft
22. The average of six scores is 83. If the highest score is removed, the
average of the remaining scores is 81.2. Find the highest score.
A. 91
C. 93
B. 92
D. 94
SOLUTION:
Given: 𝐴𝑣𝑒 𝑜𝑓 𝑠𝑖𝑥 𝑠𝑐𝑜𝑟𝑒𝑠 = 83
𝐴𝑣𝑒 𝑜𝑓 𝑓𝑖𝑣𝑒 𝑠𝑐𝑜𝑟𝑒𝑠 = 81.2
Find: Highest score
𝑥
= 83
6
𝑥 = 498
𝑥−𝑦
= 81.2
5
𝑥 − 𝑦 = 407.5
𝑦 = 498 − 407.5
𝒚 = 𝟗𝟎. 𝟓 𝒐𝒓 𝟗𝟏
23. A coat of paint of thickness 0.01 inch is applied to the faces of a cube
whose edges is 10 inches, thereby producing a slightly larger cube.
Estimate the number of cubic inches of paint used.
A. 3
B. 6
C. 2
D. 4
SOLUTION:
V=x3
Dv=3x2dx
Dv=3(10)2(0.01)
Dv=3
24. The area in the second quadrant of the circle x^2+y^2=36 is revolved
about the line y+10=0. What is the volume generated?
A. 2932 c.u
C. 2229 c.u
B. 2392 c.u
D. 2292 c.u
SOLUTION:
y’=4r/3π
y’=4(6)/3π
Second prop of pappus
V=Ax2πxd’
=1/4(πr2)(2π)(10+y’)
V=2228.83 cubic units
25. Find the equation of the parabola whose vertex is the origin and whose
directrix is the line x=4
A. y^2=16
B. y^2=-16x
C. x^2=16y D. x^2=-16y
SOLUTION:
a=4
y2=-4ax
y2=-4(4)x
y2=-16x
26. A solid has a circular base of radius 4 units. Find the volume of the solid
if every plane section perpendicular to a fixed diameter is an equilateral
triangle.
A. 147.80
B. 256
C.148.96
D. 86
SOLUTION:
2r=d
2(4)=d
D=8
d/6(0+4am+0)=V
d/6(0+4(Sqrt of ¾ a2)+0)=V
V=147.80 cubic units
27. From past experience, it is known 90% of one year old children can
distinguish their mother’s voice from the voice of a similar sounding
female. A random sample of 20 one year’s old are given this voice
recognize test. Find the probability that all 20 children recognize their
mother’s voice.
A. 0.122
B. 0.500
C. 1.200
D. 0.222
SOLUTION:
.9022=0.122
28. If Jose is is 10% taller than Pedro and Pedro is 10% taller than Mario,
then Jose taller than Mario by _______%.
A. 18
B.20
C.21
D.23
SOLUTION:
Jose
Pedro
Mario
1.1(1.1x)
1.1x
x
1.1(1.1x)-x=0.21x= 21%
29. The area of circle is six times it’s circumference. What is the radius of
the circle?
A. 10
B. 11
C. 12
D. 13
SOLUTION:
(πr2)= 6(2πr)
r=6
30. Find the orthogonal trajectories of the family of parabolas y^2=2x+C
A. y=Ce^x
C.y=Ce^(2x)
B.y=Ce^(-x)
D.y=Ce^(-2x)
SOLUTION:
Y2=2X+C
2ydy=2dx+0
dy/dx= 2/2y
dy/dx=1/y
dy/dx=-dx/dy
dy/dx=-y
∫ 𝑑𝑦/𝑦 = − ∫ 𝑑𝑥
Lny=-x+c
e^lny=e^-x+c
y=ce-x
31. A pole which lean 11 degrees from the vertical toward the sun cast a
shadow 12m long when the angle of the elevation of the sun is 40
degrees. Find the length of the pole.
A. 15.26 m
B. 14.26 m
C. 13.26 m
D. 12.26 m
SOLUTION:
X= 180 - 40 – 90 – 11=39
𝑧
12
=
𝑠𝑖𝑛40
𝑠𝑖𝑛39
z = 12.6
32. A tree stands vertically on a sloping hillside. At a distance of 16 m down
the hill, the tree subtends an angle of 34 degrees. If the inclination of
the hill is 20 degrees. Find the height of the tree.
A. 12.5 m
B. 13.4 m
C. 14.3 m
D. 15.2 m
SOLUTION:
16
ℎ
=
𝑠𝑖𝑛56
sin 14
= 𝟏𝟒. 𝟑 𝒎
33. Robin flies to San Francisco from Santa Barbara in 3 hours. He flies
back in 2 hours. If the wind are blowing from the north at the velocity of
40mph going but changed in 20mph from north returning. What was the
air speed of the plane.
A. 140mph
B. 150mph
C. 160mph
D. 170mph
SOLUTION:
(x-3) (40) = (x+2)(20)
40x-120 = 20x+40
x= 40+ 120
x= 160mph
34. What would happen in the volume of a sphere if the radius is tripled?
A. Multiplied by 3
C. Multiplied by 27
B. Multiplied by 9
D. Multiplied by 6
SOLUTION:
V=4/3 πr^3
V(3) = 4/5π(3)^3 = 4/3π
= 27
therefore: multiplied by 27
35. The distance between the center of the 3 circles which are mutually
tangent each other are 10,12, and 14 units. Find the area of the largest
circle.
A. 72pi
B. 64pi
C. 23pi
D. 16pi
SOLUTION:
A= πr^2
A= π(8)^2
=64π
36. What is the vector which is orthogonal both to 9i + 9j and 9i + 9k?
A. 81i+ 81j – 81k
C. 81i - 81j + 81k
B. 81i - 81j – 81k
D. 81 i+ 81j + 81k
SOLUTION:
(9i + 9j) (9i + 9k)
= 81(i + j) (i + k)
= 81 (i –j –k)
= 81 -81i -81k
37. Good costs in merchants P72 at what price should he mark them so that
he may sell the at discount of P10 form marked price and still make a
profit of 20% on the selling price?
A. P150
B. P200
C. P100
D. P250
SOLUTION:
Capital
P72
Worked price
X
Selling price
Profit
0.20X
0.20(0.90)
Profit = Income + capital
= 0.20 (0.90) = 0.90X-72
X= 100
38. A ranch has cattle and horses in a ratio of 9.5. If there are 80 more heads
of a cattle than horses. How many animals are on the ranch?
A. 140
B. 150
C. 238
D. 280
SOLUTION:
(9/5) = (x+80/x)
9x = 5(x+80)
9x-5x = 400
X = 100 + 80
Y = 100
Total= 180+100 = 280
39. A group of students plan to pay equal amount in hiring a vehicle for an
excursion trip at a cost of P6000. However, by adding two more students
to the original group, the cost of each student will be reduced by P150.
Find the number of students in the original group.
A. 10
B. 9
C. 8
D. 7
SOLUTION:
𝑆𝑛 =
(𝑛 − 1)(𝑑)
𝑛
(2𝑎1 +
)
2
𝑛
6000 =
6000= n
(2a1 + (n-1) ( 600 )
2
n= 8
n
40. The volume of the sphere is 36pi cu.m. The surface area of the sphere
in sq. m is.
A. 36pi
B. 24pi
C. 18pi
D. 12pi
SOLUTION:
V = 4/3 πr^3
36π = 4/3 πr^3
r=3
A = 4π(3)^2
A = 36π
41. The logarithm of MN is 6 and the logarithm of N/M is 2. Find the value
of logarithm of N.
A. 3
B. 4
C. 5
SOLUTION:
Given:
log 𝑀𝑁 = 6
log
𝑁
=2
𝑀
log 𝑀 + log 𝑁 = 6
log 𝑀 = 6 − log 𝑁 → ①
log
𝑁
=2
𝑀
log 𝑁 − log 𝑀 = 2
log 𝑁 − 2 = log 𝑀 → ②
D. 6
Equate 1 & 2
6 − log 𝑁 = log 𝑁 − 2
2 log 𝑁 = 8
𝐥𝐨𝐠 𝑵 = 𝟒
42. Peter can paint a room for 2 hrs and John can paint the same room in
1.5 hrs. How long can they do it together in minutes?
A. 0.8571
B. 51.43
C. 1.1667
D. 70
SOLUTION:
Given:
Peter =
John =
1 𝑟𝑜𝑜𝑚
2 ℎ𝑟𝑠
1 𝑟𝑜𝑜𝑚
1.5 ℎ𝑟𝑠
1
=
=
2
1
1.5
1
1
1
+
=
2 1.5 𝑥
60 𝑚𝑖𝑛𝑠
𝑥 = 0.86 ℎ𝑟𝑠 (
1ℎ𝑟
) = 53.43 mins.
43. An airplane has an airspeed of 210 mph the bearing of N 30deg E a
wind is blowing due west at 30 mph. Find its ground speed rounded to
the nearest degree.
A. 201
B. 187
C. 197
D. 175
SOLUTION:
𝐻𝑜𝑟𝑖𝑧𝑎𝑜𝑛𝑡𝑎𝑙 ∶ 30 sin 30 = −15 𝑚𝑝ℎ
𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 ∶ 30 cos 30 = 25.98 𝑚𝑝ℎ
𝑝𝑙𝑎𝑛𝑒 𝑠𝑝𝑒𝑒𝑑 = 210 𝑚𝑝ℎ
∑ 𝐹𝐻 = 210 + (−15) = 195
∑ 𝐹𝑉 = 25.98
∑ 𝑅𝑒𝑠𝑢𝑙𝑡𝑎𝑛𝑡 = √(195)2 + (25.98)2 = 𝟏𝟗𝟔. 𝟕 𝒎𝒑𝒉 ≈ 𝟏𝟗𝟕 𝒎𝒑𝒉
44. Find the area of a regular hexagon circumscribing a circle with an area
of 289pi sq. cm.
A. 2,002 sq. cm. B. 1,001 sq. cm. C. 550 sq.cm. D. 328 sq. cm.
SOLUTION:
𝐴2 = 289 𝜋 𝑐𝑚2
𝜋𝑟 2 = 289 𝜋
r = 17
𝐴 = 𝑛𝑟 2 tan
𝐴 = 6(17)2 tan
180
6
180
= 𝟏, 𝟎𝟎𝟏 𝒄𝒎𝟐
6
45. If y = 4cosx + sin2x, what is the slope of the curve when x = 2?
A. -2.21
SOLUTION:
B. -4.94
C. -3.25
D. 2.21
y = 4cosx + sin2x, x=2 rad
𝑦 ′ = 4(−sin 𝑥) + 2 cos 2𝑥 = 2 cos 2𝑥 − 4 sin 𝑥
@ 𝑥 = 2 𝑟𝑎𝑑
180
180
𝑦 ′ = 2 cos 2 (2 (
)) − 4 sin (2 (
))
𝜋
𝜋
𝑦 ′ = 2 cos 229.183 − 4 sin 114.591
𝒚′ = −𝟒. 𝟗𝟒
46. A rectangular plate of 6 m by 8 m is submerged vertically in a water.
Find the force on one face if the shorter side is uppermost and lies in the
surface of the liquid.
A. 941.76 kN
C. 3,767.04 kN
B. 1,883.52 kN
D. 470.88 Kn
SOLUTION:
ℎ̅ =
ℎ
8
+ 6 = + 6 = 10
2
2
𝐹 = (𝐷𝐻2 0)(ℎ̅)(𝐴)
= (981)(10)(6(8))
𝑭 = 𝟒𝟕𝟎. 𝟖𝟖 𝒌𝑵
47. A steel ball at 120 deg C cools in 20 minutes to 80 deg C in a room at
25 deg C. Find the temperature of the ball after half an hour.
A. 40.96 deg C
C. 66.85 deg C
B. 45.96 deg C
D. 55.96 deg C
SOLUTION:
𝑻𝒕 − 𝑻𝒔 = (𝑻𝒐 − 𝑻𝒔 )𝒆−𝒌𝒕
80 − 25 = (120 − 25)𝑒 −𝑘(20)
50 = 95 (−20𝑘) ln 𝑒
𝑘 = 0.02733
@𝑡 =0
𝑇𝑡 − 25 = (120 − 25)𝑒 −0.02733(30)
𝑻𝒕 = 𝟔𝟔. 𝟖𝟓 ℃
10
48. Evaluate the inverse Laplace transform of 𝑠+50
A. 10𝑒 −5𝑡
B. 10𝑒 −𝑡
C. 𝟏𝟎𝒆−𝟓𝟎𝒕
D. 10𝑡𝑒 −50𝑡
SOLUTION:
𝟏𝟎
𝓛−𝟏 (𝒔+𝟓𝟎) = 𝟏𝟎𝒆−𝟓𝟎𝒕
49. In a printed circuit board may be purchased from 5 suppliers in how
many ways can 3 suppliers can be chosen from the 5?
A. 20
B. 5
C. 10
SOLUTION:
5C3
5!
= 3!(5−3)! = 𝟏𝟎
50. Find the length of the vector (2, 4, 4).
D. 68
A. 5
B. 6
C. 4
D. 8
SOLUTION:
|𝒂
̅| = √𝒂𝟐 + 𝒃𝟐 + 𝒄𝟐
= √22 + 42 + 42
|𝒂
̅| = 𝟔
51. What is the perimeter of a regular 15-sided polygon inscribed in a circle
with radius 10 cm?
A. 63.77 cm
B. 62.37 cm
C. 64.52 cm
D. 68.48 cm
SOLUTION:
180
𝑃 = 2𝑛𝑟 𝑠𝑖𝑛
𝑛
180
𝑃 = 2(15)(10)sin
= 𝟔𝟐. 𝟑𝟕𝒄𝒎
15
52. Find the area bounded by the curve (y square) – 3x + 3 = 0 and x = 4.
A. 12
B. 9
C. 16
D. 8
SOLUTION:
𝑦2
𝑦 2 − 3𝑥 + 3 = 0 ⟺ 𝑥 = ⁄3 + 1
𝑥=4
Intersection points are
𝑦 2 − 3(4) + 3 = 0 ⟺ 𝑦 = ±√9
𝑦 = ±√9 ⟺ 𝑦 = ±3
3
3
𝑦2
𝑦2
∫ 4 − ( ⁄3 + 1) 𝑑𝑦 ⟺ ∫ 3 − ⁄3 𝑑𝑦
−3
−3
3
∫ 3−
−3
[3𝑦 −
2
3
𝑦 ⁄
𝑦 3⁄
𝑑𝑦
⟺
[3𝑦
−
3
9]−3
3
3
𝑦 3⁄
𝑦 3⁄
]
⟺
2
[3𝑦
−
9 −3
9 ]0
3
2 [3𝑦 −
𝑦 3⁄
9]0 = 𝟏𝟐
53. A circle with a radius of 10 cm is revolved about a line tangent to it. Find
the volume generated.
A. 19, 739 𝑐𝑚3
C. 1193.24 𝑐𝑚3
3
B. 17, 843 𝑐𝑚
D. 1295.36 𝑐𝑚3
SOLUTION:
54. An inscribed angle is 𝜋⁄4 radian, and the chord of the circle subtended
by the angle is 12√2 cm. Find the radius of the circle.
A. 10 cm
B. 12 cm
C. 14 cm
D. 16 cm
SOLUTION:
𝜋
6√2
12√2⁄ = 6√2
∝= 𝜋⁄4
𝑠𝑖𝑛 4 = 𝑟
𝒓 = 𝟏𝟐𝒄𝒎
2
6√2
ɵ = (2) ∝= 𝜋⁄2
𝑠𝑖𝑛 ∝= 𝑟
55. In Jones family, each daughter has as many brothers as sisters and
each son has three times as many sisters as brothers. How many
daughters and sons are there in the Jones family?
A. 3, 2
B. 4, 2
C. 5, 2
D. 6, 3
SOLUTION:
𝐺 = 𝑛𝑜. 𝑜𝑓 𝑠𝑖𝑠𝑡𝑒𝑟𝑠
𝐺−1 =𝐵
3(𝐵 − 1) = 𝐺
𝐵 = 𝑛𝑜 𝑜𝑓 𝑏𝑟𝑜𝑡ℎ𝑒𝑟𝑠
3(𝐵 − 1) − 1 = 𝐵
3(2 − 1) = 𝐺
𝑩=𝟐
𝑮=𝟑
56. find th bounded by 𝑦 = 8 − 𝑥 3 , the x-axis and the y-axis.
A. 14
B. 10
C. 16
D. 12
SOLUTION:
57. Find the area of the square with a diagonal of 15 cm.
A. 225 𝑐𝑚2
B. 115.5𝑐𝑚2 C. 112.5 𝒄𝒎𝟐
D. 121.5 𝑐𝑚2
SOLUTION:
𝐴=
1 2
𝑑
2
1
𝐴 = (15)2 = 𝟏𝟏𝟐. 𝟓 𝒄𝒎𝟐
2
58. Find the greatest area of a rectangle inscribed in a given parabola 𝑦 =
16 − 𝑥 2 and the x-axis.
A. 24.63 s.u.
B. 49.27 s.u. C. 98.53 s.u.
D. 46.87 s.u.
SOLUTION:
A = LW
𝑦 = 16 − (
4√3 2
)
3
= 32⁄3
W = 𝑦 = 16 − 𝑥 2
𝐴(𝑥) = 2𝑥(16 − 𝑥 2 ) = 32𝑥 − 2𝑥 3
𝑑𝐴
𝑑𝑥
= 32 − 6𝑥 2 = 0
𝐴 = 2(
4√3
3
)(32⁄3)
𝑨 = 𝟒𝟗. 𝟐𝟕 𝒔. 𝒖.
=±
4√3
3
59. Evaluate Laplace transform of 𝑡 2 .
A. 2⁄𝑠
B. 1⁄𝑠 2
C. 𝟐⁄𝒔𝟑
D. 1⁄𝑠
SOLUTION:
𝑛!
𝑡 𝑛 = 𝑠𝑛+1
2!
𝑡 2 = 𝑠2+1 = 𝟐⁄𝒔𝟑
60. Two circles of different radii are concentric. If the length of the chord of
the larger circle that is tangent to the smaller circle is 40 cm, find the difference in
area of the two circles.
A. 350π sq. cm
B. 400π sq. cm
C. 500π sq. cm
D. 550π sq. cm
SOLUTION:
ɵ = 180⁄3 = 60
∝= 60⁄2 = 30
𝑟 = 20 𝑡𝑎𝑛(30) = 20√3⁄3
𝑅 = √202 + (20√3⁄3)2 = 40√3⁄3
𝐴𝐵𝑂 = 𝜋(40√3⁄3)2 = 1600⁄3 𝜋
𝐴𝑆𝑂 = 𝜋(20√3⁄3)2 = 400⁄3 𝜋
𝐴𝐵𝑂 − 𝐴𝑆𝑂 = 1600⁄3 𝜋 − 400⁄3 𝜋 = 𝟒𝟎𝟎𝝅 𝒔𝒒. 𝒄𝒎
61. Solve dy/dx = 4y divided by x(y-3)
A. 𝑥 𝑦 = 𝐶𝑒 𝑦
B. 𝐱 𝟒 𝐲 𝟑 = 𝐂𝐞𝐲
C. 𝑥 4 𝑦 2 = 𝐶𝑒 𝑦
3 4
SOLUTION:
𝑑𝑦
4𝑦
[ =
] 𝑥(𝑦 − 3)
𝑑𝑥 𝑥(𝑦 − 3)
𝑑𝑦
= 𝑥(𝑦 − 3) 𝑑𝑥 = 4𝑦
𝑑𝑦
𝑑𝑥
= [𝑥(𝑦 − 3) 𝑑𝑥 = 4𝑦] 𝑥𝑦
D. 𝑥 3 𝑦 2 = 𝐶𝑒 𝑦
=
𝑦
(𝑦−3)
𝑦
𝑑𝑦 =
3
4
𝑥
𝑑𝑥
4
= (𝑦 − 𝑦) 𝑑𝑦 = 𝑥 𝑑𝑥
3
=(1 − 𝑦) 𝑑𝑦 =
4
𝑥
𝑑𝑥
3
4
=∫ (1 − 𝑦) 𝑑𝑦 = ∫ 𝑥 𝑑𝑥
y−3 ln(𝑦) = 4ln(𝑥) + 𝐶
y + C’ = 4 ln(𝑥) + 3ln(𝑦)
𝑦 + 𝐶 ′ = ln(𝑥 4 ) + ln(𝑦 3 )
𝑦 + 𝐶 ′ = ln(𝑥 4 )(𝑦 3 )
𝑒 𝑦+𝐶 = 𝑒 ln(𝑥
4 )(𝑦 3 )
𝑪𝒆𝒚 = 𝒙𝟒 𝒚𝟑
62. The towers of a 60 meter parabolic suspension bridge are 15 m high
and the lowest point of the cable is 3 m above the roadway. Find the vertical
distance from the roadway to the cable at 15 m from the center.
A. 3 𝑚
B. 5 𝑚
C. 𝟔 𝒎
D. 8 𝑚
SOLUTION:
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 𝑦
x =0, y = 3
x = -30, y = 15
x = +30, y = 15
@ x = 0; y =15
𝑎(0)2 + 𝑏(0) + 𝑐 = 3
𝑐=3
@ x = - 30; y =15
−302 𝑎 − 30𝑏 + 3 = 15
900𝑎 − 30𝑏 + 3 = 15 → 𝑒𝑞𝑛 1
@ x = +30; y =15
302 𝑎 + 30𝑏 + 3 = 15
900𝑎 + 30𝑏 + 3 = 15 → 𝑒𝑞𝑛 2
𝐴𝑑𝑑 𝐸𝑞𝑛 1 𝑎𝑛𝑑 𝐸𝑞𝑛 2
900𝑎 − 30𝑏 + 3 = 15
+ 900𝑎 + 30𝑏 + 3 = 15
1800𝑎 + 0 + 6 = 30
𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑎
1800𝑎 = 30 − 6
1800𝑎 = 24
24
𝑎=
1800
𝑎 = 0.01333
Solve for x
𝑥 = 30 − 15
𝑥 = 15
Solve for y
𝑦 = 0.01333𝑥 2 + 3
𝑦 = 0.01333(15) + 3
𝑦 = 5.99 ≈ 𝟔𝒎
63. A target with a black circular center and a white ring of uniform width is
to be made. If the radius of the center is to be 3 cm, how wide should the ring be
so that the area of the ring is the same as the area of the center?
A. 1.232 𝑐𝑚
B. 1.263 𝑐𝑚
C. 1,252 𝑐𝑚
D. 1.243 𝑐𝑚
SOLUTION:
64. Evaluate 0.9 + 0.92 + 0.93 + ⋯ + 0.9𝑛
A. 9
B. 8
C. 7
D. 6
SOLUTION:
A. 97
65. Which of the following is a prime number?
B. 91
C. 133
SOLUTION:
Prime numbers 2, 3, 5, 7, 9, 11 …
@ 91
D. 119
=√91 = 9.53
Divide 91 by prime numbers less than the √91
91
= 13 → 𝑛𝑜𝑡 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟!
7
@ 133
=√133 = 11.53
Divide 133 by prime numbers less than the √133
133
= 19 → 𝑛𝑜𝑡 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟!
7
@ 119
=√119 = 10.91
Divide 119 by prime numbers less than the √119
119
= 17 → 𝑛𝑜𝑡 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟!
7
𝐵𝑦 𝐸𝑙𝑖𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛
Answer is 97
66. Find the sum of the interior angle of a regular hexagon?
A. 810°
B. 540°
C. 𝟕𝟐𝟎°
D. 630°
SOLUTION:
Formula:
Sum of Interior angle = (𝑛 − 2)180°
Regular hexagon; 6 sides, 6 angles
𝑛=6
(6 − 2)180° = 𝟕𝟐𝟎°
67. From a hill 600 ft high, the angles of depression to the bases in opposite
directions are 42° and 19° 23′ respectively, Find the length of the proposed tunnel
through the bases.
A. 2,589.15 ft
B. 2,371.74 ft
C. 2590.05 ft
D. 1592.20 ft
A
𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡𝑢𝑛𝑛𝑒𝑙 = 𝐴 + 𝐵
𝑡𝑎𝑛 𝜙 =
𝐴=
600
6𝑜𝑜
𝐴
tan 42°
;
= 666.37𝑓𝑡
𝛼= 19°23’
𝜃 = 42°
SOLUTION:
600 𝑓𝑡
B
𝑡𝑎𝑛 𝛼 =
6𝑜𝑜
;
600
𝐵=
= 1705.38𝑓𝑡
tan 19°23′
𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡𝑢𝑛𝑛𝑒𝑙 = 𝐴 + 𝐵
𝐵
𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡𝑢𝑛𝑛𝑒𝑙 = 666.37𝑓𝑡 + 1705.38𝑓𝑡 =𝟐𝟑𝟕𝟏. 𝟕𝟓 𝒇𝒕.
68. Find the distance of the directrix from the center of an ellipse if its major
axis is 10 and its minor axis is 8.
A. 8.1
B. 8.3
C. 8.5
D. 8.7
Given:
𝑀𝑎𝑗𝑜𝑟 𝑎𝑥𝑖𝑠 = 𝑎 = 10
𝑀𝑖𝑛𝑜𝑟 𝑎𝑥𝑖𝑠 = 𝑏 = 8
𝐹𝑜𝑐𝑖 = 𝑐
𝐷𝑖𝑟𝑒𝑐𝑡𝑟𝑖𝑥 = ?
𝑎2
𝑑 = 𝑐 ; 𝑐 = √𝑎2 − 𝑏 2
SOLUTION:
𝑐 = √102 − 82 = 6
102
𝑑=
= 16.67
6
16.67
= 𝟖. 𝟑
2
69. If the logarithm of MN is 6 and the logarithm of M/N is 2, find the
logarithm of N
A. 2
B. 3
C. 4
D. 5
SOLUTION:
Given:
log 𝑀𝑁 = 6
log
𝑁
=2
𝑀
log 𝑀 + log 𝑁 = 6
log 𝑀 = 6 − log 𝑁 → ①
log
𝑁
=2
𝑀
log 𝑁 − log 𝑀 = 2
log 𝑁 − 2 = log 𝑀 → ②
Equate 1 & 2
6 − log 𝑁 = log 𝑁 − 2
2 log 𝑁 = 8
𝐥𝐨𝐠 𝑵 = 𝟒
70. Two buildings with flat roofs are 60 m apart. From the roof of the shorter
building 40 m in height, the angle of elevation to the edge of the roof of the taller
building is 40°. How high is the taller building?
A. 60 m
B. 70 m
C. 80 m
D. 90 m
x
𝑥
tan 40 = 60
𝑥 = (tan 40)(60)
𝑥 = 50
𝐻𝑡𝑎𝑙𝑙 𝑏𝑢𝑖𝑙𝑑𝑖𝑛𝑔 = 40 + 50 = 𝟗𝟎 𝒎
40 = 𝜃
40m
60m
71. Three ships are situated as follows A is 225 mi due north of C, and B is
375 mi due to east of C. What is the bearing of B from A?
A. N 56° E
B. S 56° E
C. N 59° E
D. S 59° E
SOLUTION:
𝐭𝐚𝐧 θ =
225
375
225
= tan−1 375 = 30. 96
θ = 90° − 30. 96° = 𝟓𝟗. 𝟎𝟒
∴ Bearing of B from A is 𝐒 𝟓𝟗° 𝐄
72. The longest diagonal of a cube is 6 cm. The total area of the cube is
A. 32√2 sq. m
SOLUTION:
𝐴𝑆 = 6 𝑎2
B. 72 sq. m
C. 24√2 sq. m
D. 36 sq. m
𝑑 = √3 𝑎
𝑎=
𝑑
√3
6
=
= 2√3
√3
𝐴𝑆 = 6 (2√3)2 = 𝟕𝟐 𝒎𝟐
73. A support wire is anchored 12 m up from the base of a flagpole and the
wire makes a 15° angle with the ground. How long is the wire?
A. 12 m
B. 92 m
C. 46 m
D. 24 m
SOLUTION:
12
Tan 15° =
𝑎𝑑𝑗
𝑎𝑑𝑗 = 44. 78 𝑚
𝑐 = √44. 782 + 122 = 𝟒𝟔. 𝟑𝟓 𝒎
∴ 𝑤𝑖𝑟𝑒 𝑖𝑠 𝟒𝟔 𝒎 𝑙𝑜𝑛𝑔
74. A motorboat weighs 32000 lb and its motor provides a thrust of 5000 lb.
Assume that the water resistance is 100 pounds for each foot per second of the
𝑑𝑣
speed v of the boat. Then 1000 𝑑𝑡 = 5000 – 100 v. If the boats starts from the rest,
what is the maximum velocity that it can attain?
A. 20 ft/s
B. 25 ft/s
SOLUTION:
1000
1000
10
𝑑𝑣
𝑑𝑡
𝑑𝑣
𝑑𝑡
𝑑𝑣
𝑑𝑡
= 5000 − 100 𝑣
= 100(50 − 𝑣)
= (50 − 𝑣)
𝑑𝑣
∫ (50−𝑣) =
1
10
∫ 𝑑𝑡
𝑛𝑜𝑤 𝑢𝑠𝑒 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑤 = 50 − 𝑣
−∫
𝑑𝑣
𝑤
=
1
10
∫ 𝑑𝑡
C. 40 ft/s
D. 50 ft/s
− ln 𝑤 =
𝑡
10
+𝐶
𝑡
ln 𝑤 = − 10 − 𝐶
𝑡
ln( 50 − 𝑣) = − 10 − 𝐶
−𝑡
50 − 𝑣 = 𝐶1 𝑒 10
𝑠𝑖𝑛𝑐𝑒 𝑣0 = 0 𝑡ℎ𝑒𝑛
0
50 − 0 = 𝐶1 𝑒 10
50 − 0 = 𝐶1 = 50
𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝐶 𝑖𝑛 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛, 𝑤𝑒 𝑔𝑒𝑡
1
50 − 𝑣 = 50𝑒 10
1
𝑣(𝑡) = 50 − 50𝑒 −10
1
𝑣(𝑡) = 50(1 − 𝑒 −10 )
𝒗𝒎𝒂𝒙 = 𝟓𝟎 𝒇𝒕/𝒔
75. The base of an isosceles triangle is 20.4 and the base angles are 48°40’.
Find the altitude of the triangle
A. 11.6
B. 10.8
C. 12.7
D. 9.5
SOLUTION:
tan 48°40′ =
10.24
𝑎𝑑𝑗
𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 = 𝟏𝟏. 𝟓𝟗 𝒐𝒓 𝟏𝟏. 𝟔
76. Find the exact value of sec (-pi/6)
A. 3/√2
B. 1/√2
SOLUTION:
𝟏
𝐜𝐨𝐬−
𝝅
𝟔
=
𝟏
√𝟑
𝟐
=
𝟐
√𝟑
C. 3/√6
D. 2/√3
77. A snack machine accepts only quarters. Candy bars cost 25₵ packages
of peanuts cost 75₵ and cans of cola cost 50₵. How many quarters are needed to
buy two candy bars, one package of peanuts and one can of cola?
A. 8
B. 7
C. 6
D. 5
SOLUTION:
78. A ball is dropped from a height of 18 m. On each rebound it rises 2/3 of
the height from which it last fell. What is the total distance it travels in coming to
rest?
A. 80 m
B. 90 m
C. 72 m
D. 86 m
SOLUTION:
𝑆𝑛 = 𝑎𝑛 +
𝑆𝑛 = 18 +
𝑎𝑛
1− 𝑟
18
2
1− 3
= 72 𝑚
79. Find the work done in moving an object along the vector a=3i + 4j if the
force applied is b= 2i + j
A. 11.2
B. 10
C.12.6
SOLUTION:
𝑊 = 𝐹 𝑥 𝑣 = ( 3𝑖 + 4𝑗 )( 2𝑖 + 𝑗) = 𝟏𝟎
D. 9
80. By stringing together 9 differently colored beads. How many different
bracelets can be made?
A. 362, 880
B. 20, 160
C. 40, 320
D. 181, 440
SOLUTION:
(9−1)!
2
= 𝟐𝟎, 𝟏𝟔𝟎
81. Find the derivative of the function y=3/(x2 +1).
A. 6x/(x2 +1)2
B. 6x(x2 +1)2
C. -6x/(x2 +1)2
D. -6x(x2 +1)2
SOLUTION:
𝑦=
3
𝑥 2 +1
=
𝑢
𝑣
𝑦′ =
𝑣𝑑𝑢−𝑢𝑑𝑣
𝑣2
𝑦′ =
(𝑥 2 + 1)(0) − (3)(2𝑥)
(𝑥 2 + 1)2
∴ 𝒚′ =
−𝟔𝒙/
(𝒙𝟐 +
𝟏)𝟐
82. If 8 oranges cost Php 96, how much do 1 dozen cost at the same rate?
A. Php 144
B. Php 124
C. Php 148
D. Php 168
SOLUTION:
𝑅𝑎𝑡𝑒 =
𝑃ℎ𝑝 96
= 𝑃ℎ𝑝 12/𝑜𝑟𝑎𝑛𝑔𝑒
8 𝑜𝑟𝑎𝑛𝑔𝑒𝑠
1 𝑑𝑜𝑧𝑒𝑛 = 12 𝑝𝑖𝑒𝑐𝑒𝑠
12
@ 1 𝑑𝑜𝑧𝑒𝑛 ∶ 𝑐𝑜𝑠𝑡 = 𝑃ℎ𝑝 𝑜𝑟𝑎𝑛𝑔𝑒 𝑥 12 𝑜𝑟𝑎𝑛𝑔𝑒𝑠\
𝑐𝑜𝑠𝑡 = 𝑃ℎ𝑝 144
83. What is the slope of the linear equation 3y-x=9?
A. 1/3
B. -3
C. 3
D. 9
SOLUTION:
3𝑦 −
𝑥=9
3𝑦 =
𝑥+9
𝑥+9
3
𝑦=
𝑢
=𝑣
𝑦′ =
𝑣𝑑𝑢−𝑢𝑑𝑣
𝑣2
𝑦′ =
3(1) − (𝑥 + 9)(0)
32
′
∴ 𝒚 =𝒎=
𝟏
𝒔𝒍𝒐𝒑𝒆 = 𝟑
84. Points A and B are 100 m apart and are of the same elevation as the
foot of the building. The angles of elevation of the top of the building from points A
and B are 21 degrees and 32 degrees respectively. How far is A from the building?
A. 259.28 m
B. 265.42 m
C. 271.62 m
D. 277.92 m
SOLUTION:
tan(𝛳) =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
85. Give the degree measure of the angle 3pi/5.
A. 150 degrees
B. 106 degrees
C. 160 degrees
Solution
3𝜋 180
𝑥(
) = 108 𝑑𝑒𝑔𝑟𝑒𝑒𝑠
5
𝜋
D. 108 degrees
86. For what value of k will the line kx+5y=2k have slope 3?
B. -5
C. 15
D. -15
A. 5
SOLUTION:
𝑘𝑥 + 5𝑦 =
2𝑘
5𝑦 = 2𝑘 − 𝑘𝑥
2𝑘 − 𝑘𝑥 𝑢
𝑦=
=
5
𝑣
𝑣𝑑𝑢 − 𝑢𝑑𝑣
𝑦′ =
𝑣2
𝑦′
5(−𝑘) − (2𝑘 − 𝑘𝑥)(0)
=
52
−𝑘
𝑦′ = 5
−𝑘
3=
5
∴ 𝒌 = −𝟏𝟓
87. The cross product of vector A=4i+2j with vector B=0. The dot product A
B=30. Find B.
A. 6i+3j
B. 6i-3j
C. 3i+6j
D. 3i-6j
SOLUTION:
𝑥𝑖 + 𝑦𝑗 =?
𝑢𝑠𝑖𝑛𝑔 𝑐𝑟𝑜𝑠𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑡
42
| |=0
𝑥𝑦
4𝑦 − 2𝑥 =
0 𝑒𝑞. 1
𝑢𝑠𝑖𝑛𝑔 𝑑𝑜𝑡 𝑝𝑟𝑜𝑑𝑢𝑢𝑐𝑡
4𝑥 + 2𝑦 = 30 𝑒𝑞. 2
𝑢𝑠𝑖𝑛𝑔 𝑒𝑞. 1
𝑦
2𝑥
=
𝑒𝑞. 3
4
𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑒𝑞. 3 𝑡𝑜 𝑒𝑞. 2
4𝑥 +
2𝑥
2 ( 4 ) = 30, 𝑥 = 6
𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥 𝑡𝑜 𝑒𝑞. 3
𝑦=
2(6)
4
2𝑥
4
=
=3
∴ 6𝑖 + 3𝑗
A. 8
88. What is the discriminant of the equation 4x2=8x-5?
B. -16
C. 16
D. -8
SOLUTION:
𝑎𝑥 2 + 𝑏𝑥 +
𝑐=0
4𝑥 2 − 8𝑥 + 5 = 0
𝑑𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 = 𝑏 2 − 4𝑎𝑐
= (−8)2 − 4(4)(5)
∴ 𝒅𝒊𝒔𝒄𝒓𝒊𝒎𝒊𝒏𝒂𝒏𝒕 = −𝟏𝟔
A. 2
89. Find the slope of the curve y=x+2(x raised to -1) at (2,3)
B. ½
C. 1
D. ¼
SOLUTION:
𝑦=𝑥+
2𝑥 −1
𝑚 = 𝑦 ′ = 1 − 2𝑥 −2
∴ 𝑚 = 𝑦 ′ = 1 − 2(2)−2
𝟏
=
𝟐
90. A wheel 4 ft. in diameter is rotating at 80 r/min. Find the distance (in ft.)
travelled by a point on the rim in 1s.
A. 18.6
B. 16.8
C. 17.8
D. 18.7
SOLUTION:
ῳ = 80𝑟𝑝𝑚 =
𝑟=
= 2 𝑓𝑡.
80𝑟𝑒𝑣
min
𝑥
2𝑝𝑖 𝑟𝑎𝑑
1 rev
𝑥
1 𝑚𝑖𝑛
60s
=
8𝑝𝑖 𝑟𝑎𝑑
3
𝑠
𝑑 4
=
2 2
∴ 𝑠 = ῳ𝑡𝑟 =
8𝑝𝑖
(1)(2)
3
= 𝟏𝟔. 𝟖 𝒇𝒕.
91. A toll road averages 300,000 cars a day when the toll is $2.00 per car.
A study has shown that for each 10-cent increase in the toll, 10,000 fewer cars will
use the road each day. What toll will maximize the revenue?
A. $2.25
B. $2.75
C. $3.00
D. $2.50
SOLUTION:
Let: n = cars
P = price
R = revenue
n= no. of increment
n = 300,000 – 10,000x
P = 2.00 + 0.10x
R = nP
R = (300,000 – 10,000x)(2.00 + 0.10x)
R = 600,000 + 30,000x – 20,000x – 1000x2
𝑑𝑅
𝑑𝑥
𝑑𝑅
𝑑𝑥
Substitute x,
n = 300,000 – 10,000(5)
n = 250,000
= -1000x2 + 10,000x + 600,000
P = 2.00 + 010(5)
= -2,000x + 10,000
-2,000x + 10,000 = 0
x=5
P = $2.50
92. Find the equation of the line determined by points A(5, -2/3) and (1/2, 2)
A. 8x + y = 58
B. 8x + 27y = 58
C. 8x – 27y = 58
D. x – 2y = 58
SOLUTION:
m = 𝑋2−𝑋1
𝑌2−𝑌1
(y – y1) = m (x – x1)
m=
−2+2/3
(y + 2/3) = 27 (x – 5)
m=
1
−5
2
8
27
8
8
40
[(y+2/3) = 27 x - 27 ] 27
8x – 27y = 58
93. Find the eccentricity of a hyperbola whose transverse and conjugate
axes are equal in length.
A. √𝟐
B. √3
C. 2 √2
D. 2 √3
SOLUTION:
(x2/a2) – (y2/b2) = 1
√𝑎2 +𝑏2
e=
a=b
e=
𝑎
√2𝑎2
𝑎
e=a
√2
𝑎
e = √𝟐
A. 4
94. For what values of x is |x-3| = 1?
B. 2
C. 2, 4
D. -2, -4
SOLUTION:
By inspection and substituting all the given in the equation:
|x-3| = 1
|x-3| = 1
|2-3| = 1
|4-3| = 1
|1|= 1
|1|= 1
95. Susan’s age in 20 years will be the same as Thelma’s age now. Ten
years from now, Thelma’s age will be twice Susan’s. What is the present age of
Susan?
A. 45
B. 40
C. 50
D. 30
SOLUTION:
PRESENT
FUTURE
Thelma
X
2(x + 10)
Susan
x + 20
(x + 20) + 10
2(x + 10) = (x + 20) + 10
2x + 20 = x + 30
x = 10
Substitute:
10 + 20 = 30 years old
96. The circumference of a great circle of a sphere is 18𝜋 m. Find the volume
of the sphere.
A. 3053.6 cu. m
B. 3043.6 cu. m
C. 3033.6 cu. m
D. 3023.6 cu. m
SOLUTION:
C = 2𝜋r
18𝜋 = 2𝜋r
r = 9m
4
Vsphere= 3 𝜋𝑟 3
4
= 𝜋 (93)
3
= 3053.6 m3
97. What is the Laplace transform of f(t) = cosh at?
A. a/(s squared + a squared)
C. s/(s squared + a squared)
B. a/(s squared – a squared)
D. s/(s squared – a squared)
98. Tom inherited two different stocks whose yearly income was Php 2,100.
The total appraised value of the stocks was Php 40,000, one was paying 4% and
one 6% per year. What was the value of the stock paying 6%?
A. 27,000
B. 23,000
C. 25,000
D. 24,000
SOLUTION:
Let x = stock of value
(40,000 – x) = Appraisal value
0.06x + 0.04(40,000 – x) = 2,100
x = 25,000
99. Joe and his dad are bricklayers. Joe can lay bricks for a wall in 5 days.
With his father’s help, he can build it in 2 days. How long would it take his father to
build it alone?
A. 3-1/4 hrs
B. 3-1/3 hrs
C. 2-1/3 hrs
D. 2-2/3 hrs
SOLUTION:
1
1
1
[ + = ]10x
5
𝑥
2
2x + 10 = 5x
10
𝟏
x = 3 days or 3 𝟑 hrs
100. Find the nth term of the arithmetic sequence 11, 2, -7.
A. -6n + 12
B. -9n + 20
C. –n + 24
D. -2n + 8
SOLUTION:
d= a2 – a1= (2) – (11) = -9
a3= a3 + (n-3)d
= (-7) + (n-3)-9
= -7 - 9n + 27
= -9n + 20
REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
AUGUST 2017
MATHEMATICS
REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION
AUGUST 2017
MATHEMATICS
1. What is the value of x in Arctan 3x + Arctan 2x = 45 degrees?
SOLUTION:
.tan−1 (3𝑥) + tan−1(2𝑥) = 45
𝑡𝑎𝑛−1 (3x)(2x) = 45
𝑡𝑎𝑛−1 (6x) = 45
45
x = tan 6
x= 1/6
ANSWER: C.1/6
2. Find the volume (in cubic units) generated by rotating circle 𝑥 2 + 𝑦 2 + 6𝑥 +
4𝑦 + 12 = 0 about the y axis.
SOLUTION :
𝑥 2 + 𝑦 2 + 6𝑥 + 4𝑦 + 12 =
4
𝑣 𝑠𝑝ℎ𝑒𝑟𝑒 = 𝜋𝑟 3
3
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
𝑥 2 + 6𝑥 + 𝑦 2 + 4𝑦 = −12 + 9 + 4
(𝑥 2 + 6𝑥 + 9) + (𝑦 2 + 4𝑦 + 4) = 5
(𝑥 + 3)2 + (𝑦 + 2)2 = 1
(𝒉, 𝒌) = (−𝟑, −𝟐)
V = A2 𝜋D
V = 𝜋𝑟 2(2 𝜋)𝐷
V = (𝜋)(1)2 (2𝜋)(3)
V = 59.22 cu.units
3. If i= (-1)^1/2 find the value i^30
A.1
B.-1
C.-I
D.i
SOLUTION:
4. Solve the equation cos^2, A=1-cos^2 A
SOLUTION
cos 2𝐴 = 1 − 𝑎𝑠 2𝐴
= 2𝑎𝑠 2𝐴 = 1
𝑨𝑺𝟐 = ½
ANSWER: A. 45֯,315
5. Find the change in volume of a sphere if you increase the radius from 2 to
2.05 units.
SOLUTION
4
Vsphere =3 π3
4
= 3 π(2)3
= 33. 51
4
= 3π(2.050)3
= 36.09
ΔVsphere = 36.09-33.51
= 2.58
ANSWER: A.2.51
6. What is the general solution of (D^4 – 1)y(t) = 0?
A. 𝐲 = 𝐂𝟏𝐞𝐭 + 𝐂𝟐𝐞−𝐭 + 𝐂𝟑 𝐜𝐨𝐬 𝐭 + 𝐂𝟒 𝐬𝐢𝐧 𝐭
C. y = C1et + C2e−t
t
−t
t
−t
B. y = C1e + C2e + C3te + C4te
D. y = C1et + C2e−t
SOLUTION
𝒚 = 𝑪𝟏𝒆𝒕 + 𝑪𝟐𝒆−𝒕 + 𝑪𝟑 𝐜𝐨𝐬 𝒕 + 𝑪𝟒 𝐬𝐢𝐧 𝒕
7. What percentage of the volume of a cone is the maximum right circular
cylinder that can be
inscribed in it?
Answer: C 44 percent
8. if e^2x-3e^x + 2 = 0 , find x.
SOLUTION
𝑒 2𝑥 − 3𝑒 𝑥 + 2 = 0
𝑙𝑛𝑒 2𝑥 − 𝑙𝑛3𝑒 𝑥 + 2 = - ln 2
2x-3x = - ln 2
X (2-3) = - ln 2
2
X = - ln−1
X = ln 2
ANSWER: A. ln2
9. On a cortain day the nurses at a hospital worked the following number of
hours; nurse howard worked 8 hrs, nurse pease worked 10hrs, nurse
campbell worked 9 hrs, nurse grace worked 8 hrs, nurse mccarthy worked 7
hrs, and nurse murphy worked 12 hrs. What is the average number of hrs per
nurse on this day?
SOLUTION
Howard = 8hours
Pease = 9 hours
Campbell = 9 hours
Ave = summation of number of hours/number of
nurse
12
Grace = 8 hours
= 8+10+9+8+7+ 6 = 9
ANSWER: C. 9
10. Joy is 10 percent taller than joseph and joseph is 10 percent taller than
tom. How many percent is joy taller than tom?
A. 18%
B. 20%
C. 21%
D. 23%
SOLUTION:
JOY = JOSEPH (1+.10)
JOSEPH = TOM (1+.10)
JOY [TOM (1+.10)] (1+.10)
JOY = TOM (1+.10)2
JOY = TOM (1+.21)
.21 = 21%
11. An army food supply truck can carry 3 tons. A breakfast ration weights 12
ounces, and the other two daily meals weigh 18 ounces each assuming each
soldier gets 3 meals per day, on a ten day trip how many soldiers can be
supplied by one truck?
SOLUTION
1 ounce = 28.34g
1 ton = 100kg
3 𝑡𝑜𝑛𝑠
1 𝑑𝑎𝑦
=
0.3 𝑡𝑜𝑛𝑠
1 𝑑𝑎𝑦
12 ounce +18+18
(
48 ounces
day
28.34𝑔
1 𝑘𝑔
1 𝑡𝑜𝑛
)( 1 𝑜𝑢𝑛𝑐𝑒 )(1000 𝑔 )(100 𝑘𝑔 )
= 1.36x103
=
𝑡𝑜𝑛𝑠
𝑑𝑎𝑦
𝑑𝑎𝑦
1.36x103
𝑠𝑜𝑙𝑑𝑖𝑒𝑟𝑠
0.3
= 220 soldiers
ANSWER: C. 200 soldiers
12. Find the area enclose in the second and third quadrants by the curve x=t -1,
y= 5t^3(t^2-1)
SOLUTION
ANSWER: B. 8/7
13.csc520֯=?
SOLUTION
Csc 520 = csc (520 – 360)
Csc 520 = csc 160
Csc 160 = Csc (180 – 160)
Csc 16 = csc 20
Csc 520 = csc 20
ANSWER: B.
csc20
14. From past experience it is known 90 percent of one year old children can
distinguish their mothers voice of a similar sounding female. A random sample
of one years old are given this voice recognize test. Find the probability that
atleast 3 children did not recognize their mothers voice.
SOLUTION
0.9
= 0.3
3
ANSWER: B. 0.323
15. ln y = mx + b what is m?
ANSWER: A. slope
16.Find the area bounded by the parabola sqrt of x + sqrt of a and the line x + y
=
a
SOLUTION
𝑋2
A= ∫𝑋1 (𝑌𝑐 − 𝑌𝑙)
2
1
A= ∫2 (𝑋 − 1) − (1 − √2) 𝑑𝑥
1
A =0.8333 ≈ 3
Since a = 1
A=
𝒂𝟐
𝟑
𝟐
D.
𝒂𝟑
17. What is the integral of cosxe ^sinx dx
SOLUTION
= ∫ 𝑐𝑜𝑠𝑥𝑒 𝑠𝑖𝑛𝑋 𝑑𝑥
u = sinx
du= cosxdx
∫ 𝑒 𝑠𝑖𝑛𝑥 (𝑐𝑜𝑠𝑥)𝑑𝑥
Let u =sinx
u = cosxdx
∫ 𝑒 𝑢 𝑑𝑢 = 𝑒 𝑢 + 𝐶
= 𝒆𝒔𝒊𝒏𝒙 + 𝑪
ANSWER: B. 𝒆𝒔𝒊𝒏𝒙 + 𝑪
18. The geometric mean and the arithmetic mean of number is 0 and 10
respectively what is the harmonic mean?
SOLUTION
AM = a + b
AM =
𝐺𝑀2
+𝑏)
𝑏
(
2
82
( +𝑏)
𝑏
10 = 2
b=4
𝐺𝑀2
a=
𝑏
82
= 4
a = 16
HM =
𝑛
1 1
+
𝑎 𝑏
HM = 1
4
2
+
1
16
HM = 6. 4
ANSWER: C. 6.4
19. In how many ways can four coins be tossed once?
SOLUTION
n = 4 coins
N = 2𝑛
N = 24
N = 16
ANSWER: B. 16
20. A statue 3 m high is standing on a base of 4m high. If an observers eye is
1.5 m above the ground how far should he stand from the base in order that
the angle subtended by the statue is a maximum?
SOLUTION
X =√𝐻1𝐻2
= √(3)(4)
X = 3.71m
ANSWER: C. 3.71
21. What is the number in the series below?
3, 16, 6, 12, 12, 6,
SOLUTION
3, 16, (3x2), (16-22 ), (3x23 ), (16-2^3), (3x23 )
=(3x22 )
=24
ANSWER: D. 24
22. A man who is on diet losses 24 lb in 3 months 16 lb in the next 3 months
and so on for a long time. What is the maximum total weight loss?
A.
72
B. 64
C. 54
D. 81
SOLUTION:
23. What is the slope of the linear equation 3y-x=9?
SOLUTION
3y-x=9
3y=x+9
1
1
y= 3 𝑥 + (9)( 3 )
𝟏
m=𝟑
𝟏
ANSWER: A. 𝟑
24. Each of the following figures has exactly two pairs of parallel sides except a
A. parallelogram
B.rhombus C. trapezoid
D. square
25. A points A and B are 100 m apart and are of the same elevation as the foot
of the building. The angles of elevation of the top of the building from points A
and B are 21 degrees and 32 respectively. How far is A from the building?
SOLUTION
ℎ
ℎ
Tan32 = 𝑥
Tan 21 = 100+𝑥
𝑥𝑡𝑎𝑛32
Tan 21 = 100+𝑥
X=159.276
100+x = 100+159.276
=259.28m
ANSWER: A. 259.28
26. What is the area in sq.m.of the zone of a spherical segment having a
volume of 1470.265 cu.m if the diameter of the sphere is 30m.
A. 655.487
B.
565.487 C. 756.847 D. 465.748
SOLUTION
A = 2 πrh
V=
πh2
(3𝑟 − ℎ)
3
πh2
1470.265 = 3 (3(15) − ℎ)
h=6
A = 2 πrh
= 2π(15)(6)
A = 565.487 sq. m
ANSWER: B. 565.487
27. Which of the following numbers can be divided evenly by 19?
SOLUTION
𝟕𝟔
=𝟒
𝟏𝟗
ANSWER: C. 76
28. Where is the center of the circle x^2 + y^2 -10x + 4y – 196 = 0
SOLUTION
𝑋 2 − 10𝑋 + 25 + 𝑌 2 + 4𝑌 + 4 = 196 + 25 + 4
(𝑋 − 5)2 + (𝑌 + 2)2 = 225
𝑪(𝟓, −𝟐)
ANSWER: D. (5,-2)
29. Two ships leave from a port. Ship A sails west for 300 miles and ship B sails
north 400 miles. How far apart are the ships after their trips?
SOLUTION
𝑆 = √𝑎2 +𝑏 2
𝑆 = √3002 +4002
𝑺 = 𝟓𝟎𝟎 𝒎𝒊
ANSWER: C. 500 miles
30. if the radius of a sphere is increasing at the constant rate of 3m per second
how fast is the volume changing when the surface area is 10 sq.mm?
SOLUTION
3m/s x 10 𝑚𝑚2
=30 cu. mm per sec
ANSWER: C. 30 cu. mm per sec
31. The sum of the base and altitude of an isosceles triangle is 36cm. Find the
altitude of the ttriangle if its area is to be a maximum.
SOLUTION:
x + y = 36
x = 36 - y
1
A= 2 bh
1
A = 2 ( 36 − 𝑦 )𝑦
1
A= 2 ( 36 -𝑦 2 )
𝑦2
A = 18 −
2
0 = 18 - y
y = 18
ANSWER: C 18cm
32. An insurance policy pays 80 percent of the first P20,000 of a certain patients
medical expenses, 60 percent of the next P40,000 and 40 percent of the
P40,000 after that. If the patients total medical bill is P92,000 how much will
the policy pay?
ANSWER: C. 52,800
33. A scientist found 12mg of radioactive isotope is a soil sample. After 2 hours,
only 8.2 mg of the isotope remained. Determine the half life of the isotope?
SOLUTION:
𝑙𝑛𝑥1
𝑡1
=
𝑙𝑛𝑥
𝑡
2
2
x1 = 12 mg t2 = ?
8.2
12
6
𝑙𝑛
12
𝑙𝑛
=
2
𝑥
x = 3.64 hrs.
ANSWER: C 3.64hrs
34. find the area bounded the curves r = 2cosѲ and r = 4cosѲ.
A.
6.28 B.
9.42 C. 12.57
D. 15.72
35. Give the degree measure of angle 3pi/5
A. 150 degrees B. 106 degrees C. 160 degrees D. 108 degrees
SOLUTION:
3𝜋
5
∗
180
𝜋
= 𝟏𝟎𝟖 𝒅𝒆𝒈
ANSWER: D 108 deg
36. What is the median of the following group numbers? 14 12 20 22 14
16
SOLUTION:
1
M = 2 ( 14 +16) = 15
ANSWER: C 15
37. For what value of k will the line kx + 5y = 2k hace slope 3?
SOLUTION:
K(3) + 5(3) = 2k
k= -15
ANSWER: D. -15
38. The cross product of vector A=4i + 2j with vector B=0. The dot product
A·B=30, Find B.
ANSWER: A. 6i+3j
39. Find the length of the curve r = (1 – cos Ѳ).
ANSWER: D. 32
40. Find the equation of the curve that passes through (4,-2) and cuts at right
angles every curve of the family 𝑦 2 = 𝐶𝑥 3
ANSWER: C.𝟐𝒙𝟐 + 𝟑𝒚𝟐 = 𝟒𝟒
41.Find the area of circle with center at (1,3) and tangent to the line 5x – 12y –
8
= 0.
SOLUTION
√52 + (−12)2
A= π𝑟 2
=π(3)2 = 𝟐𝟖. 𝟐𝟕
ANSWER: B. 28.27
42. If a flat circular plate of radius r = 2 m is submerged horizontally in water so
that the top surface is at a depth of 3m, then the force on the top surface of
the plate is
SOLUTION
F= WhA
= w = 9810N
F = (9810)(3)(𝜋(2)2 )
F = 369828.29N
= 369,829.15N
ANSWER: A. 369,829.15N
43. A hemispherical tank with a diameter of 8 ft is full of water find the work
done in ft-lb in pumping all the liquid out of the top of the tank.
B. 12,546
𝑑2 𝑦
44. If 𝑥 = 3𝑡 − 1 , 𝑦 = 1 − 3𝑡 , 𝑓𝑖𝑛𝑑 𝑑𝑥 2
SOLUTION
x = 3 + 1 , y = 1-3𝑡 2
𝑥
2
2
𝑦′ = − 𝑥 −
3
3
1
Y= 1-3 (3 + 3)2
𝑦 = 1−
𝑦" = −
𝑥2 2 1
− −
3 3 6
𝟐
2
3
ANSWER: B. − 𝟑
𝑥2 2
5
𝑦 =1− − 𝑥+
3 3
6
45. if sin3A = cos 6B then:
A+2B = 30 deg
46. It takes a typing student 0.75 seconds to type one word. At this rate, how
many words can the student type in 60 seconds?
SOLUTION
0.75𝑠𝑒𝑐
𝑠𝑒𝑐
= 60
1
𝑥
X = 80
ANSWER: D. 80
47. A chord, 6 inches long from the center of a circle. Find the length of the
radius of the circle.
SOLUTION
chord = 16 in
16 2
r=√62 + ( 2 ) = 𝟏𝟎 𝒊𝒏
ANSWER: D. 10 in
48. A train is moving at the rate of 8 mph along a piece of circular track of radius
2500 Through what angle does it turn in 1 min?
SOLUTION
704𝑓𝑡
𝑚
1ℎ𝑟
.
80 ℎ ∗ 60 𝑚𝑖𝑛
𝑚𝑖𝑛
S=rѲ 2500𝑓𝑡
= Ѳ
=1.33m/in = 704 ft / min
Ѳ=0.2816 *
180
=16֯18֯
𝜋
ANSWER: A. 16 deg 8
49. The diagonal of a face of a cube is 10 ft. The total area of the cube is
SOLUTION
d= 10ft
d= √3a
10
A= 6𝑎2 = 6( )2 = 𝟑𝟎𝟎𝒇𝒕𝟐
√3
ANSWER: D. 300 sq.ft
50. The volume of the sphere is 36 pi cu. m. The surface area of this sphere in
sq.m. is:
SOLUTION
.v= 36 π𝑚3
4
A = 4𝜋𝑟 2 𝑉 = 3 𝜋𝑟 3 , 𝑟 = 3
A= 44𝜋(3)2
A= 36 π
ANSWER: B. 36pi
51. Which of the following is an exact DE?
ƔM
ƔN
SOLUTION:exact D.E Ɣy = Ɣx = 1
(2𝑥𝑦)𝑑𝑥 + ( 2 + 𝑥 2 )𝑑𝑦 = 0
𝑀 = 2𝑥𝑦, 𝑁 = 2 + 𝑥 2
ƔM 2𝑥𝑦
=
= 2𝑥
Ɣy
𝑦
Ɣ𝐍 𝟐 ∗ 𝒙𝟐 𝟐𝒙𝟐
=
=
= 𝟐𝒙
Ɣ𝐱
Ɣ𝐱
𝒙
𝟐𝒙𝒚𝒅𝒙 + (𝟐 + 𝒙𝟐 )𝒅𝒚 = 𝟎
ANSWER: C. 𝟐𝒙𝒚𝒅𝒙 + (𝟐 + 𝒙𝟐 )𝒅𝒚 = 𝟎
52. Find the value of 4sinh(pi i/3)
𝜋
SOLUTION: 4sinh (3 𝑖)
sinhjѲ= jsinѲ
=4jsinѲ
𝜋
=4jsin( ∗ 180/𝜋)
3
4𝑗√3
=
2
=2i√3
ANSWER:B. 2i(sqrt. of 3)
53. Find the coordinates of an object that has been displaced from the point (-4,
9) by the vector 4i-5j).
A. (0,4)
B. (0,-4)
C. (4,0)
D. (-4,0)
54. Find the work done in moving an object along a vector r= 3i + 2j - 5k if the
applied force F = 2i – j – k.
SOLUTION:. r= 3i +2j -5k F= 2i-j-k
(3·2)i = 6
(2·-1)i = -2
(-5·-1)k = 5
6+(-2)+5 = 9
ANSWER: A. 9
55. Find the value of k for which the line 2x + ky = 6 is parallel to the y-axis.
SOLUTION: 2(3-KY) + KY = 6
6-2KY + KY = 6
-2KY + KY = 6
K=6
ANSWER: A. k=0
56. Find the area inside one petal of the four leaved rose r = sin2theta.
SOULITON:rsin2Ѳ
𝜋/2
A= ∫0
𝜋
𝜋𝑟
𝜋
=2 ∫02 (𝑠𝑖𝑛2Ѳ(2)𝑑Ѳ)
𝜋
𝜋
=2 (− cos ( 2 ) − cos(0))
𝜋
=2 (1)
𝝅
=𝟐
ANSWER:D. pi/8
57. Which of the following is a vector?
A. kinetic energy
B. electric field intensity C. entropy D. work
58. In how many ways can 6 people be lined up to get on a bus if certain 3 persons
refuse to follow each other?
SOLUTION:. 6P3
=120 ways
ANSWER:D. 480
59. The bases of a frustum of a pyramid are 18cm by 18cm and 10cm by 10cm.
Its lateral area is 448 sq. cm. what is the altitude of the frustum?
ANSWER:B. 6.93cm
60. A store advertises a 20 percent off sale. If an article marked for sale at $24.48,
what is the regular price?
SOLUTION:20 % discount
$24.48 discounted price
$24.48 = x – 20% (x)
X = $30.60
ANSWER: C. $30.60
61. If the area of the equilateral triangle is 4 (sqrt. of 3), find the perimeter.
SOLUTION:
. A= 4√3
A= √s (s-x)(s-x)(s-x)
S=
4√3 = √(
x=4
𝑥+𝑥+𝑥
𝑥+𝑥+𝑥
2
2
𝑥+𝑥+𝑥
)(
2
− 𝑥))3
P= x+x+x
P=12
ANSWER: B. 12
62. Dave is 46 yrs old. Twice as old as rave. How old is rave?
SOLUTION:
D=46 yrs
R=2X
2x=46
X = 23yrs old
ANSWER: C. 23 yrs
63. The angles of elevation of the top of a tower at two points 30 m and 80 m
from the foot of the tower, on a horizontal line are complementary. What is the
height of the tower?
SOLUTION:
A+B = 90
A= 90-B
𝐻
tanѲ =80
𝐻
B =tan−1 30 equation no. 2
𝐻
𝐻
Tan 90- (tan−1(30)) = 80
H= 49m
𝐻
tan(90-B)=80 equation no. 1
ANSWER: C. 49m
64.A large tank filled with 500 gallons of pure water. Brine containing 2 pounds
of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well-mixed
solution is pumped out at the same rate. What is the concentration of the
solution in the tank at t = 5 min?
ANSWER: C. 0.0795 lb/ gal
65. The intensity I of light at a depth of x meters below the surface of a lake
satisfies the differential dldx = (-1/4)I. At what depth will the intensity be 1
percent of thtat at the surface?
ANSWER: B. 2.29m
66. What is the discriminant of the equation4𝑥 2 = 8𝑥 − 5?
ANSWER: B. -16
67. Find the percentage error in the area of a square of side s caused by
increasing the side by 1 percent.
ANSWER: B. 2 percent
68. What is the height of a right circular cone having a slant height of 3.162 m
and base diameter of 2 m?
SOLUTION:
H=√(3.162)2 − (1)2
H = 3m
ANSWER: C. 3
69. In how many orders can 7 different pictures be hung in a row so that 1
specified picture is at the center?
SOLUTION:
6i = 720 ways
ANSWER: D. 720
70. What is the x-intercept of the line passing through (1,4) and (4,1)?
ANSWER: B. 5
71. One ball is drawn at random from a box containing 3 red balls, 2 white balls,
and 4 blue balls. Determine the probability that is not red.
SOLUTION
# 𝑜𝑓𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑓𝑢𝑙 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑃=
# 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
6
𝑃=
9
𝟐
𝑷=
𝟑
ANSWER: B. 2/3
72. An airplane flying with the wind took 2 hours to travel 1000 km and 2.5
hours flying back. What was the wind velocity in kph?
SOLUTION
𝑆 = 𝑉𝑡
1000 = (𝑉𝑎 + 𝑉𝑤)2
𝑉𝑎 = 500 − 𝑉𝑤
1000 = (𝑉𝑎 − 𝑉𝑤)2.5
𝑉𝑎 = 400 + 𝑉𝑤
500 − 𝑉𝑤 = 400 + 𝑉𝑤
2𝑉𝑤 = 100
𝑽𝒘 = 𝟓𝟎
ANSWER: A. 50
73. In how many ways can a person choose 1 or more of 4 electrical
appliances?
SOLUTION
𝑁 = 𝑛𝐶𝑟
𝑁 = 4𝐶1 + 4𝐶2 + 4𝐶3 + 4𝐶4
𝑵 = 𝟏𝟓
ANSWER: A. 15
74. What are the third proportional to y/x and 1/x?
SOLUTION
𝑎 𝑐
𝟏
=
𝒅
=
𝑏 𝑑
𝒙𝒚
𝑐𝑏
𝑑=
𝑎
1 1
( )( )
𝑑= 𝑥𝑦𝑥
𝑥
ANSWER: C. 1/xy
75. If 7 coins are tossed together, in how many ways can they fall with most
three heads?
SOLUTION
𝑁 = 𝑛𝐶𝑟
𝑁 = 7𝐶3 + 7𝐶2 + 7𝐶1 + 7𝐶0
𝑵 = 𝟔𝟒
ANSWER: B. 64
76. If y = ln (sec x tan x). find dy/dx.
A. cot x
B. cos x
C. csc x
D. sec x
SOLUTION
1
(𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥)
𝑦′ =
𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥
1
(𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 + 𝑠𝑒𝑐 2 𝑥)
𝑦′ =
𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥
1
(𝑠𝑒𝑐𝑥(𝑡𝑎𝑛𝑥 + 𝑠𝑒𝑐𝑥))
𝑦′ =
𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥
𝒚′ = 𝒔𝒆𝒄𝒙
ANSWER: D. sec x
77. A rubber ball is made to all from height of 50 ft and is observed to rebound
2/3 of the distance it falls. How far will the ball travel before coming to rest if
the ball continues to fall in this manner?
SOLUTION
a1= 50 x 2/3 = 33.33
𝑎1
33.33
S=1−𝑟=1−2/3 = 100
St= 50 +(2)(100)
St = 250ft
ANSWER: A. 250
78. In a class of 40 students, 27 like calculus and 25 like Chemistry. How many
like calculus only?
SOLUTION
40 students, 27 like cal, 25 like chem
40 = x + 25
X = 15
ANSWER: B. 15
79. Simplify (cos θ / sin θ + 1 ) + tan θ
SOLUTION
𝑐𝑜𝑠 2 𝜃 + 𝑠𝑖𝑛2 𝜃 + 𝑠𝑖𝑛𝜃
=
𝑐𝑜𝑠𝜃(𝑠𝑖𝑛𝜃 + 1)
𝑠𝑖𝑛𝜃 + 1
=
𝑐𝑜𝑠𝜃(𝑠𝑖𝑛𝜃 + 1)
ANSWER: A. sec
1
𝑐𝑜𝑠𝜃
= 𝒔𝒆𝒄𝜽
=
80. What kind of graph is r = 2 sec θ?
A. straight line
B. parabola
C. ellipse
D. hypebola
81. Find the inclination of the line passing through (5,3) and (10,7)
SOLUTION:
p1(-5,3) p2(10,7)
𝑦2 − 𝑦1
7−3
Tan θ = 𝑥2 − 𝑥1 = 10 − (−5) = 𝟏𝟒. 𝟗𝟐𝒐
ANSWER: B. 𝟏𝟒. 𝟗𝟐𝒐
82. An ellipse has an eccentricity of 1/3. Find the distance between the two
directrix if the distance between the foci us 4.
SOLUTION:
𝑎
2ae=3.
2𝑒 =3.
2a*1/3=3.
2a=9.
9
a=2=4.5
(2)(
4.5
1
3
)
9
=1
distance between the directrix
𝑎
=2𝑒
3
=(9)(3)=27
ANSWER: A.36
83. Find the value of sin (arc cos 15/17).
SOLUTION:
. Call x the arc whose cosx=1517.
Find sin x.
sin2x=1−cos2x=1−225289=64289.
𝟖
sin x = ± 𝟏𝟕
ANSWER: D. 8/17
84. Find the area of the triangle having vertices at -4 -I, 1 +2i, 4-3i.
SOLUTION:
(−4 –𝐼)(1 +2𝑖)(4−3𝑖)
= 17
2
ANSWER: C. 17
85. Find the location of the focus of the parabola 𝑥 2 + 4𝑦 − 4𝑥 − 8 = 0.
SOLUTION:
𝑥 2 - 4x + 22 = - 4y + 8 +22
(𝑥 − 2)2 = - 4(y - 3)
(𝑥 − ℎ)2 = - 4a(y - k)
A = 1 therefore, focus is (-2,-2)
ANSWER: D.(-2,-2)
86. What conic section is2𝑥 2 − 8𝑥𝑦 + 4𝑥 = 12?
A. hyperbola B. ellipse
C. parabola
D. circle
87. A man bought 5 tickets in a lottery for aprize of P 2,000.00. If there are total
400 tickets, what is his mathematical expectation?
SOLUTION:
5
𝑥
= 2000 ; x = 25
400
ANSWER: A. P25.00
88. In what quadrants will Ѳ be terminated if cos Ѳ is negative?
SOLUTION:
Quadrant II, the x direction is negative, and both cosine and tangent become
negative
Quadrant III, sine and cosine are negative
Therefore 2,3
ANSWER: B. 2,3
89. For what value of the constant k is the lie x + y = k normal to the curve 𝑦 =
𝑥2
SOLUTION:
So the slope of the normal is -1, which means that the slope of the tangent is 1.
𝑑𝑦
𝑑𝑥
= 2x
Find out where the slope is
1:
1
2x = 1 --> x = 2
So we have the
coordinates (1/2), 1/4).
So eqn of normal is:
1
1
y - 4 = - (1)(x - 2)
1
1
y=-x+2+4
1
1
𝟑
k = 2+ 4 = 𝟒
ANSWER: A. 3/4
90. Any number divided by infinity is equal to:
A. 1
B. infinity
C. zero
D. indeterminate
91. The points Z1,Z2,Z3,Z4 in the complex plane are vertices of parallelogram
taken in order if and only if
SOLUTION
𝑧1 + 𝑧3
𝑧2 + 𝑧4
=
2
2
Therefore z1 + z3 = z2 + z4
ANSWER: C. Z1+ Z3 = Z2 + Z4
92. If the points (-1,-1,2),(2,m,5) and (3,11,6) are collinear, find the value of m.
SOLUTION
AB = (2 + 1)i + (m + 1) j + (5-2)k = 3i + (m+1)j + 3k
And
AC = (3+1)I + (11+1)j + (6-2)k = 4i + 12j + 4k
( 3i + (m+1)j = λ ( 4i + 12j + 4k )
3 = 4 λ and m + 1 = 12 λ
And
m=8
ANSWER: A. 8
93. Infinity minus infinity is:
A. infinity
B. zero
C. indeterminate
D. none of these
94. If in the fourier series of a periodic function, the coefficient aჿ = 0 and aⁿ =
0, then it must be having ____________ symmetry.
A. odd
B. odd quarter wave
C. even
D. either A or B
95. Tickets number 1 to 20 are mixed up then and then a ticket is drawn has a
number which is a multiple of 3 or 5?
SOLUTION
Here, S = {1, 2, 3, 4, ...., 19, 20}.
Let E = event of getting a multiple of 3 or 5 = {3, 6 , 9, 12, 15, 18, 5, 10, 20}.
P(E) = n(E)/n(S) = 9/20.
ANSWER: D. 9/20
96. A car travels 90 kph. What is its speed in meter per second?
SOLUTION:
90 km/hr x 1000 meter/1 km 1 hr/3600 sec. = 25
ANSWER: C. 25
97. The line y = 3x = b passes through the point (2,4) Find b.
SOLUTION:
(4 )= 3(2) - b therefore b = -2
ANSWER: C. -2
98.If y = tanh x, find dy/dx:
A. 𝐬𝐞𝐜 𝟐 𝒙
B. csc 2 𝑥
C. sin2 𝑥
D. tan2 𝑥
99.From the given values A and B, find the vector cross product of A and B, if:
A=2i – 5k, B=j
SOLUTION:
(2i – 5k)(j) = 5i + 2k
ANSWER: A. 5i + 2k
100. If a place on the earth is 12 degrees south of the equator, find its distance
in nautical miles from the north pole.
SOLUTION:
theta = 90+12 = 102o
60 𝑚𝑖𝑛
1 𝑛𝑚
102 degrees x 1 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 = 1 𝑚𝑖𝑛
= 6,120 nautical miles
ANSWER: D. 6,120
REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
MARCH 2017
MATHEMATICS
REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION
MARCH 2017
MATHEMATICS
1. The rectangular coordinate system in space is divided into eight
compartments, which are known as:
A. quadrants
B. octants
C. axis
D. coordinate
2. What is the value of x in Arctan 3x + Arctan 2x = 45 degrees?
A. -1/6 and 1
B. 1/6 and -1
C. 1/6 D. -1
SOLUTION:
Arctan 3x + Arctan 2x = 45°
𝑡𝑎𝑛−1 (3𝑥) + 𝑡𝑎𝑛−1 (2𝑥) = 45°
tan 𝐴 + 𝐵 =
𝑡𝑎𝑛 3𝑥 + 𝑡𝑎𝑛 2𝑥
= 𝑡𝑎𝑛45°
1 − 𝑡𝑎𝑛 3𝑥 𝑡𝑎𝑛2𝑥
3𝑥 + 2𝑥
=1
1 − 3𝑥(2𝑥)
5𝑥
=1
1 − 6𝑥 2
5𝑥 = 1 − 6𝑥 2
6𝑥 2 + 5𝑥 − 1 = 0
( 6𝑥 − 1 )(𝑥 + 1 ) = 0
𝒙=
𝟏
; 𝑥 = −1
𝟔
3. In delivery of 14 transformers, 4 of which are defective, how many ways
those in 5 transformers at least 2 are defective?
A. 940
B. 920
C. 900
D. 910
𝑛𝑪𝑟
SOLUTION:
(4𝑪2)(10𝐶3) + (4𝐶3)(10𝐶2) + (4𝐶4)(10𝐶1) = 𝟗𝟏𝟎
4. Sand is pouring to form a conical pile such that its altitude is always twice its
radius. If the volume of a conical pile is increasing at a rate of 25 pi cu.
ft./min, how fast is the radius is increasing when the radius is 5 feet?
A. 0.5 ft/min
B. 0.5pi ft/min
C. 5 ft/min
D. 5pi ft/min
SOLUTION:
𝟏
𝑉 = 𝟐 𝜋𝑟 2 ℎ
ℎ = 2𝑟 =
1
𝜋(𝑟 2 )(2𝑟)
2
𝑑𝑣
𝑓𝑡 3
2
= 25𝜋
= 𝜋𝑟 3
𝑑𝑡
𝑚𝑖𝑛
3
𝑑𝑟
𝑑𝑣
𝑑𝑟
= ? 𝑎𝑡 𝑟 = 5
= 2𝜋𝑟 2
𝑑𝑡
𝑑𝑡
𝑑𝑡
25𝜋 = 2𝜋(5)2
𝑑𝑟
𝑑𝑡
𝒅𝒓
𝒇𝒕
= 𝟎. 𝟓
𝒅𝒕
𝒎𝒊𝒏
𝑥+4
5. Evaluate lim 𝑥−4as x approaches to infinity.
A. 1 B. 0
C. 2
SOLUTION:
Using, 𝑥 = 109
D. infinite
𝑥+4
(10)9 + 4
=
= 𝟏
𝑥→∞𝑥 − 4
(10)9 − 4
lim
6. Describe the locus represented by the equation |𝑧 − 1| = 2>
A. circle
B. ellipse
C. parabola D. hyperbola
7. An air balloon flying vertically upward at constant speed is situated 150 m
horizontally from an observer. After one minute, it is found that the angle of
elevation from the observer is 28 deg 59 min. What will be then the angle of
elevation after 3 minutes from its initial position?
A. 63 deg 24 min
C. 28 deg 54 mi
B. 58 deg 58 min
D. 14 deg 07 min
SOLUTION:
height in 1 min.
ℎ = tan(29° 59` )(150) = 86.54 𝑚
height in 3 mins.
ℎ = 3 (86.54) = 259.63 𝑚
h
150m
tan 𝜃 =
259.63
150
259.63
; 𝜃 = 𝑡𝑎𝑛−1 (
150
)
𝜽 = 𝟓𝟗° 𝟓𝟖`
8. In how many ways can you pick 3 dogs from a pack of 7 dogs?
A. 32
B. 35
C. 30
D. 36
SOLUTION:
nCr = 7C3 = 35
9. Find the volume (in cubic units) generated by rotating a circle X2 + y2 + 6x +
4y + 12 = 0 about the y-axis.
A. 47.23
B. 59.22
C. 62.11
D. 39.48
SOLUTION:
𝑥 2 + 𝑦 2 + 6𝑥 + 4𝑦 + 12 = 0
(𝑥 2 + 6𝑥 + 9 ) + (𝑦 2 + 4𝑦 + 4 ) = −12 + 9 + 4
( 𝑥 + 3 )2 + (𝑦 + 2 )2 = 1
𝑉 = 𝐴𝐶 = 𝜋𝑟 2 (2𝜋𝑟)
𝑉 = 𝜋(1)2 (2𝜋(3)) = 𝟓𝟗. 𝟐𝟐 𝒄𝒖. 𝒖𝒏𝒊𝒕𝒔
10. Peter can paint a room in 2 hrs and John can paint the same room in 1.5
hrs. How long can they do it together in minutes?
A. 0.8571
B. 51.43
C. 1.1667
SOLUTION:
𝑟1 𝑡 + 𝑟2 𝑡 = 𝐴
1
1
𝑡+
𝑡=1
2
1.5
1
1
1
+
=
2 1.5
𝑡
𝑡 = 0.875 ℎ𝑟𝑠 (
6𝑜𝑚𝑖𝑛
) = 𝟓𝟏. 𝟒𝟑 𝒎𝒊𝒏.
1ℎ𝑟
11. Solve the differential equation 7yy’ = 5x.
A. 7x2 + 5y2 = C
B. 5x2 + 7y2 = C D. 5x2 - 7y2 = C
C. 7x2 - 5y2 = C
D. 70
SOLUTION:
7yy' = 5x
7y dy = 5x dx
∫ 7y dy = ∫ 5x dx
{(7/2)y² = (5/2)x² + C}1/2
5x2 - 7y2 = C
12. A cylindrical container open at the top with minimum surface area at a
given volume. What is the relationship of its radius to height?
A. radius = height
C. radius = height/2
B. radius = 2height
D. radius = 3height
13. A water tank is shaped in such a way that the volume of water in the tank
is V = 2y3/2cu. in. when its depth is y inches. If water flows out through a
hole at the bottom of the tank at the rate of 3(sqrt. Of y) cu. in/min. At what
rate does the water level in the tank fall?
A. 11 in/min
B. 1 in/min
C. 0.11 in/min
D. 1/11 in/min
SOLUTION:
2
𝑉 = 2𝑦 3
𝑑𝑉
= 3√𝑦
𝑑𝑡
𝑑𝑦
=?
𝑑𝑡
1 𝑑𝑦
𝑑𝑉
3
= (2)𝑦 2
𝑑𝑡
2
𝑑𝑡
1 𝑑𝑦
3√𝑦 = 3𝑦 2
𝑑𝑡
𝑑𝑦
= 1 𝑖𝑛⁄𝑚𝑖𝑛
𝑑𝑡
14. A family’s electricity bill averages $80 a month for seven months of the
year and $20 a month for the rest of the year. If the family’s bill were
averaged over the entire year, what would the monthly bill be?
A. $45
B. $50
C. $55
D. $60
SOLUTION:
80(7 𝑚𝑜𝑛𝑡ℎ𝑠)+20(5𝑚𝑜𝑛𝑡ℎ𝑠)
12 𝑚𝑜𝑛𝑡ℎ𝑠
= $ 55
15. When a baby born he weighs 8 lbs and 12 oz. After two weeks during his
check-up he gains 6 oz. What is his weight now in lbs and oz?
A. 8 lbs and 10 oz
C. 9lbs and 2 oz
B. 9 lbs and 4 oz
D. 10 lbs and 4 oz .
SOLUTION:
1 𝑙𝑏 = 16 𝑜𝑧
12 𝑜𝑧 + 6 𝑜𝑧 = 18 𝑜𝑧
8 𝑙𝑏𝑠 𝑎𝑛𝑑 18 𝑜𝑧 = 9 𝑙𝑏𝑠 𝑎𝑛𝑑 2 𝑜𝑧
16. A given function f(t) can be represented by a Fourier series if it
is periodic
is singled valued
is periodic, single valued and has a finite number of maxima and
minima in any one period
D. has a finite number of maxima and minima in any one period
A.
B.
C.
17.
A periodic waveform possessing half-wave symmetry has no
A. even harmonics
sine terms
B. odd harmonic
C.
D. cosine terms
18. N engineers an N nurses. If two engineers are replaced by nurses, 51
percent of the engineers and nurses are nurses. Find N,
A. 102
B. 100
C. 55
D. 110
SOLUTION:
0.51 (N+N) = N+2
N = 100
19.
If f(x) = 10^x + 1, then f(x + 1) – f(x) is equal to
A. 10(10^ + 1) B. 9(10^x)
C. 1
D. 9(10^x + 1)
SOLUTION:
f(x) = 10x + 1
f(x + 1) – f(x) = ?
f(x+1) = 10x + 1
= 10x . 10 + 1
f(x + 1) – f(x) = 10 . 10x – (10x + 1)
= 10 . 10x – 10x -1
= 9(10x)
20. There is a vector v = 7j, another vector u starts from the origin with a
magnitude of 5 rotates in the xy plane. Find the maximum magnitude of u x
v.
A. 24
B. 70
C. 12 D. 35
SOLUTION:
𝑢 𝑥 𝑣 = |𝑢||𝑣| sin 𝜃
𝑢 𝑥 𝑣 = (5 𝑥 7) 𝑠𝑖𝑛90
𝑢 𝑥 𝑣 = 35
21. Find the coordinates of the centroid of the plane area bounded by the
parabola y = 4 + x2 and the x-axis
A. (0, 1.5)
B. (0, 1)
D. (0, 1.6)
SOLUTION:
𝑦 = 4 + 𝑥20
𝑖𝑓 𝑥 = 0
𝑦=4
𝑖𝑓 𝑦 = 0
𝑥 ± 4
2
𝑥 =4−𝑦
𝑥 2 = −1(𝑦 − 4)
𝑉 (0,4)
𝐴=
2
∫−2 4
2
− 𝑥 𝑑𝑥
C. (0, 2)
𝐴=
2
32
3
𝐴 → = ∫ (4 − 𝑥 2 )(𝑥)𝑑𝑥
𝑥
−2
𝐴 →= 0
𝑥
→=0
𝑥
1 2
𝐴 → = ∫ (4 − 𝑥 2 )2
𝑦
2 −2
𝐴 → = 17.067
𝑦
17.067
= 1.6
𝑦
3
𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, ( 0 , 1.6)
→=
22. A long piece of galvanized iron 60 cm wide is to be made into a trough by
bending up two sides. Find the width of the base if the carrying capacity is a
maximum.
A. 30
B. 20
C. 40
D. 50
SOLUTION:
𝑥=
60
3
= 20
For max A, width on top must be equal to 3x the width below.
23. The price of gas increased by 10 percent. A consumer reacts by
decreasing his consumption by 10 percent. How does his total spending
change?
A. increase 1 percent
B. decrease 1 percent
C. no change
D. decrease 1.5 percent
SOLUTION:
a=10%, b= -10%
𝑎 + 𝑏 + 𝑎𝑏
100
10 + (−10) + (10)(−10)
=
100
= −𝟏%
= -1% (negative sign shows a decrease)
=
24. An audience of 450 persons is seated in rows having the same number of
persons in each row. If 3 more persons seat in each row, it would require 5
rows less to seat the audience. How many rows?
A. 27
B. 32
C. 24
D. 30
SOLUTION:
r - rows; n - number of persons
450 = rn = (r-5)(n+3)
rn = rn - 5n + 3r - 15
n = (3r-15)/5
450 = r*(3r-15)/5
750 = r² - 5r
r²-5r-750=0
(r-30)(r+25)=0
then r=30
25. The volume of a cube becomes three times when its edge is increased by
1 inch. What is the edge of a cube?
A. 2.62
B. 2.26
C. 3.26
D. 3.62
SOLUTION:
a3 = V
(a+1)3 = 3V
(a+1)3=3a3
a = 2.26
26. What is the angle of the sun above the horizon, when the building 150 ft
high cast a shadow of 405 ft?
A. 21.74
D. 69.68 deg
B. 68.26 deg
C. 20.32 deg
SOLUTION:
𝑦 = 150
𝑥 = 405
𝑦
𝜃 = Arc tan(𝑥 )
105
𝜃 = Arc tan(405)
𝜃 = 20.32
27. Water ir running out of a conical tunnel at the rate of 1 cu. in/sec. If the
radius of the base of the tunnel is 4 in and the altitude is 8 in, find the rate at
which the water level is dropping when it is 2 in from the top.
A. -1/9pi in/sec
B. -1/2pi in/sec
C. 1/2pi in/sec D.
1/9pi in/sec
SOLUTION:
dv/dt=1in3/sec
2r=h
V=(1/3)pi*r2h
=(1/3)pi*(h2/4)h
=(1/3)pi*(h3/4)
=(1/12)pi(3)h2 (dh/dt)
1in3/sec = (1/4)pi(6)2(dh/dt)
dh/dt= 1/9pi in/sec
28. A statistic department is contacting alumni by telephone asking for
donations to help fund a new computer laboratory. Past history shows that
80% of the alumni contacted in this manner will make a contribution of at
least P50.00. A random sample of 20 alumni is selected. What is the
probability that between 14 to 18 alumni will make a contribution of at least
P50.00?
A. 0.421
B. 0.589
D. 0.301
SOLUTION:
𝑝 = 0.8
𝑞 = 1 − 𝑝 = 1 − 0.8 0.2
𝑃 = 𝑛𝐶𝑟 𝑝𝑟 𝑞 20−𝑟
C. 0.844
18
𝑃 = ∑ 20𝐶𝑟 0.8𝑟 0.220−𝑟
14
= 0.844
29. Jun rows has banca a river at 4 km/hr. What is the width of the river if he
goes at a point 1/3 km.
A. 5.33 km
B. 2.25 km
C. 34.25 km
D. 2.44
30. Find the volume generated by revolving about the x-axis, the area bounded
by the curve y = cosh x from x = 0 to x = 1.
A. 5.34
B. 3.54
C. 4.42
D. 2.44
SOLUTION:
Use disk method
1
𝑉 = 𝜋 ∫0 𝑟 2 𝑑𝑥
1
𝑉 = 𝜋 ∫ 𝑐𝑜𝑠ℎ2 𝑥 𝑑𝑥
0
= 4.42
31.
1
A. - 4 xcos2x + C
Evaluate the integral of xsinxcosxdx
1
1
C. - 4 xsin2x + 8 xcos2x + C
B.
D. - 𝟒 xcos2x +𝟖 sin2x + C
1
8
xsin2x + C
𝟏
𝟏
SOLUTION:
1
(Sin(2x)=2sin(x)cos(x)) (2)
1
v = -(2)cos(2x)
du = (2)dx
dv = sin(2x)dx
1
= ½ sin(2x)=sin(x)cos(x)
1
= (2) x sin(2x)dx
udv = uv - vdu
1
1
1
1
= ((2)x) (-(2)cos(2x) + (2)cos(2x) (2)dx
1
1
1
= -(4)xcos(2x) + (4) (2)sin(2x) + C
𝟏
𝟏
1
u = (2)x
= -(𝟒)xcos2x + (𝟖)sin(2x) + C
32.
A cross-section of a trough is a semi-ellipse with width at the
top 18 cm and depth 12 cm. The trough is filled with water to a depth of 8 cm.
Find the width at a surface of the water.
A. 5√2 cm
B. 𝟏𝟐√𝟐 cm
C. 7√2 cm
D. 6√2 cm
SOLUTION:
𝑥2
𝑦2
Standard form : 81 + 144 = 1
Major axis: 24 = 2a ; a = 12 ; a2 = 144
Minor axis: 18 = 2b ; b = 9 ; b2 = 81
(x,y) = (x,-4)
𝑥2
16
+
=1
81
144
𝑥2
81
x2 =
3
16
= 1 – 144
81 𝑥 128
144
x = (4)*(√128) = 8.48
width of surface water = 2x = 16.97
33.
A.
cos2x
Simplify cos2x + sin2x + tan2x
B. sin2x
C. sec2x
D. csc2x
SOLUTION:
= sin2x + cos2x = 1
= 1 + tan2x = sec2x
What is the general solution of (D2 + 2)y(t) = 0?
A. y = C1cos2t + C2sin2t
C. C1cos√𝟐t + C2sin√𝟐t
B. y = C1sin2t + C2cos2t
D. C1sin√2t + C2cos√2t
SOLUTION:
(𝐷2 + 2)𝑦(𝑡) = 0
𝐷2 + 2 = 0
𝐷 = ±√−2
𝐷1 = +√2𝑖
𝐷1 = −√2𝑖
𝑦 = 𝐶1 𝑐𝑜𝑠√2𝑡 + 𝐶2 𝑐𝑜𝑠√2𝑡
34.
𝑥
35.
What is the distance between the lines, 1?
A. √6
B. 5
𝟗𝟎
C. √ 𝟕
D.
90
7
36.
What is a so that the points (-2, -1, -3), (-1, 0, -1) and (a, b,
3) are in straight line?
A. 2
B. 4
C. 3
D. 1
SOLUTION:
−2𝑖 − 𝑗 − 3𝑘 + −𝑖 − 𝑘 = 𝑎𝑖 + 𝑏𝑗 + 3𝑘
𝐴𝐵 𝑙𝑙 𝐴𝐶
−1
−2
1
𝐴𝐵 = ( 0 ) − (−1) = (1)
−1
−3
2
𝑎
𝑎+2
−2
𝐴𝐶 = (𝑏 ) − (−1) = (𝑏 + 1)
3
−3
𝑐
6
= 3 , 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝐴𝐵 𝑏𝑦 3
2
3(1) = 𝑎 + 2
𝑎 = 3−2= 1
37.
Find the volume generated when the area bounded by y = 2 x
– x and y = (x – 1)2 is revolved about the x-axis
A. 2.34
B. 3.34
C. 4.43
D. 1.34
38.
Find the centroid of a semi-ellipse given the area of semi4
ellipse as A=-ab and volume of the ellipse as V = 3 𝜋ab2
2𝑏
𝑏
A. 3𝜋
SOLUTION:
1
𝐴𝑠𝑒𝑚𝑖 = 2 𝜋𝑎𝑏
𝟒𝒃
B. 2𝜋
C. 𝟑𝝅
3𝑏
D. 4𝜋
4
𝜋𝑎𝑏 2
3
𝑉 = 𝐴 𝑥 2𝜋𝐷
4
1
𝜋𝑎𝑏 2 = 𝜋𝑎𝑏 𝑥 2𝜋𝐷
3
2
4
𝜋𝑎𝑏 2
3
𝐷= 2
𝜋 𝑎𝑏
4𝑏
𝐷=
3𝜋
𝑉=
39.
cards?
How many 5 poker hands are there in a standard deck of
A. 2,595,960
B. 2,959,960
C. 2,429,956
SOLUTION:
𝑛!⁄
𝑘!
C = (𝑛−𝑘)!
D. 2,942,955
52!⁄
5!
= (52−5)!
= 2,598,960
40.
A biker is 30 km away from his home, he travel 10 km and
rest for 30 mins. He travel the rest of the distance 2kph faster. What is his
original speed?
A. 7 kph
B. 10 kph
C. 8 kph
D. 12 kph
SOLUTION:
41. Cup A = fulll, cup B = full, cup C = full, cup D = 17
used to fill the three cups, what is left in the cup?
A. 1/2
B. 3/4
full. If the 4th cup is
C.
1/4
D. 19/36
SOLUTION:
A= 1-5/9=4/9
B= 1-5/6=1/6
C= 1-11/12=1/2
Total: 25/36
D=17/18 – 25/36 = 1/4
42. What percent of 500 is 750%
A.50
D. 125
B. 175
C.
57
SOLUTION:
(750)(100)/500= 150 or 125
43. Using power series expansion about 0, find cosx by differentiating from sinx
A. 1- (x^2/2!)+(x^4/4!)-(x^6/6!)+
B.x-(x^2/2!)+(x^4/4!)-(x^5/5!)+
C. 1-(x^3/3!)+(x^5/5!)-(x^7/7!)+
(x^7/7!)+
D.x-(x^3/3!)+(x^5/5!)-
SOLUTION:
𝑓(𝑥) = cos 𝑥
𝑓′(𝑥) = −sin 𝑥
𝑓(x)= -cos x
𝑓 3 (𝑥) = sin 𝑥
𝑓 4 (𝑥) = cos 𝑥
𝑓 5 (𝑥) = −sin 𝑥
𝑓 6 (𝑥) = −cos 𝑥
𝑓(0) = 1
𝑓′(0) = 0
f(0) = −1
𝑓 3 (0) = 0
𝑓 4 (0) = 1
𝑓 5 (0) = 0
𝑓 6 (0) = −1
𝑓′(𝑥)(𝑥 − 0)′ 𝑓"(𝑥)(𝑥 − 0)" 𝑓 3 (𝑥)(𝑥 − 0)3 𝑓 4 (𝑥)(𝑥 − 0)4
+
+
+
1!
2!
3!
4!
5
5
6
6
𝑓 (𝑥)(𝑥 − 0)
𝑓 (𝑥)(𝑥 − 0)
+
+
+⋯
5!
6!
1(𝑥 − 0)2
1(𝑥 − 0)4
1(𝑥 − 0)6
cos 𝑥 = 1 + 0 −
+0+
+0−
2!
4!
6!
𝑓(𝑥) = 𝑓(0) +
44. Find the area bounded by y = √4𝑥in the first quadrant and the lines x = and
x= 3
A. 7.8
B 6.7
C. 5.5
D.
6.5
SOLUTION:
𝑦 = √4𝑥
𝑥=3
3
𝐴 = ∫ √4𝑥 𝑑𝑥
0
𝐴 = 6.9
45. Express 2,400,000 in scientific notation
A. 2.4 x 10
D. 2.4 x 105
B. 2.4 x 106
SOLUTION:
2.40000x106
C.
24
x
10
46.An interior designer has to design two offices, each office containing 1 table,
1 chair, 1 mirror, 2 cabinets. A supplier gives him options between 4 tables,
5 chairs, 5 mirrors and 10 cabinets. In how many ways can he design the
offices assuming there is no repetition?
A. 14100
B. 2400
C.
21600
D. 1740
SOLUTION:
10𝑃2 ÷ (5𝑃1 + 5𝑃2 + 4𝑃1) = 119
1192 = 14161 ≈ 14100
47. What is the equation of a circle that passes through the vertex and the points
of latus rectum of y2 = x
A. x2 + y2 + 4x + 2y = 0
B. x2 + y2 + 10x = 0
C. x2 + y2 + 4y +2x = 0
D. x2 + y2 - 10x =0
48.Find the power series expansion of ln (1 – x)
A. 1 + x + (x^2)/2 + (x^3)/3 +
C. x + (x^2)/2 + (x^3)/3 + (x^4)/4 +
B. -1 – x – (x^2)/2 – (x^3)/3 D. –x –(x^2)/2 – (x^3)/3 – (x^4)/4 –
SOLUTION:
𝑓(𝑥) = ln(1 − 𝑥)
𝑓(0) = ln(1 − 0)
1
−1
−
1
=
= −1
(1 − 𝑥)1
1−0
1
−1
𝑓" =
−1=
− 1 = −1
2
(1 − 𝑥)
(1 − 0)2
−1
𝑓 3 (𝑥) =
= −1
(1 − 𝑥)3
−1
𝑓 3 (𝑥) =
= −1
(1 − 𝑥)3
1(𝑥 − 0)1 1(𝑥 − 0)2 1(𝑥 − 0)3 1(𝑥 − 0)4
ln(1 − 𝑥) = 0 −
−
−
−
1!
2!
3!
4!
𝑥2 𝑥3 𝑥4
ln(𝑥 − 1)𝑛 = −𝑥 − − −
2! 3! 4!
𝑓′ =
49. Evaluate 10(-20j) + 4(-4j)
A. 20
B. 20j
C. -20
D.-20j
50. Evaluate 1 = 1/(1+1/1+7)
A. 15/7
B. 13/15
C. 4/7
D. 7/4
51. The value of all the quarters and dimes in a parking meter is $18. There
are twice as many quarters as dimes. What is the total number of dimes in
the parking meter?
A. 40
B. 20
C. 60
D. 80
SOLUTION:
(. 25𝑄 + .10𝐷) = $18
2𝑄 = 𝐷
𝐷
. 25 ( ) + .10𝐷 = 18
2
𝐷 = 80
52. A ball is dropped from height of 12 m and it rebounds ½ of the distance it
falls. If it continues to fall and rebound in this way, how far will it travel
before coming to rest?
A. 36 m
B. 30 m
C. 48 m
D. 60 m
SOLUTION:
𝑆𝑛 = 𝑎𝑛 +
𝑆𝑛 = 12 +
𝑎𝑛
1− 𝑟
12
1
1− 2
= 36 m
53. At t = o, a particle starts at rest and moves along a line in such a way that
at time t its acceleration is 24t2 feet per second per second. Through how
many feet does the particle move during the first 2 seconds?
A. 32
B. 48
C. 64
D. 96
SOLUTION:
S = wot + at = 0 + 24(2) = 𝟒𝟖 𝐟𝐭.
54. If a trip takes 4 hours at an average speed of 55 miles per hour, which of
the following is closest to the time the same trip would take at an average
speed of 65 miles per hour?
A. 3.0 hours
B. 3.4 hours
C. 3.8 hours D. 4.1 hours
SOLUTION:
V1 t1 = V2 t 2
4
t2 = 55
= 3. 4 hrs
65
55. A laboratory has a 75-gram sample of radioactive materials. The half-life
of the material. The half life on the material is 10 days. What is the mass of
the laboratory’s sample remaining after 30 days?
A. 9,375 grams
B. 11.25 grams
C. 12.5 grams D. 22.5 grams
56. The unit normal to the plane 2x + y + 2z = 6 can be expressed in the
vector form as
A. i3 + j2 +k2
C. i1/3 + j1/2 + k1/2
B. i2/3 + j1/3 + k2/3
D. i2/3 + j1/3 + k1/3
c
2x =.+ y + 2z is also 2i + j + 2k
2𝑖 + 𝑗 + 2𝑘
√22 + 12 + 22
2𝑖 𝑗 2𝑘
=
+ +
3 3 3
57.
𝑑
𝑑𝑥
= (ln 𝑒 2𝑥 ) is
1
A. 𝑒 2𝑥
2
B.2𝑥
C. 2x
D. 2
SOLUTION:
𝑑
= 𝑙𝑛𝑒 2𝑥
𝑑𝑥
= 2𝑥
=2
58. Determine where, if anywhere, the tangent line to f(x) = x3 – 5x2 + x is
parallel to the line y = 4x + 23
A. x = 3.61 B. x = 3.23 C. x = 3
D. x = 3.43
SOLUTION:
𝑥 ′ = 3𝑥 2 − 10𝑥 + 1
𝑦′ = 4
2
3𝑥 − 10𝑥 + 1 = 4
59.
3𝑥 2 − 10𝑥 − 3 = 0
𝑥1 = 3.61 , 𝑥2 = −0.27
Which of the following is equivalent to the expression below?
(x2 – 3x + 1) – (4x – 2)
A. x2 – 7x – 1
B. x2 – 7x + 3
C. -3x2 – 7x + 3
SOLUTION:
(x2 − 2x + 1) − (4x − 3) = 0
D. x2+12x+2
𝐱𝟐 − 𝟕𝐱 + 𝟑 = 𝟎
60. For what value of k will x + have a relative maximum at x = -2?
𝑥
A. -4 B. -2 C. 2 D. 4
SOLUTION:
x − k/x = 0 ; x = −2
−2 − K/−2 = 0; 𝐤 = 𝟒
61. When the area in sq. units of an expanding circle is increasing twice as
fast as its radius in linear units, the radius is
A. 1/4 𝜋𝝅
C. 1 1/4
B. 0
D. 1
62. If the function f is defined by f(x)= f(0) = x5 – 1, then f-1, the inverse
function of f, is defined by f-1(x) =
A.
B.
SOLUTION:
C.
D.
f (0) = x5 – 1 =
f (x) = f-1(x) =
63. A school has 5 divisions in a class IX having 60, 50, 55, 62, and 58
students. Mean marks obtained in a History test were 56, 64, 72, 63 and 50
by each division respectively. What is overall average of the marks per
student?
A. 56.8
B. 58.2
C. 62.4
D. 60.8
SOLUTION:
Overall average = [56 + 56 + 64 + 72 + 63 + 50] ÷ 5 = 61 ≈ 60.8
64. The number n of ways that an organization consisting of twenty-six
members can elect a president, treasury, and secretary (assuming no
reason is elected to more than one position) is
A. 15600 B. 15400
C. 15200
D. 15000
SOLUTION:
26!/(26-3)! = 15600
65. Find the equation of the line that passes through (3, -8) and is parallel to
2x + 3y = 2
A. 2x + 3y = -18
C. 2x + 3y = -30
B. 2x + 3y = 30 D. 2x + 3y = 18
SOLUTION:
2x+3y=2; (3,8)
[3y= -2x+2] 1/3
y= -2x/3 + 2/3
Y= mx + b
m= - 2/3
y - y1= m (x-x1)
[y – 8 = - 2/3 (x-3)] 3
3y + 2x = 30 or 2x+ 3y =30
66. Find the center of the circle x2 + y2 + 16x + 20y + 155 = 0.
A. (-8, -10) B. (8, 10)
C. (8, -10) D. (-8, 10)
SOLUTION:
x2 +y2+16x+20y+155=0
(x2 +16x) + (y2-120y) =-155
(X2 + 16x + 64) + (y2 -120y + 100) = -155 + 64 + 100
(x+8)2 + (y+10)2 = 9
X= -8; y= -10 or h=-8 k=-10
P (-8,-10)
67. In how many ways can 5 red and 4 white balls be drawn from a bag
containing 10 red and 8 white balls?
A. 11760 B. 17640
C. 48620
D. none of these
SOLUTION:
10!/(10-5)! + 8!/(8-4)! = 31920
68. The area of a right triangle is 50. One of its angles is 45°. Find the
hypothenuse of the triangle
A. 10
B.
C. 10
D. 10
SOLUTION:
A=50
A=1/2 bh = 1/2 (h/sinǾ)(h)
Ǿ=45 sinǾ=h/b
b=h/sinǾ
h=
1
69. Each side of the square pyramid is 10inches. The slant height, H, of this
pyramid measures 12 in. What is the area in square inches, of the base of
the pyramid?
A. 100
B. 144
C. 120
D. 240
SOLUTION:
Ab= S2 =102 =100 sq. inches
70.
Find the exact value of
tan25°+tan 50°
1−tan 25° tan 50°
A. 1.732
B. 3.732
C. 2.732
D. 0.732
SOLUTION:
= 𝟑. 𝟕𝟑𝟐
71. Which term of the arithmetic sequence 2, 5, 8, … is equal to 227?
A. 74
B. 75
C. 76
D. 77
SOLUTION:
An = A1 + (n -1 )d
227 = 2 + (n -1) 3
n = 76
72. Name the type of graph represented by x2 – 4y2 – 10x – 8y + = 0
A. circle
B. parabola
C. ellipse
D. hyperbola
73. If logx 3 = ¼, then x =
A. 81
B.1/81
SOLUTION:
C. 3
D. 9
logx 3 = log 3 / log x
log 3 / log x = 1/4
log 3 (4) = log x (1)
x = 81
74. If f(x0 = -x2, then f(x + 1) =
A. –x2 + 1
B. –x2
C. –x2 – 2x
D. –x2 – 2x – 2
SOLUTION:
–(x+1)2=-(x2+2x+2-2)
=-x2-2x
75.If this graph of y = (x – 2)2 – 3 is translated 5 units up and 2 units to the right,
then the equation of the graph obtained is given by
A. y = x2 + 2
B. y = (x-2)2 + 5
C. y = (x + 2)2 + 2
D. y = (x – 4)2 + 2
SOLUTION:
(x+h)2=4a(y+k)
y-5=(x-2-2)2-3
y=(x-4)2+2
76. Which one is not a root of the fourth root of unity?
A. I
B. 1
D. –i
C. i/√𝟐
77.Find the area of the largest circle which can be cut from a square of edge 4
in.
A. 12.57
B. 3.43
C. 50.27
SOLUTION :
𝜋𝑑 2
𝐴=
4
D. 16
𝜋42
𝐴=
4
= 12.57𝑖𝑛2
78. If I = (-1)1/2, find the value of i36
A. 0
C. –I
B. I
D. 1
SOLUTION :
in = n/4
0.25 = i
0.50 = -1
0.75 = -i
1.00 = 1
therefore 36 / 4 = 9
Since 9 is a whole number i^36 = 1
79. If cot B = 5/2, find sin B
A.
/5
B.
C.
/2
D. 2/
SOLUTION :
B=cot-1(5/2) = 0.38
sin(0.38) =
2
√29
80. A man 1.60 m tall casts a shadow 4 m long. Nearby, a flagpole casts a shadow
18 m long. How high is the flagpole?
A. 6.4 m
B. 7.2 m
C. 4.5 m
D. 11.25 m
SOLUTION:
𝟐
𝟐
𝟐
𝑳𝟑 = 𝑿𝟑 + 𝒀𝟑
𝟐
𝟐
𝑳𝟐 = (𝑿𝟑 + 𝒀𝟑 )𝟑
𝑳𝟐 = 𝟓𝟖. 𝟕𝟔𝟓
𝑳 = √𝟓𝟖. 𝟕𝟔𝟓 = 𝟕. 𝟔𝟔𝒎
81. If Z1= 1-I, Z2= -2 + 4i, Z
a. B. 7.2 m
2i, Evaluate Z12+2z1-3.
C. 4.5 m
D. 11.25 mi+
SOLUTION:
(−2 + 4𝑗)2 + 2 ( 1 − 𝑗) − (√3 − 2𝑗)
= -11.73 -16j
82. A box contains 20 balls, 10 white, 7 blue, 3 red. What is the probability
that a ball drawn at random is red?
A. 3/20
B. 10/20
C. 7/20
D. 13/20
SOLUTION:
3
20
83. What is the probability of a three with a single die exactly 4 times out of 5
trials?
𝑃=
A. 25/776
B. 125/3888
C. 625/3888
D. 1/7776
84. A man is on a wharf 4 m above the water surface. He pulls in a rope to
which is attached a coat at the rate of 2 m/sec. How fast is the angle between
the rope and the water surface changing when there are 20 m of rope out?
A. 0.804 rad/sec B. 0.0408 rad/sec C. 0.0402 rad/sec D. 0.0204 rad/sec
SOLUTION:
𝑠𝑖𝑛𝜃 =
4
5
4
5
𝑑𝑢
𝑑
𝑑𝑥
−1
𝑠𝑖𝑛 =
𝑑𝑥
√1 − 𝑢2
𝑑𝑠
4 −0
𝑑𝑡
𝑑𝜃
𝑠2
=
2
𝑑𝑡
√1 − ( 4 )
20
𝜃 = 𝑠𝑖𝑛−1
4(2)
202
=
2
√1 − ( 4 )
20
𝑑𝜃
= 0.0204 𝑟𝑎𝑑/𝑠
𝑑𝑡
85. Find the area of the largest rectangle that can be inscribed in the ellipse
25x^2 + 16x^2 = 400
A. 30
B. 40
C. 10
D. 20
SOLUTION:
𝟐𝟓𝒙𝟐 + 𝟏𝟔𝒚𝟐 = 𝟒𝟎𝟎
𝟒𝟎𝟎
𝒙 𝟐 𝒚𝟐
+
=𝟏
𝟏𝟔 𝟐𝟓
𝒂 = √𝟏𝟔 = 𝟒
𝒃 = √𝟐𝟓 = 𝟓
𝑨 = (𝟒)(𝟓) = 𝟐𝟎
86. From the given values of A and B, find the vector cross product of A and B
if:
A=2i – k
B= j
A. 5i+2k
B. 4i-2k
C.3i-4j +2k
D. 3i-2j
87. The area of a lune is 30 sq. m. If the area of the sphere is 120sq. m. What is
the angle of the lune?
A. 80 degree
B. 90 degree
C. 120 degree
𝐴𝐿𝑈𝑁𝐸
𝜋𝑟 2 𝜃
=
= 30
90
𝐴𝑆𝑃𝐻𝐸𝑅𝐸 = 4𝜋𝑟 2
D. 60 degree
120
= 𝑟2
4𝜋
30(90)
= 𝑟2
𝜋𝜃
𝑟2 = 𝑟2
120
30(90)
=
4𝜋
𝜋𝜃
𝜃=
4 (30)(90)
= 90°
120
88. If tan x = ½, tan y = 1/3, what is the value of tan (x + y)?
A. 1
B. 2/3
C. 2
D. ½
SOLUTION:
𝟏
𝟏
+ 𝒕𝒂𝒏−𝟏 ) = 𝟏
𝟐
𝟑
89. Determine the distance between the foci of the curve 9x^2 + 18x + 25y^2 –
100y = 116
𝒕𝒂𝒏 (𝒕𝒂𝒏−𝟏
A. 8
B. 10
C. 12
D. 6
SOLUTION:
𝟗𝒙𝟐 + 𝟏𝟖𝒙 + 𝟐𝟓𝒚𝟐 − 𝟏𝟎𝟎𝒚 + 𝟗 + 𝟏𝟎𝟎 = 𝟐𝟐𝟓
𝟐𝟐𝟓
(𝒙 + 𝟏)𝟐 (𝒚 − 𝟐)𝟐
+
=𝟏
𝟓𝟐
𝟑𝟐
𝒂 = 𝟓 ;𝒃 = 𝟑
𝒇𝒐𝒄𝒊 𝒕𝒐 𝒇𝒐𝒄𝒊 = 𝟐(√𝟓𝟐 + 𝟑𝟐 ) = 𝟖
90. Using synthetic division, compute the remainder if we divide 2x^3 + x^2 =
18x + 7 by x -2
A. -9
B. -8
C. 7
D. 6
SOLUTION:
(x-2)
2
2
1
4
5
-18
10
-8
17
-16
-9
91. The force required to stretch a spring is proportional to the elongation. If 24
N stretches a spring 3 mm, find the force required to stretch a spring 2 mm.
A. 16
B. 18
C. 14
D.12
SOLUTION:
F= (24x 2mm)/ 3mm = 16N
92. A is 3 times as old as B. Three years ago, A is four times as old as B. Find
the sum of their ages.
A. 30
B. 36
C. 26
D. 28
SOLUTION:
4( X-3)- (X-3)=3X-X
4X- 12- X+3= 2X
3X-9= 2X
X= 9
B=X = 9
A=3X= 27
B+A= 9+27 = 36
93. The area of a rhombus is 264 sq. cm. If one of the diagonals is 24 cm long,
find the length of the other diagonal.
A. 22
B. 20
C. 26
SOLUTION:
𝑨=
𝟏
𝑫 𝑫
𝟐 𝟏 𝟐
D. 28
𝟏
(𝟐𝟒)𝑫𝟐
𝟐
𝑫𝟐 = 𝟐𝟐
𝟐𝟔𝟒 =
94. In a triangle ABC, angle A= 60 degree and angle B =45 degree. What is the
ratio of side BC to side AC?
A. 1:22
B. 1:36
C. 1:48
D. 1:19
95. Solve the equation cos^2 A= 1 – cos^A.
A. 45o, 315o B. 45o,225o
C. 45o,135o
SOLUTION:
𝑐𝑜𝑠 2 𝐴 = 1 − 𝑐𝑜𝑠 𝐴 𝑋
D. . 45o,225o
If A = 45
𝑐𝑜𝑠 2 𝐴 =
1
2
1
= 1 − 𝑐𝑜𝑠 45 𝑋
2
𝑋 = 10
360 − 45 = 315
∴ 45° , 315°
96. Find the distance from the point (6, -2) to the line 3x + 4y + 10 = 0.
A.4
B. 5.
C. 6.
D. 7
SOLUTION:
𝑑=
𝐴𝑥1 + 𝐵𝑦1 + 𝐶
±√𝐴2 + 𝐵 2
=
3(6) + 4(−2) + 10
√32 + 42
=4
97. If y = tanh x, find dy/dx :
A. sech^2 x
B. csch^2 x
C.sinh^2x
SOLUTION:
Y=tanh X
Y'= sech^2 X
98. What number exceeds its square by the maximum?
D. tanh^2 x
A. 1
B. ½
C. 1/3.
D. 1/4
SOLUTION:
D= X-X2
D=-X2+X
D=-(X2-X)
2
D=-(X -X+(1/2)2+(1/2)2
D=-(X-1/2)2-1/4
X= 1/2
99. Find the derivative of x^-8
A. -8x^-9.
B. -8x^-7
C. x^-9
D. 0
SOLUTION:
F(x)= X-8
F'(x)=-8x-9
100. Solve for x : X = (0.125)^-4/3
A. 8
SOLUTION:
X= (0.125)^-4/3
X= 16
B. 4
C. 16
D.2
REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
AUGUST 2016
MATHEMATICS
REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION
AUGUST 2016
MATHEMATICS
1. What is the area of the largest rectangle that can be inscribed in an ellipse with
equation 4x˄2+y˄2=4?
A. 3
B. 4
C. 2
D. 1
SOLUTION:
4𝑥 2 + 𝑦 2 = 4
Ellipse:
𝑥2
1
[4𝑥 2 + 𝑦 2 = 4 ] ( )
4
𝑦2
𝑥2 +
𝑦2
+ 𝑏2 = 1
𝑎2
𝑎2 = 1 , 𝑏 2 = 4 , 𝑏 = 2
=1
9
1
𝑥=
=
√2 √2
6
2
𝑦=
=
√2 √2
𝐴 = (𝑥)(𝑦)
1
2
𝐴 = ( ) ( ) = 1 sq. units Ans. 𝐃
√2 √2
4
2. Sand is pouring to form a conical pile such that its altitude is always twice its
radius. If the volume of a conical pile is increasing at rate of 25pi cu. Ft/min,
how fast is the radius is increasing when the radius is 5 feet?
A. 0.5 ft./min
B. 0.5pi ft./min
C. 5ft./min
D. 5pi ft./min
SOLUTION:
𝑉=
𝑉=
𝑉=
𝜋
3
r² h
𝜋
3
2𝜋
3
𝑑𝑣
𝑑𝑡
=
2𝜋
3
𝑑𝑡
𝑑𝑟
𝑟 2 (2𝑟)
𝑟3
𝑑𝑟
(3)(𝑟 2 )( )
25𝜋 = 2𝜋(5)2 𝑑𝑡
𝑑𝑟
𝑑𝑡
=
1
2
ft per minute Ans. 𝐀
3. An air balloon flying vertically upward at constant speed is suited 150m
horizontally from an observer. After one minute, it is found that the angle of
elevation from the observer is 28 deg 59 min. what will be then the angle of
elevation after 3 minutes from its initial position?
A. 63 deg 24 min
B. 58 deg 58 min
C. 28 deg 54 min
SOLUTION:
D.14 deg 07 min
@𝑡 = 1 𝑚𝑖𝑛 𝜃 = 28′ 59′′
ℎ(1)
𝑡𝑎𝑛 𝜃 =
= 130𝑡𝑎𝑛 (28′ 59′′ )
150𝑚
θ
ℎ = 83.089 𝑚
𝛥ℎ ℎ(1) 83.089
𝑉=
=
=
= 83.089
𝛥𝑡 1𝑚𝑖𝑛
1𝑚𝑖𝑛
@𝑡 = 3𝑚𝑖𝑛𝑠
ℎ(3) = 𝑣𝑡
83.089
=
= 249.268 𝑚
3𝑚𝑖𝑛𝑠
249.268
𝜃 = 𝑡𝑎𝑛−1 (
) = 58′ 57"Ans. 𝐁
150
4. A machine only accepts quarters. A bar of candy cost 25ȼ, a pack of peanuts
cost 50ȼ and a bottle of a coke cost 75ȼ. If Marie bought 2 candy bars, a pack
of peanuts and a bottle of coke, how many quarters did she pay?
A. 5
B. 6
C. 7
D. 8
SOLUTION:
𝐶𝑎𝑛𝑑𝑦 = 25ȼ
𝑃𝑒𝑎𝑛𝑢𝑡𝑠 = 50ȼ
𝐶𝑜𝑘𝑒 = 75ȼ
2 Candy + 1 peanut + 1 coke
2(25) + 75 + 50 = 175ȼ
Note: 1ȼ = 0.04 quarters
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 175(0.04) = 7 quarters Ans. 𝐂
5. A ball is dropped from a height of 18 m. On each rebound it rises 2/3 of the
height from which it last fell. What is the total distance it travels in coming to
rest?
A. 80m
B. 90m
C. 72 m
D. 86 m
SOLUTION:
2
𝑎𝑟 = (3) (18) = 12
𝑠4 = 𝑎1 [
1−𝑟 𝑛
1−𝑟
]
𝑎2 =
2
3
2
[ 3 (18)] = 8
𝑠4 = [
2 4
3
2
1−
3
1−( )
] = 28.89
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑡𝑜𝑡𝑎𝑙 = 18 + 2 (28.89 ) = 75.78 ≈ 72m Ans. 𝐂
6. Evaluate lim (𝑥 + 4 sin 𝑥)
𝑥→13𝑝𝑖
A. 2
B. 1
C. -1
D. 0
SOLUTION:
lim sin(𝑥 + 4 sin 𝑥 )
𝑥→13𝜋
cos (x + 4cos x)
cos (1 + 4 cos(13𝜋)) = 1 Ans. B
7. Find the length of the arc of the parabola x˄2=4y from x= ˗2 to x= 2.
A. 4.2
B. 4.6
C. 4.9
D. 5.2
SOLUTION:
1
S= 2 √16ℎ² + 𝑏² +
𝑏²
ln √4ℎ +
8ℎ
√16ℎ²+𝑏²
𝑏
2
1
4²
16(1) +4²
S= 2 16(1)² + 4² + 8(1) ln √4(1) + √
4
S= 4.6 units Ans. B
8. Find the coordinates of the centroid of the plane area bounded by the
parabola y=4 - x˄2 and the x-axis.
A. (0,1.5)
B. (0,1)
C. (0,2)
D. (0, 1.6)
SOLUTION:
2
𝑥′ = 0
2
𝑦′ = 5 ℎ
𝑦 ′ = (5) (4)
(0, 1.6) Ans. 𝐃
𝑦 ′ = 1.6
9. In how many ways can you pick 3 dogs from a pack of 7 dogs?
A. 32
B. 35
C. 30
D. 36
SOLUTION:
=
𝑃!
(𝑃 − 𝑛)! 𝑛!
7!
(7 − 3)! 3!
= 35 ways Ans. 𝐁
10. In how many ways can 4 coins be tossed?
A. 8
B. 12
C. 16
=
D. 20
SOLUTION:
2 faces of coin and 4 coins
2 x 2 x 2 x 2= 16 Ans. C
11. Which of the following is not multiple of 11?
A. 957
B. 221
C. 122
D. 1111
SOLUTION:
221
11
= 20.09 Therefore; 221 Ans. B
12. A certain rope is divided into 8 m, 7 m, 5 m. What is the percentage of 5 m
with the original length?
A. 20
B. 15
C. 10
D. 25
SOLUTION:
8+7+5=20m
20m x %= 5m=25% Ans. D
13. Nannette has a ribbon with a length of 13.4 m and divided it by 4. What is
the length of each part?
A. 3.35 m
B. 3.25 m
C. 3.15 m
D. 3.45 m
SOLUTION:
13.4m
4
= 𝟑. 𝟑𝟓𝒎 Ans A.
14. The area in the second quadrant of the circle x˄2 + y˄2 = 36 is revolved
about the line y+ 10 = 0. What is the volume generated?
A. 2208.53
B. 2218.33
C. 2228.83
D. 2233.48
SOLUTION
𝑥 2 + 𝑦 2 = 36
𝑥 2 + 𝑦 2 = 62
(0,0)𝑟 = 6
𝑦 + 10 = 0
𝑦 = −10
𝑉 = 𝐴(2𝜋𝐶 ′ )
𝜋
416
𝑉 = ( (6)2 ) (2𝜋) (10 +
)
4
3𝜋
𝑽 = 𝟐𝟐𝟐𝟖. 𝟗𝟏𝟖 𝑨𝒏𝒔. 𝑪
15. It represents the distance of a point from the y-axis.
A. Abscissa
B. Ordinate
C. Coordinate
D. Polar distance
16. In polar coordinate system, the polar angle is negative when;
A. Measured counterclockwise
C. measured at the terminal side of ϴ
B. Measured clockwise
D. none of these
17. A coin is tossed in times. If it is expected that 7 heads will occur, how
many times the coin is tossed?
A. 12
B. 14
C. 16
D. 10
SOLUTION:
One result in 2 sides of coin in every toss = ½
7 heads in every tossed coin = 7/x
Equating the equation:
1
7
=
2
x
X = 14
18. A long piece of galvanized iron 60 cm wide is to be made into a trough by
bending up two sides. Find the width of the sides of the base if the carrying
capacity is maximum?
A. 30
B. 20
C. 40
D. 50
SOLUTION:
1
𝐴 = 2 (𝑏1 + 𝑏2)ℎ
𝐴 = 𝑏ℎ
1
𝐴 = (60 − 2𝑥)(𝑥)
𝐴 = 2 (20 + 40)(10√3)
𝐴 = 60𝑥 − 2𝑥 2
𝐴 = 519.6𝑐𝑚2 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝐴 𝑖𝑠 𝑛𝑜𝑡 @ 𝒃 = 𝟐𝟎𝒄𝒎
𝑑𝐴
= 60 − 4𝑥 = 0
𝑑𝑥
𝑥 = 15
𝑏 = 60 − 2𝑥
= 60 − 2(15)
𝑏 = 30
𝐴 = 30(15)
𝐴 = 450
19. Totoy is 5 ft. 11 in. Nancy is 6 ft. 5 in. What is the difference in their
height?
A. 5 in
B. 6 in
C. 7 in
D. 8 in
SOLUTION:
5ft.and 11 in
5ft x
12 𝑖𝑛
1 𝑓𝑡
= 60in
60 in + 11 in = 71in
77in – 71 in = 6 in Ans. B
20. 5 years-old Tomas can tie his shoelace in 1.5 min and his right shoelace
in 1.6 min. How long will it take him to tie both shoe lace?
A. 2.9 min
B. 3 min
C. 3.1 min
D. 3.2 min
SOLUTION:
𝐿𝑠ℎ𝑜𝑒 = 1.5𝑚𝑖𝑛
𝑅𝑠ℎ𝑜𝑒 = 1.6𝑚𝑖𝑛
1.5min + 1.6 min = 3.1 min Ans. C
21. The area enclosed by the ellipse 4x˄2+9y˄2 = 36 is revolved about the
line x = 3, what is the volume generated?
A. 370.3
B. 360.1
C. 355.3
D. 365.1
SOLUTION:
[4x² + 9y² = 36} , 𝑥 = 3
𝑟 =3−𝑥
1
y = ±2√1 −
[4x² + 9y² = 36} 36
𝑥²
9
+
𝑦²
4
=1
𝑥2
9
therefore:
𝑥2
h = (2√1 −
h = 4√1 −
) − (−2√1 −
9
𝑥2
9
)
𝑥2
9
𝑏
V = 2π ∫ 𝑟ℎ𝑑𝑥
𝑎
3
V = 2π ∫ (3 − x)(4√1 −
−3
𝑥2
)𝑑𝑥
9
V = 355.3 cu. units
22. The equation y² = cx is the general solution of
A. y’= 2y/x
B. y’= 2x/y
C. y’= y/2x
D. y’= x/2y
SOLUTION:
𝑦² = 𝑐𝑥
2𝑦 𝑦´ = 𝑥
𝒚´ =
𝒙
𝟐𝒚
23. Solve the differential equation y’=y/2x.
A. y= cx
B. y˄2= cx
C. y= cx˄2
SOLUTION:
D. y˄3= cx
𝑦´ =
𝑑𝑦
𝑦
[𝑑𝑥 = 2𝑥]
ʃ
𝑑𝑦
𝑦
𝑑𝑥
𝑦
2𝑥
𝑦
𝑑𝑥
= ʃ 2𝑥
𝑙𝑛 𝑦 = 2 𝑙𝑛 𝑥 + 𝑐
𝑙𝑛 𝑦 = 𝑙𝑛 𝑥² + 𝑐
𝑦
𝑒 𝑙𝑛 (𝑥 2 ) = 𝑒 ln 𝑐
𝑦
𝑥2
= 𝑐
𝒚 = 𝒄𝒙²
24. In a school, 30 percent of students are involved in athletics. 15 percent of
these play football. What percent of the student in the school play football?
A. 4.5
B. 15
C. 5.4
D. 30
SOLUTION:
A = 0.35 → 5 =
A
0.03
𝐹 = 0.15𝐴
𝐹
15𝐴
= 0.
𝑥100
𝐴
3
0.3
(0.3)(0.15)𝑥100
=
= 𝟒. 𝟓%
25. Find the point along the line x = y = z that is equidistant from (3, 0, 5) and
(1, -1, 4).
A. (1, 1, 1)
B. (2, 2, 2)
C. (3, 3, 3)
D. (4, 4, 4)
SOLUTION:
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 + (𝑧2 − 𝑧1 )2
𝑑 = √(2 − 3)2 + (2 − 0)2 + (2 − 5)2 = √14
𝑑 = √(2 − 1)2 + (2 − (−1))2 + (2 − 4)2 = √14
answer: (2, 2, 2)
26. Which of the following is divisible by 6?
A. 792
B. 794
C. 790
SOLUTION:
D. 796
792
6
= 132 Therefore; 792 is divisible by 6
27. The cost of operating a vehicle is given by C(x) = 0.25x + 1600, where x is
in miles. If Jam just bought a vehicle and plan to spend between P5350 to
P5600. Find the range of distance she can travel.
A.14000 to 15000
B. 15000 to 16000
C. 16000 to 17000
D. 13000 to 14000
SOLUTION:
𝐶(𝑥) = 0.25𝑥 + 1600𝐶(𝑥) = 5350 – 5600
𝐶(𝑥) = 0.25𝑥 + 1600
5350 = 0.25𝑥 + 1600 = 𝟏𝟓𝟎𝟎𝟎
5600 = 0.25𝑥 + 1600 = 𝟏𝟔𝟎𝟎𝟎
28. A 20-ft lamp casts a 25 ft. shadow. At the same time, a nearby building
casts a 50 ft. shadow. How tall is the building?
A. 20 ft.
B. 40 ft.
C. 60 ft.
D. 80 ft.
SOLUTION:
20ft
20
𝛳 = 𝛳𝑡𝑎𝑛−1 = 25 = 38.66 ͦ
25ft
H
𝐻
𝛳 = 𝛳 𝑡𝑎𝑛(38.66 ͦ) = 25 = 𝟒𝟎𝒇𝒕
50ft
29. Three circle of radii 3, 4, and 5 inches, respectively, are tangent to each
other externally. Find the largest angle of a triangle found by joining the
centers of the circle.
A. 72.6 degrees
B. 75.1 degrees
C. 73.4 degrees
D. 73.5 degrees
SOLUTION:
𝑆=
7+8+9
= 12
2
𝐴 = √(12)(12 − 7)(12 − 8)(12 − 9)
𝐴 = 26.83
1
𝐴 = 2 𝑎𝑏𝑠𝑖𝑛
1
26.83 + (7)(8)𝑠𝑖𝑛𝜃
2
𝜽 = 𝟕𝟑. 𝟑𝟖
30. Simplify the expression cos²ϴ-sin²ϴ
A. cos 2ϴ
B. sin 2ϴ
C. sin 2ϴ
D. sec 2ϴ
SOLUTION:
𝑐𝑜𝑠²𝛳 − 𝑠𝑖𝑛² 𝛳 = 𝒄𝒐𝒔 𝟐𝜭
31. csc 520º =?
A. Cos 20º
SOLUTION:
B. csc 20º
C. sin 20º
D. sec 20º
1
csc 520 ͦ = sin 520 ͦ
1
2.92= sin 520ͦ
1
sin 20 ͦ
= 2.92
Therefore; CSC 20 ͦ =
32. Simplify x/(x – y) + y/(y –x).
A. -1
B. 1
SOLUTION:
𝟏
𝑺𝒊𝒏
C. x
𝑥
𝑥
+
𝑥−𝑦 𝑦−𝑥
𝑥(𝑦−𝑥)+𝑦 ( 𝑥−𝑦)
(𝑥−𝑦)(𝑦−𝑥)
D. y
𝑥𝑦−𝑥 2 +𝑦(𝑥−𝑦)
𝑥𝑦−𝑥 2 +𝑥𝑡−𝑦 2
−𝒙𝟐 −𝒚²
−𝒙𝟐 −𝒚²
=𝟏
cos 𝐴
33. Simplify 1−sin 𝐴 − tan 𝐴.
A. csc A
B. sec A
SOLUTION:
C. sin A
D. cos A
Assume A=30 @Radmode Trial and Error
(Cos 30/1-sin30)-Tan30=6.48
Sec 30= (1/cos 30) = 6.48
=Sec 30
34. Find the minimum distance from the point (4, 2) to the parabola y² = 8x.
A. 3 sqrt. of 3
B. 2 sqrt. of 3
C. 3 sqrt. of 2
D. 2 sqrt. of 2
SOLUTION:
LR=8
a=2
x=2
y=2
d=√22 + 22
=2√𝟐
35. From the past experience, it is known 90 percent of one year old children can
distinguish their mother’s voice of a similar sounding female. A random sample
of one year’s old are given this voice recognize test. Find the standard
deviation that all 20 children recognize their mother’s voice?
A. 0.12
B. 1.34
C. 0.88
D. 1.43
SOLUTION:
= √ℎ𝑝𝑞
= √20(0.9)(1 − 0.9)
= 𝟏. 𝟑𝟒
36. An equilateral triangle is inscribed in the parabola x² = 8y such that one of its
vertices is at the origin. Find the length of the side of the triangle.
A. 22.51
B. 24.25
C. 25.98
D. 27.71
SOLUTION:
ℎ=
𝑎√3
2
√3
𝑦 = 2𝑥 ( )
2
𝑥2
= 8√3
8
𝑥 = 8√3
9 = 2𝑥
= 2(8√3)
= 𝟐𝟕. 𝟕𝟏
37. Mary’s father is four time as old as Mary. Five years ago he was seven times
as old. How old is Mary now?
A. 8
B. 9
C. 11
D.10
SOLUTION:
Mary = x-5
Father = 4x-5
7(x-5) = 4x-5
X= 10 Ans. D
38. The lateral area of a right circular cylinder is 77 sq. cm. and its volume is 231
cu. cm. Find its radius.
A. 4 cm
B. 5 cm
C. 6 cm
D. 7 cm
SOLUTION:
A =77 cm2
V = 231 cm2
A = 2ᴨrh
V = ᴨr2h
HA = HV
A/2ᴨr = V/ᴨr2
A/2ᴨr = V/ᴨr2
71/2ᴨr = 231/ᴨr2
r= 6 Ans. C
39. A weight of 60 pounds rest on the end of an 8-foot lever and is 3 feet from the
fulcrum. What weight must be placed on the other end of the lever to balanced
60 pound weight?
A. 36 pounds B. 32 pounds
C. 40 pounds
D. 42 pounds
SOLUTION:
5x = 60 (3)
5x= 180
X = 36 lbs.
40. The average of six scores is 83. If the highest score is removed, the average
of the remaining scores is 81.2. Find the highest score.
A. 91
B. 92
C. 93
D. 94
SOLUTION:
𝒙
(81.2x5)+x/6 = 92 (𝟖𝟏. 𝟐)(𝟓) + 𝟔=9
41. A point moves on the hyperbola x²- 4y² = 36 in such a way that the xcoordinate increase at a constant rate of 20 unit per second. How fast is the ycoordinate changing at a point (10, 4)?
A. 30 units/sec
C. 30 units/sec
B. 30 units/sec
D. 30 units/sec
SOLUTION:
x 2 − 4y 2 = 36
2𝑥𝑑𝑥 8𝑦𝑑𝑦
−
=0
𝑑𝑡
𝑑𝑡
𝑑𝑦
𝑥 𝑑𝑥
=
𝑑𝑡 4𝑦 𝑑𝑡
10
(20) = 12
=
4(4)
𝐵𝑦 𝑇𝑟𝑜𝑢𝑏𝑙𝑒𝑠ℎ𝑜𝑜𝑡𝑖𝑛𝑔:
𝟏𝟎
(𝟏𝟐) = 𝟑𝟎𝒖𝒏𝒊𝒕/𝒔𝒆𝒄
𝟒
42. If the tangent of angle A is equal to the square root of 3, angle A in the 3rd
quadrant, find the square of the tangent A/2.
A. 2
B. 3
C. 4
D. 5
SOLUTION:
tan A = √3
A = tan-1(√3)
A = 180 – 60 = 120
[ tan (A/2) ]2 = [ tan(120/2) ] = (√3) = 3
43. A stone, projected vertically upward with initial velocity 112 ft./sec, moves
according to s = 112t – 16t², where s is the distance from the starting point.
Compute the greatest height reached.
A. 196 ft.
B. 100 ft.
C. 96 ft.
D. 216 ft.
SOLUTION:
dS = 112t- 16t2
𝑑𝑠
𝑑𝑡
= 112-32(t) = 0
@ t = 3.5s
S = 122 (3.5) -16 (3.5)2 = 196ft
44.) A cylinder of radius 3 is cut through the center of the base by a plane making
an angle of 45 degrees with the base. Find the volume cut off.
A. 15
B. 16
C. 17
D. 18
SOLUTION:
h
𝑉 = (𝐴12 + 𝐴13 + 𝐴14)
6
3
1
= [(0 + 0 + 4) ( ) (3)(3)]
3
2
𝑽 = 𝟏𝟖 𝒄𝒖𝒃𝒊𝒄 𝒖𝒏𝒊𝒕
45.) Find the diameter of a circle with the center at (2, 3) and passing through the
point (-1, 5).
A. 3.6
B. 7.2
C.13
D. 16
SOLUTION:
(x-h) 2 + (y-k) 2= r 2
(-1-2) 2+ (5-3) 2=r 2
√13 = √r 2
r = √13
d = 2(r) = 2 (√13) = 7.21
46.) Find the value of x for which the tangent to y = 4x-x² is parallel to the x-axis.
A. 2
B. -1
C. 1
D. -2
SOLUTION:
y = 4x – x2
y = x2 – 4x
y = (x – 2)2
if y = 0
Therefore, x= -2
47. Find the surface area generated by rotating the parabolic arc about the x-axis
from x = 0 to x = 1.
A. 5.33
B. 4.98
C. 5.73
D. 4.73
SOLUTION:
𝑑𝑥
𝑦 = 𝑥 2 = 𝑑𝑦 = 2𝑥
𝑆 = ∫ 2𝜋𝑟 𝑑𝑠q7’
𝑑𝑦 2
= ∫ 2𝜋𝑦√1 + ( ) 𝑑𝑥
𝑑𝑥
0
1
1
∫ 2𝜋𝑥 2 √1 + (2𝑥)2 𝑑𝑥 = 𝟓. 𝟐𝟕𝟗 ≈ 𝟓. 𝟑𝟑
0
48. A group of students plan to pay equal amount in hiring a vehicle for an
excursion trip at a cost of P 6, 000. However, by adding 2 more students to the
original group, the cost of each student will be reduced by P 150. Find the
number of each students in the original group.
A. 10
B. 9
C. 8
D. 7
SOLUTION:
6000
8
6000
10
= 750
= 600
750 – 600 = 150
Therefore, 8 is the no. of students in original group
49. What is the allowable error in measuring the edge of the cube that is intended
to hold 8 cu. m., if the error of the computed volume is not to exceed 0.03 cu.m.
A. 0.002
B. 0.003
C. 0.0025
D. 0.001
SOLUTION:
3
V = E3
E = √8 = 2
𝑑𝑉 = 3𝐸 2 𝑑𝐸
𝑑𝑉
0.03
𝑑𝐸 = 3𝐸2 = 3 ×22 = 0.0025
50. Find the value of x for which y = 2x³- 9x² + 12x – 2 has a maximum value.
A. 1
B. 2
C. -1
D. -2
SOLUTION:
y = 2x3- 9x2+12x – 2
y’= 6x2 – 18x + 12 = 0
By Quadratic Formula [mode, 5, 3]
x = 1, x=2
51. At a height of 23,240 ft., a pilot of an airplane measures the angle of
depression of a light at an airport as 28 deg 45 min. How far is he from the
light?
A. 20,330 ft.
B. 26,510 ft. C. 11, 180 ft.
D. 48, 330 ft.
SOLUTION:
23240
Sin Ɵ =
𝑦
Sin( 28′45′ ) =
23240
𝑦
y= 48137ft or 48,330f
52. A substance decreases at a rate which is inversely proportional to the amount
present. If 12 units of the substance are present initially and 8 units are present
after 2 days, how long will it take the substance to disappear?
A. 1.6 days
B. 2.6 days
C. 3.6 days
D.4.6 days
53. A tower 150m high is situated at the top of a hill. At a point 650m down the
hill, the angle between the surface of the hill and the line of sight to the top of
the tower is 12 deg 30 min. Find the inclination of the hill to a horizontal plane.
A. 7 deg 50 min
B. 20 deg 20 min
C. 77 deg 30 min
SOLUTION:
By Sine Law
D. 12 deg 55 min
sin(12°30′) sin 𝐶
=
150
650
C=69.70°
Answer=90° − 69.70° − 12°30′= 7°48’ ≅7°50’
54. A telephone company has a profit of $80 per telephone when the number of
telephones in exchange is not over 10,000. The profit per telephone decreases
by $0.40 for each telephone over 10, 000. Find the numbers of telephone that
will yield the largest possible profit.
A. 13,000
B. 14,000
C. 15,000
D. 16,000
55. Find the work done in moving an object along the vector a = 3i + 4j if the force
applied is b = 2i +j.
A. 11.2
B. 10
C. 12.6
D. 9
SOLUTION:
A=3i+4j , B=2i+j
5+√5=11.2
√32 + 42 =5
2
2
√2 + 1 =√5
56. A man is paid P 1, 800 for each day he works and forfeits P 300 for each day
he is idle. If at the end of 40 days, he nets P 53, 100, how many days was he
idle?
A. 6
B. 7
C. 8
D. 9
SOLUTION:
let X number of days he idle
40-X number of days he work
1800(40-X)-300X=53100
X=9
57. By stringing together 9 differently color beads, how many different bracelets
can be made?
A. 362,880
B. 20,160
C. 40,320
D. 181,440
SOLUTION:
(9!)=362,880
58. In a circle of diameter 26 cm, a chord 10 cm in length is drawn. How far is the
chord from the center of the circle?
A. 5 cm
B. 12 cm
C. 13 cm
D. 24 cm
SOLUTION:
D=26cm
L=10cm
√132 + 52 = 12
59. Find the slope of the line passing through the pair of points (-2, 0) and (3, 1).
A. 1/3
B. 1/4
C. 1/6
D. 1/5
SOLUTION:
𝒚𝟐−𝒚𝟏
𝒎=𝑿
𝟐 −𝑿𝟏
1−0
𝟏
=3+2= 𝟓
60. Find the inverse of the function f(x) = sqrt. of (2x – 3).
A.sqrt. of (2y-3)
B. 1/ sqrt. Of (2x-3)
½(x2+3)
D. ½ (y2+3)
SOLUTION:
F(x)=√2𝑋 − 3
Y=√2𝑋 − 3
𝑦 2 = 2𝑥 − 3
𝑦2 − 3
=𝑥
2
𝟏
𝟑
𝑿 = 𝒚𝟐 +
𝟐
𝟐
C.
61. If f (3) =7, f’ (3) = -2, g (3) =6 and g’ (3) = -10, find the (g/f)’ (3).
A. -82/49
B. -49/82
C. -49/58
D. -58/49
SOLUTION:
f ( g’ ) – g (f’) / f 2 (3) = 7(-10) –
6 (−2)
72
2
= -70 + 49 = -58/49
62. The length of the median drawn the hypotenuse of a right triangle is 12 inches.
Find the length of the hypotenuse.
A. 24 in
B. 20 in
C. 23 in
D. 25 in
SOLUTION:
H = 12 + 12 = 24
63. Find the derivative of the function y = 3/(x²+ 1).
A. 6x/(x2+1)2
B.6x(x2+1)2
C. -6x/(x2+1)2
D.-6x(x2+1)2
SOLUTION:
3
y = (x²+ 1)
y = 3(x²+ 1)-1
y’ = 3(x²+ 1)-2 (2x) = −
𝟔
(𝐗 𝟐 +𝟏)
𝟐
64. A passenger in a helicopter shines a light on a car stranded 45 ft from a point
just below the helicopter is hovering at 85 ft, what is the angle of depression
from the light source to the car?
A. 82 degrees
B. 80 degrees
C. 60 degrees
D.62 degrees
SOLUTION:
85
θ = tan−1 (45) = 𝟔𝟐. 𝟏𝟎
65. Find the area bounded by the curve r = 8 cos ѳ.
A. 50.27
B.12.57
C. 8
D. 67.02
SOLUTION:
A = (ᴨ / 4) (a2)
A = (ᴨ / 4) (42) = 12.57 sq.units
66. If 2log4x – log49 = 2, find the value of x.
A. 10
B. 12
C. 11
D. 9
SOLUTION:
2 log4 x – log4 9 = 2
Solving x,
x = 12
67. Find the value of 2 cos (pii/4).
A. 1.41
B. 1.41i
C. 2.65
D. 265i
SOLUTION:
𝜋
4
𝑥
180
𝜋
= 45 degrees
2 cos (45) = √2 = 1.414
68. A pole is on top of a building. At a point 240 meters from the base of the
building, the angle of elevation of the base and top of the pole are 42 degrees
and 44 degrees respectively. Find the height of the pole.
A. 15.8m
B. 18.5m
C. 16.9m
D. 19.6m
SOLUTION:
Base to top
o
tan ᴓ = a
Base to pole
tan(44) =
h
tan(42) = 240
216+x
240
x= 15.8 m
h = 216 m.
69. The volume of a hemisphere of radius 2 m is
A.14.67 cu.m
B.67.04cu.m
C.16.76cu.m
D.33.53cu.m
SOLUTION:
V=
2
3
πr 3 =
2
3
π(2)3 = 16.76 m3
70. Five scores and 4 years is equivalent to how many years?
A. 49
B. 29
C. 54
D. 104
SOLUTION:
5 scores and 4 years ; 100 years + 4 years = 104 years
71. Find the equation of one of the asymptotes of the hyperbola 𝑥 2 − 4𝑦 2 −
6𝑥 − 8𝑦 + 1 = 0.
A. x – 2y – 5 = 0 B. x – 2y + 5 = 0
C. x – 2y – 1 = 0
D. x – 2y + 1 = 0
72. The wheel of a truck is turning at 6 rps. The wheel s 4 ft in diameter. Find
the linear velocity iin fps point on the rim of the wheel.
A.75.4
B.57.4
C.150.8
D.105.8
SOLUTION:
2𝜋
ῳ = (6 rps)(1𝑟𝑒𝑣) = 37.7 rad/sec
d = 4ft; r = 2ft
v = rῳ = (2ft)(37.7 rad/sec) = 75.4 ft/sec
73. Solve the inequality 3 – 2x < 4x -5.
A. x < 4/3
B. x > 4/3
SOLUTION:
-4x-2x < -3-5
-6x < -8
x < 4/3
C. x < ¾
D. x > ¾
74. The polynomial 𝑥 2 + 4𝑥 + 4 is the area of a square floor. What is the length
of its side?
A. x + 2
B. x – 2
C. x + 1
D. x – 1
SOLUTION:
A = s2
√𝑥2 + 4𝑥 + 4 = s
s=x+2
75. If there are 2 computers for every 4 students, how many computers are
needed .For 60 students?
A.24
B.26
C.30
D.32
SOLUTION:
2:4 = 60:x
4x = (60)(2)
X = 30
76. From Pagasa island in the Spratlys, two helicopters travel to two different
islands.One helicopter travels 185 km N 65 deg E to island A and the other
travels at S 25 deg E for 120 km to island B. What is the distance between the
two islands?
A. 198.5 km
B. 187.3
C. 235.2
D. 202.5
SOLUTION:
Ɵ1 = 65°
Ɵ2 = 25°
ƟT = 90°
AB = √(185)2 + (120)² = 202.5 km
77. If x = y + 2, what is the value of (𝑥 − 𝑦)4 ?
A. 10
B. 16
C. 18
SOLUTION:
(y + 2 – y)4 = ?
24 = 16
D. 24
78. An equilateral triangle has sides of 8 inches. What us the height?
A. 6.32 in
B. 6.93 in
C. 5.66 in
D. 6.56 in
SOLUTION:
s = 8/2 = 4
h = √(8)2 − (4)2 = 6.93in
79. If in the Fourier series of a periodic function, the coefficient 𝑎0 = 0 and 𝑎𝑛 =
0, then It must be having ____________ symmetry.
A. odd
B. odd quarter-wave
C. even
D. either A or B
80. Find the area of the triangle whose vertices are (4,2,3), (7,-2,4) and (3,-4,
6).
A. 15.3
B. 13.5
C. 12.54
D. 12.45
81. Find the moment of inertia of the area bounded by the curve x^2=8y, the
line x =4 and the x-axis on the first quadrant with respect to y-axis.
A. 25.6
SOLUTION:
B. 21.8
C. 31.6
D. 36.4
4
𝑥 2 = 8𝑦
Subs. x= 4
𝑥2
𝐼𝑦 = ∫0 𝑥 2 ( 8 ) 𝑑𝑥
𝑰𝒚 = 𝟐𝟓. 𝟔
42 = 8𝑦
16
=𝑦
8
𝑦=2
82. If 8 oranges cost Php 96, how much do 1 dozen at the same rate?
A. Php 144
B. Php 124
C. Php 148
D. Php 168
SOLUTION:
8𝑥 = 96
8𝑥
8
=
96
8
1 𝐷𝑜𝑧𝑒𝑛 = 12
𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 12 𝑥 12 =144
𝑥 = 12
83. A particle moves in simple harmonic in accordance with the equation 𝑠 =
3 sin 8 𝜋 𝑡 + 4 cos 8 𝜋 𝑡, where s and t are expressed in feet and seconds,
respectively. What is the amplitude of its motion?
A. 3ft
B. 4ft
C. 5ft
SOLUTION:
𝑊ℎ𝑒𝑟𝑒 𝐴 = 3, 𝐵 = 4
𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 = √𝐴2 + 𝐵 2
𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 = √32 + 42
𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 = 𝟓 𝒇𝒕
84. If 𝑍1 = 1 – 𝑖 and 𝑍2 = −2 + 4𝑖, evaluate 𝑍12 + 2𝑍1 – 3.
D. 8ft
A. -1 +4i
B. 1 – 4i
C. 1 + 4i
D. -1 – 4i
SOLUTION:
(1 − 𝑖)2 + 2 (1 − 𝑖) − 3 = 0
1 − 2𝑖 + 𝑖 2 + 2 − 2𝑖 − 3 = 0
𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒,
𝟏 + 𝟒𝒊 = 𝟎
𝑖 2 = 4𝑖
−1 = 4𝑖
85. Identify the property of real numbers being illustrated: x + (y + z) = (x + y) +
z
A. Commutative Property of
Addition
B. Commutative Property of
Multiplication
C. Associative Property of
Addition
D. Associative Property of
Multiplication
86. The distance between -9 and 19 on the number line is
A. 28
B. -28
C. 10
D. -10
SOLUTION:
19 + 9 = 𝟐𝟖
a
a
87. If the function f is odd and ∫0 f(x)dx = 5m − 1, then ∫−a f(x)dx =?
B. 10m – 2
A. 0
C. 10m – 1
D. 10m
SOLUTION:
𝑎
𝑎
∫0 𝑓(𝑥)𝑑𝑥 = 5𝑚 − 1
∫0 𝑑𝑥 = 𝑦
𝑎
𝑎 − ( −𝑎 ) = 𝑦
∫0 𝑓(𝑥)𝑑𝑥 = ?
𝐼𝑓 𝐹 𝑖𝑠 𝑂𝐷𝐷, 𝐹 = 1
2𝑎 = 𝑦 𝑒𝑞. 2
𝑎
∫0 𝑑𝑥 = 5𝑚 − 1
2 ( 5𝑚 − 1 ) = 𝑦
𝑎 = 5𝑚 − 1 𝑒𝑞. 1
𝟏𝟎𝒎 − 𝟐 = 𝒚
88. Find the mass of a 1.5-m rod whose density varies linearly from 3.5 kg/m
from end to end
A. 3.5 kg
B. 2.5kg
C. 4.5kg
D. 5.0kg
SOLUTION:
𝑚=
(2.5
𝑘𝑔
𝑘𝑔
+3.5 )(1.5 𝑚)
𝑚
𝑚
2
= 𝟒. 𝟓 𝒌𝒈
89. Find the area bounded by the parabola 𝑦 = 𝑥 2 , the tangent line to the
parabola at the point (2, 4) and the x axis.
A. 9/2
B. 8/3
SOLUTION:
2
𝐴 = ∫0 𝑥 2 𝑑𝑥
𝑨 = 𝟖/𝟑
C. 8/5
D. 9/4
90. Find the coordinates of an object that has been displaced from the point (-4,
9) by the vector (4i – 5j)
A. (0, 4)
B. (0, -4)
C. (4, 0)
D. (-4, 0)
SOLUTION:
𝑉𝐸𝐶𝑇𝑂𝑅: (4 − 5𝑖) = 6.40 < −51.34
𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑋 𝑎𝑛𝑑 𝑌 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟.
cos(−51.34) =
𝑥
−5
4
=
𝑌2 − 𝑌1
𝑋2 − 𝑋1
𝑦−9
𝑥+4
(𝑥 + 4) =
6.40
𝑥=4
sin(−51.34) =
𝑚=
41 =
𝑦
(𝑦−9) 4
−5
[ (𝑦−4)4 ]2
−5
+ (𝑦 − 9)2
𝒚=𝟒
6.40
41 = (𝑥 + 4)2 + (4 − 9)2
𝑦 = −5
𝑑 = √(𝑥 + 4)2 + (𝑦 − 9)2
𝒙 = 𝟎 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒,
𝑑 = √41 𝑜𝑟 6.40
𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑎𝑟𝑒 (𝟎, 𝟒)
91. Find the major axis of the ellipse x2 +4y2 -2x – 8y + 1 = 0.
A. 2
B. 10
C. 4
D. 6
.SOLUTION:
𝑥 2 − 2𝑥 + 4𝑦 2 + 8𝑦 = −1
(𝑥 2 − 2𝑥 + 1) + 4(𝑦 2 − 2𝑦 + 1) = −1 + 1 + 4
1
(x -1)2 + 4(y-1)2 =( 4) (4)
(𝑥−1)2
4
+
(𝑦−1)2
1
=1
𝑎2 = 4 = 22
a=2
92. A car travels 90kph. What is its speed in meter per second?
A. 43
B. 30
C. 25
D. 50
SOLUTION:
𝑆=
𝑑
; 𝑑 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑛𝑑 𝑡 = 𝑡𝑖𝑚𝑒
𝑡
=90 kph (
1000𝑚
1𝑘𝑚
1ℎ𝑟
) (3600𝑠) = 𝟐𝟓𝒎/𝒔
93. The vertices of the base of the isosceles triangle are (-1, -2) and (1, 4). If the
third vertex lies on the line 4x + 3y = 12. Find the area of the triangle.
A. 8
B. 15
C. 12
D. 10
SOLUTION:
4𝑥 + 3𝑦 = 12
3𝑦 = 12 − 4𝑥 ; 𝑑𝑖𝑣𝑖𝑑𝑒 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 3
4
𝒚 = 4− 3𝑥
@𝑦 = 0
4
𝟎 = 4− 3𝑥
𝑥=3
; therefore the third point is (3,0)
1
𝑨 = 2 [(𝑥1 𝑦2 + 𝑥2 𝑦3 + 𝑥3 𝑦1 ) − (𝑦1 𝑥2 + 𝑦2 𝑥3 + 𝑦3 𝑥1 )]
1
𝑨 = 2 [(−1)(4) + (1)(0) + (3)(−2)] − [(−2)(1) + (4)(3) + (0)(1)] = 𝟏𝟎
94. Assume that f is a liner function. If f(4) = 10 and f(7) = 24, find f(100).
A.98
B. 144
C. 576
D.458
SOLUTION:
f (4) = 10 ; (4,10)
f (7) = 24 ; (7,24)
𝑦 −𝑦
𝑚 = 𝑥2−𝑥1
2
1
24−10
=
7−4
m=
14
3
using point slope form:
f(x) – 10 =m(x-4)
f(x) =
14
3
(x-4) -10
f(x) =
14
26
3
3
x-
f(100) =
14
3
(100) -
26
3
f(100) = 458
95. The line y = 3x +b passes thru the point (2, 4). Find b.
A. 2
B. 10
C. -2
D. -10
SOLUTION:
y = 3x + b, x = 2 ; y = 4
4 = 3(2) + b
b = 4 – 6 = -2
96. How far is the directrix of the parabola (x - 4)2 = -8(y - 2) from the x-axis?
A. 2
B. 3
C. 4
D. 1
SOLUTION:
(x – 4)2 = -8(y – 2)
2x – 8 = -8y +16
2x + 8y = 8
4𝑎 = 8; 𝒂 = 𝟐
97. Find the second derivate of y = xlnx.
A. x
B. 1/x
C. 1
SOLUTION:
𝑦 ′ = 𝑢𝑑𝑣 + 𝑣𝑑𝑢
1
= (x)(𝑥) + ln 𝑥(1) = 1 + lnx
D. x2
𝑑
Since, 𝑑𝑥 =
𝑦" =
𝑑𝑢
𝑑𝑥
𝑢
, then :
𝟏
𝒙
98. Find the point where the normal to y = x + x1/2 at (4, 6) crosses the y-axis.
A. 5.75
B. 9.2
C. 23
D. 11
SOLUTION:
𝑦 = 𝑥 + √𝑥
y '= 1 + 2
1
√𝑥
At (4, 6) slope of tangent = 1 + [2
Slope of normal =
1
√4]
= 5/4
−4
5
Normal line is:
−4
𝑦 − 6 = ( 5 ) (𝑥 − 4)
−4
𝑦 = ( 5 )𝑥 +
46−4𝑥
𝑦=(
5
16
5
+
30
5
)
Crossing y-axis means x = 0
𝑦=
46−0
5
(x, y) = (0,
𝟒𝟔
𝟓
or 9.2)
99. There are four geometric mean between 3 and 729. Find the sum of the
geometric progression.
A. 1092
B. 1094
C. 1082
D. 1084
SOLUTION:
3_,_,_,_,729
𝑎𝑡 = 𝑎1(𝑟)𝑛
729 = 3(𝑟)5
𝑟=3
𝑛(1) = (3)(3) = 9
𝑛(2) = (3)(3)2 = 27
𝑛(3) = (3)(3)3 = 81
𝑛(4) = (3)(3)4 = 243
𝑛(𝑡) = 3 + 9 + 27 + 81 + 243 + 729
𝒏(𝒕) = 𝟏𝟎𝟗𝟐
100. Find the area of a circle inscribed in a rhombus whose perimeter is 100
inches and whose longer diagonal is 40 inches.
A. 364. 43 sq. in
C. 452. 39 sq. in.
B.590. 62 sq. in.
D. 389. 56 sq. in.
SOLUTION:
the perimeter is 100 the sides are 25 each.
The diagonals are perpendicular and meet in the center of the circle.
There is a right triangle with a side of
40
2
=20 and hypotenuse of 25.
202 + b2 = 252
400 + b2 = 625
b2 = 225
b=15
15∗20
2
=150 sq cm Area of triangle.
Draw a line from the right angle, which is also the center of the circle,
perpendicular to the hypotenuse and label it x.
25x
2
= 150
25x = 300
x = 12 radius of the circle.
pi ∗ 122 = 𝟏𝟒𝟒𝐩𝐢 sq cm area of the circle
REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
MARCH 2016
MATHEMATICS
REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION
MARCH 2016
MATHEMATICS
1. Given a conic section, if B2-4AC=0, it is called?
A. circle
B. parabola
C. hyperbola
D. ellipse
2. Give a conic section, if B2-4AC >0 it is called?
A. Circle
B. parabola
C. hyperbola
D. ellipse
3. A conic section whose eccentricity is equal to one is known as
A. A parabolaB. an ellipse
C. a circle D. a hyperbola
4. A length of the latus rectum of the parabola y2 = 4px is
A. 4p
B. 2p
C. p
D. -4p
5. Two engineers facing each other with a distance of 5km from each other, the
angles of elevation of the balloon from the two engineers are 56 degrees and
58 degrees, respectively. What is the distance of the balloon from the two
engineers?
A. 4.45km,4.54km
C.4.64km,4.54km
B. 4.54km,4.45km
D. 4.46km,4.45km
SOLUTION:
(90-56) + (90-58) = 66
5/sin(66) = a/sin(58) = 4.64km
5/sin(66) = b/sin(56) = 4.54km
6. Joy is 10% taller than joseph is 10% taller than Tom. How many percent is
Joy taller than Tom?
A. 18%
SOLUTION:
B. 20%
C. 21%
D. 23%
JOY = JOSEPH (1+.10)
JOSEPH = TOM (1+.10)
JOY [TOM (1+.10)] (1+.10)
JOY = TOM (1+.10)2
JOY = TOM (1+.21)
.21 = 21%
7. In a hotel it is known than 20% of the total reservation will be cancelled in the
last minute. What is the probability that these will be fewer than 2
reservations cancelled out of 4 reservations?
A. 0.6498
B. 0.5629
C. 0.3928
D. 0.8192
SOLUTION:
Let p = probability of outcome = 0.2
Q = complement of p = 1 – 0.2 = 0.8
𝑃𝐴 = 𝑛𝐶𝑟 𝑝𝑟 𝑞 𝑛−𝑟
𝑃𝑂 = 4𝐶𝑜 0.20 0.84−1 = 0.4096
𝑃1 = 4𝐶1 0.20 0.84−1 = 0.4096
𝑃𝐴 = 𝑃𝑂 + 𝑃1 = 0.4096 + 0.4096 = 0.8192
8. Find the area of the region inside the triangle with vertices (1,1),(3,2), and
(2,4)
A. 5/2
B. 3/2
C. ½
D. 7/2
SOLUTION:
111
A= ½ {3 2 1 =5/2 Ans.
2 41
9. The cost per hour of running a boat is proportional to the cube of the speed of
the boat. At what speed will the boat run against a current of 8kph in order to
go a given distance most economically?
A. 15kph
B. 14kph
C. 13kph
D. 12kph
SOLUTION:
Let c = cost per hour
X = speed of motor boat
C1 = total cost
C =kx3
Where: k = proportionality constant
𝑑
t = 𝑥−8
Ct = Ct
𝑑
C1 = kx3 (𝑥−8)
𝑑𝐶𝑡
𝑑𝑥
= (x-8)(3kdx2)-kdx3(1)/(x-8)2 = 0
(x-8)(3x2)= x3
3x3-24x2= x3
2x3 = 24x2
X = 12kph
10. What is the unit vector which is orthogonal both to 9i+9j and 9i+9k?
A.
B.
C.
D.
SOLUTION:
9i+9j 9i+9k
A x B = (9i+9j)*( 9i+9k)
= 81(i+j)(i+k)
= 81(i-j-k)
= (a x b)/ a x b
=
=
=
81(𝑖−𝑗−𝑘)
81√3
(𝑖−𝑗−𝑘)
√3
Ans. c
11. In polar coordinate system the distance from a point to the pole is known as
A. Polar angle
B. radius vector
C. x-coordinate
D. y-coordinate
12. N engineers and N nurses. If two engineers are replaced by nurses,
51% of the engineers and nurses are nurses. Find N
A. 100
B. 110
C. 50
D. 200
SOLUTION:
{ 0.51 [ (N-s) + (N+2)] = N+2 }
= 100 Ans.
13. If sinA=
and cotB= 4, both in Quadrant III, the value of sin (A+B) is
A. -0.844 B. 0.844
C. -0.922
D. 0.922
SOLUTION:
sin 𝐴 =
−4
5
, treat A as it is in QI
4
= 53.13010235°
5
Sin A is in QII, True value of A is 180 – 53.13010235 = 126.8698976o
A = 𝑠𝑖𝑛−1
Cot B = 4 or
1
tan B =
4
4
B = 𝑡𝑎𝑛−1 = 14.0362437°
5
True value of B is 180 – 14.0362437 = 165.9637565o
sin(𝐴 + 𝐵) = sin(126.8698976° + 165.9637565° ) = −0.9216 ≅ −0.922
14. Two stores are 1 mile apart and are of the same level as the foot of the hill.
The angles of depression of the two stores viewed from the top of the hill are
5 degrees and 15 degrees respectively. Find the height of the hill
A. 109.01m B. 209.01m
SOLUTION:
Tan 5 =
Tan 15 =
;
X= 109.01m Ans
C. 409.01m
D. 309.01
15. A fair coin is tossed three times and it appeared always exactly three
heads. Find the probability in a single toss it will appear head.
A. ½
B. ¼
C. 1/6
D. 1/16
SOLUTION:
1
1
#Flip = (# 𝑜𝑓 𝐻𝑒𝑎𝑑)(#𝑜𝑓 𝑇𝑎𝑖𝑙 )
Since 1 coin = 2 outcome
𝟏
#Flip =(𝟐)
16. The product of the slopes of any two straight lines is negative 1, one of
these lines are said to be
A. Perpendicular
B. parallel
C. non intersecting
D. skew
17. When two lines are perpendicular, the slope of one is
A. Equal to the negative of the other
B. equal to the other
C. equal to the negative reciprocal of the other
D. equal to the reciprocal of the other
18. A statistic department is contacting alumni by telephone asking for
donations to help fund a new computer laboratory. Past history shows that
80% of the alumni contacted in this manner will make a contribution of at
least P50, 000. A random sample of 20 alumni is selected. What is the
probability that more than 15 alumni will make a contribution of at least
P50.00?
A. 0.4214
B. 0.5890
C. 0.6296
D. 0.3018
SOLUTION:
Let p = probability of an alumni giving contributions = 0.8
Q = complement of p = 1 – 0.8 = 0.2
𝑃𝐴 = 𝑛𝐶𝑟 𝑝𝑟 𝑞 𝑛−𝑟
Since we are looking for more than 15 out of 20 it means we are interested for
P16 + P17 + P18 + P20 since 16 to 20 is higher than 15.
We can use the summation formula
20
𝑃𝐴 = ∑(20𝐶𝑥)(0.8𝑥 )(0.220−𝑥 ) = 0.6296
16
19. If z1 =1-i , z2= -2+4i, z3= √3 − 2𝑖, evaluate Re(2z13+3z22-5z32)
A. 35
B. 35i
C. -35
D. -35i
SOLUTION:
𝑧3 = √3 − 2𝑖 = 3 − 2𝑖
20. Simplify (1-tan theta) / (1+tan theta)
A. (cos theta+ sin theta)/(cos theta- sin theta)
B. Cos theta/(cos theta-sin theta)
C. (cos theta-sin theta)/(cos theta+sin theta)
D. Sin theta/ (cos theta+sin theta)
SOLUTION:
Assume the value of 𝜃 is = 30
1−tan 𝜃
1−tan 30
=
= 2 − √3
1+ tan 𝜃 1+ tan 30
Then troubleshoot the choices,
A.
cos 𝜃+ sin 𝜃 cos 30+ sin 30
cos 𝜃−sin 𝜃
cos 𝜃
cos 𝜃− sin 𝜃 cos 30− sin 30
= cos 30−sin 30 =2 + √3 C. cos 𝜃+sin 𝜃 = cos 30+sin 30 =𝟐 − √𝟑 𝑨𝒏𝒔.
cos 30
3+√3
B. cos 𝜃−sin 𝜃=cos 30−sin 30=
2
sin 𝜃
sin 30
D. cos 𝜃+sin 𝜃=cos 30+sin 30=
−1+√3
2
21. A sinking ship makes a distance signal seen by three observers all 20m
inland from the shore. First observer is perpendicular to the ship, second
observer 100m to the right of the first observer and the third observer is
125m to the right of the first observer. How far is the ship from the shore?
A. 60m
B. 80m
C. 100m
D. 136.2m
22. A die and a coin are tossed. What is the probability that a three and a head
will appear?
A. ¼
B. ½
C. 2/3
D. 1/12
SOLUTION:
1
Probability of the die= 6
1
Probability of the coin= 2
1 1
𝟏
Total Probability = (6)(2)= 𝟏𝟐
23. A tangent to a conic is a line
A. Which is parallel to the normal
B. Which touches the conic at only one point
C. Which passes inside the conic
D. All of the above
24. If tan A = 1/3 and cot B = 4 find tan (A+B)
A. 11/7
B. 7/11
C. 7/12
D. 12/7
SOLUTION:
tan (A + B)= (tanA + tanB)/(1-tanAtanB)
tan A= 1/3
cot B= 4 ; it is also equal to tanB= 1/4
Substitute:
tan (A+B)= (.3333+.25)/(1-(.3333)(.25))
=7/11
25. What would happen to the volume of a sphere if the radius is tripled?
A. Multiplied by 3
B. multiply by 9
C. multiply by 27
D. multiply by 6
26. A container is in the form of a right circular cylinder with an altitude of 6in
and a radius of 2in. If an asbestos of 1in thick is inserted inside the container
along its lateral surface, find the volume capacity of the container.
A. 12.57 cu. in B. 12.75 cu. in C. 18.58 cu. in
SOLUTION:
D. 18.85 cu. in
Asbestos is placed inside, the thickness of it will be subtracted to the radius
since
it serves as an inside coating.
V= π(r3)(h)
= π(13)(6) = 18.85 cu.in.
27. Is it convergent or divergent? If convergent, what is the limit?
A. Convergent, π/2
B. divergent
C. convergent, π
D. convergent, π/4
28. If the sides of a right triangle is in arithmetic progression, what is the ratio of
its sides?
A. 1,2,3
B. 4,5,6
C. 3,4,5
D. 2,3,4
SOLUTION:
Since right triangle, it must satisfy the Pythagorean's theorem
29. What is the area bounded by the parabola x2 = 8y and its latus rectum?
A. 54/3 s.u.
B. 8/3 s.u
C. 16/3 s.u.
D. 31/3 s.u.
SOLUTION:
Latus rectum= 8
So that we will choose limits (-4,4)
then came up with:
Integral of (x^2/8)dx with limits -4 to 4 = 16/3 s.u.
30. Find the general solution if y’’+10y=0
A. y = 𝐶1 cos(√𝟏𝟎) 𝑥 + 𝐶2 sin(√𝟏𝟎) 𝑥
C. y = 𝐶 cos(√𝟏𝟎) 𝑥
SOLUTION:
B. y = 𝐶1 cos(√𝟓)) 𝑥 + 𝐶2 sin(√𝟓) 𝑥
D. y = 𝐶 sin(√𝟏𝟎) 𝑥
Case 3 of Conjugate Complex Roots
D²y+ 10 =0
y = e^ax ( C1 cos bx + C2 sin bx)
dx²
( D²+10 ) y = 0
y = e0x ( C1 cos √10𝑥 + C2 sin √10𝑥 )
m² + 10 = 0
y = C1 cos √𝟏𝟎𝐱 + C2 sin √𝟏𝟎𝐱 Ans.
m= + √10
31. The volume of a cube becomes three times when its edge is increased by
1inch. What is the edge of a cube?
A. 2.62
B. 2.26
C. 3.26
D. 3.62
SOLUTION:
when edge increased by 1 inch
V= a³
3V = ( a+1 )³
Dv = 3a²
3dV = 3 (a+1)²
3(3a²) = 3a² + 6a +3
6a² - 6a – 3 =0
(a-1.366) (a+0.366) = 0
A= 1.366+1 = 2.366 Ans.
32. The areas if a regular pentagon and a regular hexagon are equal to 12
sq.cm. What is the difference between their perimeters?
A. 0.02
B. 0.03
C. 0.2
SOLUTION:
Area of Pentagon
12=¼(5b²cot
Area of Hexagon
180
5
)
12=¼(6b²cot
b = 2.641 inch
180
6
)
b = 2.149 inch
Perimeter P = nb
P = 5(2.641) =13.205
= 13.205
P = 6(2.149) = 12.894
–
= 0.311 Ans.
33. Evaluate limx-
12.894
D. 0.3
A. 4
B. 6
C. 8
D. 16
SOLUTION:
Apply L’Hospital’s rule
x²-4=2x=2(2)=4 Ans.
x-2 1 1
34. The length of a rectangle is seven times of its width. If its perimeter is 72cm,
find its width
A. 3
B. 3.5
C. 4
D. 15
SOLUTION:
P= 2(w+L)
W -72= 2(w+7w)
W= 4.8 Ans
35. A family’s electricity bill averages $80 a month for seven months of the year
and $20 a month for the rest of the year. If the family’s bill were averaged
over the entire year, what would the monthly bill be?
A. $45
B. $50
C. $55
D. $60
SOLUTION:
= 55 Ans.
36. In order to pass a certain exam, candidates must answer 70% of the last
questions correctly. If there are 70 questions on the exam, how many
questions be answered correctly in order to pass
A. 46
SOLUTION:
(70)(70%) = 49 Ans.
B. 52
C. 56
D. 60
37. A firefighter determines that the length of hose needed to reach a particular
building is 131m. If the available hoses are 47m long, how many sections of
hose when connected together will it takes to reach the building?
A. 3
B. 4
C. 5
D. 6
SOLUTION:
141
47
= 3 Ans.
38. If the average person throws away 38.6 pounds of trash every day, how
much trash would the average person throw away in one week?
A. 270.2 lbs
B. 207.2 lbs
C. 290.6 lbs
D. 209.6lbs
SOLUTION:
38.6 x 7 = 270.2 Ans.
39. If the csc2∅= 1+x, find cot2∅
A. X
C. 1 – x
B. 1 + x
D. 𝑥2
SOLUTION:
From trigonometric identities: 1 + cot2ϴ = csc2 ϴ
cot2ϴ = csc2 ϴ - 1
cot2ϴ = (1 + x) -1 = x
40. A runner runs a circular track and a set of data is recorded:
Time
Distance
68 sec-----------------
400m
114 sec ----------------
600m
168 sec ---------------- 800m
209 sec ---------------- 1000m
256 sec ---------------- 1200m
322 sec ---------------- 1400m
What is the average velocity from 68 sec to 168 sec?
A. 3 𝑚/𝑠2 B. 4 𝑚/𝑠2
C. 8 𝑚/𝑠2
D. . 6 𝑚/𝑠2
Vave
6
?
A. ½
B. ¼
C. 2/5
D. 5/2
SOLUTION:
(2/3 – 1/4)=
5/12 = 2/5 Ans
(3/8 + 1/2 + 1/6)
25/24
42. Water is flowing into a conical vessel 10ft high and 2ft radius at the rate of
50 cu. Ft per minute. If the deep of the wateris 6ft, how fast is the radius
increasing?
A. 2.12 ft/mIN
B. 12 ft/min
C. 2.21 ft/min
D. 11 ft/min
SOLUTION:
V’= 50 ft3 / min
r’ at h = 6ft
𝑟
2
=
ℎ
10
1
𝑟 = ℎ, 5𝑟 = ℎ
5
1
𝑉 = 𝜋𝑟 2 (5𝑟)
3
5
𝑉 = 𝜋𝑟 3
3
𝑉′ = 5𝜋𝑟 2 𝑟′
6
@ ℎ = 6 ,𝑟 =
5
𝑉′
50
𝑓𝑡
𝑟′ =
=
= 2.21
2
2
5𝜋𝑟
5𝜋(1.2)
𝑚𝑖𝑛
43. A steel grinder 8m long is moved on rollers along a passageway 4m wide
and into a corridor at right angles with the passageway. Neglecting the width
of the girder, how wide must the corridor be?
A. 3.6 m
m
SOLUTION:
B. 1.4 m
C. 1.8 m
D. 2.8
44. If in the Fourier series of a periodic function, the coefficient a 0 is zero, it
means that the function has
A. Odd symmetry
C. odd-quarter wave symmetry
B. Even quarter-wave symmetry
D. any of the above
45. What is the general solution of (D4-1) y (t) = 0?
A. 𝑦 = 𝐶1𝑒𝑡 + 𝐶2𝑒−𝑡 + 𝐶3𝑐𝑜𝑠𝑡 + 𝐶4𝑠𝑖𝑛𝑡
C. 𝑦 = 𝐶1𝑒𝑡 + 𝐶2𝑒−𝑡
B. 𝑦 = 𝐶1𝑒𝑡 + 𝐶2𝑒−𝑡 + 𝐶3𝑡𝑒𝑡 + 𝐶4𝑡𝑒−𝑡
SOLUTION:
D. 𝑦 = 𝐶1𝑒𝑡 + 𝐶2𝑡𝑒−𝑡
46. Remy earns P10 an hour for walking the neighbor’s dog. Today she can
only walk the dog for 45. How much will Remy make today?
A. P10.00
B. P7.25
C. P7.60
D. P6.75
SOLUTION:
47. When a baby born the weighs 8 lbs. and 12 oz. After two weeks during his
checkup he gains 8 oz. What is his weight now in lbs. and oz.?
A. 8 lbs. and 10 oz.
B. 9 lbs. and 4 oz.
C. 9 lbs. and 2 oz.
D. 10 lbs. and 4 oz.
SOLUTION:
48. An equation of the form
A. An inequality
is
B. an equality
C. a proportion
D. a ratio
49. Michael’s favorite cake recipe calls for 0.75 pounds of flour, he has a 5
pound bag. He wants to make several cakes for the school bake sale. How
many cakes can he make?
A. 5
B. 6
C. 7
D. 8
SOLUTION:
50. Simplify (1+tan2x) / (1-tan2x)
A. Sin 2x
B. Cos 2x
C. Csc 2x
SOLUTION:
51.
52.
53. Find the general solution of y’’+10y’+41y=0
D. Sec 2x
A. 𝑦 = 𝑒−5 (𝐶1𝑐𝑜𝑠4𝑥 + 𝐶2𝑠𝑖𝑛4𝑥)
C. 𝑦 = 𝑒−4(𝐶1𝑐𝑜𝑠5𝑥 + 𝐶2𝑠𝑖𝑛5𝑥)
B. 𝑦 = 𝑒5(𝐶1𝑐𝑜𝑠4𝑥 + 𝐶2𝑠𝑖𝑛4𝑥)
SOLUTION:
D. 𝑦 = 𝑒4𝑥(𝐶1𝑐𝑜𝑠5𝑥 + 𝐶2𝑠𝑖𝑛5𝑥)
54. Find the general solution of y’+
C. 𝑥2 − 2𝑦2 = 𝐶
A. 𝑥2 + 2𝑦2 = 𝐶
D. 𝑥2 − 𝑦2 = 𝐶
B. 𝑥2 + 𝑦2 = 𝐶
SOLUTION:
55. Find the general solution of y’’-4y’+10y=sin x
A.
B.
C.
D. 𝒚 = 𝒆𝟐𝒙 [𝑪𝟏 𝒄𝒐𝒔√𝟔𝒙 + 𝑪𝟐 𝒄𝒐𝒔√𝟔𝒙] +
𝟗
𝟗𝟕
𝟒
𝒔𝒊𝒏𝒙 + 𝟗𝟕 𝒄𝒐𝒔𝒙
SOLUTION:
Solve for Yh
𝐷2 − 4𝐷 + 10 = 0
𝐷 = 2 ± √6𝑖
2𝑥
Let
𝑒 [ 𝐶1 𝑐𝑜𝑠√6 + 𝐶2 𝑐𝑜𝑠√6 ]
y = A sinx + Bcosx
y’ = A sinx – Bcosx
y” = - A sinx – Bcosx
𝑦" − 4𝑦 ′ + 10𝑦 = sin 𝑥
−𝐴𝑠𝑖𝑛𝑥 − 𝐵𝑐𝑜𝑠𝑥 − 4𝐴𝑐𝑜𝑠𝑥 + 4𝐵𝑠𝑖𝑛𝑥 + 10𝐴𝑠𝑖𝑛𝑥 + 10𝐵𝑐𝑜𝑠𝑥 = 𝑠𝑖𝑛𝑥
-B – 4A + 10B = 0
- A + 4B + 10A = 1
9 (−4𝐴 + 9𝐵 = 0 )
+ 4(9𝐴 + 4𝐵 = 1 )
4
97
9𝐵 9 4
9
𝐴=
= 𝑥
=
4
4 97
97
9
4
𝑦 = 𝑒 2𝑥 [𝐶1 𝑐𝑜𝑠√6𝑥 + 𝐶2 𝑐𝑜𝑠√6𝑥] +
𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥
97
97
𝐵=
56. Find the equation of the line that passes through (1,3) and tangent to the
curve y=
𝑥
A. 4x+y-7=0
B. 24x+y-27=0
C. 4x-y+7=0
D. 24x-y+27=0
SOLUTION:
𝑦′ =
1 (𝑥) − (𝑥 + 5)(1) −5
= 2
𝑥2
𝑥
𝑦 ′ (1) =
−5
= −5
12
𝑦 − 𝑦 ′ = 𝑚 (𝑥 − 𝑥1 )
𝑦 − 3 = −5 (𝑥 − 1)
𝑦 − 3 = −5𝑥 + 5
5𝑥 + 𝑦 = 8
Choose A. 4x + y – 7 = 0
57. The ceiling in a hallway 10m wide is in the shape of a semi-ellipse and is 9m
high in the center and 5m high at the side walls. Find the height of the ceiling
2m from either wall.
A. 11.7 m
B. 8.4 m
C. 6.4 m
D. 17.5 m
SOLUTION:
2m from the wall =3m from center the origin is 5m high @ side wall
𝑥
𝑦
( 5 )2 + ( 5 )2 = 1
𝑥2
25
+
𝑦2
y= 2.4m + 5= 7.4
=1
9
𝑦2
9
T.S: origin is 6m high @ side wall
𝑥2
= 1 − 25
y= 2.4 + 6 = 8.4 m Ans.
32
𝑦 = √9(1 − 25)
Y=2.4 m
58. If in the Fourier series of a periodic function, the coefficient a0=0 and a=0,
then it must be having _____ symmetry.
A. Odd
B. Odd-quarter wave
C. Even
D. Either A or B
59. If the Fourier coefficient b0 of a periodic function is zero then it must possess
______ symmetry.
A. Even
B. Even-quarter-wave
C. Odd
D. Either A or B
60. Find the area of the region between the x-axis and y=(x-1)2 from x=0 to x=2
A. 1/3
B. 2/3
C. ½
D. ¼
SOLUTION:
1
2
∫0 (𝑥2 − 2𝑥 + 1 − 0)𝑑𝑥 − ∫1 0 − (𝑥2 − 2𝑥 + 1)𝑑𝑥
= 𝟐/𝟑
61. Find the slope of the line through the points (-2,5) and (7,1)
A. 4/9
B. -4/9
C. 9/4
D. ¼
SOLUTION:
m=
62. A train is moving at the rate of 8mi/h along a piece of circular track of radius
2500 ft. Through what angle does it turn in 1min?
A. 16 deg. 8 min.
C.18 deg. 9 min.
B. 15 deg. 6 min.
D.17 deg. 10 min.
SOLUTION:
𝒔 = 𝒓𝜽
𝒅𝒔 = 𝒓𝒅𝜽
𝟖 𝒎𝒊⁄𝒉𝒓
𝒅𝒔
𝒓𝒂𝒅
𝟏 𝒉𝒓
𝒅𝜽 =
=
= 𝟏𝟔. 𝟖𝟗𝟔
𝒙
= 𝟎. 𝟐𝟖𝟏𝟔 𝒓𝒂𝒅⁄𝒎𝒊𝒏
𝟏 𝒎𝒊
𝒓
𝒉𝒓
𝟔𝟎
𝒎𝒊𝒏
𝟐𝟎𝟎 𝒇𝒕 𝒙
𝟓𝟐𝟖𝟎 𝒇𝒕
𝟏𝟖𝟎°
𝒅𝜽 = 𝟎. 𝟐𝟖𝟏𝟔 𝒓𝒂𝒅⁄𝒎𝒊𝒏 =
= 𝟏𝟔°𝟏𝟖′
𝝅𝒓𝒂𝒅
63. An artist wishes to make a sign in the shape of an isosceles triangle with a
42 degrees vertex angle and a base of 18m. What is the area of the sign?
A. 109 sq. m
B. 209 sq. m
C. 112 sq. m
D. 211 sq. m
SOLUTION:
tan = opp/adj
tan= opp/9
opp= height = 9tan(69) = 23.4
A=1/2bh
211 sq. m Ans.
A=1/2(18)(24.4) =
64. If x2-y2=1 find y’’’
A. −2𝑥/𝑦5
SOLUTION:
B. 2𝑥/𝑦5
C. −𝑥/𝑦4
D. 𝑥/𝑦4
65. A second hand scientific calculator was sold to Michael for P600. The
original price of the item was P800. How many percent discount was given to
him?
A. 25
B. 35
C. 40
D. 20
SOLUTION:
Discount = 800-800(X%)= 600
Discount = 25%
66. Find the volume of a cube if its total surface area is 54 sq. cm.
A. 21 cu. m
B. 30 cu. m
C. 27 cu. m
D. 54 cu. m
SOLUTION:
S =6a^2
54 = 6(a)^2
a=3
V = a^3
V = 3^3 = 27cu.m
67. A girl is flying a kite which is at a height of 120ft. The wind is carrying the
kite horizontally away from the girl at a speed of 10ft/sec. How fast must be
kite sizing be let out when the sizing is 150 feet long?
A. 4 ft/s
SOLUTION:
B. 5 ft/s
C.8 ft/s
D. 6 ft/s
SOLUTION:
63(-15)+12 = 63
6.3 =6.3
X = 6 Ans.
70. Robert has 50 coins all in nickels and dimes amounting to $3.50. How many
nickels does he have?
A. 20
B. 30
C. 15
D. 35
SOLUTION:
0.05 + 0.01d = 3.5
n = 30
d = 20 Ans.
71. The equation of the folium of Descartes is x2+y2=34xy. Find the area
enclosed by the loop
A.
B.
C.
D.
72. Find the acute angle of intersection of the curves x2+y2=5 and x2-y26x=15
A. 53.14
֯
B. 52.13
֯
C. 36.86
֯
D. 37.87 ֯
73. For what value of k will the line kx+5y=2k have y-intercept 4?
A. -10
SOLUTION:
B. 10
C. 9
D. -9
Kx+5y=2k; y-int =4
5y = -kx + 2k; When x= 0, y= 4
5y= -k(0) + 2k
y=
4(5) = 2k
K =10 Ans.
74. Find the volume formed by revolving the triangle whose vertices are
(1,1),(2,4) and (3,1) about the line 2x-5y=10
A. 49
B. 94
C. 65
D. 56
SOLUTION:
Centroid @ (2,2)
Perpendicular length to 2x – 5y = 10
𝑑=
2(2) − 5(2) − 10
√22 + 52
= 2.97
1
𝑉 = 𝐴2𝜋𝑑 = [ (2)(3)] (2𝜋)(2.97) = 55.98 ≈ 56
2
75. A tank contains 760 liters of fresh water. Brine containing 2.5N/liter of salt
enters the tank at 15 liter/min, and the mixture kept uniform by stirring runs
out at 10liters/min. Find the amount of salt in the tank after 30 minutes?
A. 1028.32 N
B. 649.52 N
C. 949.75 N
SOLUTION:
Ds/Dt = (2.5)(15) – [s(10)/v]
V= 760+(15-10t)
V= 760 + 5t
Ds/Dt = 37.5 – [s(10)/760+5t]
(Ds/Dt) + [s(10)/760+5t] = 37.5
D. 864.88 N
Integrating Factor (IF) = e^(10 integral of (dt/760+5t))
IF = e^[ln(760+5t)]^2
IF = (760+5t)^2
IF(s) = C integral of IF + C
(760+5t)^2 = integral of (760+5t)^2 + C
[(760+5t)^2 = (1/15) (760+5t)^3 + C
@ t= 0; s=0
C= -1097440000
@ C= -7097440000; t=30 mins
S=949.749 N = 949.75 N (ANS)
76. Find the volume of the solid generated when the region bounded by y=x24x+6 and y=x+2 is revolved about the x-axis
A. 100.89
B. 104.60
C. 103.04
D. 101.79
SOLUTION:
y = X2-4x +6
y = x+2
revolved in x – axis x+2 =
9x2 -4x + 6 x2- 5x – 4 =0
(x-4) (x-1) =0
X =4 & 1
y = 4+2 = 6
y = 1+2 = 3 intersection (4,6)
& (1,3)
v=
v = 101.79
X = 101.79 Ans.
77. The rate at which a body cools is proportional to the difference in
temperature between it and the surrounding atmosphere. If in air at 60 deg. C
a body cools from 90 deg. C to 80 deg. C in 10min, find its temperature 10
minutes later?
A. 80 deg. C
deg. C
D. 64.4 deg. C
SOLUTION:
B. 73.3 deg. C
C. 90
Tb @ 20mins = 73.33oC
30 – 60 = (90 – 60) ek(10)
ln 3
K=
10
= 0.0405
Tb – 60 = (90-60)e-( 0.0405)(20)
= 73.33 Ans.
78. A sector of a circle has a central angle of 80 degrees and radius of 5m.
What is the area of the sector?
A. 16.5 sq. m
B. 17.5 sq. m
C. 15.8 sq. m
D. 18.8 sq. m
SOLUTION:
2
= 25π = 78.534 (
A=
Aa= 17.4533 Ans.
79. A grocer bought a number of cans of corn for $14.40. Later the price
increased 2 cents a can and as a result she received 24 fewer cans for the
same amount of money. How many cans were in his first purchase?
A. 142
B. 140
C. 144
D. 143
SOLUTION:
xy=14.40
(y+0.02)(x-24)=14.40
x=?
y=
24) = 14.40 ; x=144
80. Find the area inside the cardiod r=1+costheta and outside the circle r=1.
A. 2.79
SOLUTION:
A = (1/2
B. 2.97
C. 3.98
D. 3.89
A= 2.79 Ans.
81. If 2log4x-log49=2, find the value of x
A. 10
B. 12
C. 11
D. 9
SOLUTION:
2log4x-log49 = 2 2log4x
= 2-log 49 X = 12
Ans.
82. If 7 coins are tossed together in how many ways can they fall with at most 3
heads?
A. 63
B. 64
C. 65
D. 62
SOLUTION:
(7C3)+(7C2)+(7C1)
= 63
83. The eccentricity if the hyperbola having the rectangular equation 3x24y2-24x+16y+20=0
A. 1.12
B. 1.22
C. 1.32
SOLUTION:
3x2-4y2-24x+16y+20=0
Ax2-Cy2+Dx+Ey+F=0
e=c/a
or
e=a/d
3x2-24x -4y2+16y = -20
3(x2-8x+16)-4(y2-4y+4)=-20+16(3)+4(-4)
3(x2-8x+16)-4(y2-4y+4)=12
(x-4)2/4 – (y-2)2 = 1
D. 1.42
STANDARD EQUATION
a=sq.4
b=sq.3
c=sq (a2+b2)
c=sq(22+(sq.3)2)
c= sq.7
e=c/a = sq.7/2 = 1.32
84. Find the slope of the tangent line to the parabola y2=4x+1 at the point (2,3)
A. 1/3
B. 2/3
C. ¼
D. ¾
SOLUTION:
y=
y=
y’
𝑦=
2
3
85. If x=3t-1, y=1-3t^2, find d^2y/dx^2
A. -1/3
B. -2/3
C. -1
D. -4/3
SOLUTION:
y = 1-3t^2
t = (x+1/3)
y = 1-3(x+3/3)^2
y= 1-(x^2/3)-(2x/3)-(1/3)
yI = 0 – (2x/3) – (2/3) -0
yII= -2/3 ans.
86. Find the equation of the line through the point (3,4) which cuts from the first
quadrant a triangle of maximum are
A. 4x+3y-24=0
C. 3x+4y-25=0
B. 4x-3y+24=0
SOLUTION:
y=m=rise/run=4/-3
y-y1=m(x-x1)
y-4=4/-3 (x-3)
(-3)(y-4)=4(x-3)
-3y+12 = 4x-12
4x+3y-24=0 ans.
D. 3x-4y+25=0
87. Find the moment of inertia with respect to the y-axis of the plane area
between the parabola y=2-x^2 and the x-axis
A. 243/5
324/5
B. 234/5
C. 342/5
D.
88. A man drives 500ft along a road which is inclined 20 degrees to the
horizontal. How high above the starting point is he?
A. 171 ft.
B. 182 ft.
C. 470 ft.
D. 162 ft
SOLUTION:
tan Ø = h/500
h= 500 tan(20)
h= 181.985ft = 182 ft
89. An angle is 30 degrees more than one-half its complement. Find the angle
A. 20 degrees
B. 50 degrees
C. 60 degrees
D. 75 degrees
SOLUTION:
Complementary Angle = 90 degrees
An angle is 30 degree more than one half its complement
Angle = 45 + 30 = 75 degrees Ans.
90. How many ways can 5 keys be placed on a key ring?
A. 8
SOLUTION:
nPn= ( n-1 )!
B. 12
C. 20
D. 24
= ( 5-1 )!
= 24
91. What is the diameter of a sphere for which its volume is equal to its surface
area?
A. 4
B. 6
C. 5
D. 7
SOLUTION:
V= 4/3πr^3
A= 4πr^2
r = d/2
4/3π(d/2)^3 = 4π(d/2)^2
d/24 = 1/4
d = 24/4
d=6
92. Find the area of the triangle whose vertices are A(4,2,3) B(7,-1,4) and (3,4,6)
A. sqrt of 156 B. sqrt of 155
C. 13.5
D. 15.5
93. If the second term of a geometric progression is 6 and the fourth term is 64.
How many terms must be taken for their sum to equal 242?
A. 4
B. 6
SOLUTION:
a1 r = a2
a1 r r^2 = a4
6r^2 = 64
r = 3.2699
a1 = 6/3.26599
a1 = 1.83712
Sn = a1 (1 – r^n-1) / (r – 1)
C. 5
D. 7
242 = 1.83712 [ 1 – ( 3.26599^n-1 ) / (1-3.26599) ]
-298.494 = 1- 3.26599^n-1
3.26599^n-1 = 299.494
n-1 log(3.26599) = log(299.424)
n = 5.81 = 5
94. Convert the point (r, , Φ) = (10, pi/2, 0) from spherical to Cartesian
coordinates
A. (10, 0, 1)
(10, 0, 0)
B. (10, 1, 1)
C. (10, 1, 0)
D.
SOLUTION:
( r , α , Φ ) = ( 10. π/2, 0 )
x = r sin α cos Φ
= 10 (1) (1)
x = 10
y = r sin α sin Φ
=(10)(1)(0)
y=0
z = r cos α
= (10)(0)
z=0
( 10, 0, 0 )
95. The probability that A can solve a given problem is 4/5 that B can solve it is
2/3 and that C can solve it is 3/7. If all three try, compute the probability that
the problem will be solved.
A. 101/305
B. 101/105
C. 102/305
SOLUTION:
𝑃(𝐴) =
4
5
𝑃(B) =
2
3
D. 102/105
𝑃(𝐶) =
3
7
𝑃′ (𝐴) = 1 −
4
1
=
5
5
𝑃′ (𝐵) = 1 −
2
1
=
3
3
𝑃′ (𝐶) = 1 −
3
4
=
7
7
1 1 4
101
𝑃(𝑠𝑜𝑙𝑣𝑒𝑑) = 1 − [ 𝑥 𝑥 ] =
5 3 7
105
96. A club of 40 executives, 33 like to smoke Marlboro, and 20 like to smoke
Philip Morris. How many like to smoke Philip Morris only?
A. 33
B. 13
C. 20
D. 7
SOLUTION:
(33-x) + x + (20-x) = 40
33 + 20 – x = 40
x = 13
From equation of Philip Morris
20 – 13 = 7 ans.
97. Find the value of 4 sinh (pi i/3)
A. -2i (sqrt of 3)
B. 2i (sqrt of 3)
C. -4i (sqrt of 3)
D. 4i (sqrt of 3)
SOLUTION:
4sinh (πi/3) = 4sinh i (π/3)
4sinh i (π/3) = 4isin (π/3) * (π/180)
= 2i (sqrt. of 3) Ans.
98. An equilateral triangle has an altitude of 5(sqrt of 3) cm long. Find the area
in sq. m.
A. 5(sqrt of 3)
B. 25(sqrt of 3)
SOLUTION:
C. 100(sqrt of 3)
D. 50(sqrt of 3)
b = 5 sqrt. of 3 /tan(60) = 5 cm
A = 2(½ (5 cm)(5 sqrt. of 3 cm))
“troubleshoot sq. m to cm”
A = 25 (sqrt. of 3)
99. The line y = 3x + b passes through the point (2, 4). Find b.
A. 2
B. 10
C. -2
D. 10
SOLUTION:
y = 3x + b
Substitute (2,4)
4 = 3(2) + b
b=4–6
b = -2
100. If f(x) = sinx and f(pi) = 3 then f(x) =
A. 4 + cosx
SOLUTION:
“troubleshoot f(x) to f’(x)”
∫ 𝑓 ′ (𝑥) = ∫ sin 𝑥
f(x) = - cos x + c ( g.e.)
solve for c :
f(pi) = - cos (pi) + c
3=1+c
c=2
from g.e. :
f (x) = - cos x + 2 or 2 - cosx
B. 3 + cosx
C. 2 – cosx D. 4 – cosx
REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
AUGUST 2015
MATHEMATICS
REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION
AUGUST 2015
MATHEMATICS
1. Given a conic section, if B2 - 4AC= 0, it is called?
A. Circle
B. Parabola
C. Hyperbola
D. Ellipse
2. Given a conic section, if B2 -4AC>0, it is called?
A. Circle
B. Parabola
C. Hyperbola
D. Ellipse
3. Describe and graph the locus represented by lm{z4} =4.
A. Circle
B. Parabola
C. Hyperbola
D. Ellipse
4. A tangent to conic is a line
A. which is parallel to the normal
B. which touches the conic at only one point
C. which passes inside the conic
D. all of the above
5. All circle having the same center but with unequal radii are called
A. encircle
B. tangent circles
C. concyclic
D. concentric circles
6. If z = 6eiπ/3, evaluate |eiz|,
A. e-3(sqrt. of 3)
B. e3(sqrt. of 3)
C. e-2(sqrt. of 2)
D. e2(sqrt. of 2)
7. Simply (cosβ - 1)(cosβ + 1)
A. -1/sin2β
B. -1/cos2β
C. -1/csc2 β
D. -1/sec2 β
sin2 β + cos 2 β = 1
cos 2 β − 1 = −sin2 β
−𝟏
= −sin2 β
𝐜𝐬𝐜 𝟐 𝛃
8. Find the height of a right circular cylinder of maximum volume which can be
inscribed in a sphere of radius 10 cm.
A. 11.55 cm
B. 14.55 cm
C. 12.55 cm
D. 18.55cm
9. A bus leaves Manila at 12 NN for Baguio 250 km away, traveling an average
of 55 kph. At the same time, another bus leaves Baguio for Manila traveling 65
kph. At what distance from manila they will meet?
A. 135.42 km
B. 114.58 km
C. 129.24 km
D. 120.76 km
250 = 55𝑡 + 65𝑡
𝑡 = 2.0833
𝑑 = 𝑟1 𝑡 = 55(2.0833) = 𝟏𝟏𝟒. 𝟓𝟖
10. A waiter earned tips for a total of $17 for 4 consecutive days. How much he
earned per day?
A. $4.25
B. $4.50
C. $3.25
D. $3.50
$17
= $𝟒. 𝟐𝟓
4 days
11. What is the value of x in Arctan 2x + Arctan x = pi/4 ?
A. 0.28 and -1.78 B. -0.28 and 1.78 C. 0.28
D. -1.78
tan−1(2x) + tan−1(x) =
π
4
𝑆ℎ𝑖𝑓𝑡 𝑠𝑜𝑙𝑣𝑒 𝑋 = 𝟎. 𝟐𝟖
12. The length of the latus rectum of the parabola y2=4px is:
A. 4p
B. 2p
C. p
D.-4p
𝐿𝑅 = 𝟒𝒑
13. A post office can accept for mailing only if the sum of its length and its girth
(the circumference of its cross section) is at most 100 in. What is the maximum
volume of a rectangular box with square cross section that can be mailed?
A. 5432.32in3
B. 1845.24in3
C. 2592.25in3
D. 9259.26in3
14. Water is running out of a conical funnel at the rate of 1 cu. In/sec. If the
radius of the base of the funnel is 4 in. and the altitude is 8 in, find the rate at
which the water level is dropping when it is 2 in. from the top.
A. -1/9pi in/sec
B. -1/2pi in/sec
C. 1/2pi in/sec
D. 1/9pi in/sec
15. A ball is dropped from a height of 18m. On each rebound it rises 2/3 of the
height from which it last fell. What distance has it traveled at the instant it
strikes the ground for the 5th time?
A. 37.89m
B. 73.89m
C. 75.78m
D. 57.78m
16. 3 randomly chosen senior high school students was administered a drug
test. Each student was evaluated as positive to the drug test (P) or negative
(N). Assume the possible combinations of the three student’s drug test
evaluation as PPP, PNP,NPN,NNP,NNN. Assuming each possible
combination is equally likely, what is the probability that all 3 students get
positive results?
A. 1/8
B. 3/4
C. 1/4
D. 1/2
17. The cost per hour of the running the boat is proportional to the cube of the
speed of the boat. At what speed will the boat run against a current of 4 kph in
order to go a given distance most economically?
A. 6kph
B. 12kph
C. 20kph
D. 24kph
18. Ben is two years away from being twice Ellen’s age. The sum of Ben’s age
and thrice Ellen’s age is 66. Find Ben’s age now.
A. 19
B. 20
C. 18
D. 21
by inspection: 2(𝑋) + 3(10) = 66; 𝑥 = 𝟏𝟖
19. The cable of suspension bridge hangs in the form of a parabola when the
load is uniformly distributed horizontally. The distance between towers is 150
m, the points of the cable on the towers are 22m above the roadway, and the
lowest point on the cable is7 m above the roadway. Find the vertical distance
to the cable form a point in the roadways 15m from the foot of a tower.
A. 16.6m
B. 9.6m
C. 12.8m
D.18.8m
20. If z is directly proportional to x and inversely proportional to the square of y
and that y= 2 when z=4 and x= 2. Find the value of z when x= 3 and y=4.
A. 2/3
B. 3/2
C.3/4
D.4/3
𝑥
. 𝑧 = 𝑦2
𝑧1 𝑦1 2 𝑧2 𝑦2 2
=
𝑥1
𝑥2
4(2)2 𝑧2 (4)2
𝟑
=
∴ 𝑧2 =
2
3
𝟐
21. Find a∙b if lal = 26 and lbl =17 and the angle between them is pi/3.
A. 221
B. 212
C. 383
D.338
𝜋
𝜋
𝐴𝐵 cos ( ) = 26(17) cos ( ) = 𝟐𝟐𝟏
3
3
22. The side of a square is 5 cm less than the side of the other square. If the
difference of their areas is 55cm2, what is the side of the smaller square?
A. 3
B. 4
C. 5
D. 6
by inspection: 82 − 𝑋 2 = 55; 𝑥 = 𝟑
23. The area bounded by the curve y2= 12x and the line x= 3 is revolved about
the line x= 3. What is the volume generated?
A. 186
B. 179
C. 181
D. 184
6
𝐴 = 𝜋∫ |
−6
𝑥2
− 3| 𝑑𝑥 = 𝟏𝟖𝟎. 𝟗𝟔
12
24. Evaluate the integral of (sinx) raised to the 6th power and the limits from 0
to pi/2.
A. 0.49087
B. 0.48907
C. 0.96402
D. 0.94624
5(3)(1) 𝜋
( ) = 𝟎. 𝟒𝟗𝟎𝟖𝟕
6(4)(2) 2
25. How many ounces will she make to serve 25 half-cup?
A. 25
B. 50
C. 12.5
D. 75
5
𝑎
=
; 𝑎 = 𝟒. 𝟔𝟒
sin(66) sin(58)
5
𝑏
=
; 𝑏 = 𝟒. 𝟓𝟒
sin(66) sin(56)
26. Two engineers facing each other with a distance of 5km from each other,
the angles of elevation of the balloon from the two engineers are 56 degrees
and 58 degrees, respectively. What is the distance of the balloon from the two
engineers?
A. 4.46 km, 4.54km
B. 4.64, 4.45km
C. 4.64km, 4.54km D.4.46km, 4.45km
SOLUTION:
27. Evaluate the line integral from (0,0) to (1,1)
.∫[√𝑦𝑑𝑥 + (𝑥 − 𝑦)𝑑𝑦]
A. 5/3
B.4/3
C. 2/3
D. 1/3
28. Find the area of the triangle having vertices at -4-I, 1+2i, 4-3i.
A. 15
B. 16
C. 17
D. 18
29. How many even numbers of three digits each can be made with the digits
0,2,3,5,7,8,9 if no digit is repeated?
A. 102
B. 126
C. 80
D. 90
4 × 5 × 2 = 40
Case II: including 0
5 × 1 × 2 = 10
5 × 6 × 1 = 30
∴ 40 + 10 + 30 = 𝟖𝟎
30. What is the angle subtended in mils of arc length of 10 yards in a circle of
radius 5000 yards?
A. 1.02
B. 2.40
C. 4.02
D. 2.04
31. How many 5 poker hands are there in a standard deck of cards?
A. 2,598,960
B. 2,958,960
C. 2,429,955
D. 2,942,955
Using Calculator: 𝑛𝐶𝑟 = 52𝐶5 = 𝟐𝟓𝟗𝟖𝟗𝟔𝟎
32. In delivery of 14 transformers, 4 of which are defective, how many ways
those in 5 transformers at least 2 are defective?
A. 940
B. 920
C. 900
D. 910
33. A point is chosen at random inside the circle of diameter 8 in. What is the
probability that it is at least 1.5 in away from the center of the circle?
A. 53/64
B. 55/64
C. 52/64
D. 56/64
34. A student did not study for his upcoming examination on which is 15 multiple
choice test questions, with five possible choices of which only one is correct,
what is the expected number of correct answers he can get?
A. 2
B. 3
C. 4
D. 5
35. Evaluate (1+i) raised to (1-i).
A. 2.82+i1.32
B. 2.82-i1.32
C. -2.82-j1.32
D. -2.82+i1.32
36. A boy, 1.20m tall, is walking directly away from the lamp post at the rate of
0.90 m/sec. If the lamp is 6m above the ground, find the rate at which his
shadow is lengthening.
A. 2.25 m/sec
B. 0.225 m/sec
C. 1.125 m/sec
D. 0.235 m/sec
37. A painter needs to find the area of the gable end of the house. What is the
area of the gable if it is a triangle with two sides of 42.0 ft. that meet at a 105
degrees angle?
A. 852 sq. ft.
B. 825 sq. ft.
C. 892 sq. ft.
D. 829 sq. ft.
38. A sector of a circle has a central angle of 50 degrees and an area of 605
sq. cm. Find the radius of the circle
A. 34.6cm
B. 36.4cm
C. 37.2 cm
D. 32.7cm
1
𝐴 = 𝑟 2𝜃
2
50° (
𝜋
) = 0.8727
180°
1
605 = 𝑟 2 (0.8727)
2
𝑟 = 𝟑𝟕. 𝟐
39. If f(x)= sin x and f(𝜋)= 3, then f(x)=
A. 4+cos x
B. 3+cos x
C. 2-cos x
D. 4-cos x
40. If f(x) = 32x, then f(x)=
A. 2(32x)
B. 62x
D. 9(ln9)
C. 9(ln6)
𝑑(9𝑥 ) = 𝑎𝑢 𝑙𝑛𝑎𝑑𝑢 = 𝟗𝒙 𝒍𝒏𝟗
41. Find the slope of the line tangent to 3y2-2x2= 5xy at the point(1,2).
A. -1
B.-2
C. 1
D.2
42. The volume V in3 of unmelted ice remaining from the melting ice cube after
t seconds is given by V(t)=2000-40t+0.2t2. How fast is the volume changing
when t= 40 seconds?
A.-26 in3 /sec
B. -24in3/sec
C. -20in3 /sec
D. -8in3/sec
43. The radius of a circle is measured to be 3 cm correct to within 0.02cm.
Estimate the propagated error in the area of the circle.
A. 0.183cm
B. 0.213cm
C. 0.285cm
D. 0.377cm
44. What is the area within the curve r2= 16cos𝜃.
A. 26
B. 28
C. 30
D. 32
2
𝑟 = 𝑘 cos(𝜃) = 16 cos(𝜃) ; 𝐴 = 2𝑘 = 2(16) = 𝟑𝟐
45. A solid is formed by revolving about the axis, the area bounded by the
curve x3 = y, the y-axis and the line y = 8. Find its centroid.
A. (0,4.75)
B. (0, 4)
C.(0, 5.25)
D. (0,5)
46. Find the area in the first quadrant that is enclosed by y= sin 3x and the xaxis from x=0 the first x-intercept on the positive x-axis.
A. -1/4
B. 2/3
C. 1
D.2
47. Let f(x)= x3+x+4 and let g(x)= f-1 (x). Find g’(6)
A. -1/4
B. -4
C. 1/4
48. 2 gallons is how many quartz?
A. 2
B. 4
C. 6
D. 4
D. 8
4 𝑞𝑢𝑎𝑡𝑠
48. 2𝑔𝑎𝑙𝑙𝑜𝑛𝑠 × 1 𝑔𝑎𝑙𝑙𝑜𝑛 = 𝟖 𝒒𝒖𝒂𝒓𝒕𝒔
49. A recipe calls for 1 cup of milk for every 2-1/2 cups of flour to make a cake
that would feed 6 people. How many cups of both flour and milk need to be
measured to make a similar cake for 8 people?
A. 1-1/3
B. 2-1/3
C. 1-1/2
D.2-1/2
50. Find the vertex of the parabola y2-8x +6y+1=0
A. (3, -1)
B.(-3, 1)
C. (3, 1)
𝑦 2 + 6𝑦 + 9 = 8𝑥 − 1 + 9
D. (-3,-1)
(𝑦 + 3)2 = 8(𝑥 + 1)
𝑽(−𝟏, −𝟑)
51. Find the volume of a cone to be constructed from a sector having a diameter
of 72 cm and a central angle of 150 degrees.
A. 7711.82
B. 5533.32
C. 6622.44
D. 8866.44
52. A and B are points on the opposite sides of a certain body of water. Another
point C is located such that AC= 200 meters, BC= 160 meters and angle BAC=
50 degrees. Find the length of AB.
A. 164.67m
B. 174.67m
C. 184.67m
D.194.67m
160
200
=
; 𝐶 = 73.25
sin(50) sin(𝐶)
90 − 50 = 40
73.25 − 40 = 33.25
𝐴 = 90 − 33.25 = 56.75
160
𝑎
=
; 𝑎 = 𝟏𝟕𝟒. 𝟔𝟕
sin(50) sin(56.75)
53. Find the area of the ellipse 4x2 + 9y2=36.
A. 15.71
B. 18.85
C. 12.57
D. 21.99
. 4𝑥 2 + 9𝑦 2 = 36
4𝑥 2 9𝑦 2 36
+
=
36
36
36
𝑥2 𝑦2
+
=1
32 22
𝐴 = 𝜋𝑎𝑏 = 𝜋(3)(2) = 𝟏𝟖. 𝟖𝟓
54. A couple plans to have 7 children. Find the probability of having at least one
boy.
A. 0.1429
B. 0.1667
C. 0.9922
D. 0.8571
55. A person has 2 parents, 4 grandparents, 8 great grandparents and soon.
How many ancestors during the 15 generations preceding his own, assuming
no duplication?
A. 131070
B. 65534
C. 32766
D. 16383
56. A vendor buys an apple for Php 10 and sells it for Php 15. What percent of
the selling price of apple is the vendor’s profit?
A. 50
B. 33.33
C. 25
D. 66.67
15 − 10
× 100 = 𝟑𝟑. 𝟑𝟑%
15
57. What is the numerical coefficient of the term next to 240x2y2?
A. 220
B. 240
C. 320
D. 340
𝐴𝐵
. 𝐷 = 𝐶+1 =
240(4)
2+1
= 𝟑𝟐𝟎
58. Determine the sum of the first 12 terms of the arithmetic sequence: 3,8,13,..
A. 366
B. 363
C. 379
D. 397
. 𝑑 = 𝑎2 − 𝑎1 = 8 − 3 = 5
𝑎𝑛 = 𝑎𝑚 + (𝑛 − 𝑚)𝑑 = 3 + (12 − 1)(5) = 58
𝑆=
𝑛
12
(𝑎1 + 𝑎𝑛 ) =
(3 + 58) = 𝟑𝟔𝟔
2
2
59. In how many ways can 5 letters be mailed if there are 3 mailboxes
available?
A. 60
B. 80
C. 243
D. 326
60. James is 20 years old and john is 5 years old. In how many years will James
be twice as old as john?
A. 15
B. 10
C. 12
D. 8
20 + 𝑋 = 2(5 + 𝑋); 𝑋 = 𝟏𝟎
61. The diagonal of square is 6 cm. Find its area.
A. 18
B. 24
C. 28
. 𝑑 = 𝑎√2; 6 = 𝑎√2; 𝑎 = 3√2
𝐴 = 𝑎2 = (3√2)2 = 𝟏𝟖
D. 16
62. If cos A = 4/5 and angle A is not in Quadrant I, what is the value of sin A?
A. 0.6
B. -0.6
C. 0.75
D. -.75
4
. 𝐴 = cos −1 (5) = 36.87 𝑏𝑢𝑡 𝑖𝑠 𝑛𝑜𝑡 𝑖𝑛 𝑄𝑢𝑎𝑑𝑟𝑎𝑛𝑡 𝐼 𝑠𝑜 𝐴 𝑖𝑠 − 36.87
sin(− − 36.87) = −𝟎. 𝟔
63. Find the area of a circle inscribed in a rhombus whose perimeter is 100 in.
and whose longer diagonal is 40 in.
A. 116 pi in2
B. 128 pi in2
C. 144 pi in2
D. 188 pi in2
64. A ranger’s tower is located 44m from a tall tree. From the top of the tower,
the angle of elevation to the top of the tree is 28 degrees, and the angle of
depression to the base of the tree is 36 degrees. How tall is the tree?
A. 48 m
B. 62 m
C. 55 m
D. 99 m
65. In an ellipse, a chord which contains a focus and is in line perpendicular to
the major axis is a:
A. latus rectum
B. minor
C. focal width
D. Conjugate axis
66. Find the force on one end of a parabolic trough full of water, if depth is 2ft,
and with across the top is 2 ft. Use 𝜔= 62.5 lb/ft3
A. 125 lbs
B. 133.33 lbs
C. 200 lbs
D. 208.33 lbs
67. Find the Laplace transform of f(t)= e raised to (3t+1).
A. e/(s+3)
B. e/(s-3)
C. e/(s2+ 3)
D. e/(s2-3)
68. If the half-life of a substance is 1,200 years, find the percentage that
remains after 240 years.
A. 76%
B. 77%
C. 87%
D. 97%
. 𝑞1 = 𝑄0 𝑒
1
ln( )
2 )(𝑡)
ℎ𝑙
(
= 240𝑒
(
1
ln( )
2 )(240)
1200
= 208.93
208.93
× 100 = 𝟖𝟕. 𝟎𝟓%
240
69. Robin flies to San Francisco from Santa Barbara in 3 hours. He flies back
in 2 hours. If wind was blowing from north at velocity of 40 mph going, but
changed to 20 mph from the north returning, what was the airspeed of the
plane?
A. 140 mph
B. 150 mph
C. 160 mph
D. 170 mph
3(𝑋 − 40) = 2(𝑋 + 20); 𝑋 = 𝟏𝟔𝟎
70. A tree is broken over by a windstorm. The tree was 90 feet high and the top
of the tree is 25 feet from the foot of the tree. What is the height of the standing
part of the tree?
A. 48.47 ft.
B. 41.53 ft.
C. 45.69 ft.
D. 44.31 ft.
71. In a frustum of cone of revolution the radius of the lower base is 11 in, the
radius of the upper base is 5 in, and the altitude is 8 in. Find the total area in
square inches.
A. 80pi
B. 160pi
C. 226pi
D. 306pi
72. A cask containing 20 gallons of wine emptied on one-fifth of its content and
then is filled with water, if this is done 6 times, how many gallons of wine remain
in the cask?
A. 5.242
B. 5.010
C. 5.343
D. 5.121
73. Goods cost a merchant $ 72. At what price should he mark them so that he
may sell them at a discount of 10% from his mark price and still make a profit
of 20% on the selling price?
A. $ 150
B. $ 200
C. $ 100
D. $ 250
74. Determine the length of the latus rectum of the curve r= 4(1-sin theta).
A. 6
B. 9
C. 8
D. 7
75. Find the radius of the curvature of r= tan theta at theta= 3pi/4.
A. sqrt. of 3
B. sqrt. of 5
C. sqrt. of 6
D. sqrt. of 2
76. Given A= 5i+3j and B=2i+kj where k is a scalar, find k such that A and B are
parallel.
A. 3/5
B. 3
C. 6/5
D. 6
77. What is the x-intercept of the line whose parametric equations are x= 2t -1
and y= 6t+11?
A. -2/3
B. -5/3
C. -7/3
D. -14/3
𝑥 = 2𝑡 − 1 𝑒𝑞. 1; 𝑦 = 6𝑡 + 11 𝑒𝑞. 2; 𝑡 =
𝑦−11
6
𝑒𝑞. 3
𝑦 − 11
2𝑦 22
𝑠𝑢𝑏𝑡. 𝑒𝑞. 3 𝑡𝑜 1: 𝑥 = 2 (
)−1=
−
−1
6
6
6
[𝑥 =
2𝑦 22
−
− 1] 6
6
6
6𝑥 = 2𝑦 − 22 − 6 = 2𝑦 − 28
6𝑥 = 2(0) − 28 ∴ 𝑥 =
−28 −𝟏𝟒
=
6
𝟑
78. What is the coefficient of the (X-1)3 term in the Taylor series expansion of
f(x)=lnx expanded about x= 1?
A. 1/6
B. 1/4
C. 1/3
D. ½
𝑓(𝑥) = 𝑓(𝑎) + 𝑓
′ (𝑎)
(𝑥 − 𝑎)2
(𝑥 − 𝑎)3
(𝑥 − 𝑎)
+ 𝑓′′(𝑎)
+ 𝑓′′′(𝑎)
1!
2!
3!
𝑓(𝑥) = 𝑙𝑛𝑥 = ln(1) = 0
𝑓 ′ (𝑥) =
𝑓 ′′ (𝑥) = −
1 1
= =1
𝑥 1
1
1
=
−
= −1
𝑥2
12
𝑓 ′′′ (𝑥) =
2
2
=
=2
𝑥 3 13
(𝑥 − 1)
(𝑥 − 1)2
(𝑥 − 1)3
𝑓(𝑥) = 0(1) + 1(1)
− 1(1)
+ 2(1)
1!
2!
3!
𝟏
𝑓(𝑥) = 0 + (𝑥 − 1) − (𝑥 − 1)2 + (𝑥 − 1)3
𝟑
79. The position of a particle moving along the x-axis at any time t is given by
x(t)= 2t3 - 4t2+2t-1. What is the slowest velocity achieved by the particle?
A. 17/4
B. 3
C. -2/3
D. -3/2
80. For what value of k will the line kx +5y= 2k have y-intercept 4?
A. 8
B. 9
C. 10
D. 11
k𝑥 + 5𝑦 = 2k
10𝑥 + 5𝑦 = 2(10)
10
5
20
𝑥+ 𝑦=
20
20
20
𝑥 𝑦
+ = 1; ∴ 𝑘 = 𝟏𝟎
2 4
81. Find the circumference of the circle x2+y2-12x+10y+15=0
A. 75.40
B. 57.40
C. 96.12
D. 69.12
𝑥 2 + 𝑦 2 − 12𝑥 + 10𝑦 + 15 = 0
(𝑥 2 − 12𝑥 + 36) + (𝑦 2 + 10𝑦 + 25) = −15 + 36 + 25
2
(𝑥 − 6)2 + (𝑦 + 5)2 = √46
𝐶 = 2𝜋𝑟 = 2𝜋 = 𝟒𝟐. 𝟔𝟏
82. Find the slope of the curve x=t2+et, y=t+et. At the point (1,1).
A. 1
B. 2
C. 3
D. 4
83. Which of the following is true?
A. sin(-θ)=sin θ
B. tan(-θ)=tan θ
C. cos(-θ)=cosθ
D. csc(-θ)=cscθ
Trigonometry Identities (Negative Relations): 𝐜𝐨𝐬(−𝜽) = 𝐜𝐨𝐬 𝜽
84. The hypotenuse of a right triangle is 34 cm. Find the length of the two legs,
if one leg is 14 cm longer than the other.
A. 15 and 29
B. 16 and 30
C. 18 and 32
D. 17 and 31
by inspection 𝑐 = √𝑎2 + 𝑏 2 = √162 + 302 = 34; 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑎 = 𝟏𝟔 𝑎𝑛𝑑 𝑏 = 𝟑𝟎
85. John’s factory has 60 workers. If 4 out of 5 workers are married, how many
workers are not married?
A. 12 workers
B. 24 workers
C. 48 workers
D. 60 workers
SOLUTION:
60 – [(60/5)(4)] = 12 workers
86.
Find the equation of the line whose slope is-3 and the x-intercept is 5.
A. y= -3x+5
B. 3x-y=5
C. 3x+y=15
D. y=3x+15
87. The positive value of k which will make 4x2-4kx+4k+5 a perfect square
trinomial is
A. 6
B. 5
C. 4
D. 3
by inspection 4𝑥 2 − 4𝑘𝑥 + 4𝑘 + 5 = 4𝑥 2 − 4(5)𝑥 + 4(5) + 5 = 4𝑥 2 − 20𝑥 + 25
mode 5 − 3 ∶ roots x =
88. If ln x=2 and ln y= 3, find ln(x3/y1/2).
A. 3.5
B. 4.5
89. If 3x3y= 27 and 2x + y=5, find x.
A. 3
B. 4
C. 2
5
∴ 𝑘=𝟓
2
C. 2.5
D. 1.5
D. 1
90. The area of a circle is six time its circumference. What is the radius of the
circle?
A. 10
B. 11
C. 12
D. 13
𝐴 = 6𝐶; 𝜋𝑟 2 = 6(2𝜋𝑟 2 ); 𝑟 = 12
91. Twelve round holes are bored through a piece of steel plate. Their centers
are equally spaced on the circumference of a circle 18 cm in diameter. What
is the difference between the centers of two consecutive holes?
A. 4.71 cm
B. 4.66 cm
C. 4.32 cm
D. 4.55 cm
92. What is the minimum possible perimeter for a rectangle whose area is 100
sq. in?
A. 50 in.
B. 60 in.
C. 30 in.
D. 40 in.
SOLUTION:
By trial and error
A = lw
100 = 20 x 5
100 = 25 x 4
100 = 50 x 2
Let l = 20
w=5
P = 2l + 2w
P= 2(20) + 2(5)
P = 50 in
93. Find the work done by the force of F= 3i + 10j newton’s in moving an object
10 meters north.
A. 104.40J
B. 100J
C. 106J
D. 108.60J
94. Find the abscissa of a point having an ordinate of 4 of a line that has a yintercept of 8 and slope of 2.
A. -2
B. +2
C. -3
D. +3
95. Find arch of an underpass semi-ellipse 60ft wide and 20ft high. Find the
clearance at the edge of a lane if the edge is 20 ft. from the middle.
A. 18.2 ft.
B. 12.8 ft.
C. 14.9 ft.
D. 16.8 ft.
96. Find the moment of inertia with respect to the y-axis of the first-quadrant
area bounded by the parabola x2= 4y and the line y=x.
A. 34/5
B. 24/5
C. 54/5
D. 65/5
97. What is the length of the transverse axis of the hyperbola whose equation
is 9y2-16x2=144?
A. 6
B. 9
C. 8
D. 7
9 2
16 2 144
𝑥 −
𝑦 =
144
144
144
𝑥2 𝑦2
−
=1
42 32
𝑇𝐴 = 2𝑎 = 2(4) = 𝟖
98.
Find the mass of lamina in the given region and density function:
pi
D[(x, y)], 0 ≤ x ≤ , o ≤ y ≤ cosx and ρ = 7x
2
A. 2
B. 3
C. 4
D. 5
99. How many cubic inches of lumber does a stick contain if it is 4 in. by 4 in. at
one end, 2 in. by 2 in. at the other end, and 16ft long?
A. 1729
B. 1927
C. 1972
D. 1792
100. A goat is tied to a corner of 30ft by 35ft building. If the rope is 40ft and the
goat can reach 1ft farther than the rope length, what is the maximum area the
goat can cover?
A. 4840.07
B. 4084.07
C. 4804.07
D. 4408.07
REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
MARCH 2015
MATHEMATICS
REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION
MARCH 2015
MATHEMATICS
1. Sand is pouring to from a conical pile such that its radius is always twice its
height. If the volume of a conical pile is increasing at the rate of 2 cu. m/sec.
how fast is the height is increasing when the height is 4m?
A. 1/16pi m/s
B. 1/32 pi m/s
C. 1/64 pi m/s
D. 1/8 pi m/s
SOLUTION:
𝝅𝒓𝟐
𝟑
𝑽=
𝒅𝒗
=𝟐
𝒅𝒕
𝒓 = 𝟐𝒉
𝒓=
𝑽=
𝑽=
𝑹𝒉
𝑯
𝝅𝒓𝟐 𝒉
𝟑
𝝅𝑹𝟐 𝒉𝟐 𝒓𝟐
𝟑𝑯𝟐
𝒅𝒗 𝝅𝑹𝟐 𝒉𝟐
=
𝒅𝒕
𝑯𝟐
𝟐=
𝝅𝑹𝟐 𝒉𝟐 𝒓𝟐
𝑯𝟐 𝑹 𝟐
𝟐 = 𝝅(𝟐(𝟒))
𝒉=
𝟐
𝝅
𝟑𝟐
2. A triangular corner lot has perpendicular sides of lengths 90 m and 60 m. find
the dimension of the largest rectangular building that can be constructed on
the lot with sides parallel to the streets.
A. 30 m x 30 m
B. 24 m x 24 m
C. 25 m x 40 m
D. 45m x 30 m
3. Joy is 10% taller than Joseph and Joseph is 10% taller than Tom. How many
percent is Joy taller than Tom?
A. 18%
B. 20%
C. 21%
D. 23%
SOLUTION:
JOY = JOSEPH (1+.10)
JOSEPH = TOM (1+.10)
JOY [TOM (1+.10)] (1+.10)
JOY = TOM (1+.10)2
JOY = TOM (1+.21)
.21 = 21%
4. What is the length of the shortest line segment in the first quadrant drawn
tangent to the ellipse b2 x2 + a2 y2 = a2 b2 and meeting to the coordinate axes?
A. a/b
B. a + b
C. ab
D. b/a
5. What is the area of largest rectangle that can be inscribed in an ellipse with
equation 4x^2+y^2=4?
A. 3
B. 4
C. 2
D. 1
6. A company hires 30 new employees today. It increases their workforce by 5%.
How many workers now?
A. 610
B. 600
C. 630
D. 620
SOLUTION:
𝒘𝒐𝒓𝒌𝒆𝒓𝒔 𝒏𝒐𝒘 =
𝟑𝟎
+ 𝟑𝟎
. 𝟎𝟓
𝒘𝒐𝒓𝒌𝒆𝒓𝒔 𝒏𝒐𝒘 = 𝟔𝟑𝟎
7. Find the radius of the circle inscribed in the triangle determined by the lines
y=x+4, y=-x-4 and y=7x-2.
A.5/sqrt of 2
B. 5(2sqrt of 2)
C. 3/R
D. 3/(2sqrt of 2)
8. What is the ratio of the surface area of a sphere to its volume?
A. 5/R
B. 4/R
C. 3/R
D. 2/R
SOLUTION
𝟒𝝅𝟐
𝒓𝒂𝒕𝒊𝒐 =
𝟒𝝅𝑹𝟑
𝟑
𝟑
𝒓𝒂𝒕𝒊𝒐 =
𝑹
9. Using original diameter, d, what is the new diameter when the volume of the
sphere is increased 8 times?
A. 2d
B.3d
C.4d
D. 5d
SOLUTION
𝒗𝟐 = 𝟖 𝒕𝒊𝒎𝒆𝒔
𝒅 𝟑
𝟒𝝅 (𝟐)
𝟖=
𝟑
𝟒𝟖
= 𝒅𝟑
𝝅
𝟏
𝟔 𝟑
𝒅 = 𝟐( )
𝝅
𝒗𝟏 = 𝒐𝒓𝒊𝒈𝒊𝒏𝒂𝒍
𝟏
𝟔 𝟑
𝒅=( )
𝝅
Therefore, 2d
10.
In a hotel it is known that 20% of the total reservation will be cancelled in
the last minute. What is the probability that there will be less than 2
reservations cancelled out of 4 reservations?
A. 0.6498
21.
B. 0.5629
C. 0.3928
D. 0.4096
A political scientist asked a group of people how they felt about two political
policy statements. Each person was to respond A (agree) (N) neutral or (D)
disagree to each NN, NA, DD, DN, DA, AA, AD and AN. Assuming each
response combination is equally likely, what is the probability that the person
being interviewed agrees with exactly one of the political policy statements?
A. 1/9
SOLUTION:
B. 2/5
C. 2/9
D. 4/9
NA NN
ND
AA
AN
AD
DA DN
DD
=
𝟒
𝟗
22.
Evaluate Laplace transform of t^n
A. n!/s^n
23.
B. n!/s^(n+1)
C. n!/s^(n-1)
D. n! s^(n+2)
Find the area of a quadrilateral having vertices at (2,-1), (4.3), (-1,2) and (-
3,-2)
A. 16
B. 18
C. 17
D. 14
SOLUTION:
1
𝑆𝐴 = √𝐴𝐵 + 𝐵𝐶 + 𝐶𝐷 + 𝐷𝐴
2
𝐴𝐵 = √(4 − 2)2 + (3 − (−1))2 = 4.47
𝐵𝐶 = √(−1 − 4)2 + (2 − 3)2 = 5.1
𝐶𝐷 = √(−3 − (−10))2 + (−2 − 2)2 = 4.47
𝐷𝐴 = √(2 − (−3))2 + (−1 − (−2))2 = 5.1
1
𝑆𝐴 = 2 √4.47 + 5.1 + 4.47 + 5.1 = 18 sq. unit
24.
In a 15 multiple choice test questions with five possible choices of which
only one is correct, what is the standard deviation of getting a correct answer?
A. 1.55
B. 1.07
C. 1.50
D. 1.65
SOLUTION:
15(5 − 1)
5(1)
𝑆𝐷 =
𝟏. 𝟓𝟓
5
25.
In polar coordinate system the distance from a point to the pole is known
as:
A. Polar angle
B. radius vector
C. x- coordinate
D. y-coordinate
26.
Evaluate Laplace transform of cos2kt.
A. s/s(s2 -2k2 )
B. s/(s2+2k2)
C. s/(s2-4k2)
D. s/(s2+4k2)
SOLUTION:
𝐶𝑜𝑠𝑏𝑡 =
𝑠2
𝑠 2 + 𝑏2
𝐶𝑜𝑠𝑏𝑡 =
𝑠2
𝑠 2 + (2𝑘)2
𝒔𝟐
𝑪𝒐𝒔𝒃𝒕 = 𝟐
𝒔 + 𝟒𝒌𝟐
27.
Find the power series of tan-1 (t2)
A. T2+t6/2 +t12/6 +t24/12+…
C. t2+t6/3+t10/5+t14/7+…
B. T2 - t6/2 + t12/6 – t24 /12+…
D. t2-t6/3+t10/5-t14/7+…
28.
Simplify (1+tanx)/(1-tanx)
A. Sec x + tan x
B. cos x + tan x
C. cos 2x+ tan 2x
D. sec 2x+tan 2x
SOLUTION:
𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥
1 + 𝑡𝑎𝑛𝑋 1 + 𝑐𝑜𝑠𝑥
𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥
𝑐𝑜𝑠𝑥
=
=
=
1 − 𝑡𝑎𝑛𝑥 1 − 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥
𝑐𝑜𝑠𝑥
𝑐𝑜𝑠𝑥
=
(cos 𝑥)2 + 2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥 + (𝑠𝑖𝑛𝑥)2
𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥
(
)=
(cos 𝑥)2 − (𝑠𝑖𝑛𝑥)2
𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥
=
1 + 𝑠𝑖𝑛2𝑥
1
𝑠𝑖𝑛𝑥
=
+
= 𝒔𝒆𝒄𝟐𝒙 + 𝒕𝒂𝒏𝟐𝒙
𝑐𝑜𝑠2𝑥
𝑐𝑜𝑠2𝑥 𝑐𝑜𝑠𝑥
29.
A. 1
Evaluate lim x+4/x-4 as x approaches to infinity
B. 0
C. 2
D. infinite
SOLUTION:
LIM𝑋=∞
𝑋+4
𝑋−4
LIM𝑋=∞
∞+4
= INDETERMINATE
∞−4
Apply L′ Hospital,
LIM𝑋=∞
30.
1
=𝟏
1
It represents the distance of a point from the y-axis
A. Ordinate B. coordinate
C. abscissa
D.polar distance
31. A and B can do piece of work is 5 days, B and C in 4 days while A and C in
2.5days in how many days can all of them do the work together?
A. 40/11
B. 30//11
C. 30/17
D. 40/17
SOLUTION:
Workers
Hours per
day
Ratio of work
per day
1/A + 1/B
5
2/5
1/B + 1/C`
4
2/4
1/C + 1/A
2.5
2/2.5
2/A + 2/B + 2/C =x/0.825
X = 1/0.425
X = 40 / 17
32.
Chona the golden retriever gained 5.1 pounds is one month. She weighs
65.1 pounds now. What is the percent weight gain of Chona in one month?
A. 7.3%
B. 8.2%
D. 7.8%
SOLUTION:
Weight gain = 5.1
Present weight = 65.51
65.51- 5.1 = x
X = 60.41
D. 8.5%
% of weight gain = (weight gained / original weight) x 100
% = (5.1 / 60.42) x 100
% = 8.44 %
33.
What is the center and radius of a circle with an equation x2+y2-1/4x-
1/4y=1/64?
A. C (1,1/2) R=4
B. C(1,1), R=sqrt5/9
C. C(1/2-1/2 R=sqrt2/5
D. C (1/8, 1/8) R=sqrt 3
SOLUTION:
X^2 + y^2 – 1/4x – 1/4y = 1/64
X^2-1/4x + y^2-1/4y = 1/64
(x^2 – 1/4x + 1/64) + (y^2 -1/4y+1/64) = 1/64 +1/64 + 1/64
(x – 1/80)^2 + (y-1/8)^2 = 3/64
C ( 1/8 , 1/8) r= sqrt.of 3/8
34.
A machine only accepts quarters. A bar of candy cost 25c a pack of peanuts
cost 50c and the bottle of coke cost 75c. If Marie bought 2 candy bars a pack
of peanut and a bottle of coke how many quarters did she pay?
A. 5
B. 6
C.7
D. 8
SOLUTION:
Cost of
Food
Number
of
Total
Cents
Pieces
Candy Bar
2pc
50c
1pc
50c
1pc
75c
25c
Pack of
Peanuts
50c
Bottle of
Coke 75c
50c = 2quarter
50c = 2quarter
75c = 3quarter
Total number of quarters is
2 + 2 + 3 = 7 quarters
35.
Solve for x and y in xy +8+j (x2y+y)=4x+4+j(xy2+x)
A. 2, 2
B. 2,3
C. 3,1
D. 3,4
SOLUTION:
Real number :xy + 8 = 4x + 4
Imaginary number: x^2y + y = xy^2 = x
Get y @
eq. of real no.
Y= 4x-4 / x
X^2y + y = xy^2+y
X^2(4x-4)/x) + 4x-4 / x = x(4x-4 / x)2 +x
X= 2
Y= 4x-4 / x
= 4x-4 / x = 2
x,y (2,2)
36.
There are a set of triplets. If there are 11 generations how many ancestors
do they have if duplication is not allowed?
A. 4095
B. 4065
C.59,049
D. 265,719
SOLUTION:
37.
Carmela and Marian were hired on a summer job. Each of them work 15
hours a week. Carmela was absent for one week and Marian has to take her
shift. If they work for 8 weeks, what is the total number of hours did Marian
works?
A. 120
SOLUTION:
B. 135
C. 67.5
D. 60
#
Carmel
Number
Number
hour
of
of
s
week
Hours
per
s
rendere
wee
worke
d
k
d
15
7 (1
a
x-15
week
absen
t)
Marian
15
8 weeks
x
+1
Solve for number of hours Marian worked:
15(8) +15 = 135 hrs.
38.
From the top of a building the angle of depression of the floor of a pole is
48 deg 10min. from the foot of a building the angle of elevation of the top is 18
deg 50 min, both building and pole are on a level ground. If the height of a pole
is 4m, how high is the building?
A. 13.10m B. 12.10
C. 10.90
SOLUTION:
TanƟ =h/x
h=height of building/pole
x= distance between
tan (18°50°) = 4 / x
x = 11.73
x=11.73
tan (48°10°) = h / x
h = 11.73 (tan 48°10°)
h= 13.10m
D. 11.60
39.
The towers of a parabolic suspension bridge 300m long are 60 m high and
the lowest point of a cable is 20m above the roadway. Find the vertical distance
from the roadway to the cable at 100m from the center.
A. 17.78
B.37.78
C.12.86
D. 32.86
SOLUTION:
Y= ax^2+bx+c
@(0,20) lowest point of the cable
20= a(0)^2 + b(0) +c
20= c
Solving a and b
@ P (150 , 60)
60= a(150)^2 + b(150) + 20
60-20 = 150^2 a + 150b
40 = 150 (150a +b)
4/15 = 150 a+b
4/15 – 150a = b
eq.1
@ P(-150, 60)
60 = a(-150)^2 + b(-150) + 20
60-20 = (-150)^2 a + (-150)b
40 = -150 (b-150a)
-4/15 = b-150a
Subst 1 to 2
-4/15 = 4/15 -150a – 150a
300a = 8/15
A = 2/1125
B= 4/5 – 150(2/1125)
B= 4/5 – 4/5
B= 0
@ x=100 find y=?
Y = ax^2 +bx + c
Y= 2/1125(100)^2 + 0(100) + 20
Y =37.78
40.
Find the centroid of the plane area bounded by the parabola y=4 - x^2 and
the x-axis
A. (0 3/2)
B. (0,1)
C. (0 , 12/5)
D. (0,8/5)
SOLUTION:
Y = 4-x^2 when x=0
Y= 4
When y= 0
X^2 = 4
X= +- (2)
2
A= ∫−2(4 − 𝑋^2)𝑑𝑥
A= 32/3 ,𝑥̅ =0
2
A𝑦̅ = ∫−2 𝑦𝑐 𝑑𝐴
2
32/3𝑦̅ = ∫−2(4 − 𝑋 2 )/2)𝑑𝑥 (4 − 𝑥 2 )𝑑𝑦)
𝑦̅ =
8
5
C ( 0 , 8/5)
41.
Evaluate the double integral 1/(x-y) dxdy with inner bounds of 2y to
3y and outer bounds of 0.2.
A. Ln3
B. ln4
C. ln2
SOLUTION:
2
3𝑦
∫ ∫
0
2𝑦
1
𝑑𝑥 𝑑𝑦
𝑥−𝑦
Le 𝑢 = 𝑥 − 𝑦
𝑑𝑢 = 𝑑𝑥
3𝑦
∫
2𝑦
3𝑦
1
𝑑𝑢
𝑑𝑥 = ∫
𝑥−𝑦
2𝑦 𝑢
D ln8
3𝑦
=∫
2𝑦
𝑑𝑢
𝑢
= ln 𝑢
3𝑦
= ln 𝑥 − 𝑦|2𝑦
= (ln 3𝑦 − 𝑦) − (ln 2𝑦 − 𝑦)
= (ln 2𝑦) − (ln 𝑦)
3𝑦 1
Sub the value of ∫2𝑦
𝑥−𝑦
𝑑𝑥
2
∫ (ln 2𝑦) − (ln 𝑦) 𝑑𝑦
0
2
2
= ∫ ln 2𝑦 𝑑𝑦 − ∫ ln 𝑦 𝑑𝑦
0
2
0
2
= ∫ ln 2𝑦 𝑑𝑦 − ∫ ln 𝑦 𝑑𝑦
0
0
2
2𝑦𝑙𝑛 2𝑦 − 2𝑦
= (
) − (𝑦𝑙𝑛 𝑦 − 𝑦)|
2
0
= (𝑦𝑙𝑛 2𝑦 − 𝑦) − (𝑦𝑙𝑛 𝑦 − 𝑦)|20
= 𝑦𝑙𝑛 2𝑦 − 𝑦 − 𝑦𝑙𝑛 𝑦 + 𝑦|20
= 𝑦𝑙𝑛 2𝑦 − 𝑦𝑙𝑛 𝑦|20
= 𝑦(𝑙𝑛 2𝑦 − 𝑙𝑛 𝑦)|20
= 2(𝑙𝑛 2(2) − 𝑙𝑛 2)
= 2(𝑙𝑛 4 − 𝑙𝑛 2)
4
= 2 (ln )
2
= 2(ln 2)
= ln 22
= 𝐥𝐧 𝟒
42.
Write the equation of a line with x-intercepts a=8 and y intercept b=-1
A. x+8y-8=0 B. x-8y+8=0
C. x+8y+8=0
D. x-8y-8=0
SOLUTION
𝑥 𝑦
+ =1
𝑎 𝑏
𝑥
𝑦
+
=1
8 −1
𝑥 𝑦
− =1
8 1
𝑥 − 8𝑦 = 8
𝒙 − 𝟖𝒚 − 𝟖 = 𝟎
43.
Solver for x; 125x-5=5x-4
A. 11/2
B. 15/2
C. 17/2
D. 19/2
SOLUTION:
ln 125𝑥−5 = ln 5𝑥−4
(𝑥 − 5) ln 125 = (𝑥 − 4) ln 5
ln 5
ln 125
1
(𝑥 − 5) = (𝑥 − 4)
3
1
4
(𝑥 − 5) = 𝑥 −
3
3
1
4
𝑥− 𝑥 =5−
3
3
2
11
𝑥=
3
3
𝟏𝟏
𝒙=
𝟐
(𝑥 − 5) = (𝑥 − 4)
44.
Find the ratio of the surface area of a cube to its volume if the side is s.
A. 3/s
B. 4/s
C. 6/s
SOLUTION:
𝐴 6𝑠 2
= 3
𝑉
𝑠
D.5/s
𝑨 𝟔
=
𝑽 𝒔
45.
Solve the equation y’ = y/2x
A. Y^2=cx^3
B. y=cx^2
C. y^2=cx
SOLUTION:
Rewriting the equation
𝑦
=0
2𝑥
𝑑𝑦 𝑦
−
=0
𝑑𝑥 2𝑥
𝑑𝑦
1
−𝑦
=0
𝑑𝑥
2𝑥
𝑦′ −
Using linear DE
∅ = 𝑒 ∫ 𝑃(𝑥)𝑑𝑥
1
∅ = 𝑒 ∫ −2𝑥𝑑𝑥
1
∅ = 𝑒 −2 ln 𝑥
∅=
1
ln
√
𝑒 𝑥
∅=
1
√𝑥
𝑦∅ = ∫ ∅𝑄(𝑥)𝑑𝑥
𝑦
1
=∫
√𝑥
𝑦
=𝐶
√𝑥
1
√𝑥
(0)𝑑𝑥
𝑦 = 𝐶 √𝑥
(𝑦)2 = (𝐶 √𝑥)
2
𝑦2 = 𝐶2𝑥
𝒚𝟐 = 𝑪𝒙
D. y=cx
46. The sum of the first 7 terms of an A.P is 98 and the sum of the first 12 terms
is 288. Find the sum of the first 20 terms
A. 980
B. 800
C. 880
D. 980
SOLUTION:
𝑛
[2𝑎1 + (𝑛 − 1)𝑑]
2
7
98 = [2𝑎1 + (7 − 1)𝑑]
2
98 = 2𝑎 + (𝑛 − 1)𝑑
1
⁄7
2
28 − 2𝑎1
𝑑=
6
𝑆𝑛 =
288 =
12
[2𝑎1 + (12 − 1)𝑑]
2
288
⁄12 = 2𝑎1 + (12 − 1)𝑑
2
48 − 2𝑎1
𝑑=
11
Equate d
48 − 2𝑎1 28 − 2𝑎1
=
11
6
288 − 12𝑎1 = 308 − 22𝑎1
10𝑎1 = 20
𝑎1 = 2
48 − 2𝑎1
11
48 − 2(2)
𝑑=
11
𝑑=
𝑑=4
𝑆20 =
20
[2(2) + (20 − 1)(4)]
2
𝑺𝟐𝟎 = 𝟖𝟎𝟎
47.
When the sun is 20 degrees above the horizon, how long is the shadow
cast by a building 200 ft high?
A. 550 ft
B. 580ft
C. 405ft
D. 450ft
SOLUTION:
ℎ
𝑥
200
tan 20 =
𝑥
200
x=
tan 20
tan 𝜃 =
𝐱 = 𝟓𝟓𝟎 𝒇𝒕
48.
A central angel of a circle of radius 30 in intercepts an arc of 6 in is how
many radian?
A. 1/3
B. 1/5
C. ¼
D. ½
SOLUTION:
𝑆 = 𝑟𝜃
𝑆
𝑟
6
𝜃=
30
𝟏
𝜽=
𝟓
𝜃=
49.
A, B and C work independently on a problem. If the respective probabilities
that they will solve it are ½, 1/3, 2/5 find the probability that the problem will be
solved.
A. 1/5
B. 2/5
C. 3/5
D. 4/5
SOLUTION:
𝑃(𝐴 ∪ 𝐵 ∪ 𝐶) = ?
𝑃(𝐴 ∪ 𝐵 ∪ 𝐶) = 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶) − 𝑃(𝐴𝐵) − 𝑃(𝐵𝐶) − 𝑃(𝐶𝐴) + 𝑃(𝐴𝐵𝐶)
𝑃(𝐴 ∪ 𝐵 ∪ 𝐶) =
1 1 2
1 1
1 2
2 1
1 1 2
+ + −( × )−( × )−( × )+( × × )
2 2 5
2 3
3 5
5 2
2 3 5
𝑷(𝑨 ∪ 𝑩 ∪ 𝑪) =
50.
𝟒
𝟓
A car goes 14kph faster than a truck and requires 2 hours and 20 minutes
less time to travel 300km. Find the rate of the car.
A. 40kph
B. 50kph
C. 60kph
SOLUTION:
Let 𝑥 = 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑏𝑢𝑠
𝑥 + 14 = 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐𝑎𝑟
𝑦 = 𝑡𝑖𝑚𝑒 𝑡𝑟𝑎𝑣𝑒𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑎𝑟
𝑦+
7
= 𝑡𝑖𝑚𝑒 𝑡𝑟𝑎𝑣𝑒𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑢𝑠
3
𝑑 = 300 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑑 = (𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐𝑎𝑟)(𝑡𝑖𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑎𝑟)
𝑑 = (𝑥 + 14)(𝑦)
𝑑 = (𝑟𝑎𝑡𝑒 𝑜𝑓 𝑏𝑢𝑠)(𝑡𝑖𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑢𝑠)
7
𝑑 = (𝑥)(𝑦 + )
3
Equate d
7
(𝑥 + 14)(𝑦) = (𝑥)(𝑦 + )
3
𝑥𝑦 + 14𝑦 = 𝑥𝑦 + 2.333𝑥
14𝑦 = 2.333𝑥
𝑥=
Subst. the value of x
𝑑 = (𝑥 + 14)(𝑦)
14𝑦
300 = (
+ 14) 𝑦
2.333
300 = 6𝑦 2 + 14𝑦
0 = 6𝑦 2 + 14𝑦 − 300
14𝑦
2.333
D.70kph
0 = 3𝑦 2 + 7𝑦 − 150
To find y
𝑦=
−𝑏 + √𝑏 2 − 4𝑎𝑐
2𝑎
𝑦=
−7 + √72 − 4(3)(150)
2(3)
𝑦= 6
Solving the rate of the car
𝑥 + 14 = 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐𝑎𝑟
14(6)
+ 14 = 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐𝑎𝑟
2.333
𝟓𝟎𝒌𝒑𝒉 = 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒄𝒂𝒓
51.
Find the area bounded by 𝑥 = 2𝑦 − 𝑦 2 and the y-axis
A. 4/3
B. 5/3
C. 2/3
D. 1/3
SOLUTION: Get the limit.
Let X = 0; 0 = 2𝑦 − 𝑦 2
𝑦 2 − 2𝑦 = 0
−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎
2±√−22 −4(1)(0)
→
2(1)
= (2,0)
2
A = ∫0 ( 2𝑦 − 𝑦 2 ) 𝑑𝑦
2
A = 2 – 2y∫0
A = 4/3
52. A steel ball at 120 deg C cools in 20 minutes to 80 deg C in a room at 25
deg C. Find the temperature of the ball after half an hour.
A. 40.96 deg C
B. 45.96 deg C
C. 66.85 deg C D. 55.96
deg C
SOLUTION:
𝑑𝑇
𝑑𝑡
= −𝑘(𝑡 − 25)
∫ 𝑑𝑇⁄𝑇 − 25 = −𝑘 ∫ 𝑑𝑡
Ln(T-25) = -kt +C
𝑒 ln(𝑇−25) = 𝑒 −𝑘𝑡 𝑒 𝐶
𝑇 − 25 = 𝑒 −𝑘𝑡 𝑒 𝐶
𝑇 = 𝐶𝑒 −𝑘𝑡 + 25, solve for C and k
120 = 𝐶𝑒 0 + 25
120 = C + 25
C = 95
, solve for k
80 = 95𝑒 −𝑘(20) + 25
55 = 95𝑒 −20𝑘
11
= 𝑒 −20𝑘
19
Ln (
11
) = 𝑙𝑛𝑒 −20𝑘
19
11
Ln(19) = −20𝑘
K = 0.027, then solve for the temperature after 30 min.
𝑇 = 95𝑒 −0.027(30) + 25
T = 67 deg C ≅ 66.85 deg C
53. If the line kx+3y+8=0 has a slope of 2/3, determine k.
A. -3
B. -2
SOLUTION:
C. 3
D. 2
Kx+3y+8=0
3y = kx + 8
−3𝑦
−3
𝑘𝑥
8
= −3 + −3
𝑦=−
2
𝑘𝑥
8
𝑘
3
3
− ; - = 𝑠𝑙𝑜𝑝𝑒
3
𝑘
= − 3, therefor k = - 2
3
54.Find the numerical coefficient of the term involving 𝑋 20 of (3𝑥𝑦 2 − 𝑥 4 )3
without expanding.
A. 21.402
B. 22.104
C. 20.412
D. 23.214
55. A rock is dropped down a well that is 256 feet deep. When will it hit the
bottom of the well?
A. 1 sec
B. 2 sec
C. 3 sec
D. 4 sec
SOLUTION:
1𝑚
1
𝑚 2
256𝑓𝑡 (
) = (9.81
)𝑡
3.281𝑓𝑡
2
𝑠𝑒𝑐 2
𝑡2 =
78.02
𝑚
)
𝑠𝑒𝑐2
2(9.8
t = 4 sec
56. If the side of a cube is measured with an error of at most 3 percent, estimate
error in the volume of the cube.
A. 3 percent
SOLUTION:
B. 6 percent
C. 9 percent
let x be the side of the cube
D. 12 percent
V = 𝑥3
V = 1+0.03 𝑥 3
V = (1.03 𝑥)3
V = 1.092727𝑥 3 , subtracting the volume
it is supposed to have,
We have an error of 0.092727 or 9.27 percent
57. Find the k so that A = <3, -2> and B = <1, k > are parallel.
A. 2/3
B. -2/3
C. 3/2
D. -3/2
SOLUTION:
𝐹𝑜𝑟 𝑙𝑖𝑛𝑒 1:
𝑦 +2
2
𝑚1 = 𝑥2 −3 = − 3
2
Since parallel: 𝑚1 = 𝑚2
𝑦 −𝑘
𝑚1 = 𝑚2 = 𝑥2 −1
2
2
0−𝑘
− 3 = 0−1
𝟐
𝑘 = −𝟑
58. Find the slope of the curve x =3t, y = 9𝑡 2 – 3t when t=1.
A. 4
SOLUTION:
x = 3t
B. 5
C. 6
D. 3
Dx = 3;
Dy= 18t-3
𝑑𝑦
Y’ = 𝑑𝑥 =
Y’ =
(18𝑡−3)
3
(18(1)−3)
=5
3
59. Find the area of a triangle having vertices at -4-i, 1+2i, 4-3i
A. 15
B. 16
C. 17
D. 18
SOLUTION:
60. Find the acute triangle between the vectors 𝑧1 = 3 − 4𝑖 and 𝑧2 = 4 + 3𝑖.
A. 18deg 18 min
B. 15deg 15 min C. 17deg 17 min D. 16deg 16 min
61. A chord is 36 cm long and its midpoint is 36 cm from the midpoint of the
longer arc. Find the radius of the circle.
A. 22.5 cm
B. 28.5
C. 20.5
D. 24.5
SOLUTION:
 36 
R 2  (36  R) 2   
 2
2
R = 22.5
62. Sarah leaves seattle for New York in her car, averaging 80 mph across open
country. One hour later a plane leaves seattle for New York following the same
route and flying 400 mph. How long it be before the plane overtakes the car?
A. 1 hr
B. 1/3 hr
C. 1/2 hr
D. 1/4 hr
SOLUTION:
80(x-1) = 400(x)
T
1
hrs
4
63. What is the length of the latus rectum of the parabola x^2 = -16y
A. 8
B. -8
C. 16
D. -16
SOLUTION:
LR = 4a = 16
64. Mr. Santos owns a jewelry store. He marks up all merchandise 50 percent of
cost. If he sells a diamond ring for P15,000, what did he pay the wholesaler for
it?
A. P 5,000
B. P 12,000
C. P 10,000
D. P 20,000
SOLUTION:
15,000 = x(1+0.05)
X = 10,000
65. What is the equation of the normal to the curve X^2 +y^2 = 25 at (4,3)?
A. 3x-4y=0
B. 5x+3y=0
C. 5x-3y=0
D. 3x+4y=0
SOLUTION:
X 2  Y 2  25
2 X  2Y  0
 2X  X
Y' 

2y
Y
dy  4
M1  
dx 3
1
3
M2

4 4
 
 3 
3
Y
4X
4y = 3x
3x - 4y = 0
66. For what values of X is I x-3 I = 1?
A. 4
B. 2
C. 2,4
D. -2,-4
SOLUTION:
I 4-3 I = 1
I 2-3 I = 1
= 2,4
67. If 3x = 4y then 4y^2/3x^2 is equal to:
A. 3/4
B. 4/3
C. 2/3
D. 3/2
SOLUTION:
3x = 4y then
4 y2
?
3x 2
 3Y 
3X  
'
4Y * Y
 4 

2
3X
3X 2
= 3/4
68. A wall is 15 ft high and 10 ft from a house. Find the length of the shortest
ladder which will just touch the top of the wall and reach a window 20.5 ft above
the ground.
A. 11.4 ft
B. 42.5 ft
C. 14.1 ft
D. 54.2 ft
SOLUTION:
Tan Ɵ = 20.5 Xa
Tan Ɵ = 15 Xb
20.5
15

10  Xb Xb
Xb = 27.27
2
3
2
3
L  20.15  10
2
3
L = 42.25
69. A bag contains 3 white and 5 red balls. If two balls are drawn at random, find
the probability that both are white.
A. 3/28
B. 3/8
C. 2/7
D. 5/15
70. Determine the eccentricity of the hyberbola xy = 8
A. 1.368
B. 1.414
C. 1.521
D. 1.732
SOLUTION:
xy = 8
x=y
X
8
2
X 
8
2
= 1.414
71. Which term of the arithmetic sequence 2, 5, 8, . . . is equal to 227?
A. 20
B. 120
C. 76
D. 36
SOLUTION:.
An = A1 + (n -1 )d
227 = 2 + (n -1) 3
n = 76
72. Name the type of graph represented by x2-4y2-10x-8y=0
A. Ellipse
B. Parabola
C. Hyperbola
1
73. If logx3=4 then x=
A. 91
B. 81
C. 42
D. 50
SOLUTION:.
Log3
=0.25
LogX
0.477=0.25logx
D. Circle
101.908=x
x=81
74. If f(x0=-x2, then f(x+1)=
B. –x3-2x
A. -x2-2x
C. 3x2-2x
D. -x-2x
SOLUTION:
–(x+1)2=-(x2+2x+2-2)
=-x2-2x
75. If this graph of y=(x-2)2-3 is translated 5 units up and 2 units down to the right,
then the equation of the graph obtained is given by
A. x=(x-4)2+2
B. y=(x-4)2+2
C. y=(x-4)4+2
D. y=(x-4)2+5
SOLUTION:
(x+h)2=4a(y+k)
y-5=(x-2-2)2-3
y=(x-4)2+2
76. Which one is not a root of the fourth root of unity?
A.
𝒊
√𝟐
𝑖
B. √20
𝑖
2
C. √40
D. √2
77. Find the area of the largest circle which can be cut from a square of edge 4
in.
A. 14
B. 12.57
C. 20
SOLUTION :
𝜋𝑑 2
𝐴=
4
𝜋42
𝐴=
4
D. 11.25
= 12.57𝑖𝑛2
78. If I=(-1)1/2, find the value of i36
A. 3
B. 4
C.1
D. 2
SOLUTION
(-1)36/2=(-1)18=1
79. If cot B=5/2, find sin B
𝟐𝟐
22
A. sin B=√𝟐𝟗
B. cos B=√29
22
C. tan B=√29
22
D. cot B=√29
SOLUTION :
B=cot-1(5/2) = 0.38
sin(0.38) =
2
√29
80. A man is 1.6 m tall casts a shadow 4 m long. Nearby, a flagpole casts a
shadow 18 m long. How high is the flagpole?
A. 8.1
B. 9.2
C. 7.2
D. 6.2
SOLUTION:
x:1.6=18:4
4x=28.8
x=7.2 m
81. The rotary Club and the Jaycees Club had a joint party 120 members of the
rotary Club and 100 members of the Jaycees Club also attended but 30 of
those attended are members of both clubs. How many Jaycees attended the
party?
A. 150
B.250
C. 190
D. 22
SOLUTION
Members
Rotary
120
Club
Jaycee
100
Club
X-100=220-30
X=290, but Jaycees have 100 members
290-100=190
82. Find the work done by a force F= -2j( pounds) applied to a point that moves
on a line from (1, 3) to (4, 7).Assume that distance is measure in feet.
A. 8ft.lb
B. -10ft.lb
C. -12 ft.lb
SOLUTION
Given: distance= [(1-3)(4-7)]j
Force= (-2j)
work done= F.d
Required: work
work done= F.d
d= [(1-3)(4-7)]j=-6j
work done= ( -2j)(-6j)
D. 15ft.lb
Work done= -12ft.lb
83. A particle has a position vector<2 cos2t, 1-3sint>. What is the speed of the
particle at time t= pi/4?
A. 5.427
B. 7.245
C. 1.879
D. 4.528
SOLUTION:
84. Evaluate Γ(-3/2)
2
A. 3(sqrt.of pi)
3
1
B. 4( sqrt of pi)
C. 2(sqrt of pi)
𝟒
D. 𝟑(sqrt of pi)
SOLUTION:
n Γ n= Γ( n-1)
−3
−3
3
( 2 )Γ( 2 )= Γ((2+1)
[-3/2Γ -3/2= Γ( -1/2)] -2/3
3
2
1
Γ2=-3 Γ(− 2)
1
1
1
=-2Γ(-2)=Γ⟦− 2 + 1⟧-2
−3
=Γ( 2 ) =
−2
3
(−2)Γ
1
2
𝟒
=𝟑π√𝟐
85. Evaluate tan2(j 0.78)
A. 0.533
B. -0.653
C. 0.426
SOLUTION:
sin(j0.78)
= ⌊cos(𝑗0.78)⌋2
D. -0.426
=⌊jsinh(0.78)⌋2
𝒄𝒐𝒔𝒉(𝟎.𝟕𝟖)
=-0.426
86. A store advertised dresses on sale at 20 percent off. The sale price $76. What
was the original price of the dress?
A. $95
B. $60.80
C. $ 59
D. $80.60
SOLUTION:
X- 76= 0.20(X)
X= 95
87. A woman is paid $20 for each day she works and forfeits $5 for each day she
is idle. At the end of 25 days and nets $450. How many days did she work?
A. 20
B. 21
C. 22
D. 23
SOLUTION:
450𝑝𝑒𝑠𝑜𝑠
20𝑝𝑒𝑠𝑜𝑠/𝑑𝑎𝑦
= 23 days
88. What do you call a radical expressing an irrational number?
A. Surd
B. Radix
C. Complex number
D. Index
89. The arc of a sector is 9 units and its radius is 3 units. What is the area at the
sector in square units?
A. 12.5
SOLUTION:
1
A=2𝑟𝐶
B. 13.5
C. 14.5
D. 15.58
1
A= 2×3×9
A=13.5 sq.units
90. The base radius of a right circular cone is 4m while is slant height is 10m.
What is the surface area?
A. 127.5 sq.m
B. 125.7 sq.m
C. 139.5 sq.m
D. 135.9sq.m
SOLUTION:
C= 2πr
C=2π (4)
C=25.13
𝐶𝐿
25.13(10)
A= 2 =
2
A=125.65 sq.m
91. A line with equation y=mx+b passes through (-1/3, -6) and (2,1). Find the
value of m.
A. 1
B. 3
C. 4
D. 2
SOLUTION:
𝑚=
=
𝑦2 − 𝑦1
𝑥2 − 𝑥1
1+6
=𝟑
2 + 1/3
92. The vertical end of a water trough is an isosceles triangle with width of 6 feet
and depth of 3 feet. Find the force on one end when the trough field with water.
A. 638 lbs B. 683 lbs
SOLUTION:
C. 562 lbs
D. 526 lbs
3
6
𝐹 = ∫ 62.4 (6 − 𝑋) 𝑋𝑑𝑋 = 𝟓𝟔𝟏. 𝟔 𝒐𝒓 𝟓𝟔𝟐
3
0
93. A lady gives a dinner party for six guests. In how many ways the be selected
from among 10 friends?
A. 110
B. 220
C. 105
D. 210
SOLUTION:
10𝐶6 = 𝟐𝟏𝟎
94. For a complex number Z=2=2(sqrt of 3) i. The modulus is.
A. 2
B. 3
C. 4
D. 5
SOLUTION:
2
𝑟 = √22 + 2√3 = 𝟒
95.Which of the following has no middle term?
A. (x-2y)6
B. (x+y)8
C. (x-y)5
D. (x+2y)4
96. A sports car 2 m long overtakes a 12 m van which is traveling at the rat of 36
kph. How fast must the car travel to overtake the van in 3 seconds if their rear
ends are aligned initially?
A. 46 kph
B. 47 kph
C. 48 kph
SOLUTION:
16 12
=
𝑋
36
𝑥 = 𝟒𝟖
D. 49 kph
97. A tank in an ice plant is to contains 3,000 liters of brine. It is constructed to be
4 m long and 1.5 wide. Find the height of the tank.
A. 0.3
B. 0.4
C. 0.5
D. 0.6
SOLUTION:
𝑉 = 𝐿𝑥𝑊𝑥𝐻;
3𝑚3 = 4𝑥1.5𝑥𝐻;
𝐻=
3
= 𝟎. 𝟓
4 ∗ 1.5
98. The eccentricity of the hyperbola having the rectangular equation 3x 2-4y224x+16y+20=0 is
A. 1.12
B. 1.22
C. 1.32
D. 1.42
SOLUTION:
3𝑥 2 − 4𝑦 − 24𝑥 + 16𝑦 + 20 = 0
3(𝑥 2 − 8𝑥 = 16) − 4(𝑦 2 -16y+64) =16+64-20
(𝑥 2 − 4)2 (𝑦 2 − 8)2
−
= 60
4
3
𝑒=
𝑐
;
𝑎
𝑐 = √4 + 3 ;
𝑐 = 𝟏. 𝟑𝟐
99.Find the equation of the parabola whose vertex is the origin and whose focus
is the point (0,2)
A. x2=10y
B. x2=8y
C. x2=-10y
D. x2=-8y
100. Find the equation of the family of curves at every point which the tangent
line has a slope of 2y.
A. x=Cey
B. y=Cex
C. x= Ce2y
D.y=Ce2
REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
AUGUST 2014
MATHEMATICS
REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION
AUGUST 2014
MATHEMATICS
1. What percentage of the volume of a cone is the maximum right circular cylinder
that can be inscribed in it?
A. 24%
B. 34%
C. 44%
D. 54%
2. A railroad curve is to be laid out on a circle. What radius should be used if the
track is to change direction by 30 degrees in a distance of 300 m?
A. 566 m
B. 592 m
C. 573 m
SOLUTION:
30C = 300x360; C = 3600 = 2πr;
D. 556 m
r = 𝟓𝟕𝟐. 𝟗𝟔 𝐦 (C)
3. Express in polar form: -3-4i
𝟒
𝟒
A. 𝟓𝐞−𝐢(𝐩𝐢+𝐭𝐚𝐧−𝟏𝟑)
C. √𝟓𝐞−𝐢(𝐩𝐢+𝐭𝐚𝐧−𝟏𝟑)
𝟒
𝟒
B. 𝟓𝐞𝐢(𝐩𝐢+𝐭𝐚𝐧−𝟏𝟑)
D. √𝟓𝐞𝐢(𝐩𝐢+𝐭𝐚𝐧−𝟏𝟑)
4. Find the values of z for which 𝒆𝟒𝒛 = 𝒊.
A. 1/6 pi i + ½ kpi i
C. 1/8 pi i + ½ kpi i
B. -1/6 pi i + ½ kpi i
D. -1/8 pi i + ½ kpi i
5. A ladder leans against the side of a building with its foot 12 ft. from the building.
How long is the ladder if it makes of 70 degrees with the ground?
A. 32 ft
SOLUTION:
12
cos70° = x ;
B. 33 ft
C. 34 ft
D. 35 ft
𝐱 = 𝟑𝟓. 𝟎𝟖 𝐟𝐭 (𝐃)
6. If the 5th term in arithmetic progression is 17 and the 3rd is 10, what is the 8th
term?
A. 27.5
B. 24.5
C. 36
D. 38
SOLUTION:
𝐀 𝐧 = 𝐀 𝟏 + (𝐧 − 𝟏)𝐝; 𝐀 𝟓 = 𝐀 𝟑 + (𝟐 − 𝟏)𝐝; 𝟏𝟕 = 𝟏𝟎 + (𝟏)𝐝; 𝐝 = 𝟑. 𝟓𝐀 𝐧
= 𝐀 𝐦 + (𝐧 − 𝐦)𝐝; 𝐀 𝟖 = 𝟏𝟕 + (𝟖 − 𝟓)(𝟑. 𝟓) = 𝟐𝟕. 𝟓 (𝐀)
7. A balloon is released of eye level and rises at the rate of 5 ft/s. An observer 50
ft away watches the balloon rise. How fast is the angle of elevation measuring
6 seconds after the moment of release?
A. 0.007 rad/s
B. 0.07 rad/s
C. 0.008 rad/s
D. 0.08 rad/s
SOLUTION:
𝐝𝛉
𝐝𝐭
𝟏
= 𝐭𝐚𝐧−𝟏 𝟏𝟎 𝐭 =
𝟏
𝟏𝟎
𝟏 𝟐
𝟏+( 𝐭)
𝟏𝟎
=
𝟏
𝟏𝟎
𝟐
𝟏
𝟏+( (𝟖))
𝟏𝟎
𝐫𝐚𝐝
= 𝟎. 𝟎𝟕𝟑𝟓 𝐬𝐞𝐜 (𝑩)
8. If cos z = 2, find cos 3z.
A. 7
B. 17
C. 27
D. 37
9. Points (6,-2) and (a,6) are on a line with a slope of 4/3. What is the value of a?
A. -2
B. 4.5
C. 9
D. 12
SOLUTION:
𝐦=
𝐲𝟐 − 𝐲𝟏
;
𝐱𝟐 − 𝐱𝟏
𝟒 𝟔+𝟐
=
; 𝐱 = 𝟏𝟐 (𝐃)
𝟑 𝐱−𝟔
10. The foci of an ellipse are on the points (4,0) and (-4,0) and its eccentricity
is 2/3. Find the equation of the ellipse.
A. x^2/36 + y^2/20 = 1
C. x^2/20 + y^2/16 = 1
B. x^2/20 + y^2/36 = 1
D. x^2/16 + y^2/20 = 1
11. The plate number of a vehicle consists of 5 alphanumeric sequence is
arranged
such that the first 2 characters are alphabet and the
remaining are 3 digits. How
many arrangements are possible if the first
character is a vowel and repetition is
not allowed?
A. 90
B. 900
C. 9,000
D. 90,000
Vowel – Letter – Digit – Digit – Digit and repetition is not allowed
5 X 25
X 10
X
9
X
8 = 90,000
12. One end of a 32-meter ladder resting on a horizontal plane leans on a
vertical wall. Assume the foot of the ladder to push towards the wall at the rate
of 2 meters per minute. When will be the top and bottom of the ladder move at
the same rate?
A. 30.4 m
B. 22.6 m
C. 17.75 m
D. 26.6 m
13. A triangle is inscribed in a circle of radius 10. If two angles are 70 degrees
and
50 degrees, find the length of the side opposite to
the third angle.
A. 15.32
B. 16.32
C. 17.32
D. 18.365
SOLUTION:
Third angle = 180˚- 70˚- 50˚ = 60˚
The central angle intercepting the same chord is 120˚. Joining the endpoints of
the chord to the circle centre results in an isosceles triangle has equal legs10
and odd angle 120.
Let x= bet the length of the chord.
By Cosine Law:
𝑥 2 = 102 +102 -2(10)(10) cos(120˚)
𝑥 2 = 300
𝑥 = √300
𝑥 = 10√3 or 17.32
14 Find the volume generated by revolving the area cut off from the parabola
y=4x-x^2 by the axis about the line y=6.
A. 295
B. 340
C. 286
D. 362
SOLUTION:
The outer radius, R, is the difference between y=6 and the x-axis (y=0) or 6.
The inner radius r, is the difference between y=6
y= 4𝑥 − 𝑥 2 ; at x= 0, x=4
The volume, V is given by the integral
4
V =𝜋 ∫0 ((6)2 − (6 − (4𝑥 − 𝑥 2 ))²)𝑑𝑥 = 294.89 ≈ 𝟐𝟗𝟓
cubic unit
15. The axis of the hyperbola through its foci is known as:
A. conjugate axis
B. transverse axis
C. major axis
D. minor axis
16. From past experience it is known 90% of one year old children can
distinguish
their mother’s voice of similar sounding female. A random sample of 20
one
year’s old given this voice recognize test. Find the probability that all
children
recognize mother’s voice.
A. 0.122
B. 0.500
C. 1.200
D. 0.222
SOLUTION:
The probability that all 20 will recognize Mom is p=.90; 𝑛 = 20
𝑝𝑛 =. 9020 = 𝟎. 𝟏𝟐𝟐
17. If the equation of the directrix of a parabola is x-5=0 and its focus is at (1,0),
find the length of its latus rectum.
A. 6
18.
B. 8
D. 12
Describe the locus represented by |𝒛 − 𝒊| = 𝟐.
A. circle
19.
C. 10
B. parabola
C. ellipse
Evaluate lim (( z – 1 – I )/( z2 -2z+2))
D. hyperbola
2
z―›1+i
A. ¼
B. -1/4
C. ½
D. -1/2
20. Nanette has a ribbon with a length of 13.4 m and divided it by 4. What is the
length of each part?
A. 3.35 m
B. 3.25 m
C. 3.15 m
SOLUTION:
Length of ribbon = 13.4m and divided by 4
Length of each part =
𝟏𝟑.𝟒
𝟒
= 𝟑. 𝟑𝟓𝒎
D. 3.45m
21. Simplify 1 (csc x + cot x) 1(csc x – cot x).
A. 2 cos x
B. 2 sec x
C. 2 csc x
D. 2 sin x
SOLUTION:
= 1/ (1/sinx − cosx/sinx) + 1/ (1/cscx + cosx/sinx)
= 1/ 1−cosx/sinx + 1/ 1+cosx/sinx
= Sinx/ 1−cosx + sinx/ 1+cosx
= (sinx(1+cosx)+sinx(1−cosx)) / (1−cosx)(1+cosx)
= (sinx+sinxcosx+sinx−sinxcosx) /1−cos2x
= (sinx+sinx) / sin2x
= 2sinx / sin2x
= 2 / sinx
= 2cscx
22. If the area of a sector of a circle is 248 sq. m and the central angle is 135
degrees. Find the diameter of the circle.
A. 29 m
SOLUTION:
L=
23.
B. 26 m
C. 32 m
D. 39 m
qπr 135(π)(r)
1
1 135πr
=
; Asector = rL = r (
) ; r = 14.51;
180
180
2
2
180
d = 𝟐𝟗. 𝟎𝟏 𝐦 (A)
In how many ways can two lines intersect from given 6 lines?
A. 14
SOLUTION:
B. 15
C. 16
D. 17
n(n-1)/2= 6(6-1)/2 = 15
24. Find the half line of a radioactive substance if 20 percent of it disappears in
40 years.
A. 123.25 yrs.
SOLUTION:
B. 124.25 yrs.
C. 125.25 yrs.
D. 126.25 yrs
40
0.8 = 1(0.5) 𝑥
= 124.25 yrs
25.
Find the area of curvature of 𝐲 = 𝐞𝐱 − 𝟐𝐱 at the point (0,1).
A. 2.91
B. 2.83
C. 2.72
D. 2.63
26. 3 randomly chose high school students were administered a drug test. Each
student was evaluate as positive to the drug test (P) or negative to the drug
test (N). Assume the possible combination of the 3 students drug test
evaluation as PPP, PPN, PNP, NPN, NNP, NNN. Assume the possible
combination is equally likely and knowing that 1 student get a negative results,
what is the probability that all 3 students get a negative result?
A. 1/8
B. 1/7
C. 7/8
D. ¼
27. A bridge is 1.4 kilometers long. A bus 10 meters long is crossing the bridge
at 30 kph. How many minutes will it take the bus to completely cross the
bridge?
A. 1.82 min
SOLUTION:
B. 2.82 min
C. 3.82 min
D. 4.82 min
1.4
10×10-3
(60)+
(60)= 2.82 mins
30
30
28.
Find the fifth term of the sequence 16, 4, 1, -1/4,…
A. 4
B. 16
C. ¼
D. 1/16
SOLUTION:
𝟏
𝟏
𝐀 𝐧 = 𝐀 𝟏 𝐫 𝐧−𝟏 = 𝟏𝟔(𝟒)𝟓−𝟏 = 𝟏𝟔 (D)
29.
A. pi
Find the area of the three-leaved rose r = 2 sin 2 theta.
B. 2 pi
C. 3 pi
D. 4 pi
SOLUTION:
𝐀 = 𝐩𝐢
30.
𝐚𝟐
𝟒
= 𝐩𝐢
𝟐𝟐
𝟒
= 𝐩𝐢 (𝑨)
Evaluate lim (x-6) tan (pix/12)
x―›6
A. -3.82
B. 0
C. -1.91
D. -2.64
31. What is the area of an isosceles triangle whose base is 10 and its base
angle is
60 degrees?
A. 25 (sqrt of 3)
B. 50 (sqrt of 3)
C. 25
D. 50
SOLUTION:
A =½ a² sin 60
A = ½ (10) (10) sin 60
A=25 √3
32. If y = 2x + sin 2x, what is the value of x so that y' =0?
3 pi/2
B. pi/2
C. pi/3
D. 2 pi/3
SOLUTION:
Y= 2x = sin 2x x = ? y= 0
Y = 2x + sin 2x
Y^1 = 2+2 cos 2x
0 =2 + 2 cos 2x choose
X = pie/2
Then @ rad mode
0 = 2 + 2 cos (2x pie/2)
0=0
=pi/2
33. What is the vector length 2 and direction 150 degrees in the form ai + bj.
1.73i + j
B. -1.73i – j
SOLUTION:
@ COMPLEX MODE
C.1.73i - j
D. -1.73i + j
Z = 2 cis 150
Z = -1.73i + j
34. If a man works at an average speed of 4 kph, what is the time consume to
reach 250 m.
0.25 min
B. 2.50 min
C. 3.75 min
SOLUTION:
V= s/t
T =0.25 km/4km/hr x hr/60min
D. 4.25 min
T = 3.75 min
35. N engineers and N nurses, if two engineers are replaced by nurses, 51% of
the
engineers and nurses are nurses. Find N.
A. 100
B. 110
C.50
D.200
SOLUTION:
N ENGINEERS, N NURSES
0.51 (2x) = x + 2
1.02x = x + 2
X = 100
36. A house has assessed value of P 720,000.00 worth which is 60% of the
market
value. If the tax is P 3.00 for P 1,000.00 market value, how much
is the tax?
P 3, 200.00 B. P 3,800.00
C. P 3,600.00
D. 3,400.00
SOLUTION:
ASSESSED VALUE =720,000
Tax = 3 FOR EVERY 3 Php of Market Value
X =market value = 0.6 of as V.
720,000 = 0.6 (x)
X =1,200,000/1000 x3
X = 3600Php
𝟏
37. 𝟔 is what percent of ¾?
37.5
B. 66.67
C. 50
D.75
SOLUTION:
𝟑
½ =(x)(𝟒)
X =66.7%
38. In a hotel it is known that 20% of the total reservation will be cancelled in the
last minute. What is the probability that out of 15 reservations there will be
more than 8 but less than 12 cancelled?
0.00784
B. 0.0784
C. 0.000784
D. 0.784
SOLUTION:
N = 9,10,11
Pr = n Cr (p)(𝒒)𝒏−𝒓
N =15 reservations
Pa =15 C9 (𝟎. 𝟐)𝟗 (𝟎. 𝟖)𝟏𝟓−𝟗 = 6.718x𝟏𝟎−𝟒
Pb =15 C10 (𝟎. 𝟐)𝟏𝟎 (𝟎. 𝟖)𝟏𝟓−𝟗 = 1x 𝟏𝟎−𝟒
Pc =15C 11(𝟎. 𝟐)𝟏𝟏 (𝟎. 𝟖)𝟏𝟓−𝟗 =1.1145 x 𝟏𝟎−𝟓
Pt =Pa +Pb +Pc Pt =0.000784
39. If 16 is more than 4x, find x.
1.4
B. 3
C. 12
D. 5
SOLUTION:
16 =4 + 4x
𝟏𝟐
X =𝟒
; X =3
40. Locate the midpoint of the line segment joining point 1(2,15,4)and point 2
(6,3,-12)
A. (4,9,4)
B. (4,-9,4)
C. (4,94)
D. (-4,9,4)
SOLUTION:
P1 (2,15,4)
P2 (6,3,-12)
𝟔
𝟑
𝟏𝟐
𝟐
𝟐
𝟐
MP (2 + ,15 + , 4 -
)
MP (4,9,-4)
41. A conic section whose eccentricity is greater than one (1) is known as?
A. A parabola
B. an ellipse
C. a circle
D. a hyperbola
42. Find the distance travelled by the tip of a pendulum if the distance of the first
swing
is 8 cm and the distance of each succeeding is 0.75 of the distance
of the previous swing.
A. 32 cm
B. 28 cm
C. 27 cm
D. 30 cm
43. Describe the locus represented by the curve |𝒛 + 𝟐𝒊| + |𝒛 − 𝟐𝒊| = 𝟔.
A. circle
B. parabola
C. ellipse
D. hyperbola
44. Find the area bounded by the curve 𝒚𝟐 = 𝟑𝒙 − 𝟑 and the line x = 4.
A. 10
B. 16
C. 15
D. 12
SOLUTION:
𝐲 𝟐 = 𝟑𝐱 − 𝟑 = 𝟑(𝟒) − 𝟑 = 𝟗; 𝐲 = 𝟑 & 𝐲 = −𝟑
𝟑
𝐲𝟐 + 𝟑
∫ (
) 𝐝𝐲 = 𝟏𝟐 (𝑫)
𝟑
−𝟑
45. Helium is escaping a spherical balloon at the rate of 2 cm3 /min. When the
surface area is shrinking at the rate of 1/3 cm2 /min, find the radius of the
spherical balloon.
A. 14 cm
B. 12 cm
C. 16 cm
D. 8 cm
46. What is the maximum area of the rectangle whose base is on the z-axis and
whose upper two vertices lie on the parabola 𝒚𝟐 = 𝟏𝟐 − 𝒙𝟐 .
A. 30
B. 32
C. 36
D. 40
SOLUTION:
A(x)=2x(12−x2)
A(x)=24x−2x3A(x)=24x−2x3
A′(x)=24−6x2.A′(x)=24−6x2.
Solving A′(x)=0A′(x)=0 gives x=2x=2
and A(x)=2⋅2⋅(12−22)=2⋅2⋅8=32A(x)=2⋅2⋅(12−22)=2⋅2⋅8=32
47. A car racer covers 225 km in 2.5 hrs. How far can he go in 1.75 hrs?
A. 267.5 km
B.168.75 km
C. 394 km
D. 157.5 km
SOLUTION:
225km/2.5hrs =x/1.75hrs
X=157.5 (D)
48. Find the area of the triangle with vertices A (0,1), B (5,3), and C (-2,-2)
A. 19
B. 19/2
C. 15
D. 15/2
49. What is the sum of coefficients of the expansion of (𝟐𝒙 − 𝟏)𝟐𝟎 ?
A. 0
B. 1
C. 2
D. 3
SOLUTION:
((2)(1)-1)20-(-1)20=0
50. The parabola defined by the equation 𝟑𝒚𝟐 + 𝟒𝒙 = 𝟎 opens ____________.
A. upward
B. downward
C. to the left
D. to the right
51. How many tiles 10 cm on a side are needed to cover a rectangular wall 3 m
by 4 m?
A. 1500
B. 1000
C. 1200
D. 1600
SOLUTION:
𝐀 𝐫𝐞𝐜 = 𝟑𝐱𝟒 = 𝟏𝟐 = 𝐱𝐀 𝐬𝐦𝐚𝐥𝐥 𝐫𝐞𝐜 ; 𝐱 = 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐬𝐦𝐚𝐥𝐥 𝐫𝐞𝐜;
𝐱=
𝐀 𝐫𝐞𝐜
𝐀 𝐬𝐦𝐚𝐥𝐥 𝐫𝐞𝐜
=
𝟏𝟐𝐦𝟐
= 𝟏𝟐𝟎𝟎 (𝐂)
𝟏𝐦
𝟏𝟎 𝐜𝐦𝐱 𝟏𝟎𝟎 𝐜𝐦
52. Find the equation of the line whose slope is -3 and the x-intercept is 5.
A. 𝒚 = −𝟑𝒙 + 𝟓
B. 𝟑𝒙 − 𝒚 = 𝟓
B. C. 𝒚 = 𝟑𝒙 + 𝟏𝟓
D. 𝟑𝒙 + 𝒚 = 𝟏𝟓
SOLUTION:
𝐲 = 𝐦(𝐱 − 𝐱 𝟏 ) = −𝟑(𝐱 − 𝟓) = −𝟑𝐱 + 𝟏𝟓 = 𝐲 𝐨𝐫 𝟑𝐱 + 𝐲 = 𝟏𝟓
53. In how many ways can the letters of the word “CHACHA” be arranged by
taking the letters all at a time?
A. 120
B. 720
C. 85
D. 90
54. Find the equation of the horizontal line though (-4,3).
A. x = 4
B. x = -4
C. y = 3
D. y = -3
SOLUTION:
(4,3)
X=-4 vertical line
Y=3 horizontal line (C)
55. If g(x) = 9f(x) and f(-6), find g’(-6)
A. -54
B. -40
C. -36
D. -28
SOLUTION:
g(x)=9f(x);f(-6)=-6,find g’(-6)
g’(-6) =9f(x)
g’(-6) =9f(-6) =-54
56. Determine k so that the points A (7,3), B (-1,0), and C (k,-2) are the vertices
of a right triangle with right angle at B.
A. -1
B. 1
C. -1/4
D. ¼
57. The radius of the circle 𝒙𝟐 + 𝒚𝟐 + 𝟒𝒙 − 𝟔𝒚 − 𝟑 = 𝟎 is _______
A. 2
B. 3
C. 4
SOLUTION::
𝒙𝟐 + 𝒚𝟐 + 𝟒𝒙 − 𝟔𝒚 − 𝟑 = 𝟎
𝒙𝟐 + 𝟒𝒙 + 𝟒 𝒚𝟐 − 𝟔𝒚 + 𝟗 = 𝟑 + 𝟒 + 𝟗
(𝒙 + 𝟐)𝟐 + (𝒚 − 𝟑)𝟐 = 𝟒𝟐
(𝒙 + 𝒃)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐
𝒓 = 𝟒 (C)
D. 5
58. If the logarithm of MN is 6 and the logarithm N/M is 2, find the logarithm of N.
A. 3
B. 4
C. 5
D. 6
SOLUTION::
𝐥𝐨𝐠 𝐌𝐍 = 𝐥𝐨𝐠 𝐌 + 𝐥𝐨𝐠 𝐍 = 𝟔; 𝐥𝐨𝐠 𝐌 = 𝟔 − 𝐥𝐨𝐠 𝐍
𝐥𝐨𝐠 𝐍
𝐥𝐨𝐠 𝐍 − 𝐌 =
= 𝟐; 𝐥𝐞𝐭 𝐱 = 𝐥𝐨𝐠 𝐍;
𝐥𝐨𝐠 𝐌
𝐥𝐨𝐠 𝐍
𝐱
=
= 𝟐; 𝐱 = 𝟒 (𝐁)
𝐥𝐨𝐠 𝐌
𝟔−𝐱
59. If 4 electricians earn x pesos in 7 days, how much can 14 carpenters paid of
the same rate, earn in12 days?
A. 3x
B. 4x
C. 5x
D. 6X
60. Write the differential equation of the family of circle with center at the origin.
A. 𝐱𝐝𝐲 + 𝐲𝐝𝐱 = 𝟎
B. 𝐱𝐝𝐲 − 𝐲𝐝𝐱 = 𝟎
C. 𝐱𝐝𝐱 + 𝐲𝐝𝐲 = 𝟎
D. 𝐱𝐝𝐲 − 𝐲𝐝𝐱 = 𝟎
No Answer!
61. Find the volume of a spherical segment, the radii of whose bases are 4 m and
5 m respectively with an altitude of 6 m.
A. 159 pi
B. 165 pi
C. 150 pi
D. 145 pi
62. A taxpayer’s state and the federal income taxes plus an inheritance tax totaled
$ 14,270. His California state income tax was $ 5,780 less than his federal
tax. His inheritance tax was $ 2, 750. How much did he pay in state tax?
A. $ 8,560
B. $ 2,870
C. $ 8,650
D. $ 2,780
SOLUTION::
𝐱 = 𝐬𝐭𝐚𝐭𝐞 𝐢𝐧𝐜𝐨𝐦𝐞 𝐭𝐚𝐱; 𝐲 = 𝐟𝐞𝐝𝐞𝐫𝐚𝐥 𝐢𝐧𝐜𝐨𝐦𝐞 𝐭𝐚𝐱; 𝐳 = 𝐢𝐧𝐡𝐞𝐫𝐢𝐭𝐚𝐧𝐜𝐞
𝐱 + 𝐲 + 𝐳 = 𝟏𝟒, 𝟐𝟕𝟎; 𝐱 = 𝐲 − 𝟓, 𝟕𝟖𝟎; 𝐳 = 𝟐, 𝟕𝟓𝟎
𝟏𝟒, 𝟐𝟕𝟎 = (𝐲 − 𝟓, 𝟕𝟖𝟎) + 𝐲 + 𝟐, 𝟕𝟓𝟎; 𝐲 = 𝟖𝟔𝟓𝟎;
𝐱 = 𝟖, 𝟔𝟓𝟎 − 𝟓, 𝟕𝟖𝟎 = $ 𝟐, 𝟖𝟕𝟎 (B)
63. The first term of a geometric sequence is 160 and the common ratio is 3/2.
How many consecutive terms must be taken to give a sum of 2110?
A. 3
B. 4
C. 5
D. 6
SOLUTION::
𝐒𝐧 =
𝐚𝟏 (𝐫 𝐧 −𝟏)
𝐫−𝟏
=
𝟏𝟔𝟎(
𝟑𝐗
−𝟏)
𝟐
𝟑
−𝟏
𝟐
;
𝐗 = 𝟓. 𝟎𝟖
64. The total area of a cube is 150 sq. in. A diagonal of the cube is ______ in.
A. 5(sqrt of 2)
SOLUTION::
B. 4(sqrt of 3)
C. 5(sqrt of 3)
D. 4(sqrt of 2)
𝐀 𝐜𝐮𝐛𝐞 = 𝟔𝐚𝟐 ;
𝟏𝟓𝟎
𝐚=√
𝟔
= 𝟓;
𝐝 = 𝐚√𝟑 = 𝟓√𝟑 (C)
65. In triangle ABC, sin (A+B) = 3/5. What is the value of sin C?
A. 2/5
B. 2/3
C. 3/5
D. ½
66. Find the slope of the curve whose parametric equations are x = -1 +t and
y=2t.
A. 2
B. 3
C. 1
D. 4
67. Find the length of the latus rectum of the ellipse 25x^2 + 9^2 – 300x – 144y +
1251 = 0.
A. 3.4
B. 3.2
C. 3.6
D. 3.0
SOLUTION:
25x2 + 9y2 -300x+144y+1251=0
A = 25, C = 9
a =√A= √25=5
b = √C= √9=3
2b2 2(3)2
LR=
=
= 3.6
a
5
68. A triangular trough whose the edges are 5, 5, and 8 m long is place vertically
in
water with its longest edge uppermost, horizontal, and 3 m below the
water level. Calculate the force on a side of the plate.
A.235.2 kN
B. 470.4 kN
C. 940.8 kN
D. 1,881.6 Kn
69. Find the area of the ellipse whose eccentricity is 4/5 and whose major axis
is 10.
A. 12 pi
B. 13 pi
C. 14 pi
SOLUTION::
𝑒=
4
5
Major axis = 10 = 2a
D. 15 pi
a=5
e=
c
4
=
a 5
b = √a2 -c2
= √52 -42
b=3
A =πab= π(3)(5)= 15π
70. Find the average rate of change of the area of a square with respect to its
side x as x changes from 4 to 7.
A. 14
B. 11
SOLUTION:: 𝐫𝐚𝐭𝐞 𝐨𝐟 𝐜𝐡𝐚𝐧𝐠𝐞 =
C. 12
∆𝐀
∆𝐱
=
𝟕𝟐 −𝟒𝟐
𝟕−𝟒
D. 13
= 𝟏𝟏
71. Find the moment of inertia with respect to the y-axis of the area bounded by
y = x2 and y = 2x.
A. 11/5
72.
B. 9/5
C. 7/3
D. 8/5
Find the length of the arc of r = 4 sin u from u = 0 to u = pi/2.
A. pi
B. 2 pi
C. 3 pi
D. 4 pi
SOLUTION::
𝐫 = 𝟒𝐬𝐢𝐧𝐮;
73.
𝛑
𝐫 = 𝟒; 𝛉 = ;
𝟐
𝛑
𝐒 = 𝐫𝛉;
𝛑
∫𝟎𝟐 𝟒𝐬𝐢𝐧 𝐮 𝐫 = 𝟒 𝟐 ;
𝐒 = 𝟐𝛑
What is the angle between -2.5 + j4.33 and 4.33 – j2.5?
A. 0 deg
SOLUTION::
B. 30 deg
C. 120 deg
D. 150 deg
−𝟐. 𝟓 + 𝐣. 𝟑𝟑 = 𝟓∠𝟏𝟐𝟎;
74.
𝟏𝟐𝟎 − (−𝟑𝟎) = 𝟏𝟓𝟎 (D)
If A = (2, 4) and B = (4,3), find |𝟕𝑨 − 𝑩|.
A. sq. rt. of 21
B. sq. rt. of 1061
B. C. sq. rt. of 41
D. sq. rt. of 949
75.
Find the initial poin of v = -3i + j +2k if the terminal point is (5, 0, -1).
A. (8,1, -3)
SOLUTION:
B. (8, -1, 3)
C. (8,-1,-3)
D. (8,1,3)
v = 3i + k + 2 k
( -3, 1, 2 )
( 5, 0, -1)
= 5 (-2) = 8
= 0 – 1 = -1
= (-1) - 2 = -3
= ( 8, -1, -3)
76.
What is the laplace transform of 1/sqrt of t?
A. (sqrt of pi)/s^2
B. (sqrt of pi)/s
C. pi/sqrt of s
D. sqrt of (pi/s)
SOLUTION:
Pi (n+1)/ s^n+1
= sqrt (pi/s)
77.
A pair of dice is tossed. Find the probability of getting at most a total of 5.
A. 5/9
B. 5/16
SOLUTION:
𝑬
PE= 𝑺=
(𝟓)(𝟐)
𝟔𝟐
𝟏𝟎
= 𝟑𝟔 = 5/18
C. 5/18
D. 5/36
78. On a day when the temperature is 30 deg C. a cool drink is taken from a
refrigerator whose temperature is 5 deg. C. If the temperature of the drink is
20 deg C after 10 minutes, what will its temperature be after 20 minutes?
A. 21 deg C
SOLUTION:
B. 24 deg C
C. 28 deg C
Tbo = 5
Tb1 = 20
Tm = 30
=
=
D. 26 deg C
Tb2 = x
t2 = 20
t1 = 10
Tb2 -Tm
Tb0-Tm
Tb1 -Tm
ln
Tb0-Tm
ln
ln(
ln(
x-30
)
5-30
20-30
)
5-30
20
= 10
X = 26
79. The positive value of k which will make 4x^2 – 4kx + 4k +5 a perfect square
trinomial is
A. 6
B. 5
C. 4
D. 3
SOLUTION::
4𝒙𝟐 – 4(5)x + 4(5) + 5
4𝒙𝟐 – 20x + 25
Therefore = 5
80. A stone advertises a 20 percent-off sale. If an article is marked for the sale
at $24.48, what is the regular price?
A. $30.60
B. $34.80
C. $36.55
D. $28.65\
SOLUTION::
𝟐𝟒. 𝟒𝟖 = 𝐱 + 𝟎. 𝟐𝟎𝐱;
𝐱 = 𝟑𝟎. 𝟔 (𝐀)
81. For a given arithmetic series the sum of the first 50 terms is 200, the sum
of the next 50 terms is 2700. The first term of the series is:
A. -12.2
B. -21.5
C. -20.5
D. -25.2
SOLUTION::
The sum of the first n terms of an arithmetic sequence is given by:
S(n) .= .(n/2)[2a + (n-1)d]
The sum of the first 50 terms is 200.
S(50) = (50/2)[2a + 49d] = 200 → 2a + 49d = 8 [1]
The sum of the next 50: (sum of the first 100) - (sum of the first 50)
(100/2)[2a + 99d] - 200
Hence, we have: .50[2a + 99d] - 200 = 2700 → 2a + 99d = 58 [2]
Subtract [1] from [2]: .50d = 50 → d = 1
Substitute into [1]: .2a + 49(1) = 8 → a = -41/2
82.
The total area of a cube is 150 sq. in. A diagonal of the cube is:
A. 4 in
𝐀 𝐜𝐮𝐛𝐞 = 𝟔𝐚𝟐 ;
B. 5 in
𝟏𝟓𝟎
𝐚=√
𝟔
C. 7.07 in
= 𝟓;
D. 8.66 in
𝐝 = 𝐚√𝟑 = 𝟓√𝟑 = 𝟖. 𝟔𝟔 (D)
83. A tree is broken over by a windstorm. The tree was 90 feet high and the top
of the tree is 25 feet from the foot of tree. What is the height of the standing
part of the tree?
A. 48.47 ft
B. 41.53 ft
C. 45.69 ft
D. 44.31 ft
Soln:
𝟐𝟓 = √(𝟗𝟎 − 𝒙)𝟐 − (𝒙)𝟐
√𝟖𝟏𝟎𝟎 − 𝟏𝟖𝟎𝒙 + 𝒙𝟐 − 𝒙𝟐
(𝟐𝟓)𝟐 = √𝟖𝟏𝟎𝟎 − 𝟏𝟖𝟎𝒙
𝟔𝟐𝟓 = 𝟖𝟏𝟎𝟎 − 𝟏𝟖𝟎𝒙
𝟏𝟖𝟎𝒙
𝟖𝟏𝟎𝟎 − 𝟔𝟐𝟓
=
𝟏𝟖𝟎
𝟏𝟖𝟎
𝒙 = 𝟒𝟏. 𝟓𝟑
84. Goods cost a merchant & 72. At what price should he mark them so that he
may sell them at a discount of 10% from his marked price and still make a profit
of 20% on the selling price?
A. $ 150
B. $ 200
C. $ 100
SOLUTION::
𝐱 = 𝟕𝟐 + 𝟕𝟐(𝟎. 𝟏𝟎 + 𝟎. 𝟏𝟎 + 𝟎. 𝟐𝟎) = 𝟏𝟎𝟎. 𝟖 (C)
D. $ 250
85. An edge of the base of a regular hexagonal prism is 4 in. and a lateral edge
is 9 in. Find the lateral area of the prism.
A. 216 sq. in.
B. 299 sq. in.
C. 206 sq. in.
D. 288 sq. in.
SOLUTION::
B= 4in
h= 9
s=6
𝑨 = 𝟒 ∗ 𝟔 ∗ 𝟗 = 𝟐𝟏𝟔 𝒔𝒒𝒎. 𝒊𝒏.
86. In a potato race, 8 potatoes are place 6 ft apart on a straight line, the first
being 6 ft from the basket. A contestant starts from the basket and puts one
potato at a time into the basket. Find the total distance must run in order to
finish the race.
A. 423 ft
B. 432 ft
C. 428 ft
D. 436 ft
SOLUTION::
a1=6x2=12, n=8, d=12, an=a1+(n-1)d an=12+(8-1)(12)= 96
𝑛
S = ( 2) ∗ (𝑎1 + 𝑎𝑛)
8
S = (2) ∗ (12 + 96)
S = 432 ft
87.
Given that sin theta = 3/5 and theta is acute, find cos 2theta.
A. -7/25
B. -4/5
C. 7/25
SOLUTION::
𝐬𝐢𝐧−𝟏 (𝒔𝒊𝒏𝜽 =
𝟑
) 𝐬𝐢𝐧−𝟏
𝟓
𝜽 = 𝐬𝐢𝐧−𝟏 𝟑⁄𝟓
𝐜𝐨𝐬 𝟐 𝐬𝐢𝐧−𝟏
= 𝟕/𝟐𝟓
𝟑
𝟓
D. 4/25
88. A side and a diagonal of a parallelogram are 12 inches and 19 inches,
respectively. The angle between the diagonals, opposite the given side is 124
degrees. Find the length of the other diagonal.
A. 7.48 in
B. 7.84 in
C. 8.47 in
D. 8.74 in
SOLUTION::
𝟏𝟐
𝟗. 𝟓
=
; 𝛃 = 𝟒𝟏. 𝟎𝟐°
𝐬𝐢𝐧𝟏𝟐𝟒 𝐬𝐢𝐧𝛃
𝛂 = 𝟏𝟖𝟎° − 𝛃 − 𝟏𝟐𝟒° = 𝟏𝟖𝟎° − 𝟒𝟏. 𝟎𝟐° − 𝟏𝟐𝟒° = 𝟏𝟒. 𝟗𝟖°
𝐛𝐲 𝐬𝐢𝐧𝐞 𝐥𝐚𝐰:
𝐛𝐲 𝐜𝐨𝐬𝐢𝐧𝐞 𝐥𝐚𝐰: 𝐚 = √𝟗. 𝟓𝟐 + 𝟏𝟐𝟐 − 𝟐(𝟗. 𝟓)(𝟏𝟐)𝐜𝐨𝐬𝟏𝟒. 𝟗𝟖° = 𝟑. 𝟕𝟒𝟏𝟒
𝟐𝐧𝐝 𝐝𝐢𝐚𝐠𝐨𝐧𝐚𝐥 = 𝟐𝐚 = 𝟐(𝟑. 𝟕𝟒𝟏𝟒) = 𝟕. 𝟒𝟖𝟑 𝐢𝐧𝐜𝐡𝐞𝐬 (𝐀)
89. A window in Mr. Royce’s house is stuck. He takes an 8-inch screwdriver to
pry open the window. If the screwdriver rests on the still (fulcrum) 3 inches from
the window and Mr. Royce has to exert a force of 10 pounds on the other end
to pry open the window, how much force was the window exerting?
A. 12-2/3
B. 14-2/3
C. 18-2/3
D. 16-2/3
90. A boat, propelled to move at 25 mi/hr in still water, travels 4.2 mi against the
river current in the same time that it can travel 5.8 mi with the current. Find the
speed of the current in mi/hr.
A. 4
B. 5
C. 3
D. 2
91. An open-top cylindrical tank is made of metal sheet having an area of 43.82
square meter. If the diameter is 2/3 the height, what is the height of the tank?
A. 3.24 m
B. 2.43 m
C. 4.23 m
SOLUTION::
𝐴 = 43.82 𝑠𝑞. 𝑚
2
𝑑= ℎ
3
𝑑 = 2𝑟
2
1
2𝑟 = ℎ , 𝑟 = ℎ
3
3
𝐴 = 2𝜋𝑟ℎ
1
43.82 = 2𝜋 ( ℎ) (ℎ)
3
𝒉 = 𝟒. 𝟓𝟕
D. 5.23 m
92. How much water must be added to 8 gallons of 80% boric SOLUTION: to
reduce it to a 50% SOLUTION:?
A. 4 gal
B. 4-4/5 gal
C. 5 gal
D. 5-3/5 gal
SOLUTION::
𝑙𝑒𝑡 𝑥 = 𝑎𝑛𝑜𝑢𝑛𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑡𝑜 𝑏𝑒 𝑎𝑑𝑑𝑒𝑑
+
=
80%
8 gal.
.8(8) + 0(x)= .70(8+𝑥)
x
50%
8+𝑥
𝟒
x= 𝟒 𝟓
93. The line y = 3x + b passes through the point (2, 4). Find the b.
A. 2
B. 10
C. -2
D. -10
SOLUTION::
𝑦 = 3𝑥 + 𝑏
4 =3×2 +𝑏
𝑏 = −2
94. The simplest form of in (𝒆𝟑𝒙 ) is ______
A. 3
B. 𝒆𝒙
C. e
SOLUTION::
𝑙𝑛(33𝑥 )
ln(𝑒 𝑛 )
𝑙𝑛(𝑒 3𝑥 ) = 𝟑𝐱
D.3x
95. Thirty degrees is how many radius?
A. pi/3
B. pi/6
C. pi/4
D.
pi/2
SOLUTION::
𝜋
30°(180°)
𝝅
𝟏𝟖𝟎
96. If the measure of one angle of a regular polygon is 135 degrees, then the
number of sides of that polygon is ______.
A. 4
SOLUTION:
B. 6
C. 8
D. 9
(𝑛 − 2)180
= 135
𝑛
n=8
97. What is the Laplace transform of 𝒆−𝟐𝒕
A. 1/s-2
B. 1/s+2
C. 1/s-1
D. 1/s+1
SOLUTION:
𝑒 −2𝑡
𝐿[𝑒 ±𝑎𝑡 ] =
1
𝑠∓𝑎
𝟏
𝒔+𝟐
98. The area in the second quadrant of the circle 𝒙𝟐 + 𝒚𝟐 = 𝟑𝟔 is revolved about
the line y+10=0. What is the volume?
A. 228.63
B. 2228.83
SOLUTION:
𝑥 2 + 𝑦 2 = 36
y + 10 = 0
4𝑟
4(6)
24
y’ = 3𝜋 = 3𝜋 = 3𝜋
v = A (2𝜋)(𝑑′ )
1
v = 4 (𝜋)(6)2 (2𝜋)(10 +
v = 2228.83 cu. units
r=6
d = 10 + y’
24
3𝜋
)
C. 2233.43
D. 2208.53
99. The average of six scores is 83. If the highest score is removed, the average
of the remaining scores is 81.2. Find the highest score.
A. 91
SOLUTION:
B. 92
C. 93
D. 94
6 𝑥 83 = 498
5 𝑥 81.2 = 406
498 − 406 = 𝟗𝟐
100. The sum of the base and altitude of an isosceles triangle is 36 cm. Find the
altitude of the triangle if its area is to be a maximum.
A. 18 cm
B. 16 cm
C. 9 cm
SOLUTION:
1
𝐴 = (𝑏1 + 𝑏2 )ℎ
2
𝑏1 + 𝑏2 + ℎ = 36
1
𝐴 = (36 − ℎ)ℎ
2
ℎ2
𝐴 = 18 −
2
Taking the derivative
𝑑𝐴
= 18 − ℎ
𝑑ℎ
𝒉 = 𝟏𝟖
D. 17 cm
REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
MARCH 2014
MATHEMATICS
REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION
MARCH 2014
MATHEMATICS
1.
What is the differential equation of the family of parabolas having their
vertices at the origin and their foci on the x-axis?
A. 2xdy – ydx = 0
C. 2ydx –xdy = 0
B. ydx + ydx = 0
D. dy/dx – x = 0
SOLUTION:
𝑦 2 = 4𝑎𝑥
4𝑎 =
𝑦2
𝑥
Differentiating
0=
𝑥(2𝑥𝑦𝑑𝑦)−𝑦 2 𝑑𝑥
𝑥2
[0 = 2𝑥𝑦𝑑𝑦 − 𝑦 2 𝑑𝑥]
1
𝑦
.𝑶 = 𝟐𝒙𝒅𝒚 − 𝒚𝒅𝒙
2. Find the rthogonal trajectories of the family of parabolas y^2 = 2x + C.
A. y = Ce^x B. y = Ce^(-x)
C. y = Ce^(2x)
D. y = Ce^(-2x)
SOLUTION:
𝑦 2 = 2𝑥 + 𝐶
𝑑𝑦
2𝑦 𝑑𝑥 = 2
𝑑𝑦
𝑑𝑥
1
=𝑦
Slope of orthogonal trajectories
𝑑𝑦
𝑑𝑥
1
𝑑𝑥
= − 𝑑𝑦 = − 𝑑𝑦
𝑑𝑥
Subs.
𝑑𝑦
𝑑𝑥
= −𝑦
𝑑𝑦
= − ∫ 𝑑𝑥
𝑦
ln 𝑦 = −𝑥 + 𝑐
𝑒 ln 𝑦 + 𝑒 −𝑥+𝑐
𝑦 = 𝑒 −𝑥 (𝑒 𝑐 )
𝐲 = 𝐂𝐞−𝐱
∫
3. A reflecting telescope has a parabolic mirror for which the distance from the
vertex to the focus is 30 ft. If the distance across the top of the mirror is 64 in.,
how deep is the mirror of the center?
A. 32/45 in.
B. 30/43 in.
C. 32/47 in.
D. 35/46 in.
SOLUTION:
1. 𝑥 2 = −4𝑎𝑦
𝑉 𝑡𝑜 𝐹 = 𝑎 = 30𝑓𝑡 = 360 𝑖𝑛
𝐿𝑅 = 4𝑎 = 1440
𝑥 2 = 4𝑎𝑦
322 = 1440𝑦
𝟑𝟐𝟐
𝟑𝟐
𝒚 = 𝟏𝟒𝟒𝟎 = 𝟒𝟓 𝐢𝐧
4. Simplify (1 – tan2x) / (1 + tan2x)
A. sin 2x
B. cos 2x
SOLUTION:
1−tan2 𝑥
1+tan2 𝑥
=
1−tan2 𝑥
1
= sec2 𝑥 −
sec2 𝑥
C. sin x
D. cos x
C. n!/s^(n-1)
D. n!/s^(n+2)
sin2 𝑥
cos2 𝑥
sec2 𝑥
sin2 𝑥
(cos2 𝑥)
cos2 𝑥
= cos2 𝑥 − sin2 𝑥
= 𝐜𝐨𝐬 𝟐𝒙
= cos2 𝑥 −
5. Evaluate L { t^n }.
A. n!/s^n
B. n!/s^(n+1)
SOLUTION:
𝒏!
.∫(𝒕𝒏 ) = 𝑺𝒏+𝟏
6. Simplify 12 cis 45 deg + 3 cis 15 deg.
A. 2 + j
B. sqrt. of 3 + j2
C. 2 sqrt. Of 3 + j2 D. 1 + j2
SOLUTION:
12cis45/3cis15
=
12
3
𝑐𝑖𝑠(45 − 15)
= 𝟐√𝟑 + 𝒋𝟐
arcsin 9𝑥
7. Evaluate lim (
𝑥→0
2𝑥
)
B. π
A. 9/2
SOLUTION:
log 𝑥=0
C. ∞
D. -∞
sin−1 𝑎𝑥
2𝑥
log x=0
sin−1 (9) 0.0001
𝟒. 𝟓 𝒐𝒓
2(0.0001)
𝟗
𝟓
8. Find the area of the lemniscate r2 = a2cos2θ
A. a2
B. a
C. 2a
D. a3
SOLUTION:
𝑟 2 = 𝑎2 cos 2𝜃
1
𝜃2
𝐴 = 2 ∫𝜃1 𝑟 2 𝑑𝜃
1
𝜋
𝐴 = [2 ∫04 cos 2𝜃𝑑𝜃 ]
𝑨 = 𝒂𝟐
9. Find the area bounded by the parabola sqrt. of x + sqrt. of y = sqrt. of a and
the line x + y = a.
A. a2
B. a2/2
C. a2/4
D. a2/3
SOLUTION:
√𝑥 + √4 = √𝑎
𝑎𝑠𝑠𝑢𝑚𝑒 𝑎 = 1
(√𝑦 = 1 − √𝑥)
𝑦 = (1 − 𝑥)2
2
𝑥+𝑦 =1
𝑦 =1−𝑥
2
1
𝐴 = ∫0 (1 − 𝑥) − (1 − √𝑥) 𝑑𝑥
𝑨 = 𝟎. 𝟑𝟑𝟑 𝒐𝒓
𝒂𝟐
𝟑
10. Ben is two years away from being twice Ellen’s age. The sum of twice Ben’s
age and thrice Ellen’s age is 66. Find Ben’s age now.
A. 19
B. 20
C. 16
D. 21
SOLUTION:
2𝑥 + 3(2𝑥) = 66
X=8.25
Age of ben =2X
=2(8.25)
= 16.5
11. What percentage of the volume of a cone is the maximum volume right
circular cylinder that can be inscribed in it?
A. 24%
B. 34%
C. 44%
D. 54%
SOLUTION:
4
𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 9 𝑉𝑐𝑜𝑛𝑒
0.4444 𝑉𝑐𝑜𝑛𝑒
= 𝟒𝟒. 𝟒𝟒%
12. A balloon rising vertically, 150 m from an observer. At exactly 1 min, the
angle of elevation is 29 deg 28 min. How fast is the balloon using at that
instant?
A. 104m/min
B. 102m/min
C. 106m/min
D. 108m/min
SOLUTION:
@𝑡 = 1 𝑚𝑖𝑛, 𝜃 = 29.28
𝑑𝜃 29.28
=
𝑑𝑡 1 𝑚𝑖𝑛
28
29 + 60
𝑑𝜃
𝜋
=(
)𝑥
𝑟𝑎𝑑
𝑑𝑡
1 𝑚𝑖𝑛
180
𝑑𝜃
𝜋
= 0.5143
/𝑚𝑖𝑛
𝑑𝑡
180
𝑡𝑎𝑛 𝜃 =
𝑦
190
𝑑(𝑡𝑎𝑛 𝜃) = 𝑑(
𝑦
)
190
𝑑𝑦
𝑑𝜃
sec 2 𝜃
= 𝑑𝑡
𝑑𝑡
190
𝑑𝑦
1
= 190 ( 2
) (0.5143)
𝑑𝑡
cos 29.28
𝒅𝒚
= 𝟏𝟎𝟐 𝒎/𝒎𝒊𝒏
𝒅𝒕
13. A conic section whose eccentricity is less than one (1) is known as:
A. a parabola B. an ellipse
C. a circle
D. a hyperbola
14. A tangent to a conic is a line
A. which is parallel to the normal
B. which touches the conic at only one point
C. which passes inside the conic
D. all of the above
15. A die and a coin are tossed. What is the probability that a three and a head
will appear?
A. 1/4
B. 1/2
C. 2/3
D.1/12
1 1
𝟏
𝑥 =
6 2
𝟏𝟐
16. Find the integral of 12sin5xcos5xdx if lower limit = 0 and upper limit = pi/2.
A. 0.8
B.0.6
C.0.2
D.0.4
𝑃=
SOLUTION:
𝜋
2
∫ 12 sin5 𝑥 cos 2 𝑥 𝑑𝑥
0
Wallis Formula
(4)(2)(4)(2)
1
12 = [
] = 𝑜𝑟 𝟎. 𝟐
(10)(8)(6)(4)(2)
5
17. 12 oz of chocolate is added to 10 oz of flavoring is equivalent to
A.1 lb and 8 oz
B. 1 lb and 6 oz
C.1 lb and 4 oz
and 10 oz
SOLUTION:
22 𝑜𝑧 𝑥
28.30 𝑔
1 𝑘𝑔
2.2 𝑙𝑏
𝑥
𝑥
= 0.37𝑙𝑏
1 𝑜𝑧
1000𝑔
1 𝑘𝑔
0.37𝑙𝑏 =
Therefore: 1 lb and 6 oz
1𝑘𝑔 1000𝑔
1 𝑜𝑧
𝑥
𝑥
= 6 𝑜𝑧
2.2𝑙𝑏 1𝑘𝑔 28.35𝑔
D.1 lb
18. The Ford company increased its assets price from 22 to 29 pesos. What is
the percentage of increase?
A.24.14%
B.31.82%
C.41.24%
D.28.31%
SOLUTION:
% 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 =
29.22
𝑥 100
22
=31.82 %
19. Find the area bounded by outside the first curve and inside the second
curve, r = 5, r = 10sinθ
A. 47.83
B.34.68
C.73.68
D.54.25
SOLUTION:
𝜋
r= 10sin𝜃 r=s 6
1
10 sin 𝜃
1
sin 𝜃 = 2
𝜋
A = [2] ∫𝜋2 (10 sin 𝜃)2 − 𝑠 2 )𝑑𝜃)
6
= 47.83 sq.u.
20. In two intersecting lines, the angles opposite to each other are termed as:
A. opposite angles
C. horizontal angles
B. vertical angles
D. inscribed angles
21. The area in the second quadrant of the circle x^2 + y^2 = 36 is revolved
about the line y + 10 = 0. What is the volume generated?
A. 2932 c.u.
B. 2392 c.u.
C. 2229 c.u.
D. 2292 c.u.
22. A cardboard 20 in x 20 in is to be formed into a box by cutting four equal
squares and folding the edges. Find the volume of the largest box.
A.592 cu.in.
B.529 cu.in.
C.696 cu.in.
D.689 cu.in.
SOLUTION:
V = (20-2x)(20-2x)(x)
V= (400 – 40x – 40x +4x^2) x
V = 400x – 80 X^2 + 4x^3
𝑑𝑉
𝑑𝑥
= 12x2 – 160x + 400 = 0
X1 = 10 ---reject
X2 = 3.33 ---accept
Subs.
V= ((20-2)(3.33))(20-2(3.33))(3.33)
V = 592 cu. in
23. A retailer bought a number of ball pens for P90 and sold all but 3 at a profit
P2 per ball pen. With the total amount received she could buy 15 more ball
pens than before. Find the cost per ball pen.
A. P2
B. P3
C.P4
D.P5
24. What is –i^i?
A.4.81
B.-4.81
C.0.21
D.-0.21
25. A balloon travel upwards 6m, North and 8m, East. What is the distance
traveled from the starting point?
A. 7
B. 10
C.14
D. 20
SOLUTION:.
x=8
y=6
d=?
d=√𝑥 2 + 𝑦 2
d = √82 + 62 = 10
26. What do you call the integral divided by the difference of the abscissa?
A. average value
C. abscissa value
B. mean value
D. integral value
ANSWER: A. average value
27. Water is running out of a conical funnel at the rate of 1 cubic inch per sec.
If the radius of the base of the funnel is 4 in. and the altitude is 8 in., find the
rate at which the water level is dropping when it is 2 in. from the top.
` A. -1/pi in./sec B. -2/pi in./sec
C. -1/9pi in./sec D.-2/9pi in./sec
SOLUTION:.
1
V = 3 𝜋𝑟 2 ℎ
𝑅
𝑟
1
ℎ
=ℎ=2
𝐻
;𝑟 = 2
𝜋 ℎ 2
𝜋
V = 3 (2) ℎ = 12 ℎ3
𝑑𝑉
𝑑𝑡
=
3𝜋
12
−1 =
𝑑ℎ
𝑑𝑡
𝑑ℎ
ℎ2 𝑑𝑡
3𝜋
(2)2
12
𝑑ℎ
𝑑𝑡
𝟏
= − 𝟗 𝝅/𝒔𝒆𝒄
28. How many inches is 4 feet?
A. 36
B. 48
SOLUTION:
4ft x 12inch / 1ft = 48inch
C. 12
D. 56
29. A rectangular trough is 8 ft. long, 2 ft. across the top, and 4 ft. deep. If water
flows in at a rate of 2 cu. ft./min., how fast is the surface rising when the water
is 1 ft. deep?
A. 1/5 ft./min
B. 1/8 ft./min
C. 1/6 ft./min
D. 1/16 ft./min
SOLUTION:.
V = (8)(2)(1)h
𝑑𝑉
𝑑𝑡
𝑑ℎ
𝑑𝑡
𝑑ℎ
= 16 𝑑𝑡
𝟏
= 𝟖 𝒇𝒕/𝒎𝒊𝒏
30. Five tables and eight chairs cost $115; three tables and five chairs cost
$70. Determine the total cost of each table.
A. $15
B. $30
C. $25
D. $20
SOLUTION:.
5 tables + 8 chairs = 115
3 tables + 5 chairs = 70
(5T + 8C = 115) 5
(3T + 5C = 70 ) -8
T=15
31. Find the 16th term of the arithmetic sequence; 4, 7, 10,……..
A. 47
B. 46
C. 49
D. 48
SOLUTION:
A15 = ?
d =3 a =4
A15 = A1 + (n-1) d
A15 = 4 + (15 – 3)(3) = 49
32. Find the slope of the line through the points (-2, 5) and (7, 1).
A. 9/4
B. -9/4
C. 4/9
D. -4/9
SOLUTION:
𝑌2− 𝑌1
1−5
m = 𝑋2−𝑋1 = 7+2 =
−𝟒
𝟗
33. For what value of k will the line kx +5y = 2k have a y-intercept 4?
A. 8
B. 7
C. 9
D.10
SOLUTION:
Kx + 5y = 2k
K =?
@ y=4
@x=0
K(0) + 5y = 2k
5y = 2k
@ y=4
5 (4) = 2k
K = 20/2 = 10
34. If a bug moves a distance of 3pi cm along a circular arc and if this arc
subtends a central angle of 45 degrees, what is the radius of the circle?
A. 8
B. 12
C. 14
D. 16
SOLUTION:
C= r𝜃
R=
3𝜋
45 𝑥
𝜋
180
= 12 cm
35. Two vertices of a rectangle are on the positive x-axis. The other two
vertices are on the lines y = 4x and y = -5x + 6. What is the maximum possible
area of the rectangle?
A.2/5
B.5/2
C.5/4
D. 4/5
SOLUTION:
Since AD should be equal to BC
4a=-5b+6
A=(base)(height)
A=(b-a)(4a)
4𝑎−6
A=(
−5
4
− 𝑎)(4𝑎)
A’=(− 5 − 1)(4𝑎) + (
4
A’=(− 5)(4𝑎) + (
−36𝑎
0=
5
+
4𝑎−6
−5
4𝑎−6+5𝑎
−5
− 𝑎)(4)
)(4) = 0
36𝑎−24
−5
0=-72a+24
a=1/3
b=14/15
A=(b-a)(4a)
14
1
1
A=(15 − 3)(4(3))
A=4/5
36. Find the length of the arc of 6xy = x^4 + 3 from x = 1 to x = 2.
A.12/17
B.17/12
C.10/17
D.17/10
SOLUTION:
6𝑥𝑦 = 𝑥 4 + 3
𝑥4 = 3
𝑦=
6𝑥
4
6(𝑥 + 3) − (6𝑥)(4𝑥 3 )
𝑑𝑦 =
36𝑥 2
4
= 6𝑥 + 18 − 24𝑥 4
−18𝑥 4 + 18
36𝑥 2
2
𝑠 = ∫ √1 + (𝑑𝑦)2
1
2
𝑠 = ∫ √1 + (
1
−18𝑥 4 + 18 2
)
36𝑥 2
S = 17/12
37. A certain radioactive substance has half-life of 3 years. If 10 grams are
present initially, how much of the substance remain after 9 years?
A.2.50g
B.5.20g
C. 1.25g
D.10.20g
SOLUTION:
ln(2) 10
=
9
9
ln(𝑥)
=1.25g
38.
A cubical box is to built so that it holds 125 cu. cm. How precisely should
the edge be made so that the volume will be correct to within 3 cu. cm.?
A.0.02
B.0.03
C.0.01
D.0.04
SOLUTION:
V=125cm3
Dv= 3cm3
V=s3
3
𝑠 = √125
S= 5
Dv=3s2 ds
3
𝑑𝑠 = 3𝑠2
3
𝑑𝑠 = 3(5)2
𝒅𝒔 = 𝟎. 𝟎𝟒
39. Find the eccentricity of the ellipse when the length of its latus rectum is 2/3
of the length of its major axis.
A.0.62
B. 0.64
C.0.58
D.0.56
40. Find k so that A = <3, -2> and B =<1, k> are perpendicular.
A. 2/3
B.3/2
C.5/3
D.3/5
41. Find the moment of inertia of the area bounded by the curve x^2 = 8y, the
line x = 4 and the x-axis on the first quadrant with respect to y-axis.
A.25.6
B. 21.8
C.31.6
D.36.4
42. Find the force on one face of a right triangle of sides 4m and altitude of 3m.
The altitude is submerged vertically with the 4m side in the surface.
A.62.64 kN
B.58.86 kN
C.66.27 kN
D.53.22 kN
43. In how many ways can 6 people be seated in a row of 9 seats?
A. 30,240
B. 30,420
C.60,840
D. 60,480
SOLUTION:
9P6 = 60,480
44. The arc of a sector is 9 units and its radius is 3 units. What is the area of
the sector?
A.12.5
B.13.5
C.14.5
D.15.5
SOLUTION:
1
A = 2 𝑟𝐶
1
A = 2 (3)(9)
A = 13.5
45. The sides of a triangle are 195, 157, and 210, respectively. What is the area
of the triangle?
A.73,250
B.10,250
C.14,586
D.11,260
SOLUTION:
S=
195+157+210
2
= 281
A = √281 (281 − 195(281 − 157)(281 − 210)
A = 14586.21
46. A box contains 9 red balls and 6 blue balls. If two balls are drawn in
succession, what is the probability that one of them is red and the other is
blue?
A.18/35
B.18/37
C.16/35
D.16/37
47. A car goes 14 kph faster than a truck and requires 2 hours and 20 minutes
less time to travel 300 km. Find the rate of the car.
A.40 kph
B.50 kph
C.60 kph
D.70 kph
48. Find the slope of the line defined by y – x = 5.
A.1
B.1/4
C.-1/2
SOLUTION:
D.5
y = mx+b
y–x=5
y=x+5
by inspection, the slope is equal to 1
49. The probability of John’s winning whenever he plays a certain game is 1/3.
If he plays 4 times, find the probability that he wins just twice.
A.0.2963
B.0.2936
C.0.2693
D.0.2639
SOLUTION:
nCrpq
n = 4,
p = 1/3
r=2 ,
q = 2/3
therefore :
1 2 2 2
4C2 x (3) (3) = 0.2963
50. A man row upstream and back in 12 hours. If the rate of the current is 1.5
kph and that of the man in still water is 4 kph, what was the time spent
downstream?
A.1.75 hr
B.2.75 hr
C.3.75 hr
D. 4.75 hr
SOLUTION:
d=d
(V + c)(t) = (V – c)(t)
(4 + 1.5)(x) = (4 – 1.5)(12 - x)
x = 3.75 hrs
51. If cot A = -24/7 and A is in the 2nd quadrant, find sin 2A.
A.336/625
B.-336/625
C.363/625
D. -363/625
SOLUTION:
Cot A =
1
tan 𝐴
=
−24
7
−24
7
7
tan A = −24
7
A = tan−1 (−24)
A = -16.260
sin 2A = sin (2x = 16.250)
=
−336
625
52. The volume of a square pyramid is 384 cu. cm. Its altitude is 8 cm. How
long is an edge of the base?
A.11
B.12
C.13
D.14
SOLUTION:
V = 384 cm^3
h= 8 cm
1
V = 3Abh
1
384 = 3Ab (8)
Ab = 144
A = a2
√𝑎2 = √𝐴 = √144 = 12
53. The radius of the circle x^2 + y^2 – 6x + 4y – 3 = 0 is
A.3
B.4
C.5
D.6
SOLUTION:
𝑥 2 + 𝑦 2 − 6𝑥 + 4𝑦 − 3 = 0
(𝑥 2 − 6𝑥 + 9) + (𝑦 2 + 4𝑦 + 4)
(𝑥 + 3)2 + (𝑦 + 2)2 = 16 = 42
54. If the planes 5x – 6y - 7z = 0 and 3nx + 2y – mz +1 = 0
A.-2/3
B. -4/3
C.-5/3
D.-7/3
55. If the equation of the directrix of the parabola is x – 5 = 0 and its focus is at
(1, 0), find the length of its latus rectum.
A.6
B.8
C.10
D.12
SOLUTION:
d=x–5=0 d=5
f(1,0) = a = 1
LR = 2a
d=F
2a = 4
a=2
LR = 2a = 8
56. If tan A = 1/3 and cot B = 4, find tan (A + B).
A. 11/7
B. 7/11
C. 7/12
D. 12/7
SOLUTION:
1
A = tan−1 ( ) = 18.43
B=
3
−1 1
tan ( 4)
= 14.04
tan (18.43 + 14.04) = 0.636
7
= 11
57. A club of 40 executives, 33 like to smoke Marlboro, and 20 like to smoke
Philip Morris. How many like both?
A. 13
B. 10
C. 11
D. 12
SOLUTION:
(33 - x) + x + (20 - x) = 40
x=13
58. The area of the rhombus is 264 sq. cm. If one of the diagonals is 24 cm
long, find the length of the other diagonal.
A. 22
B. 20
C. 26
D. 28
SOLUTION:
A=
1
2
d1 d2
1
264 = 2 (26) d2
d 2 = 22 cm
59.
How many sides have a polygon if the sum of the interior angles is 1080
degrees?
A. 5
B. 6
C. 7
D. 8
SOLUTION:
S = (n - 2)(180)
1080 = (n - 2)(180)
n=8
60. The line segment connecting (x, 6) and (9, y) is bisected by the point (7, 3).
Find the value of x and y.
A. 5, 0
B. 4, 0
C. 5, 2
D.4,1
SOLUTION:
Let Xm and Ym the coordinates of the midpoint
Xm =
𝑋1+𝑋2
Ym =
2
𝑥+9
7=
3=
2
𝑌1+𝑌2
2
6+𝑦
2
x=5
y=0
61. What is the height of the parabolic arch which has span of 48 ft. and having
a height of 20 ft. at a distance of 16 ft. from the center of the span?
A. 30 ft.
B. 40 ft.
C. 36 ft.
D.34ft.
SOLUTION:
62. Determine B such that 3x + 2y – 7 = 0 is perpendicular to 2x –By + 2 =0.
A. 2
B. 3
C. 4
D.5
SOLUTION:
y=
−3
2
7
𝑥+2
y=
−2
𝐵
2
𝑥+𝐵
m1 = - 3/2
m2 = -2/B
Since perpendicular, m2 = - 1/m1
−2
𝐵
=
1
−3
2
=3
63. The value of x + y in the expression 3 + xi = y + 2i is;
A. 5
B. 1
C. 2
D.3
SOLUTION:
64. If sin3A = cos6B then:
A. A + B = 180 deg
B. A + 2B = 30 deg
SOLUTION:
Sin 3A = cos 6B
Sin 3A = sin (90 – 6B)
3A = 90 – 6B
(3A + 6B = 90) 1/3
A + 2B = 30
C. A - 2B = 30 deg
D. A + B = 30 deg
65. What is the area between y = 0, y = 3x^2, and x = 2?
A. 8
B. 12
C. 24
D.6
SOLUTION:
2
2
A = ∫0 𝑦𝑑𝑦 = 3∫0 𝑥 2 𝑑𝑥
=3
=
x3
𝑥3
3
= (2)3 = 8
66. The volume of the sphere is 36pi cu. m. The surface area of this sphere in
sq. m is:
A. 36pi
B. 24pi
C. 18pi
D.
12pi
SOLUTION:
Vs = 36π
4
V = 3 𝜋𝑟 3
36π = 4/3πr3
r=3
As = 4πr2
As = 4π(3)2 = 36π m2
67. The vertex of the parabola y^2 – 2x + 6y + 3 = 0 is at:
A. (-3, 3)
B. (3, 3)
C. (3, -3)
D.
(-3,
-3)
SOLUTION:
𝑦 2 − 2𝑥 + 6𝑦 + 3 = 0
𝑦 2 + 6𝑦 + 9 = 2𝑥 − 3
(𝑦 + 3)2 = 2𝑥 − 3 + 9
(𝑦 + 3)2 = 2𝑥 + 6
(𝑦 + 3)2 = 2(𝑥 + 3)
(𝑦 − 𝑘)2 = 4𝑎(𝑥 − ℎ)
= - 3, - 3
68. Add the following and express in meters: 3 m + 2 cm + 70 mm
A. 2.90 m
B. 3.14 m
C. 3.12 m
D.3.09m
SOLUTION:
3+(𝑐𝑚 𝑥
1𝑐𝑚
100𝑐𝑚
) + (70𝑚𝑚 𝑥
1𝑚
1000𝑚𝑚
)
= 3.09m
69. A store advertised on sale at 20 percent off. The sale price was $76. What
was the original price?
A. $95
SOLUTION:
76 = .80(x)
B. $96
C. $97
D.$98
X = 95
70. Find the equation of the straight line which passes through the point (6, -3)
and with an angle of inclination of 45 degrees.
A. x + y = 8
B. x – y = 8
C. x + y = 9
D. x – y = 9
SOLUTION:
n = tan 𝜃
n = tanus = 1
y + 3 = 1(x-6)
y=x–6–3
x–y=9
71. A freight train starts from Los Angeles and heads for Chicago at 40 mph.
Two hours later a passenger train leaves the same station for Chicago
traveling at 60 mph. How long will it be before the passenger train overtakes
the freight train?
A. 3 hrs.
B. 5 hrs.
C. 4 hrs.
D. 6 hrs.
SOLUTION:
Time Rate distance
X 40
CHI-Lo
40x
x-2
60
60(x-2)
d1 = d2
40x =60(x-2)
x=6
x -2 = 6 – 2 = 4 hrs
72. The number of board feet in a plank 3 inches thick, 1 ft. wide, and 20 ft.
long is:
A. 30
B. 60
C. 120
D. 90
SOLUTION:
V= 3(1)(20) = 60 inch
73. Boyles’s law states that when a gas is compressed at constant
temperature, the product of its pressure and volume remains constant. If the
pressure gas is 80 lb/sq.in. when the volume is 40 cu.in., find the rate of change
of pressure with respect to volume when the volume is 20 cu.in.
A. -8
B. -10
C. -6
D.-9
SOLUTION:
74. Find the average rate of change of the area of a square with respect to its
side x as x changes from 4 to 7.
A. 8
B. 11
C. 6
D. 21
SOLUTION:
A= x^2
Limits 7-4=3 lim
A’= 2x = 2(3) = 6
75. How many cubic feet is equivalent to 100 gallons of water?
A. 74.80
B. 1.337
C. 13.37
D. 133.7
SOLUTION:
100L = 1m 3
1m = 3.28 ft
1L = 0.2642 gal
1𝐿
1𝑚3
3.18 𝑓𝑡 3
100 gal = 0.2642 𝑥 1000𝐿 𝑥 (
1𝑚
) = 13.37
76. A merchant purchased two lots of shoes. One lot he purchased for $32 per
pair and the second lot he purchased for $40 per pair. There were 50 pairs in
the first lot. How many pairs in the second lot if he sold them all at $60 per pair
and made a gain of $2800 on the entire transaction?
A. 50
B. 40
C. 70
D. 60
SOLUTION:
PB=50(32)=1600
PS= 60(50)=3000
PR=PS –PR
= 3000-1600
= 1400
PRT = PR1 + PR2
2800 = 1400 + PR2
PR2 = 1400
PB=40(Y) PS= 60(Y)
PR=PS –PR
1400 = 40(Y) -40(Y)
1400 = 20Y
Y = 70
77. The diagonal of a face of a cube is 10 ft. The total area of the cube is
A. 300 sq. ft.
B. 150 sq. ft.
C. 100 sq. ft.
D. 200 sq. ft.
SOLUTION:
√2𝑎 = 10
5
a = √2
total Area = 6𝑎2
A total = 300 ft2
78. A ship is sailing due east when a light is observed bearing N 62 deg 10 min
E. After the ship has traveled 2250 m, the light bears N 48 deg 25 min E. If the
course is continued, how close will the ship approach the light?
A. 2394 m
B. 2934 m
C. 2863 m
D. 1683 m
SOLUTION:
79. If f(x) = 1/(x – 2), (f g)’(1) = 6 and g’(1) = -1, then g(1) =
A.-7
B. -5
C. 5
D. 7
SOLUTION:
f(1)g’(1)+g(1)f’(1)=6
−1
(-1)(-1) + g(1) ((1−2)^2)=6
g(1) (-1) = 5
g(1) = -5
80. Find the work done by the force F = 3i + 10j newtons in moving an object
10 meters north.
A.104 40 J
B. 100 J
C.106 J
D. 108.60 J
SOLUTION:
F = 3 + j10
d = 10m
W = Fd
W = 10j(10) = 100 cis 90
81. The volume of a frustum of a cone is 1176pi cu.m. If the radius of the lower
base is 10m and the altitude is 18m, compute the lateral area of the frustum of
a cone
A.295pi sq. m.
B. 691pi sq. m.
C.194pi sq. m.
D. 209pi sq. m.
SOLUTION:
V = 1176πm3
ℎ
V = 3 [𝐴1 + 𝐴2 + √𝐴1𝐴2]
1176π =
18
3
[𝜋𝑟 2 + 𝜋(10)2 ]+ √𝜋𝑟 2 (10)2
r=6
𝐿2 = (10 − 6)2 + (18)2
𝐿 = 2√85
2√85
AL =
2
(2π(6) + 2π(10))
AL = 926.77 m2
82. In an ellipse, a chord which contains a focus and is in a line perpendicular
to the major axis is a:
A.latus rectum
B. minor axis
C. focal width
D. major axis
83. With 17 consonant and 5 vowels, how many words of four letters can be
four letters can be formed having 2 different vowels in the middle and 1
consonant (repeated or different) at each end?
A.5780
B. 5785
C. 5790
D. 5795
2
84. Evaluate tan (j0.78).
A.0.653
B.-0.653
C.0.426
D. -0.426
SOLUTION:
tan(𝑗0.78)2 =
sin(𝑗 0.78)2
cos(𝑗0.78)
= -0.426
85. A particle moves along a line with velocity v = 3t^2 – 6t. The total distance
traveled from t = 0 to t = 3 equals
A.8
B. 4
C. 2
D. 16
86. An observer at sea is 30 ft. above the surface of the water. How much of
the ocean can he sea?
A.124.60 sq. mi.
C. 154.90 sq. mi.
B.142.80 sq. mi.
D. 132.70 sq. mi.
SOLUTION:
1 mile = 1.609344km=1609.344m
1𝑚
30 ft x 3.28 𝑓𝑡 𝑥
1𝑚𝑖𝑙𝑒
1609.344𝑚
=5.68x10^-3
87. There are three consecutive integers. The sum of the smallest and the
largest is 36. Find the largest number.
A.17
B. 18
C.19
D. 20
SOLUTION:
x + x + 2 =36
x = 17 –small
x + 2 = 19 - largest
88. If y = sqrt. of (3 – 2x), find y.
A.1/sqrt. of (3 – 2x)
B. -1/sqrt. of (3 – 2x)
C. 2/sqrt. of (3 – 2x)
D. -2/sqrt. of (3 – 2x)
Y = √(3 − 2𝑥)
1
𝑦 = (3 − 2𝑥)2
−1
1
Y’ = 2 (3 − 2𝑥) 2 (−2)
y’ =
y=
1
1
2
(3−2𝑥)^
𝟏
√((𝟑−𝟐𝒙)
89. The logarithm of MN is 6 and the logarithm of N/M is 2, find the value of
logarithm of N.
A.3
B. 4
C. 5
D.6
SOLUTION:
log MN =6
log M/N = 2
log M + log N = 6
Log MN – log M = 2
M+N=6
-M + N =2
M =2
N=4
90. A woman is paid $20 for each day she works and forfeits $5 for each day
she is idle. At the end of 25 days she nets $450. How many days did she work?
A.21 days
B. 22 days
C. 23 days
D.24 days
91. Francis runs 600 yards in one minute. What is his rate in feet per second?
A.25
B. 30
C.35
D.40
SOLUTION:
1yd = 0.914m
1m = 3.28ft
1min = 60 seconds
600𝑦𝑑
𝑚𝑖𝑛
92.
A.3
1𝑚𝑖𝑛
𝑥 60𝑠𝑒𝑐𝑜𝑛𝑑𝑠 𝑥
0.914𝑚
1𝑦𝑑
𝑥
3.28𝑓𝑡
1𝑚
= 30ft/sec
For a complex number z = 3 + j4 the modulus is:
B. 4
C. 5
D. 6
SOLUTION:
Z = a+jb
z = 3+j4
z = 5 cis 53.13
r=5
93. Which of the following is an exact DE?
A. (x^2 + 1)dx – xydy = 0
C. 2xydx + (2 + x^2)dy = 0
B. xdy + (3x – 2y)dy = 0
D. x^2 ydy – ydx = 0
94. There are 8 different colors, 3 of which are red, blue and green. In how
many ways can 5 colors be selected out of the 8 colors if red and blue are
always included but green is excluded?
A.12
B.11
C. 10
D.9
SOLUTION:
n=8
since, green is excluded
n=7
red and blue are always included
therefore: n = 7-2 =5
n=5
,r=3
nCr
= 5C3
= 10
95. Five cards are drawn from a pack of 52 well – shuffled cards. Find the
probability that 3 are 10’s and 2 are queens.
A. 1/32
B. 1/108,290
C. 1/54,350
D.1/649,740
SOLUTION:
P1 = 4/52
P2 = 3/51
P3 = 2/50
P4 = 4/49
P5 = 3/48
P = P 1 X P2 X P3 X P4 X P 5
= 4/52 + 3/51 + 2/50 + 4/49 + 3/48
P = 1/108,290
7
7
7
If ∫1 𝑓(𝑥)𝑑𝑥 = 4 and ∫1 𝑔(𝑥)𝑑𝑥 = 2, find ∫1 [3𝑓(𝑥) + 2𝑔(𝑥) + 1]𝑑𝑥.
96.
A. 23
B. 22
C. 25
D. 24
SOLUTION:
𝟕
∫𝟏 [𝟑𝒇(𝒙) + 𝟐𝒈(𝒙) + 𝟏]𝒅𝒙
=3f(x)dx + 2g(x)dx + x ]17
=3(4) + 2(2) + (7-1)
=12 + 4 + 6
=22
97. When the ellipse is rotated about its longer axis, the ellipsoid is
A. spheroid
B. oblate
C. prolate
D. paraboloid
98. If the distance between points A(2, 10, 4) and B(8, 3, z) is 9.434, what is
the value of z?
A. 4
B. 3
C. 6
D. 5
SOLUTION:
D = √(𝑿𝟐 − 𝑿𝟏)𝟐 + (𝒀𝟐 − 𝒀𝟏)𝟐 + (𝒁𝟐 − 𝒁𝟏)𝟐
9.434 = √(8 − 2)2 + (3 − 10)2 + (𝑧 − 4)2
Z= 6
99. A line with equation y = mx + b passes through (-1/3, -6) and (2, 1). Find
the value of m.
A. 1
B. 3
C. 4
D. 2
SOLUTION:
y=mx+b
m=
1+6
2+
1
3
m=3
100. For the formula R = E/C, find the maximum error if C = 20 with possible error
0.1 and E = 120 with a possible error of 0.05.
A. 0.0325
B. 0.0275
C. 0.0235
D. 0.0572
REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
AUGUST 2013
MATHEMATICS
REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION
AUGUST 2013
MATHEMATICS
1
1
1. Simplify (csc 𝑥+1) + (csc 𝑥−1)
A. 2 sec x tan x
B. 2 csc x cot x
C. 2 sec x
D. 2 csc x
SOLUTION:
(𝑐𝑠𝑐𝑥−1)+(𝑐𝑠𝑐𝑥+1)
(𝑐𝑠𝑐𝑥+1)(𝑐𝑠𝑐𝑥−1)
2𝑐𝑠𝑐𝑥
2𝑐𝑠𝑐𝑥
2𝑐𝑠𝑐𝑥
2𝑠𝑖𝑛𝑥
𝑠𝑖𝑛𝑥
1
= 𝑐𝑠𝑐 2 𝑥−1 = 𝑐𝑜𝑡 2 𝑥 =( 𝑐𝑜𝑠2𝑥 ) = 𝑐𝑜𝑠2 𝑥 = 2(𝑐𝑜𝑠𝑥) (𝑐𝑜𝑠2 𝑥) = 𝟐 𝐬𝐞𝐜 𝒙 𝐭𝐚𝐧 𝒙
𝑠𝑖𝑛2 𝑥
2. A bus leaves Manila at 12NN for Baguio 250 km away, traveling an average of
55 kph. At the same time, another bus leaves Baguio for Manila traveling
65kph. At what distance from Manila they will meet?
A. 135.42 km
B. 114.56km
C. 129.24km
D. 181.35km
SOLUTION:
T
R
D
x
55
55x
x
65
65x
55x + 65x = 250
D = TR
120x = 250
D = (2.0833)(55)
x = 2.0833
D = 114.56 km
3. Simplify (cos β -1)(cos β+1)
A. -1/sin2β
B. -1/cos2β
C. -1/csc2β
D. -1/sec2β
SOLUTION:
cos2β – 1
1
𝟏
(csc2 β)2 – 1 = − 𝐜𝐬𝐜𝟐𝛃
4. Simplify 1/(csc x + cot x) + 1 /(csc x – cot x).
A. 2 cos x
B. 2 sec x
C. 2 csc x
SOLUTION:
D. 2 sin x
𝑐𝑠𝑐𝑥−𝑐𝑜𝑡𝑥+𝑐𝑠𝑐𝑥+𝑐𝑜𝑡𝑥
2𝑐𝑠𝑐𝑥
= 𝑐𝑠𝑐 2 𝑥−𝑐𝑜𝑡 2 𝑥 =
(𝑐𝑠𝑐𝑥+𝑐𝑜𝑡𝑥)(𝑐𝑠𝑐𝑥−𝑐𝑜𝑡𝑥)
2𝑐𝑠𝑐𝑥
1
1
−
𝑠𝑖𝑛2 𝑥 𝑡𝑎𝑛2 𝑥
= 2cscx
5. From past experience, it is known 90% of one year old children can distinguish
their mother’s voice from the voice of a similar sounding female. A random
sample of 20 one year’s old are given this voice recognize test. Find the
probability that all 20 children recognize their mother’s voice.
A. 0.122
B. 1.500
C. 1.200
D. 0.222
SOLUTION:
Let X - number of children who recognize their mother’s voice
X has Binomial distribution (n=20, p= 0.90)
E(X)=m= np= 20* 0.90=18
P(x = 20) = P(x ≤ 20) – P(x ≤ 19) =
= 1 – 0.878 = 0.122
6. Find the differential equation of the family of lines passing through the origin.
A. xdx – ydy = 0
C. xdx – ydy = 0
B. xdy – ydx = 0
D. ydx – xdy = 0
SOLUTION:
Let y = mx be the family of lines through origin.
Therefore, dy dx = m
Eliminating m,
x dy – ydx = 0.
7. A chord passing through the focus of the parabola y2 = 8x has one end at the
point (8, 8). Where is the other end of the chord?
A. (1/2, 2)
B. (-1/2, -2)
C. (-1/2, 2)
D. (1/2, -2)
8. Find the radius of the circle inscribed in the triangle determined by the line y=
2
x+4, y= -x -4, and y = 7x + 2.
A. 2.29
B. 0.24
C. 1.57
D. 0.35
9. What would happen to the volume of a sphere if the radius is tripled?
A. Multiplied by 3
C. Multiplied by 27
B. Multiplied by 9
D. Multiplied by 6
SOLUTION:
𝐕𝟏
𝐫𝟏
𝐫𝟏
= (𝐫𝟐)3 = (𝟑𝐫𝟏)3
𝐕𝟐
Therefore: V2= 27V1
10. Six non- parallel lines are drawn in a plan. What is the maximum number of
point of intersection of these lines?
A. 20
B. 12
C. 8
D. 15
SOLUTION:
N=6
𝑁(𝑁−1)
2
=
6(6−1)
2
= 15
11. In a triangle ABC where AC=4 and angle ACB=90 degrees, an altitude t is
drawn from C to the hypotenuse. If t = 1, what is the area of the triangle ABC?
A. 1.82
B. 1.78
C. 2.07
D. 2.28
SOLUTION:
Using sine law:
4
x
(sin45) = (sin90)
X=AB=4.2
Side CB= √(4.22 − 42 )
CB= 1.289
1
Area=(2)(b)(h)sin Ɵ
1
= (2)(1.289)(4)sin90
= 2.07
12. In a 15 multiple choice test questions, with five possible choices if which only
on is correct, what is the standard deviation of getting a correct answer?
A.1.55
B. 1.65
C. 1.42
D. 1.72
SOLUTION:
1 4
√[(15) ( ) ( )]
5 5
= 1.55
13. What is the area bounded by the curve y = tan2 x and the lines y = 0 and x =
pi/2?
A. 0
B. infinity
C. 1
D. Ɵ
14. What is the power series of (e^x)/(1-x) about x = 0?
A. 1-2x+(5/2)x^2-(8/3)x^3
C. 2x-(5/2)x^2+(8/3)x^3
B. 1+2x+(5/2)x^2+(8/3)x^3
D. 2x+(5/2)x^2+(8/3)x^3
SOLUTION:
𝐶𝑛𝑋^𝑛 = 𝐶𝑜 + 𝐶1𝑋 + 𝐶2𝑋^2+. . . . 𝐶𝑛𝑋 𝑛
= 1 + C1(X − 0) + C2(X − 0)2 + C3(X − 0)3
8
= 1 + 2X + (5/2)X^2 + (3) X 3
15. What is the vector which is orthogonal both to 9i + 9j and 9l + 9k?
A. 81l + 81j – 81k
C.81l - 81j + 81k
B. 81l – 81j – 81k
D.81l+81j – 81k
16. 24 is 75 percent of what number?
A. 16
B. 40
SOLUTION:
32×0.75
=24
Therefore 24 is 75 percent of 32
Ans. =32
C. 36
17. Evaluate lim (x^2-4)/(x-4), when X is approaches to 4.
A. 4
B. 2
C. 16
SOLUTION:
𝑥2 − 4
𝑥−4
The derivative of the numerator is 2x
The derivative of the denominator is 1
Therefore,
2𝑥
= 1
D. 32
D. 8
2(4)
= 1
=8
18. If sin A = and cot B = 4, both in Quadrant III, the value of sin ( A + B) is
A. -0.844
B. 0.844
C. -0.922
D. 0.922
SOLUTION:
4
3
sin( A + B ) = (− 5) (4) + (5) (1)= 0.922
19. A fence of 100 m perimeter such that its width is 6m less than thrice its length.
Find the width?
A. 28 m
B. 14 m
C. 36 m
D. 40 m
SOLUTION:
P=100m
W=3L-6
P=2(W+L)
100=2(3L-6+L)
L=14
Therefore,
W=3(14)-6
W=36
20. Evaluate log (2 – 5i)
A. 0.7 – 0.5iB. -0.7 + 0.5i
D. -0.5 – 0.7i
C. 0.7 + 0.5i
21. An air balloon flying vertically upward at constant speed is situated 150m
horizontally from an observer. After one minute, it is found that the angle of
elevation from the observer is 28 deg 50 min. what will be then the angle of
elevation after 3 minutes from its initial position?
A. 48 deg
B. 56 deg
C. 61 deg
D. 50 deg
22. If m is jointly proportional to G and x, where a,b,c and d are constant.
Therefore.
A. M = aG + bx
C. m = aG
B. m = aGz
D. m = bG
23. In how many ways can a student going to abroad accompanied by 3 teachers
selecting from 6 teachers?
A. 16
B. 24
C. 20
D. 12
SOLUTION:
Permutation
Using calculator(6-shift-divide sign(nCr)-3)
6C3=20
24.If a man travels 1 km north, 3 km west, 5 km south, and 7 km east, what is his
resultant displacement vector?
A. 5.667 km, 45 deg above + x-axis
C. 5.667 km, 225 deg above + x-axis
B. 5.667 km, 45 deg above – x-axis
D. 5.667 km, 225 deg above – x-axis
SOLUTION:
N
3km
W
E
1km
5km
Resultant vector
b
S
7km
a=7km-3km=4km
b=5km-1km=4km
c=? resultant vector
Using Pythagorean theorem
C2=42+42
=5.6568 km, 225 deg above – X axis
25.
a
What is the general solution of (D4 – 1) y(t) = 0?
A. y = c1Ɵt + c2Ɵ-t +c3 cost + c4 sint
B. y = c1Ɵt + c2Ɵt +c3 Ɵ-t + c4t Ɵ-t
C. y = c1Ɵt + c2Ɵ-t
D. y = c1Ɵt + c2tƟt
SOLUTION:
It is a homogeneous linear differential equation of IV order with constant
coefficients. The corresponding auxiliary equation is m 4 + 1 = 0, whose roots
are the four complex 4th roots (-1) = cost + isint
26. Marsha is 10 years older than John, who is 16 years old. How old is
Marsha?
A. 24 yrs. B. 26 yrs.
C. 6 yrs.
D. 12 yrs.
SOLUTION:
Marsha: 10 + age of john (x)
John(x): 16 y.o
Marsha = 10 + 16 = 26 yrs.
27. Seven times a number x increased by 2 is expressed as
A. 7(x + 2) B. 2x + 7
C. 7x + 2
D. 2(x + 7)
28. The plane rectangular coordinate system is divided into four parts which are
known as:
A. octants
B. quadrants
C. axis
D. coordinates
29. A student already finished 70% of his homework in 42 minutes. How many
minutes does she still have to work?
A. 18
B. 15
C. 20
D. 24
SOLUTION:
Equation; 0.70 x total time(t) = 42min
Total time(t) = 60min
60 – 42 = 18min
30. In how many ways can 5 people be lined up to get on a bus, if a certain 2
persons refuse to follow each other?
A. 36
B. 48
C. 96
D. 72
SOLUTION:
Using calculator
3!(3)(4)= 72
31. Water is being pumped into a conical tank at the rate of 12 cu.ft/min. The
height of the tank is 10 ft and its radius is 5ft. How fast is the water level rising
when the water height is 6ft?
A. 2/3 pi ft/min
B. 3/2 pi ft/min
C. ¾ pi ft/min
D. 4/3 pi ft/min
32. Write the equation of the line with x-intercept a = -1, and y intercept b = 8
A. 8x + y – 8 = 0
C. 8x + y + 8 = 0
B. 8x – y + 8 = 0
D. 8x – y – 8 = 0
SOLUTION:
x
a
y
8x – y = -8
+ b= 1
x
−1
y
8x – y + 8 = 0
+ 8= 1
33. In a single throw of pair of dice. Find the probability that sum is 11.
A. 1/12
B. 1/16
C.
1/36
D. 1/18
SOLUTION:
P = no. of successful trials / total no. of trials
2
Total no. of trials = 36
P = 36
𝟏
No. of trials w/ sum 11 = 2
P = 𝟏𝟖
34. Find the area bounded by one arch of the companions to the cycloid x = a
theta, y = a (1- cos theta) and the y-axis.
A. 2pi a^2
B. 4pi a^2
C. pi a^2
D. 3pi a^2
35. A rectangular plate 6m by 8m is submerged vertically in a water. Find the
force on one face if the shorter side is uppermost and lies in the surface of the
liquid.
A. 941.76 kN
B. 1,583.52 kN
C. 3,767.04 kN
D. 470.88 kN
36. Michael is four times as old as his son Carlos. If Michael was 18 years old
when Carlos was born, how old is Michael now?
A. 36 yrs.
B. 20 yrs.
C. 24 yrs.
D. 32 yrs.
SOLUTION:
Given:
+ 18
Then
x + 4x = x + x
X – Carlo’s age
x – Carlo’s age was born
5x = 2x + 18
4x – Michael’s age
x + 18 – Michael’s age
x=6
Substitute value of x=6 to x + 18: x + 18 = 24 yrs.
37. In polar coordinate system the distance from a point to the pole is known as:
A. polar angle
C. X-coordinates
B. radius vector
D. Y-coordinates
38. A certain man sold his ballot at Php 1.13 per piece. If there 100 balots sold
all in all, how much is his total collection?
A. Php 113.00
B. Php 115.00
C. Php 112.00
D. 116.00
SOLUTION:
X = 1.13(100)
= Php 113.00
39. A certain population of bacteria grows such that its rate of change is always
proportional to the amount present. It doubles in 2 years. If in 3 years there are
20,000 of bacteria present, how much is present initially?
A. 9,071
B. 10.071
C. 7,071
D. 8,071
SOLUTION:
1
Q=2𝑄𝑜
Q = 𝑄𝑜 22𝑡
Q = 𝑄𝑜 𝑒 𝑟𝑡
20000 = 𝑄𝑜 (2)3/2
2𝑄𝑜 = 𝑄𝑜 𝑒 2𝑟
𝑄𝑜 = 20000 / (2)3/2
2 = (𝑒 2𝑟 )1/2
𝑸𝒐 = 7,071
𝑒 𝑟 = 21/2
40. In throwing a pair of dice, what is the probability of getting of 5?
A. 1/36
B. 1/9
C. 1/16
SOLUTION:
P = no. of successful trials / total no. of trials
D. 1/6
4
Total no. of trials = 36
P = 36
No. of trials 5 = 4
P=𝟗
𝟏
41. What is the distance between at any point P(x ,y) on the ellipse b 2x2 + a2y2
= a2b2 to its focus.
A. by ±ax
B. b ± ay
C. ay ± bx
D. a ± ex
42. Calculate the eccentricity of an ellipse whose major axis and latus rectum
has length of 10 and 32/5, respectively.
A. 0.4
B. 0.5
C. 0.8
D. 0.6
43. Evaluate (3 + j4)(3 – j4)
A. 9 – j16
B. 9 + j16
SOLUTION:
C. 25
D. 36
9-j12+j12-j216
= 9+16
= 25
44. What is the area bounded between y = 6x^2 and y = x^2 + 7?
A. 9
B. 10
C. 11
D. 12
SOLUTION:
x2 + 7 = 6x2
x2
-
6x2
7
±√5 = x
7
5
√
+7 = 0
∫− 7(𝑥 2 + 7) − (6𝑥 2 )𝑑𝑥
√
5
7 = 5x2
= 11
45. Two vertical poles are 10 m apart. The poles are 5 m and 8 m, respectively.
They are to be stayed by guy wires fastened to a single stake on the ground
and attached to the tops of the poles. Where should the stake be placed to use
the least amount of wire?
A. 6.15 m from 5 m pole
C. 6.51 m from 5 m pole
B. 6.15 m from 8 m pole
D. 6.51 m from 8 m pole
SOLUTION:
ab
x =b + c
a – x = 10 – 3.85
10(5)
x= 8 + 8
x = 3.58
= 6.15m from 8m pole
46. A and B are points on circle Q such that triangle AQB is equilateral. If AB =
12, find the length of arc AB.
A. 15.71
B. 9.42
C. 12.57
D. 18.85
47. The area under the portion of the curve y = cosx from x = 0 to x = pi/2 is
revolved about the x-axis. Find the volume of the solid generated.
A. 2.47
B. 2.74
C. 3.28
D. 3.82
48. Find the length of arc of r = 2/(1 +costheta) from theta = 0 to theta = pi/2.
A. 2.64
B. 3.22
C. 2.88
D. 3.49
49. Find the equation of the straight line which passes through the point (6, -3)
and with an angle of inclination of 45 degrees.
A. x + y = 3 B. 4x – y =27
C. x- 2y = 12
D. x – y = 9
SOLUTION:
m = tan Ɵ
(y-y1) = m (x-x1)
= tan 45
(y+3)=1(x-6)
x–y=9
m=1
50. The equation of the directrix of the y^2 = 6x is
A. 2x – 3 = 0
B. 2x + 3 = 0
C. 3x – 2 = 0
SOLUTION:
4a = 6
(x + 3/2 = 0)2
a = 3/2
2x + 3 = 0
51. Find the area bounded by r = 4(sq.rt. of cos 2 theta).
A. 16
B. 8
C. 4
D. 12
SOLUTION:
𝜋
𝜋
(𝑟 = 4√𝑐𝑜𝑠2𝜃)2
−4 < 𝜃 < 4
𝑟 2 = 16𝑐𝑜𝑠2𝜃
1
𝐴 = 2 ∫ 𝑟 2 𝑑𝑟
𝐴=
1
2
𝜋
4
𝜋
−
4
∫ 16𝑐𝑜𝑠2𝜃𝑑𝜃
𝜋
𝐴 = 8 ∫ 4𝜋 𝑐𝑜𝑠2𝜃 2𝑑𝜃
−
4
𝜋
4
𝜋
−
4
𝐴 = 4 ∫ 𝑐𝑜𝑠2𝜃 2𝑑𝜃
𝜋
4
𝐴 = 4 sin 2𝜃]
−
𝜋
𝜋
𝜋
4
𝜋
= 4 sin(2 4 ) − 2 sin(2 (− 4 ))
𝜋
𝐴 = 4 sin( 4 ) − 4 sin (− 4 ) = 4(1) − 4(−)
𝑨=𝟖
D. 3x + 2 = 0
52. In an arithmetic progression whose first term is 5, the sum of 8 terms is 208.
Find the common difference.
A. 3
B. 4
C. 5
D. 6
SOLUTION:
𝑛
𝑆 = 2 [2𝑎1 + (𝑛 − 1)𝑑]
8
208 = 2 [2(5) + (8 − 1)𝑑]
𝒅=6
53. If 3x = 7y, then 3x2/7y2 = ?
A. 1
B. 3/7
C. 7/3
SOLUTION:
𝟕𝒚
𝒙= 𝟑
𝟑𝒙𝟐
𝟕𝒚𝟐
=
𝟕𝒚
𝟑
𝟕𝒚𝟐
𝟑( )𝟐
=
𝟐𝟏
𝟗
D. 49/9
𝟕
=𝟑
54. What is the area of the ellipse whose eccentricity is 0.60 and whose major
axis has a length of 6?
A. 40.21
B. 41.20
C. 42.10
D. 40.12
SOLUTION:
2𝑎 = 6
𝑎=3
𝑐
𝑒=𝑎
𝑐 = .6 ∗ 3 = 1.8
𝑏 = √𝑎2 − 𝑐 2
𝑏 = √32 − 1.82 = 2.4
𝐴 = 𝜋𝑎𝑏
𝐴 = 𝜋(3)(2.4)
𝐴 = 22.61
55. Tickets to the school play sold at $4 each for adults and $1.50 each for
children. If there were four times as many adult’s tickets sold as children’s
tickets, and the total were $3500. How many children’s tickets were sold?
A. 160
B. 180
C. 200
D. 240
SOLUTION:
𝑥 = 𝑎𝑑𝑢𝑙𝑡 ; 𝑦 = 𝑐ℎ𝑖𝑙𝑑𝑟𝑒𝑛
𝑦 = 1100 − 𝑥
4𝑥 + 𝑦 = 3500
4𝑥 + (1100 − 𝑥) = 3500
𝑥 = 800
𝑦 = 1100 − 800
𝑦 = 300
4(800) + 300 = 3500
300
= 𝟐𝟎𝟎
1.5
56. If the line kx + 3y + 8 = 0 has a slope of 2/3, determine k.
A. -3
B. -2
C. 3
D. 2
SOLUTION:
3𝑦 = −𝑘𝑥 − 8
−𝑘𝑥−8
𝑦= 3
−𝑘𝑥
𝑦=
𝑚=
2
3
−𝑘
8
−3
3
−𝑘
= 3
3
𝒌 = −𝟐
57. The Rotary Club and the Jaycees Club had a joint party. 120 members of
the Rotary Club attended and 100 members of the Jaycees Club also attended
but 30 of those who attended are members of both parts. How many persons
attended the party?
A. 190
B. 220
C. 250
D. 150
SOLUTION:
120 + 100 = 220
220 − 30 = 𝟏𝟗𝟎
58. Find the value of k for which the graph of y = x^3 + kx^2 + 4 will have an
inflection point at x = -1.
A. 3
B. 4
C. 2
D. 1
SOLUTION:
𝑦 ′ = 3𝑥 2 + 𝑘𝑥+ 0
𝑦 ′′ = 6𝑥 + 2𝑘
2𝑘 = −6𝑥
𝑘 = −3𝑥
𝑘 = −3(−1)
𝒌=𝟑
59. Solve for x if log4x = 5.
A. 2048
B. 256
SOLUTION:
45 = 𝑥
𝒙 = 𝟏𝟎𝟐𝟒
C. 625
D. 1024
60. An observer wishes to determine the height of a tower. He takes sights at
the top of the tower from A and B, which are 50 ft apart at the same elevation
on a direct line with the tower. The vertical angle at point A is 30 degrees and
at point B is 40 degrees. What is the height of the tower?
A. 85.60 ft B. 143.97 ft
C. 110.29 ft
D.92.54 ft
SOLUTION:
ℎ
𝑡𝑎𝑛 𝜃 =
𝑥
ℎ
ℎ
𝑡𝑎𝑛 30 = (50+𝑥) 𝑡𝑎𝑛 40 = 𝑥
𝑡𝑎𝑛 30(50 + 𝑥) = 𝑡𝑎𝑛 40𝑥
50+𝑥
𝑥
𝑡𝑎𝑛40
= 𝑡𝑎𝑛30
𝑥 = 110.29
ℎ = 𝑡𝑎𝑛 40𝑥
ℎ = 𝑡𝑎𝑛 40(110.29)
𝒉 = 𝟗𝟐. 𝟓𝟒 𝒇𝒕.
62. If four babies are born per minute, how many babies are born in one hour?
A. 230
B. 250
C. 240
D. 260
𝑀=
4
𝑚𝑖𝑛
𝑥 1 min 𝑥 60
= 𝟐𝟒𝟎 𝒃𝒂𝒃𝒊𝒆𝒔
min
ℎ𝑟
63. What was the marked price of a shirt that sells at P 225 after a discount of
25%?
A. P 280
B. P 300
C. P 320
D. P 340
x - 0.25 x = 225
x = P300
64. Which number is divisible by both 3 and 5?
A. 275
B. 445
C. 870
870
3
D. 955
= 290
870
= 174
: (3, 5)
5
65. If s = t^2 – t^3, find the velocity when the acceleration is zero
A. 1/4
B. 1/2
C. 1/3
D. 1/6
S = t2 – t1 find when a = 0
𝑣=
𝑎=
ds
= 2t - 3t2
dt
ds"
d"t
= 2 – 6t
1
1
𝑣 = 2 (3) − 3 (3)²
@ a=0
a = 2 – 6t
𝒕 =
𝟏
𝟑
𝑣=
𝟏
𝟑
66. Find k so that A = (3, -2) and B = (1, k) are parallel
A. 3/2
B. -3/2
C. 2/3
A = ( 3, -2) B = ( 1 , k )
1
𝑘
= −2
3
k=
D. -2/3
an parallel
−𝟐
𝟑
67. A lady gives a dinner party for six guest. In how many may they be selected
from among 10 friends?
A. 110
B. 220
C. 105
D. 210
r=6
n = 10
P = 10 C6 =
𝟏𝟎!
(𝟏𝟎−𝟔)!(𝟔)!
= 𝟐𝟏𝟎 𝒘𝒂𝒚𝒔
68. A wheel 4 ft in diameter is rotating at 80 rpm. Find the distance (in ft) traveled
by a point on the rim in 1 s.
9.8 ft
B. 19.6 ft
C. 16.8 ft
D. 18.6 ft
d = 4ft
s = cv =
s = 16.76 ft
v = 80
4
𝐬𝐞𝐜
𝐦𝐢𝐧
𝟏𝒎𝒊𝒏
𝟒
𝒙 𝟔𝟎𝒔𝒆𝒄 = 𝟑 𝒓𝒑𝒔
= 𝜋(4ft) (3 𝑟𝑝𝑠)( 1 𝑠𝑒𝑐)
69. If f(x) = 6x – 2 and g(x) = 4x + 3, then f(g(2)) = ____?
52 B. 53
C. 50
D. 56
f (6x) = 5x -2
________
g (2) = 4 (2) + 3 = 11
f(g(2)) = 5 (11) – 2 = 53
f (g(2)) = 53
g(x) = 4x+3
find
f(g
(2)
=
70. From the top of lighthouse, 120 ft above the sea, the angle of depression of
a boat is 15 degrees. How far is the boat from the lighthouse?
A. 444 ft
B. 333 ft
C. 222 ft
D. 555 ft
h = 120ft
θ = 1s
ℎ
tan (15) = 𝑑
𝒅=
𝟏𝟐𝟎
𝟎.𝟐𝟕
= 𝟒𝟒𝟒. 𝟒𝟒 𝒇𝒕
71. If 8 men take 12 days to assemble 16 machines, how many days will it take
15 men to assemble 50 machines?
16 B. 24
C. 16
D. 20
𝑟𝑎𝑡𝑒 =
𝟏𝟔
𝟖 (𝟏𝟐)
𝒎𝒂𝒄𝒉𝒊𝒏𝒆
𝒎𝒆𝒏 𝒙 𝒅𝒂𝒚𝒔
𝟓𝟎
= 𝟏𝟓(𝒙)
X = 20 days
72. Find the coordinate of the highest point of the curve x = 90t, y = 96t – 16t^2.
A. (288, 144)
B. (144, 288)
C. (288, -144)
D.(-144, 288)
x = 96t
y = 96t – 16t2
𝒅𝒚
𝟑𝟐𝒕
=
96
−
=𝟎
𝒅𝒙
𝟗𝟔
t=3
dy = 96 – 32t
dx = 96
x = 96 (3) = 288
y = 96 (3) – 6 (32) = 144
288, 144
73. The vertex of parabola y = (x – 1)^2 + 2 is _____.
(-1, 2)
B. (1, 2)
C. (1, -2)
D. (-1, -2)
y = ( x-1) 2 + 2
( x-1) 2 =y-2
V ( 1,2)
74. Two angles measuring p deg and q are complementary. If 3p – 2q = 40 deg,
then the smaller angle measures
40 deg
B. 44 deg
C. 46 deg
D. 60 deg
p and q are complementary
q = 90 – p
3p -2 (90-p) = 40
p = 44° q = 90 – 44 = 46°
smaller angle is 44°
75. In an ellipse, a chord which contains a focus and is in a line perpendicular to
the major axis is a:
A. latus rectum
C. focal width
B. minor axis
D. conjugate axis
76. Determine the rate of a woman rowing in still water and the rate of the river
current, if it takes her 2 hours to row 9 miles with the current and 6 hours to
return against the current.
1 mph
B. 2 mph
C. 3 mph
D. 4 mph
d1 = d 2
V 1 t1 = V 2 t2
( V1 + VR) (2) = ( V – VR ) 6
V + VR = 3 ( V – VR)
2V – 4VR = 0
V = 2VR
VR = ½ V
d = 9 miles
2 (V + VR ) = 9
2V + 2VR = 9
2V + ½ V = 9
3V = 9
V = 3 mph
77. If f(x) = sin x and f(pi) = 3, then f(x) =
4 + cos x B. 3 + cos x
C. 2 – cos x
f(x) = sin x
f (x) = 2- c0s x
f(x) = 3
D. 4 – cos x
then f(x) = ?
78. What is the value of the circumference of a circle at the instant when the
radius is increasing at 1/6 the rate the area is increasing?
A. 3
B. 3/pi
C. 6
D. 6/pi
C= 2 𝜋𝑟
Error question
79. A ball is thrown from the top of a 1200-foot building. The position function
expressing the height h of the ball above the ground at any time t is given as
h(t) = -16t^2 – 10t + 1200. Find the average velocity for the first 6 seconds of
travel.
A. -202 ft/sec
B. -106 ft/sec
C. -96 ft/sec
D. -74 ft/sec
h (t) = - 16t2 – 10t + 1200
@ t = 6 h = -16(6) 2 – 10 (t) + 1200 = 564
@ t= 0
h = 1200
𝐻₁−𝐻₂
= 564−1200 = -106ft/sec
VA = t1 +t₂
6−0
−1
80. ∫−2 |𝑥 3 |𝑑𝑥 =
A. -7/8
−1
∫−2 |𝑥 3 |𝑑𝑥 =
1
4
x⁴
4
B. 7/8
D. 16/4
−1
|𝑥 3 | ∫−2
[(−2)4 − (−1)4 ] =
=
C. -15/4
1
4
[16 − 1]
𝟏𝟓
𝟒
81. The distance covered by an object falling freely rest varies directly as the
square of the time of falling. If an object falls 144 ft in 3 sec, how far will it fall
in 10 sec?
A. 1200 ft B. 1600 ft
C. 1800 ft
D. 1400 ft
82. For what values(s) of x will the tangent lines to f(x0 + ln x and g(x) = 2x^2 be
parallel?
A. 0
B. 1/4
C. 1/2
D. ±1/2
83. What kind of graph has r =2 sec theta?
A. Straight line
B. parabola
C. ellipse
D. hyperbola
84. The probability of A’s winning a game chess against B is 1/3. What is the
probability that A will win at least 1 of a total 3 games?
A. 11/27
B. 6/27
C. 19/27
D. 16/27
85. If f(x) = 2^(x^3 + 1), then to the nearest thousandth f(1) =
A. 2.000
B. 2.773
C. 4.000
D. 8.318
𝑎
𝑎
86. If line function f is even and ∫0 𝑓(𝑥)𝑑𝑥 = 5𝑚 − 1, then ∫−𝑎 𝑓(𝑥)𝑑𝑥 =
A. 0
B. 10m – 2
C. 10m – 1
D. 10m
87. What is the slope of the line through (-1, 2) and (4, -3)?
A. 1
B. -1
C. 2
D. -2
88. The axis of the hyperbola through its foci is known as:
A. Conjugate axis
B. major axis
C. transverse axis
D. minor axis
89. Determine a point of inflection for the graph of y = x^3 + 6x^2
A. (-2, 16) B. (0, 0)
C. (-1, 5)
D. (2, 32)
SOLUTION:
yI = 3x2 + 12x
x = -2
II
y = 6x + 12
y = (-2)3 + 6(-2)2
6x = -12
y = 16
POI = (-2, 16)
90. Clarify the graph of the equation x^2 + xy + y^2 – 6 = 0.
A. circle
B. parabola
C. ellipse
D. hyperbola
91. What is the coefficient of the (x – 1)^3 term in the Taylor series expansion of
f(x) = ln x expanded about x = 1?
A. 1/6
B.1/4
SOLUTION:
f(x) = ln(x)
f’(x) = 1/x
f’’(x) = −
2
C. 1/3
ln(x) = 0 + 1 (𝑥 − 1)’ −
f(1) = 0
f’(1) = 1
1
ln(x) =
f’’(1) = -1
x2
f’’’ (x) =x3
ln(x) =
(x−1)3
D. ½
1(𝑥 − 1)2
2!
+
2(𝑥 − 1)3
3!
3
(2−1)3
3
= 1/3
f’’’ (1) = 2
92. If x varies directly as y and inversely as z, and x = 14, when y = 7 and z = 2,
find x when y = 16 and z = 4.
A. 4
B. 14
C. 8
D. 16
SOLUTION:
Y
X=kxZ
Where: k is constant and when x = 14, y = 7 & z = 2
7
14 = (k) ( )
2
k=4
Y
X = (4)( Z)
Where: y = 16 & z = 4
X = (4 )(
16
4
)= 16
93. Solve the differential equation
A. y = cx
SOLUTION:
1
B. y = 𝑥 + 𝑐
𝑑𝑦
𝑑𝑥
𝑦
+𝑥 =2
C. y = 3x + c
𝑐
D. y = x + 𝑥
94. In triangle ABC, AB = 40 m, BC = 60 m and AC = 80m. How far from a will
the other end of the bisector angle B located along the line AC?
A. 40
B. 32
C. 38
D. 35
SOLUTION:
𝑋
𝐴𝐵
− 𝑋 = 𝐵𝐶
X =32
𝐴𝐶
𝑋
40
−𝑋 =
80
60
60X = 40 (80-X)
60X + 40X = 3200
100𝑋 3200
=
100
100
95. What amount should an employee receive a bonus so that she would net
$500 after deducting 30% from taxes?
A. $ 714.29 B. $814.93
C. $ 624.89
D. $ 538.62
SOLUTION:
96 A rectangular trough us 8ft long, 2ft across the top, and 4 ft deep. If water
flows in at a rate of 2 cu. Ft per min. how fast is the surface rising when the
water is 1ft deep ?
A. 1/4 ft/min
B. 1/6 ft/min
C. 1/3 ft/min
D. 1/5 ft/min
SOLUTION:
Volume of water :
V = ½ (xy)(8) = 4xy
By similar triangle :
𝑥 2
=
𝑦 4
x=½y
1
Y= 8( )y = 4y2
2y
𝑑𝑣
𝑑𝑦
= 8𝑦
𝑑𝑡
𝑑𝑡
When y=1ft
2ft3/min = 8 (1) dy/dt
2 𝑑𝑦
=
8 𝑑𝑡
𝑑𝑦
𝒇𝒕
= ¼
𝑑𝑡
𝒎𝒊𝒏
97. If the parabola y = x^2 + C is tangent to the line y = 4x + 3, find the value of
C.
A. 4
B. 7
C. 6
D. 5
SOLUTION:
y = x2 + c
y = 4x + 3
4x + 3 = x2 + c
x2 - 4x + (c - 3) = 0
√ 𝑏2 − 4𝑎𝑐= 0
√(−4)2 − 4(1)(𝑐 − 3)= 0
−
16 − (4𝑐 − 12) = 0
C=7
4𝑐
−
4 =
28
4
Squared both sides:
√16 − (4c − 12) = √0
16 - 4c + 12 = 0
-4c = 28
98. A parabola having its axis along the x-axis passes through (-3, 6). Compute
the length of latus rectum if the vertex is at the origin.
A. 12
B. 8
C. 6
D. 10
SOLUTION:
Formula
4p
4(3)
12
99. If the average value of the function f(x) = 2x^2 on the interval (0, c) is 6, then
c=
A. 2
B. 3
C. 4
D. 5
SOLUTION:
100. Find the volume of the tetrahedron bounded by the coordinate planes and
the plane z = 6 – 2x + 3y.
A. 4
B. 5
C. 6
D. 3
SOLUTION:
REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
MARCH 2013
MATHEMATICS
REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION
MARCH 2013
MATHEMATICS
1. If the man sleeps from 6:48 PM up to 7:30 AM. The number of hours and
minutes he sleeps is.
A. 11 hrs and 42 min
B. 12 hrs and 42 min
C. 13 hrs and 42 min
D. 10 hrs and 42 min
SOLUTION:
6:48PM – 7:30AM = 12hrs and 42mins
2. The price of a ballpen rises from Php 4.00 to Php 12.00. What is the percent
increase
in.price?
A. 100 percent
C. 150 percent
B. 120 percent
D. 200 percent
SOLUTION:
8
- 4 = 8; 4 x 100% = 200%
𝜋𝑥
3. Evaluate: lim(2 − 𝑥)^tan( 2 ).
𝑥→1
A. e^(2/pi)
B. e^(pi/2)
C. e^(2pi)
D. 0
SOLUTION:
𝜋(0.09999)
180
2
𝜋
(2– 0.09999) tan(
)(
) = 1.89 or e2/𝝅
4. Thirty is 40 percent of what number?
A. 60
SOLUTION:
30 = 40% (X)
X = 75
B. 70
C. 75
D. 80
5. Roll a pair of dice. What is the probability that the sum of two numbers is 11?
A. 1/36
B. 1/9
C. 1/18
D. 1/20
SOLUTION:
Pair of dice = 2
Possible rolls = 36
Two ways to roll 11 = (5,6) (6,5)
2
36
𝟏
= 𝟏𝟖
6. If the logarithm of MN is 6 and the logarithm of N/M is 2, find the logarithm of
M.
A. 2
B. 3
C. 4
D. 6
SOLUTION:
Log N = 6 -Log M
6 – 2(LogM) = 2
-2Log M = 2- 6
LogM = 2
7. The mean duration of television commercials on a given network is 75
seconds, with a standard deviation of 20 seconds. Assume that duration time
are approximately normally distributed. What is the approximate probability
that
a
commercial
will
last
less
than
35
seconds?
A. 0.055
B. 0.025
C. 0.045
D. 0.035
8. In how many ways can 5 people be lined up if two particular people refuse to
follow
each
other?
A. 52
B. 62
C. 72
D. 82
SOLUTION:
5! – 2(4!) = 72
9. Which of the following is not included?
10.
A. 0.60
B. 60%
C. 0.06
Which of the following is not included?
A. 0.60
11.
B. 60%
C. 0.06
D. 3/5
D. 3/5
The area of the circle is 89.42 sq. in. What is its circumference?
A. 32.25 in B. 33.52 in
C. 35.33 in
D. 35.55 in
SOLUTION:
89.24 = 𝜋 𝑟2
R = 5.3351
C = 2𝜋 (5.3351) = 33.52in
12. If a truck parks in at 1 PM in a parking lot and leaves at 4 PM. Find the
number of hours it stayed at the parking lot
A. 1
13.
B. 2
C. 3
D. 4
C. 4
D. 2
If (x+3): 10=(3x-2): 8, find 2x-1.
A. 1
SOLUTION:
B. 3
((x+3))/10 = ((3x-2))/8
8x+24 = 30x-20
30x-8x = 24+20
22x = 44
x=2
2(2) – 1 = 3
14. Evaluate the Laplace transform of t^n
A. n!/s^n
B. n!/s^(n+1)
C. n!/2s^n
D. n!/2s^(n+1)
SOLUTION:
𝑡 𝑛 𝑒 −𝑠𝑡
∞
Laplace {t^n} = ∫0 𝑡 𝑛 𝑒 −𝑠𝑡 𝑑𝑡 = -
𝑠
Let du = ntn-1 ; v =
Laplace
{tn}
Laplace {tn} =
=
𝑛
𝑠
0-0n
+
∞
− ∫0 𝑛𝑡 𝑛−1 −
𝑠𝑒 −𝑠𝑡
𝑠
𝑑𝑡
−𝑠𝑒 −𝑠𝑡
𝑠
∞ 𝑛−1 −𝑠𝑡
𝑒 𝑑𝑡
∫ 𝑡
𝑠 0
𝑛
1
ւ [𝑡 𝑛−1] ; 𝑡1 = 𝑠2 then s>0
𝐧!
Laplace {tn} = 𝐬𝐧+𝟏
15.
Find the volume generated by rotating a circle x^2+y^2+6x+4y+12=0
about the y-axis.
A. 58.24 B. 62.33
C. 78.62
D. 59.22
SOLUTION:
x2+y2+6x+4y+12=0
(x2+6x+9) +(y2+4y+4) =-12+9+4
(x + 3)2 + (y + 2)2 = 1
(x - h)2 + (y - k)2 = r2
r=1
C = (-3, -2)
By inspection: d = 3
Using second proposition of Pappus
V = A x 2πd
V = π (1)2 x 2π (3)
V = 59.22 cubic units
16.Determine all the values of 1^sqrt. of 2.
A. sin (sqrt. of 2 kpi) + icos (sqrt. of 2 kpi)
B. cos (sqrt. of 2 kpi) + isin (sqrt. of 2 kpi)
C. sin (2sqrt. of 2 kpi) + icos (2sqrt. of 2 kpi)
D. cos (2sqrt. of 2 kpi) + isin (2sqrt. of 2 kpi)
17. The slope of the curve y^2-xy-3x=1 at the point (0, -1) is
A. -1
B. -2
C. 1
D. 2
SOLUTION:
𝑦+3
Y1 = 2𝑦−𝑥 =
18.
−1+3
2(−1)−0
= -1
Express Ten million forty-three thousand seven hundred seventy-one.
A. 10,403,771
C. 10,430,771
19.
A. 8
B. 10,433,771
D. 10,043,771
Find the length of the curve r = 8 sin theta.
B. 4
C. 8 pi
SOLUTION:
r2 = 64sin2ϴ
𝑑𝑟
(𝑑𝛳)2 = 64cos2ϴ
𝑏
𝑑𝑟
L = ∫𝑎 √𝑟 2 + (𝑑𝛳)2 𝑑𝜃
π
D. 4 pi
Note: sin2ϴ + cos2ϴ = 1
π
L = ∫0 √64(1) 𝑑𝜃
π
L = ∫0 8 𝑑𝜃 = 8 [ϴ − ϴ]
L = ∫0 √64𝑠𝑖𝑛2 ϴ + 64𝑐𝑜𝑠 2 ϴ 𝑑𝜃
L = 8 [π − 0] = 8𝛑
20. A pole which leans 11 degrees from the vertical toward the sun cast a
shadow 12 m long when the angle of elevation of the sun is 40 degrees. Find
the
length
of
the
pole.
A. 15.26 m
B. 14.26 m
C. 13.26 m
D. 12.26 m
SOLUTION:
x = 90 + 11 = 101 Sine Law
B = 180 – (101 +40)
B= 39
𝑆𝑖𝑛(40)
𝑋
=
𝑆𝑖𝑛(39)
12
X = 12.26m
21. How long is the latus rectum of the ellipse whose equation is 9x^2+16y^2576=0?
A. 7
B. 9
C. 10
D. 15
SOLUTION:
𝑥2
𝑦2
+ 36 = 1
64
L.R =
2(6)2
8
;
a=8 b=6
=9
22. If the initial and final temperatures of an object are 97.2 and 99 deg F
respectively, find the change in temperature.
A. 1.7 deg F
B. 1.8 deg F
C. 1.9 deg F
D. 1.6 deg F
SOLUTION:
99 – 97.2 = 1.8 F
23. A rectangular plate 6 m by 8 m is submerge vertically in a water. Find the
force on one face if the shroter side is uppermost and lies in the surface of
the
liquid.
A. 941.76 kN
C. 3,767.04 kN
B. 1,883.52 kN
D. 470.88 kN
SOLUTION:
F = (8)(6)(4)(9.81) = 1,883.52
24.
Find the area enclosed by the loop y^2 = x(x-1) ^2
A. 8/15
B. 8/17
C. 7/15
D. 7/17
25. The GCF of two numbers is 34, and their LCM is 4284. If one of the
number is 204, the other number is
A.714
B. 716
C. 2124
D. 3125
SOLUTION:
Other Number =
(34)(4284)
204
= 714
26. Jonas, star player of Adamson University has free throw shooting of 83%.
The game is tied at 87-87. He is fouled and given 2 free throws. What is the
probability that the game will go overtime?
A. 0.3111 B. 0.6889
C. 0.0289
D. 0.9711
SOLUTION:
𝑋 = 100% − 83% = 17 %
𝑁 = 2 𝑡𝑖𝑚𝑒𝑠
𝑃 = .17 𝑥 .17 = 0.0289
27. Find the work done in moving an object along a vector a = 31 + 4j if the
force applied is b = 21 + j.
A. 8
B. 9
C. 10
D. 12
SOLUTION:
A = 3i +4j
B = 2i+j
;
w = (3) (20) +(4)(1)
w = 10
28. If 3z + 5 = 7z-7. Find Z
A. 3
B. 5
SOLUTION:
3z -7z = -7 – 5
−4𝑧
−4
=
C. 7
D. 9
−12
−4
Z=3
29. Where does the normal line of the curve y = x - x^2 at the point (1,0) intersect
the curve a second time?
A. (-3, -12)
30.
Simplify
B. (0,0)
1+tan2 𝑥
1+cot2 𝑥
C. (-2, -6)
D. (-1, -2)
A. sec2x
B. tan2x
C. csc2x
D. cot2x
SOLUTION:
1+tan2 𝑥
1+cot2 𝑥
sec2 𝑥
1
= csc2 𝑥 = cos2 𝑥 =
sin2 𝑥
1
= tan2X
31. Jodi wishes to use 100 feet of fencing to enclose a rectangular garden.
Determine the maximum possible area of her garden.
A. 850 sq. ft.
C. 625 sq. ft.
32.
B. 1250 sq. ft.
D. 1650 sq. ft.
Simplify 1/(csc x + 1) + 1/(csc x – 1).
A. 2 sec x tan x
C. 2 sec x
B. 2 csc x cot x
D. 2 csc x
SOLUTION:
2𝑐𝑠𝑐𝑥
2𝑐𝑠𝑐𝑥
2
sin2 𝑥
1/(csc x + 1) + 1/(csc x – 1) = 𝑐𝑠𝑐𝑥−1 = cot2 𝑥 = 𝑠𝑖𝑛𝑥 ∗ cos2 𝑥 = 2secx tanx
33. A certain chemical decomposes exponentially. Assume that 200 grams
becomes 50 grams in 1 hour. How much will remain after 3 hours?
A. 1.50 grams
B. 6.25 grams
C. 4.275 grams
D. 3.125 grams
34. The locus of a point that moves so that the sum of its distances between
two fixed points is constant called:
A. a parabola
B. a circle
C. an elipse
D. a hyperbola
35. Michael’s age is seven-tenths of Richard’s age. In four years Michael’s
age will be eight-elevenths of Richard’s age. How old is Michael?
A. 26 yrs.
B. 28 yrs.
C. 40 yrs.
D. 48 yrs.
SOLUTION:
7
8
x +4 = 11 (x+4)
10
X = 40
36.
7
; 10 (40) = 28
A conic section whose eccentricity is equal to one (1) is known as:
A. a parabola
B. an elipse
C. a circle
D. a hyperbola
37. The angle of a sector is 30 degrees and the radius is 15 cm. What is the
area
of
a
sector?
A. 59.8 sq. cm.
C. 89.5 sq. cm.
B. 58.9 sq. cm.
D. 85.9 sq. cm.
SOLUTION:
1
𝜋
A sector = 2 (15)2 (30)( 180) = 58.90
38.
In a conic section, if the eccentricity is greater than (1), the locus is:
A. a parabola
39.
B. an elipse
C. a circle
D. a hyperbola
If f’(x) = sin x and f(pi) = 3, then f(x) =
A. 4 + cos x
C. 2 – cos x
B. 3 + cos x
D. 4 – cos x
40. Two stones are 1 mile apart and are of the same level as the foot of a hill.
The angles of depression of the two stones viewed from the top of the hill are
5 degrees and 15 degrees respectively. Find the height of the hill.
A. 109.1 m B. 209.1 m
C. 409.1 m
D. 309.1 m
SOLUTION:
1 mile = 1609.75m
ℎ
Tan 15 = 1609.75+𝑋 = eq.1
ℎ
Tan 15 = 𝑥
H = xtan15 = eq. 2
(1606.75+x) tan15 = xtan15
X = 780.425m
H = 780.425 (tan15) = 209.11m
41. What is the equation of the line, in the xy-plane, passing through the point
(6, 4) and parallel to the line with parametric equations x = 5t + 4 and y = t –
7?
A. 5y – x = 14
B. 5x – y = 26
C. 5y – 4x = -4
D. 5x – 4y = 14
42. Evaluate (8+7i) ^2
A. 15 + 112i
B. 15 – 112i
C. -15 + 112i
D. -15 – 112i
SOLUTION:
(8+7i)(8+7i) = 15 + 112i
43. How far is the directrix of the parabola (x-4)^2 = -8(y-2) from the x-axis?
A. 2
B. 3
C. 4
D. 1
SOLUTION:
1
y = − (𝑥 − 4)2 + 2
8
1
Where: a = − 8 , b = 1, c = 0
y=k–p
4𝑎𝑐−𝑏 2 −1
𝑦=
4𝑎
y =4
44. A weight W is attached to a rope 21 ft long which passes through a pulley
at P, 12 ft above the ground. The other end of the rope is attached to a truck
at a point A, 3 ft above the ground. If the truck moves off at the rate of
10ft/sec, how fast is the weight rising when it is 7 ft above the ground?
A. 9.56 ft/sec
B. 7.82 ft/sec
C. 8.27 ft/sec
D. 6.25 ft/sec
45. The first farm of GP is 160 and the common ratio is 3/2. How many
consecutive terms must be taken to give a sum of 2110?
A. 5
B. 6
C. 7
D. 8
SOLUTION:
2𝑛
2110 =
160( 1− )
3
1−3
2
n=5
46. Steve earned a 96% on his first math test, a 74% his second test, and
85% on 3 tests average. What is his third test?
A. 82%
B. 91%
SOLUTION:
0.96+0.74+𝑋
3
= 0.85
X = 0.85 * 100 = 85%
C. 87%
D. 85%
47. The base radius of a right circular cone is 4 m while its slant height is 10
m. What is the surface area?
A. 124.8 sq. m.
C. 226.8 sq. m.
B. 128.6 sq. m.
D.125.7 sq. m
SOLUTION:
Surface area = 𝜋 (4)(10) = 40𝜋 or 125.66 m2
48. Ian remodel a kitchen in 20 hrs and Jack in 15 hours. If they work together,
how many hours to remodel the kitchen?
A. 8.6
B. 7.5
C. 5.6
D. 12
SOLUTION:
1
20
1
+ 15 =
1
t
T = 8.6hrs
49. If 15% of the bolts produced by a machine will be defective, determine the
probability that out of 5 bolts chosen at random, at most 2 bolts will be
defective.
A. 0.9754 B. 0.9744
C. 0.9734
SOLUTION:
1 – 0.15 = 0.85
P (0) = 0.852 = 0.04437
P (1) = (5) (0.15) (0.85)4 = 0.3915
D. 0.9724
1
P(2) = (2) (5) (4)(0.15)2(0.85)3 = 0.138178
P (0 or 1 or 2)
= 0.9734
50. Find the average rate of the area of a square with respect to its side x as
x changes from 4 to 7.
A. 9
B. 3
C. 11
D. 18
51. The equations for two lines are 3y – 2x = 6 and 3x + ky = -7. For what
value of k will the two lines be parallel?
A. -9/2
SOLUTION:
B. 9/2
C. -7/3
D. 7/3
x2/y2= x1/y1
52.
-3/k= 2/3
k = -9/2 = A.
5pi/18 rad is how many deg?
A. 60
B. 50
SOLUTION:
C. 30
D. 90
5 180
𝜋
(
𝜋
) = 50 deg
53. Find the point of infection of the curve y = x^3 + 3x^2 – 1.
A. (-1, 1)
B. (-2, 3)
C. (0, -10)
D. (-3, -1)
SOLUTION:
Y1 = 3x2 + 6x
Y2 = 6x + 6
X = -1
y = (-1)3 + 3(1)2 -1
y=1
P (-1,1)
54. A fair coin is tossed three times. Find the probability that there will appear
three heads.
A. 1/4
B. 1/2
C. 1/8
D. 1/6
SOLUTION:
You have a fair coin: this means it has a 50% chance of landing heads up and
a 50% chance of landing tails up.
pH=pT=1/2
pHxpTxpH=1/2×1/2×1/2 = 1/8 = C.
1
1
𝟏
P3H = C(3,3) (2)3 (2)3-3 = 𝟖
55. A spherical balloon inflated with r = 3(cube root of t) as t is greater than
zero and t is less than equal or equal to 10. Find the rate of change of volume
in cubic cm at t = 8.
A. 37.70
B. 150.80
C. 113.10
SOLUTION:
r= 3 (t) 1/3 ; @ t=8:
r= 3 (8) 1/3 = 6
r’= 3 (1/3) t -2/3 ; @ t=8; r’= 8-2/3 = ¼
v= 4/3pi r3
v’= 4pi r2r’ = 4pi (6)2(1/4)
v’= 113.10 = C.
D. 75.40
56. Joe and his dad are bricklayers. Joe can lay bricks for a well in 5 days.
With his father’s help, he can build it in 2 days. How long would it take his
father to build it alone?
A. 3-1/4 days
B.3-1/3 days
C. 2-1/3 days
D.2 -2/3 days
SOLUTION:
1
1
2((5 + 𝑥)) = 1
x = 3.33 = 3 -
𝟏
𝟑
days
57. Find x so that the line containing (x, 5) and (3, -4) has a slope of 3.
A. 3
B. 4
C. 5
D. 6
SOLUTION:
3=
5−(−4)
𝑋−3
;x=6
58. Find the length of the chord of a circle of radius 20 cm subtended by a
central angle of 150 degrees.
A. 49 cm
B. 42 cm
C. 39 cm
D. 36 cm
SOLUTION:
COSINE LAW
C = √202 + 202 − 2 (20)(20)cos(15)
C = 38.64 or 39
59. Find the area of the ellipse 4x^2 + 9y^2 =36.
A. 15.71
B. 18.85
C. 21.99
D. 25.13
SOLUTION:
A = 2 and b = 3
A = 𝜋 (2) (3) = 18.85
60. Convert Cartesian coordinates (9, -9, 2) into cylindrical coordinates.
A. (-9sqrt. of 2, pi/4, 2)
B. (9sqrt. of 2, pi/4, 2)
C. (-9sqrt. of 2, 7pi/4, 2)
D. (9sqrt. of 2, 7pi/4, 2)
SOLUTION:
X = r = √92 + −92 = 9√𝟐
Y= tan-1 (
−9
9
𝟏
)=-𝟒𝝅
Z=2
Rectangular Coordinates: 9, -9, 2
r = sqrt(x2+y2)
r = sqrt((9)2+(-9)2)
r = 9 sqrt 2
Ɵ = tan-1 (y/x)
Ɵ = tan-1 (-9/9)
Ɵ = -45 = -45+360 = 315 degrees = 7pi/4 rad
z=2
Cylindrical Coordinates (9sqrt. of 2, 7pi/4, 2) = D.
61. The area of a square is 32 square feet. Find the perimeter of the square.
A. 27. 71 feet
B. 55. 43 feet
C. 45. 25 feet
D. 22.63 feet
SOLUTION:
√𝟑𝟐 = √𝒂𝟐
a = 4 √2
P = 4(4√2 ) = 22.63
62. If cos theta = -3/4 and tan theta is negative, the value of sin theta is
A. -4/5
B. – (sqrt. of 7)/4
C. (4 sqrt. of 7)/7 D. (sqrt. of 7)/4
SOLUTION:
3
𝜽 = cos-1 ( - 4 ) = 2.42
; sin𝜃 = sin (2.42) = 0.66 or
√7
4
63. What is the numerical coefficient of the term containing x^3y^2 in the
expansion of (x+2) ^5?
A. 10
B. 20
C. 40
D. 80
SOLUTION:
5c(x)(1)5-x (2)x = 5c(2)(1)3 (2)2 = 40
64.
Find the area bounded by y = 6x – x^2 and y = x^2 -4x.
A. 125/3
SOLUTION:
6x – x2 = x2 – 4x
B. 125/2
C. 100/3
5
D. 100/9
∫0 ( 𝑥 − 2𝑥 2 + 10𝑥 ) dx
X2 – 10x = 0
-
2 (5)3
3
+
10 (5)2
2
X = 0 and (x-5) =0
X=5
= 41.67 or
𝟏𝟐𝟓
𝟑
65. Find the second derivative of y = x ln x.
A. x
B. 1/x
C. 1
SOLUTION:
D. x squared
1
Y1 = x ( 𝑥 ) + ln x
𝟏
Y2 = 0 + 𝒙
66.
What is 30% of 293?
A. 87.9
B. 89.7
C.92.8
D. 98.2
SOLUTION:
(293) (0.30) = 87.9
67. The height (in feet) at any time t (in seconds) of a projectile thrown
vertically is h(t) = -16t^2 + 256t. What is the projectile’s average velocity for
the first 5 seconds of travel?
A. 48 fps
B. 96 fps
C. 176 fps
D. 192 fps
SOLUTION:
H(t) =
16 (5)2 +256 (5)
5
= 176 fps
68. Find the general solution of y” + 6y’ + 9y = x+ 1.
A. y = (C1x + C2x2) e-3x + 1/27 + x/9
C. y = (C1x + C2x2) e3x + 1/27 + x/9
B. y = (C1 + C2x) e-3x + 1/27 + x/9
D. y = (C1 + C2x) e3x + 1/27 + x/9
69. For a complex number z = 3 + j4 the modulus is
A. 3
B. 4
C. 5
D. 6
SOLUTION:
X = √𝑎2 + 𝑏 2 = √32 + 42 = 5
70.
A. 3
Evaluate lim
x →3
B. 0
sqrt.of (x2 −9)
2𝑥−6
C. infinity
D. Undefined
SOLUTION:
2𝑥
2√(𝑥 2 −9) (2)
=∞
71. The probability that a man, age 60, will survive to age 70 is 0.80 the
probability that a woman of the same age will live up to age 70 is 0.90. What
is the probability that only one of the survives?
A. 0.72
B. 0.26
C. 0.28
D. 0.0
72. Simplify 1(sec theta -1) + 1/ (sec theta + 1).
A. 2 sec theta tan theta
C. 2 sec theta
B. 2 csc theta cot theta
D. 2 csc theta
SOLUTION:
1
=
sec2 𝜃−1
2
𝑐𝑜𝑠𝜃
tan2 𝜃
2
= 𝑐𝑜𝑠𝜃 *
cos2 𝜃
sin2 𝜃
= 2csc𝜽 𝒄𝒐𝒕𝜽
73. Find the base of an isosceles triangle whose vertical angle is 65 degrees
and whose equal sides are 415 cm.
A. 530 cm
B. 464 cm
C. 350 cm
D. 446 cm
SOLUTION:
Cosine Law
B = (415)2 (415)2 -2(415) (415) cos65
B = 446
74.
Find the general solution of y” + 10y = 0.
A. y = C1 cos (sqrt. of 10x) + C2 sin (sqrt. of 10x)
B. y = C1 cos (sqrt. of 5x) + C2 sin (sqrt. of 5x)
C. y = C cos (sqrt. of 10x)
D. y = C sin (sqrt. of 10x)
75.
Evaluate the inverse Laplace transform of 6 over (s^2 + 4).
A. 3 sin 2t
C. 3 sinh 2t
SOLUTION:
B. 3 cos 2t
D. 3 cosh 2t
6
𝑠2 +4
6
=2∫
1
𝑏
𝑠2 +22
= 𝑠2 +𝑏2 = 3sin2t
76. Evaluate L {sin t cos t}
A. 1/2 (s^2 + 4)
C. 1/ (s^2 + 1)
B. 1/ (s^2 + 4)
D. 1/2 (s^2 + 1)
SOLUTION:
1
𝟏
L ( sint cost) =(𝑠2 +1 )2 = 𝒔𝟐 +𝟒
77. Determine the moment of inertia of the area enclosed by the curved x^2
+ y^2 = 36 with respect to the line y = 8.
A. 8628
B. 8256
C. 7642
D. 7864
78. A man sleeps on Monday, Tuesday, Wednesday, Thursday and Friday for
8, 6, 7, 4, and 5 hours, respectively. Find the number of hours he slept for 5
days.
A. 35
B. 31
C. 30
D. 25
SOLUTION:
8 + 6 + 7 + 4 + 5 = 30
79. Find A fir which y = Ae^x will satisfy y” - 2y’ = 4e^x.
A. -1
B. -2
C. -3
D. -4
SOLUTION:
Aex -2 (Aex ) – Aex = 4ex
Aex (1- 2- 1 ) = 4ex
A=-2
80. Simplify 1/csc2 theta.
A. sin2 theta
C. cot2 theta
1
Sin2𝜃 = csc2 𝜃 =
B. cos2 theta
D. tan2 theta
1
1
sin2 𝜃
81. Timothy leaves home for Legaspi City 400 miles away. After 2 hours, he
has to reduce his speed by 20 mph due to rain. If he takes 1 hour for lunch
and gas and reaches Legaspi City 9 hours after left home, what was his initial
speed?
A. 63 mph B. 62 mph
C. 65mph
D. 64 mph
82. How many arrangements of the letters in the word “VOLTAGE” begin with
a vowel and end with a consonant?
A. 1490
B.1440
C.1460
D.1450
SOLUTION:
3! (4!) (10) = 1440
83. An airplane flying with the wind, took 2 hours to travel 1000 km and 2.5
hours in flying back. What was the wind velocity in kph?
A. 50
B. 60
C. 70
D. 40
SOLUTION:
100
2
–x=
1000
2.5
+x
X = 50 mph
84. A woman is paid $ 20 for each day she works and the forfeits $ 5 for each
day she is idle. At the end of 25 days she nets $ 450. How many days did
she work?
A. 21
B. 22
C.23
D.24
SOLUTION:
P/day = $20 – 5 = $15
20x – 5 = 450
X = 22.75 or 23days
85. Find the centroid if the solid formed by revolving about x = 2 bounded by
y = x^3, X = 2 and y = 0.
A. (2, 10/30)
B. (2, 10/7)
C. (2, 10/9)
D. (2, 10)
86. What is the lowest common factor of 10 and 32?
A. 320
B. 2
C. 180
D. 90
87. The positive value of k which make 4x^2 – 4kx + 4k + 5 a perfect square
trinomial is
A. 6
B. 5
C. 4
D. 3
88. A tree is broken over by a windstorm. The tree was 90 feet high and the
top of the tree is 25 feet from the foot of the tree. What is the height of the
standing part of the tree?
A. 48.47 ft B. 41.53 ft
C. 45.69 ft
D. 44.31 ft
89. The Rotary Club and the Jaycee Club had a joint party. 120 members of
the Rotary Club and 100 members of the Jaycees Club also attended but 30
of those attended are members of both clubs. How many persons attended
the party?
A. 190
B. 220
C. 250
D. 150
SOLUTION:
120 -x + x + 100 – x = 30
X = 190
90. If sin 3A = cos 6B, then
A. A + B = 90 deg
B. A + 2B = 30 deg
C. A + B = 180 deg
D. A +2B = 60 deg
SOLUTION:
Cos6B = sin (30 – 6B)
Sin3A = Sin (90 – 6B)
3𝐴
3
=
90−6𝐵
3
A = 30 – 2B or A +2B = 30
91. MCM is equivalent to what number?
A. 1000
B. 2000
C. 1800
D.1900
SOLUTION:
M = 1000
C= 100
MCM = 1000 + (1000-100) = 1900
92. What is the discriminant of the equation 5x^2 – 6x + 1 = 0?
A. 12
B. 20
C. 16
D. 18
SOLUTION:
a=5
b = -6
c=1
D = (-6)2 – 4(5)(1) = 16
93. The number of ways can 3 nurses and 4 engineers be seated in a bench
with the nurses seated together is
A. 144
B.258
C. 720
D. 450
SOLUTION:
N = Total no. of ways
N = (3!)(4!)(No. of patterns)
N = (3!)(4!)(5)
N = 720 ways
94. Find the distance from the plane 2x + y – 2z + 8 = 0 to the point (-1, 2, 3).
A. 1/3
B. 2/3
C. 4/3
D. 5/3
SOLUTION:
D=
2(−1)+(2)−(2)(3)+8
√22 +12 +22
=
2
√9
=
𝟐
𝟑
95. Find the value of x if log x base 12 = 2.
A. 144
B. 414
C. 524
D. 425
SOLUTION:
Log12 x = 2
X = 122 = 144
96. If f(x) = x^3 – 2x – 1, then f (-2) =
A. -17
B. -13
C. -5
SOLUTION:
D. -1
X3 – 2x – 1 = 0
F (-2) = (-2)3 – 2(-2) -1 = - 5
97. A particle moves along a line with acceleration 2 + 6t at time t. When t =
0, its velocity equals 3 and it is at position s = 2. When t =1, it is at position s
=
A. 2
B. 5
C. 6
D. 7
SOLUTION:
@t = 0
A = 2 +6(0)
A=2
@t = 1
A = 2 + 6(1)
A=8
at = 10
S = 10 - 3 = 7
98. The edge of a cube has length 10 in., with a possible error of 1 %. The
possible error, in cubic inches, in the volume of cube is
A. 3
B. 1
C. 10
D. 30
SOLUTION:
v = s3
dv/ds = 3s2
dv/v = (3s2ds)/s3
= 30
99. What is the rate of change of the area if an equilateral triangle with respect
to its side s when s = 2?
A. 0.43
B. 0.50
C.10
D. 1.73
SOLUTION:
A=
1
4
s2 √3 ;
𝑑𝑎
=
𝑑𝑠
1
2
s √3
@s=2
𝑑𝑎
𝑑𝑠
=
1
2
(2)(√3 ) = √𝟑 or 1.73
100. If ∫ ˥ f(x)dx = 4 and ∫ ˥ g(x)dx = 2, find ∫ ˥ [3f(x) + 2g (x) + 1]dx.
A. 22
B. 23
C. 24
D. 25
SOLUTION:
7
7
7
∫1 𝑓(𝑥)𝑑𝑥 = 4 ∫1 𝑔(𝑥)𝑑𝑥 = 2 ∫1 (3𝑓(𝑥) + 2𝑔(𝑥) + 1)𝑑𝑥 = 4
= 3(4) + (2)(2) + (7-1) = 22
REGISTERED ELECTRICAL ENGINEERS
PRE-BOARD EXAMINATION
AUGUST 2016
MATHEMATICS
REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION
AUGUST 2012
MATHEMATICS
AUGUST 2012
QUESTIONS WITH SOLUTION:
1. The equation y^2 = cx is the general equation of:
A.
y’ = 2y/x
B. y’ = 2x/y
D. y’ = x/2y
C. y’ = y/2x
SOLUTION:
y 2 = cx
c=
y2
x
Differentiate:
0 = x(2yy ′ ) − y 2 )/x^2
y 2 = 2xyy′
y′ =
y2
2xy
= y/2x
2. A line segment joining two points on a circle is called:
A.
arc B. tangent
C. sector
D. chord
SOLUTION:
An arc is a portion of the circumference of a circle. A straight line
is drawn between the end points of the arc would be
a chord of the circle
A line which touches a circle or ellipse at just one point is called
tangent.
A sector is a "pie-slice" part of a circle - the area between two
radiuses and the connecting arc of a circle.
Any straight line segment connecting two points on a circle or ellipse is called
a chord.
3. Sand is pouring to form a conical pile such that its altitude is always twice its
radius. If the volume of a conical pile is increasing at the rate of 25 pi cu.ft/min,
how fast is the radius is increasing when the radius is 5 feet?
A. 0.5 ft/min
B. 0.5 pi ft/min
C. 5 ft/min
D. 5 pi ft/min
SOLUTION:
h = 2r , r = 5ft,
dv
dt
= 25π
ft³
min
Vcone =
dv
dt
dr
dt
1
1
2
πr²h = 3 πr 2 (2r) = 3 πr³
3
2
dr
dr
= 3 π (3)(π)(r 2 )(dt ) = 25π = 2π(5)² dt
25π
= 2π(25) = 𝟎. 𝟓 𝐟𝐭/𝐦𝐢𝐧
4. Evaluate ʃ ʃ 2r²sin Ө dr dӨ, 0 > r >sin Ө, > Ө > pi/2
A.
pi/2
B. pi/8
D. pi/48
C.
pi/24
SOLUTION:
π
sin θ
∫02 ∫0
π
sin θ
= ∫02 ∫0
π
2r² sin θ cos ²θ drdθ
2
2r² dr sin θ cos ²θ dθ
sin θ
= ∫02 3 r² ∫0
π
2
=∫0
2
2
3
sin θ cos ²θ dθ
(sin θ)³ sin θ cos ²θ dθ
π
= 3 ∫02 sin4 θ cos²θ dθ
2
(3)(1)(1) π
= 3 [(6)(4)(2)] 2 =
π
48
5. A shopkeeper offers a 25% discount on the marked price on an item. In order
to now cost $ 48, what should the marked price be?
A. $ 12
C. $ 60
B. $ 36
D. $ 64
SOLUTION:
48 = (1 − 0.25)X
48
x = 0.75 = $ 64
6. An observer wishes to determine the height of a tower. He takes sights at the
top of the tower from A to B, which are 50 ft. apart, at the same elevation on a
direct line with the tower. The vertical angle at point A is 30 degrees and at
point B is 40 degrees. What is the height of the tower?
A.
ft
85.60 ft
D. 92.54 ft
B. 143.97 ft
C. 110.29
SOLUTION:
β = 180 − 40 = 140°
50
α = 180 − 30 − 140 = 10°
x
x = sin θ = sin 30 ; x = 143.969621
h = 143.969621 sin(40) = 92.54 ft
7. A tangent to a conic is a line
A. which is parallel to the normal
B. which touches the conic at only one point
C. which passed inside the conic
D. all of the above
SOLUTION:
There exists a tangent at every point of a point conic. Further, the lines
corresponding to the common line of the projectivity determining a point conic
are tangents.
8. Find the area of the triangle which the line 2x – 3y + 6 = 0 forms with the
coordinate axes.
A.
3 B. 4
C. 5
D. 2
SOLUTION:
2x − 3(0) + 6 = 0
x=
−6
2
= −3
6
y=3=2
1
A = 2 (3)(2) = 3 sq. units
2(0) − 3y + 6 = 0
9. Find the general solution of (D² - D + 2)y = 0
A. y = e^x/2 (C1 sin sqrt. 7/2 x + C2 cos sqrt. 7/2 x)
B. y = e^x/2 (C1 sin sqrt. 7/2 x - C2 cos sqrt. 7/2 x)
C. y = e^x/2 (C1 cos sqrt. 7/2 x + C2 sin sqrt. 7/2 x)
D. y = e^x/2 (C1 cos sqrt. 7/2 x - C2 sin sqrt. 7/2 x)
SOLUTION:
(D2 − D + 2)y = 0
1
−7
7
m − 2 = √ 4 = √2 i
m² − m + 2 = 0
1 2
7
(m − 2) + 4 = 0
m=
1 √7
+
i
2
2
y = eAx (C1 cosBx + C2 sinBx)
𝒙
𝟕
∴ y = 𝐞𝟐 (C1 cos 𝟐 x + C2 sin sqrt.
𝟕
x)
10.
If 10 is subtracted from the opposite of a number, the
difference is 5. What is the number?
A.
5
B.15
C.-5 D.
-15
𝟐
SOLUTION:
x - 10 = 5
Opposite of x – 10 = 5
15 – 10 = 5
∴ −𝟓
If y = 5 – x, find x when y = 7
12
B.-12
11.
A.
C. 2 D. -2
SOLUTION:
y = 5 – x, find x when y = 7
7=5–x
x = -7 + 5 = −𝟐
12.
A ranch has a cattle and horses in a ratio of 9:5. If there
are 80 more head of cattle than horses, how many animals are on the ranch?
A.140
B. 168
C. 238
D. 280
SOLUTION:
Cattle → x
Horses → y
x
y
9
= 5 ; x = y + 80
y=
5x
Substitute:
5x
9
+ 80 = 180
y = 180 − 80 = 100
x + y = 180 + 100 = 𝟐𝟖𝟎
9
13.
Martin bought 3 pairs of shoes at P240 each pair and
3 pieces of t-shirts at P300 each. How much did he spent?
A. P720
B. P900
C. P22,500
D. P 1,620
SOLUTION:
3 pairs of shoes = P240
3 pcs. of t-shirt = P300
3(240) + 3(300) = 𝐏𝟏, 𝟔𝟐𝟎
14.
Find the standard equation of the circle with the center
at (1,3) and tangent to the line 5x – 12y -8 =0.
A. (x-1)2 + (y-3)2 = 8
C. (x-1)2 + (y-3)2 = 9
2
2
B. (x-1) + (y-3) = 12
D. (x-1)2 + (y-3)2 = 23
SOLUTION:
5x -12y – 8 = 0, center of the circle C (1,3)
d=r=
A(x)+B(y)+C
|√A²+B²|
=
5(1)−12(3)−8
|√5²+12²|
=3
(x – h)² + (y – k)² = r
(𝐱 − 𝟏)𝟐 + (𝐲 − 𝟑)𝟐 = 𝟗
15.
Find the volume of the solid formed by revolving the
area bounded by the curve y2 = (x3)(1-x) in the first quadrant about x-axis.
A.
0.137
B. 0.147
C.
0.157
D.0.167
SOLUTION:
y 2 = (x 3 )(x − 1)
LR = 4
y² = (x 3 − x 4 )
π ∫0 (x 3 − x 4 ) dx = 𝟎. 𝟏𝟓𝟕
1
a=1
16. In the pile of logs, each layer contains one more log than the layer above
and the top contains just one log. If there are 105 logs in the pile, how many
layers are there?
A. 11
B. 12
C. 13
D. 14
SOLUTION:
Sn =
n
[a + (n − 1)d]
2 1
a1 = 1
n=1
n
105 = 2 [2(1) + (n − 1)(1)]
∴ n = 𝟏𝟒 𝐥𝐚𝐲𝐞𝐫𝐬
a2 = 2
Sn = 105
17. A wall 8 feet high is 3.375 feet from a house. Find the shortest ladder that
will reach from the ground to the house when leaning over the wall.
A. 16.526 ft
B. 15.625 ft
C. 14.625 ft
D. 17.525 ft
SOLUTION:
H= 8ft
2
x= 3.375ft
2
2
L3 = h3 + x 3
2
2
2
L3 = 83 + 3.3753 ∴ L = 𝟏𝟓. 𝟔𝟐𝟓 𝐟𝐭
18. If f(x) = 10x + 1, then f(x+1) is equal to
A. 10(10x )
B. 9(10x)
C. 1
D. 9(10x+1)
SOLUTION:
if f(x) = 10x + 1, then f(x + 1) − f(x) =?
let x = 1
f(1) = 101 + 1 = 11
f(1 + 1) = 101+1 + 1 = 101
then f(1 + 1) − f(1) = 10 − 12 = 90
test from the choices, set x = 1
b = 9(101 ) = 90 ∴ 𝟗(𝟏𝟎𝐱 )
19. A particle moves on a straight line with a velocity v = (4 – 2t)3 at time t. Find
the distance traveled from t = 0 to t = 3.
A. 32
B. 36
C. 34
D. 30
SOLUTION:
t1 = 0
t2 = 3
v = (4 – 2t)3
V = dx/dt
dx = Vdt
3
∫ dx = ∫0 (4 − 2t)3 dt = 𝟑𝟎
20. The area enclosed by the ellipse 4x2 + 9y2 = 36 is revolved about the line x
= 3, what is the volume generated?
A. 370.3
B. 360.1
C. 355.3
D. 365.10
SOLUTION:
A = 4x² + 9y² = 36
[4x² + 9y² = 36]
x²
1
V = AC, A = πab, C = 2πR
36
V = π(3)(2)(2π)(3) = 𝟑𝟓𝟓. 𝟑𝟏
y²
+ 2² = 1
3²
21. If the vertex of y = 2x2 + 4x + 5 will be shifted 3 units to the left and 2 units
downward, what will be the new location of the vertex?
A. (-2, 1)
B. (-5, -1)
C. (-3,1)
D. (-4,1)
SOLUTION:
[y = 2x² + 4x + 5]
y
5
1
2
(x + 1)2 =
x² + 2x − 2 + 2 = 0
y
5
y
3
(x + 1)2 =
(x + 1)2 − + − 1 = 0
2
2
2
y 3
+
2 2
1
(y − 3)
2
C(−1,3)
∴ (−𝟒, 𝟏)𝐢𝐬 𝐭𝐡𝐞 𝐚𝐧𝐬𝐰𝐞𝐫
(x + 1) − + = 0
2
2
22. A coat of paint of thickness 0.01 inch is applied to the faces of a cube whose
edge is 10 inches, thereby producing a slightly larger cube. Estimate the
number of cubic inches of paint used.
A. 4
B. 6
C. 3
D. 5
SOLUTION:
V = s²
Vpoint = |Vold − Vnew |
Snew = 10 + (0.01x2) = 10.02
= 1006.01 − 1000 =
𝟔. 𝟎𝟏𝐢𝐧³ ≅ 𝟔
Vold = 10³ = 1000 in³
Vnew = 10.02³ = 1006.01 in³
23. Find the mass of lamina in the given region and density function:
π
D[(x, y)], 0 ≤ x ≤ 2 , 0 ≤ y ≤ cos x and ρ = 7x
A. 2
B. 3
C. 4
D. 5
24.
Find the area of the region bounded by the curves y = x2 – 4x and x + y
=0
A. 4.5
B. 5.5
C. 6
D. 5
SOLUTION:
x 2 − 4x = y ,
(x − 2)2 = y + 4
a=1
LR = 4
x+y=0,
y = −x
V(2, −4)
3
A = ∫0 (−x − x 2 + 4x)dx = 𝟒. 𝟓
25. A conic section whose eccentricity is less than one is known as:
A. circle
B. parabola
C. hyperbola
D. ellipse
26. The plate number of a vehicle consists of 5-alphanumeric sequence is
arranged such that the first 2 characters are alphabet and the remaining 3 are
digits. How many arrangements are possible if the first character is a vowel
and repetitions are not allowed?
A. 90
B. 900
C. 9,000
D.
90,000
SOLUTION:
Vowel = a , e , i , o , u = 5 ; =(5)(25)(10)(9)(8) = 𝟗𝟎, 𝟎𝟎𝟎
27. The axis of the hyperbola, which is parallel to its directrices, is known as:
A. conjugate axis
B. transverse axis
C. major axis
D.
minor axis
SOLUTION:
The transverse axis is the axis that crosses through both vertices and foci, and
the conjugate axis is perpendicular to it which is parallel to its directrices.
28. The minute hand of a clock is 8 units long. What is the distance traveled by
the tip of the minute hand in 75 minutes.
A. 10pi
B. 20pi
C. 25pi
D. 40pi
SOLUTION:
1 min =
6°
6°
π
75 min (1min) = 450° (180) =
s = rθ = 8 x
5π
2
5π
2
= 𝟐𝟎𝛑
29. Find k so that A = (3, -2) and B = (1, k) are perpendicular.
A. 2
B. 3
C. 1/2
D. 3/2
SOLUTION:
0−y1
mA = 0−x1 =
−1
0−(−2)
0−3
2
0−k
= −3
0−k
3
mB = 0−x2 = 0−1 = 2
3
mB = m = 2
−2k = −3 = k =
A
𝟑
𝟐
30. The probability of a defect of a collection of bolts is 5%. If a man picks 2
bolts, what is the probability that does not pick 2 defective bolts?
A. 0.950
B. 0.9975
C. 0.0025
D. 0.9025
SOLUTION:
1 − (0.05)(0.05) = 𝟎. 𝟗𝟗𝟕𝟓
1
31. If f(x) = x−2 ,(f·g)’*(1) = 6 and g’(1) = -1, then g(1) =
A. -7
B. -5
C. 5
D. 7
SOLUTION:
𝑓(𝑔(𝑥)) =
1
𝑔(𝑥) − 2
−1
=6
(𝑔(1) − 2)
1
− = 𝑔(1) − 2
6
𝑓(𝑥) =
(𝑓°𝑔)(𝑥) =
(𝑓°𝑔)′(𝑥) =
(𝑓°𝑔)′(𝑥) =
1
𝑥−2
1
𝑔(𝑥) − 2
−𝑔′(𝑥)
[𝑔(𝑥) − 2]2
−𝑔′ (𝑥)
= −6 + 1 = −5
𝑔2 (1) − 4𝑔(1) + 4
32. 3 randomly chosen senior high school students were administered a drug
test. Each student was evaluated as positive to the drug test (P) or negative to
the drug test (N). Assume the possible combinations of the 3 students drug
test evaluation as PPP, PPN, PNP, NPP, PNN, NPN, NNP, NNN. Assuming
each possible combination is equally likely, what is the probability that at least
1 student gets a negative result?
A. 1/8
B. 1/2
C. 7/8
D. ¼
SOLUTION:
no. s of N → 12
3 students
total outcomes → 24
12
∴ 24 = 0.5 possible
→ 1 − (0.5)(0.5)(0.5) =
𝟕/𝟖
33. The tangent line to the function h(x) at (6, -1) intercepts the y-axis at y = 4.
Find h’ (6).
A. -1/6
B. -2/3
C. -4/5
D. -5/6
SOLUTION:
y−y1
y - y1 = m ( x - x1 ) m = x−x1
m=
−1−4
6−0
= −𝟓/𝟔
34. The cable of a suspension bridge hangs in the form if a parabola when the
load is uniformly distributed horizontally. The distance between two towers is
150m, the points of the cable on the towers are 22 m above the roadway, and
the lowest point on the cables is 7 m above the roadway. Find the vertical
distance to the cable from a point in the roadway 15 m from the foot of a tower.
A. 16.6 m
B. 9.6 m
C. 12.8 m
D. 18.8 m
SOLUTION:
y = a2 + bx + c
22 = a(0)2 + b(0) + c → eq. 1
7 = a(75)2 + b(75) + c
a=
1
2
,b = −
375
5
∴ the parabola equation is ∶ y
1
2
=(
) x2 − x
375
5
→ eq. 2
22 = a(150)2 + b(150) + c
→ eq. 3
from eq. 1, c
= 22, substitute it from eq 2 and 3
5625a + 75b = −15 → eq. 2
22500a + 150b = 0 → eq. 3
+ 22
the point of the parabola is (15, y)
plugging x = 15
y=(
1
2
) (152 ) − (15) + 22
375
5
= 𝟏𝟔. 𝟔𝐦
Solving the equation gives the
value of:
35. In how many ways different orders may 5 persons be seated in a row?
A. 80
B. 100
C. 120
D. 160
SOLUTION:
5! = 5 x 4 x 3 x 2 x 1 = 𝟏𝟐𝟎
36. The symbol “/” used in division is called.
A. modulus
B. minus
C. solidus
D. obelus
37. Find the area of one loop r2 = 16 sin 2theta.
A. 16
B. 8
C. 4
D. 6
SOLUTION:
r² = 16 sin 2 θ
or
r² = a sin 2 θ
1
A2loops = a ;
A(one loop) =
π
= ∫02 16 sin 2θdθ = 𝟖
2
A = 16
16
2
=8
38. Find the centroid of the upper half of the circle x2 + y2 = 9.
A. (0, 3/pi)
B. (0, 4/𝐩𝐢)
C. (0, 5/pi)
D.(0, 6/pi)
SOLUTION:
x 2 + y 2 = 32 → r
h = 0, k = 0, r =3
4r
y = 3π (centroid)
y=
4(3)
3π
4
=π
𝟒
x = 0 ∴ (𝟎, 𝛑)
39. In polar coordinate system, the distance from a point to the pole is known
as
A. polar angle
C. radius vector
B. x-coordinate
D.y-coordinate
SOLUTION:
R is the radial distance or radius vector from the origin, and is the
counterclockwise angle from the x-axis.
40. The number that is subtracted in subtraction.
A. minuend
C. dividend
B. subtrahend
D. quotient
SOLUTION:
The first number in a subtraction. The number from which another number
(the Subtrahend) is to be subtracted. minuend − subtrahend = difference.
Subtrahend is a quantity or number to be subtracted from another.
41. In how many ways can a person choose 1 or more of a 4 electrical
appliances?
A. 12
B. 13
C. 14
D. 15
SOLUTION:
c = 2n − 1
= 24 − 1 = 𝟏𝟓 𝐰𝐚𝐲𝐬
42. The surface area of a spherical segment.
A. lune
B. Zone
C. Wedge
D. sector
SOLUTION:
A spherical segment is the solid defined by cutting a sphere with a pair
of parallel planes. It can be thought of as a spherical cap with the top truncated,
and so it corresponds to a spherical frustum. The surface of the spherical
segment (excluding the bases) is called a zone.
43. A particle has a position vector (2cos2t, 1+3sint). What is the speed of the
particle at time t = pi/4?
A. 1.879
B. 4.5
C. 5.427
D. 7.245
SOLUTION:
(2cos2t, 1 + 3sint)
dx =
v = √dx 2 + dy 2
dv
dv
(2cos2t)dy =
(1 + 3sint)
dt
dt
dx = −2sin(2)
dy = 3cost
dx = −4sin2t
t=
π
4
2
π
π 2
v = √(9 − 4sin ( )) + (3cos )
4
4
𝐯 = 𝟒. 𝟓𝟐𝟖
44. If the equation is unchanged by the substitution of –x for x, its curve is
symmetric with respect to the
A. y-axis
C. origin
B. x-axis
D. line 45 degrees with the axis
SOLUTION:
If an equation is unchanged by the substitution of −y for y, the curve is
symmetrical with respect to the X-axis.
If an equation is unchanged by the substitution of −x for x and −y for y, the curve
is symmetrical with respect to the origin.
45. Find the number of sides of a regular polygon if each interior angle
measures 108 degrees.
A. 7
B. 8
C.5
D. 6
SOLUTION:
(n−2)(180)
n
= 108
n= 𝟓
46. The integer part of common logarithm is called the________.
A. radicand
B. root
C. characteristic
D. mantissa
SOLUTION:
The whole number part of a logarithm and the decimal part have been given
separate names because each plays a special part in relation to the number
which the logarithm represents. The whole number part of a logarithm is called
the CHARACTERISTIC
47. The constant “e” is named in honor of:
A. Euler
B. Eigen
C. Euclid
D. Einstein
SOLUTION:
The number e is also known as Napier's constant, but Euler's choice of the
symbol e is said to have been retained in his honor. The constant was
discovered by the Swiss mathematician Jacob Bernoulli while studying
compound interest.
48. A man rows upstream and back in 12 hours. If the rate of the current is 1.5
kph and that of the man in still water is 4 kph, what was time spent
downstream?
A. 1.75 hrs
B. 2.75 hrs.
C. 3.75 hrs
D. 4.75 hrs
SOLUTION:
T = Tup + Tdown
Tdown =
C = 1.5kph, v = 4kph
Tdown =?
S = vt
Tup =
T=
S
S
+
= 20.625 km
2.5 5.5
Tdown =
S
S
=
V − C 2.5
S
S
=
V + C 5.5
20.625
= 𝟑. 𝟕𝟓 𝐡𝐫𝐬
5.5
49. The probability that A can solve a given problem is 4/5, that B can solve it
is 2/3, and that C can solve it is 3/7. If all three try, compute the probability that
the problem will be solved.
A. 101/105
B. 102/105
C. 103/105
D. 104/105
SOLUTION:
1 – P(a fails to solve) P(b fails to solve) P(c fails to solve)
4
2
3
= 1 − (5) (3) (7) =
𝟏𝟎𝟏
𝟏𝟎𝟓
50. A steel ball at 110 deg C cools in 8 min to 90 deg c in a room at 30 deg C.
Find the temperature of the ball after 20 minutes.
A. 58.97 °C
B. 68.97 °C
C. 78.97 °C
D.
88.97 °C
SOLUTION:
Tb0 = 110
t1
t2
=
Tb1 = 90
Tb1 −Tm
)
Tb0 −Tm
Tb2 −Tm
ln(
)
Tb0 −Tm
ln(
8
= 20 =
Tm = 30
t1 = 8
t 2 = 20
90−30
)
110−30
Tb2 −30
ln(
)
110−30
ln(
Tb2 = 𝟔𝟖. 𝟗𝟕℃
51. A freight train starts from Los Angeles and head for Chicago at 40 mph.
Two hours later passenger train leaves the same station for Chicago traveling
at 60 mph. How long will it be before the passenger train overtakes the freight
train?
A. 3 hrs
B. 4 hrs
C. 5 hrs
D. 6 hrs
SOLUTION:
S = vt
Vpt = 80 + 40(Vft )(t)
Sft = (40)(20) = 80 miles
60(t) = 80 + 40(t)
Spt = (80 + Sft )
T = 𝟒 𝐡𝐫𝐬
52. Given the triangle ABC in which A = 30 deg 30 min, b = 100 m and c = 200
m. Find the length of the side a.
A. 124.64 m
B. 142.24 m
C. 130.50 m
D.
103.00
SOLUTION:
By cosine law: a2 = b2 + c2 – 2(c)(b) cos θ
a = √200² + 100² − 2(200)(100) cos(30°30´)
a = 𝟏𝟐𝟒. 𝟔𝟒 𝐦
53. Lines that intersect in a point are called______.
A. Skew lines
B. Intersecting lines C. Agonic lines D. Coincident lines
SOLUTION:
The point where the lines intersect is called the point of intersection. If the
angles produced are all right angles, the lines are called perpendicular lines.
If two lines never intersect, they are called parallel lines.
54. Find the average rate of change of the area of a square with respect to its
side x as x changes from 4 to 7.
A. 14
B. 6
C. 17
D. 11
SOLUTION:
A = s2
da = 2s ds
da
ds
= 2s
Vave =
Vo+Vf
2
Vave =
[(2)(4)−(2)(7)]
2
= 𝟏𝟏
55. If the distance x from the point of departure at time t is defined by the
equation x = -16t2 + 5000t + 5000, what is the initial velocity
A. 20000
B. 5000
C. 0
D. 3000
SOLUTION:
x − 16t 2 + 5000t + 500
x´ = −32t + 5000, @t = 0
x´ = −32(0) + 500 = 𝟓𝟎𝟎𝟎
56. What conic section is represented by 2x2 + y2 – 8x + 4y = 16?
A. parabola
B. ellipse
C. hyperbola
D. circle
SOLUTION:
Circle.
When x and y are both squared and the coefficients on them are the same —
including the sign.
Parabola.
When either x or y is squared — not both.
Ellipse.
When x and y are both squared and the coefficients are positive but
different.
Hyperbola.
When x and y are both squared, and exactly one of the coefficients is negative
and exactly one of the coefficients is positive.
57. If 9 ounces of cereal will feed 2 adults or 3 children, then 90 ounces of
cereal, eaten at the same rate, will feed 8 adults and how many children?
A. 8
B. 12
C.15
D. 18
SOLUTION:
rate of children and adult
formulate an equation:
9oz
(8)(4.5) + (x)(3) = 90
2
= 4.5 oz/adult
9oz
3
𝐱 = 𝟏𝟖 𝐜𝐡𝐢𝐥𝐝𝐫𝐞𝐧
= 3oz/children
58. Mary is twice as old as Helen. If 8 is subtracted from Helen’s age and 4 is
added to Mary’s age, Mary will then be four times as old as Helen. How old is
Helen now?
A. 24
B. 36
C. 18
D. 16
SOLUTION:
Mary = x
Helen = y
X = 2(y)
If y – 8 and x + 4, then x = 4y
Find y.
(x + 4 ) = 4 (y – 8)
( x + 4 ) = 4y – 32
4y − x = 36 → eq. 1
x = 2y
Substitute:
4y − 2y = 36
36
y = 2 = 𝟏𝟖
59. A point on the curve where the second the derivative of a function is equal
to zero is called.
A. maxima B. minima
C. point of inflection
D. point of intersection
SOLUTION:
If the second derivative is positive at a point, the graph is concave up. If the
second derivative is negative at a critical point, then the critical point is a local
maximum. An inflection point marks the transition from concave up and
concave down. The second derivative will be zero at an inflection point.
60. Find the area of the triangle whose sides are 25, 39, and 40.
A. 46
B. 684
C. 486
D. 864
SOLUTION:
a = 25, b = 39, c = 40
A = √s(s − a)(s − b)(s − c)
s=
a+b+c
2
=
25+39+40
2
= 52
A = √52(52 − 25)(52 − 39)(52 − 40) = 𝟒𝟔𝟖 𝐬𝐪. 𝐮𝐧𝐢𝐭𝐬
61. A/An_______triangle is a triangle having three unequal sides.
A. oblique
B. scalene
C. equilateral
D. isosceles
62. Find the length of the arc of 6xy = x4 + 3 from x = 1 to x = 2.
A. 1.34
B. 1.63
C. 1.42
D. 1.78
SOLUTION:
y=
x4 +3
6x
vdu−udv
s=
v2
dy
=
dx
2
2
dy
s = ∫1 √1 + (dx)
[(6x)(4x3 )−(x4 +3)(6)]
36x2
[(24x
2
∫1 √1 + (
4 −(6)(x4 +3)]
36x2
2
) dx
𝐬 = 𝟏. 𝟒𝟐
63. Give the degree measure of angle 3pi/5 radians.
A. 108
B. 120
C. 105
SOLUTION:
3π
rad
5
D. 136
180°
= π rad = 𝟏𝟎𝟖°
64. What do you call a radical expressing an irrational number?
A. surd
B. radix
C. complex number
D. index
SOLUTION:
Surd is a radical that is not evaluated, or cannot be precisely evaluated. The
radicand is often a constant, such as the square root of two.
65. Find the derivative of the function f(x) = (2x – 3x)2.
A. 2x - 4
B. 2x - 3
C. 6x - 8
SOLUTION:
D. 8x -12
f(x) = (2x − 3)²
x´ = 2(2x − 3)(2)
= 4(2x − 3) = 𝟖𝐱 − 𝟏𝟐
66. What is the length of the line with a slope of 4/3 from a point (6, 4) to the yaxis?
A. 10
B. 25
C. 50
D. 75
SOLUTION:
4
y−4
m = 3 = 0−6
y = −4
d = √(−4 − 4)2 + (0 − 6)²
d = 𝟏𝟎
67. The inclination of the line determine by the points (4, 0) and (5√3) is
A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 90 degrees
SOLUTION:
P = (4,0) and P(5, √3)
θ = tan−1 m
=m=
√3−0
5−4
= √3
θ = tan−1 (√3) = 𝟔𝟎°
68. A sequence of numbers where the succeeding term is greater than the
preceding term is called:
A. dissonant resonance
C. Isometric series
B. convergent series
D.divergent series
SOLUTION:
Divergent series is an infinite series that is not convergent, meaning that the
infinite sequence of the partial sums of the series does not have a finite limit.
A series is convergent if the sequence of its partial sums tends to a limit; that
means that the partial sums become closer and closer to a given number when
the number of their terms increases.
69. Find the value of x for which y = 4 + 3x – 3x3 will have a maximum value.
A. 0
B. -3
C. -2
D. 1
SOLUTION:
dy
dx
3
x = √3 = 𝟏
= 4 + 3x − x³
= 3 − 3x 2 = 0
70. How many cubic meters is 500 gallons of liquid?
A. 4.8927
B. 3.0927
C. 2.8927
SOLUTION:
1 gal = 3.78 li ∶ 500 gal x
3.785li
1gal
x
1m³
10³li
D. 1.8927
= 𝟏. 𝟖𝟗𝟐𝟓 ≈ 𝟏. 𝟖𝟗𝟐𝟕 𝐦³
71. A certain radioactive substance has a half-life of 3 years. If 10 grams are
present initially, how much of the substance remains after 9 years?
A. 1.50 grams
B. 1.25 grams
C. 2.50 grams
D. 1.75 grams
SOLUTION:
t1 = 3 t 2 = 9
q1 = 0.5 Q0
Q0 = 100
t1
t2
=
q
ln 1
Q0
q
ln 2
Q0
3
∴9=
(0.5Q0 )
Q0
q
ln 2
100
ln
= q 2 = 𝟏. 𝟐𝟓 𝐬𝐪. 𝐮𝐧𝐢𝐭𝐬
72. A statement of the truth of which is admitted without proof is called:
A. an axiom
B. a postulate
C. a theorem
D. a corollary
SOLUTION:
An axiom or postulate is a statement that is taken to be true, to serve as a
premise or starting point for further reasoning and arguments. The term has
subtle differences in definition when used in the context of different fields of
study. As defined in classic philosophy, an axiom is a statement that is so
evident or well-established, that it is accepted without controversy or
question. As used in modern logic, an axiom is simply a premise or starting
point for reasoning.
73. A rectangular trough is 8 feet long, 2 feet across the top and 4 feet deep. If
water flows in at a rate of 2 ft3/min, how fast is the surface rising when the
water is 1 ft deep?
A. ¼ ft/min
B. ½ ft.min
C. 1/8 ft/min
D. 1/6 ft/min
SOLUTION:
V = LWH
dv
dt
2
= H′ =
16
𝟏
𝟖
𝐟𝐭/𝐦𝐢𝐧
= (8)(2)(4)H′
2 = (8)(2)(4)H′
74. Find the point(s) on the graph of y = x2 at which the tangent line is parallel
to the line y = 6x -1.
A. (3, 17)
B. (3, 9)
C. (1, 2)
D. (2, 4)
SOLUTION:
y1 ´ = 2x
y2 ´ = 6
since tangent, the M or slope are equal
y1 ´ = y2 ´
2x = 6
y=3
y = x² ; y = 3² = 9
= 𝐏(𝟑, 𝟗)
75. How many petals are there in the rose curve r = 3 cos 5theta?
A. 5
B. 10
C. 15
D. 6
SOLUTION:
r = cos5θ
↓
odd ∴ 𝐧 = 𝟓
r = cos(kθ)
Because when k is odd and θ = pi (halfway from 0 to 2pi), cos(k) = -1,
which put the graph back at the same point as r = 1. [Note that the polar
coordinate (1,0) = (-1,pi)]. The rest of the graph (from pi to 2pi) just repeats
itself.
However, when k is even, the cos(kθ) = 1when θ = pi, which is on the oppositve
side of the origin. The rest of the graph (from pi to 2pi) follows the same
pattern but mirrored, created an entire different set of loops, resulting it twice
as many as before.
76. Find the acute angle between the vectors z1 = 3 – 4i and z2 = -4 + 3i.
A. 17 deg 17 min
C. 15 deg 15 min
B. 16 deg 16 min
D. 18 deg 18 min
SOLUTION:
Z1 = 3 − 4i = 5∠ − 53.13
= 143.13 + 53.13 = 196.26
Z2 = −4 + 3i = 5∠143.13
θ = 196.26 − 180
ZT = Z2 − Z1
(5∠143.13) − (5∠ − 53.13)
θ = 𝟏𝟔. 𝟐𝟔 = 𝟏𝟔°𝟏𝟔′
77. If z1 =1 – i and z2 = -2 + 4i evaluate z12 + 2z1 – 3.
A. -1 + 4i
B. 1 - 4i
C. -1 – 4i
SOLUTION:
D. 1 + 4i
z1 = 1 − i → √2 < −45
solve for Z1 ² + 2Z1 − 3
√2 < −45)² + 2√2 ∠ − 45) − 3 = −𝟏 − 𝟒𝐢
78. A motorboat moves in the direction N 40 deg E for 3 hours at 20 mph. How
far north does it travel?
A. 58 mi
B. 60 mi
C. 46 mi
D. 32 mi
SOLUTION:
3hrs @ 20mph
S1 = vt = (20)(3) = 60 miles
S2 = 60 cos 40 = 𝟒𝟓. 𝟗𝟔 ≈ 𝟒𝟔 𝐦𝐢𝐥𝐞𝐬
79. Find the value of 4 sinh(pi i/3)
A. 2i (sqrt. of 3)
B. 4i (sqrt. of 3)
SOLUTION:
C. i (sqrt. of 3)
D. 3i (sqrt. of 3)
= Sinh (θi) = i sin θ
πi
= 4sinh( 3 )
π
= i4 sin ( 3 ) = 𝟐√𝟑𝐢
80. Find the upper quartile in the set (0, 1, 3, 4)
A. 0.5
B. 0.25
SOLUTION:
C. 2
D. 3.5
𝟎 , 𝟏, 𝟑 , 𝟒
𝟎.̌𝟓 𝟏.̌𝟓 𝟑.̌𝟓 → 𝐮𝐩𝐩𝐞𝐫 𝐪𝐮𝐚𝐫𝐭𝐢𝐥𝐞 = 𝟑. 𝟓
81. In debate on two issues among 32 people, 16 agreed with the first issue, 10
agreed with the second issue and of these 7 agreed with both. What is the
probability of selecting a person at random who did not agree with either issue?
A. 1/32
B. 13/32
C. 3/8
D. 3/10
SOLUTION:
32 people
1st issue → (16 Agreed), (7 agreed)
2nd issue → (10 Agreed), (Both)1st issue = 16 − 7 = 9
both = 7
2nd issue = 10 − 7 = 3
19 agrees
32 − 19 = 13 disagreed
∴
𝟏𝟑
𝟑𝟐
82. From the top of the lighthouse, 120 m above the sea, the angle of
depression of a boat is 15 degrees. How far is the boat from the lighthouse?
A. 448 m
B. 428 m
C. 458 m
D. 498 m
SOLUTION:
Tan θ =
opposite
adjacent
θ = 15°
120
x = tan15 = 𝟒𝟒𝟕. 𝟖𝟓 ≈ 𝟒𝟒𝟖𝐦
83. The cross section of a certain trough is inverted isosceles triangles with
height 6 ft and base 4 ft. Suppose the trough contains water to a depth of 3 ft.
Find the total fluid force on one end.
A. 187.2 lb
B. 178.2 lb
C. 192.4 lb
D. 129.4 lb
SOLUTION:
y= 6ft b = 4ft
h= 3ft
62.4 (specific weight of water in pounds per cubic foot)
γ=
F = γh
F = [62.4
lb
ft3
][3ft]
F = 𝟏𝟖𝟕. 𝟐 𝐥𝐛/𝐟𝐭 𝟐
84. Two lines are not coplanar.
A. Parallel lines
B. Skew lines C. Secant lines
D. Straight lines
SOLUTION:
Two lines are parallel lines if they are coplanar and do not intersect. Lines that
are not coplanar and do not intersect are called skew lines. Two planes that
do not intersect are called parallel planes.
2
85. Find the inverse Laplace transform of − s−3.
A. 2 e-3t
B. 2e3t
C. 3e-2t
D. 3e2t
SOLUTION:
2
Inverse Laplace of {s−3}
1
1
= 2 [s−3] = e±at = s∓a
2
Inverse laplace of {s−3} = 2e3t
86. Find the length of the latus rectum of the curve rcos 2 theta – 4cos theta =
16sin theta.
A. 4
B. 16
C. 12
D. 18
SOLUTION:
[rcos²θ − 4cosθ = 16sinθ]
rcos²θ = 16sinθ + 4cosθ
1
rcos²θ = 16sinθ + 4 cosθ
↓
4a → LR ∴ LR = 16
87. A quadrilateral with no pair of parallel sides.
A. Trapezoid
B. Trapezium
C. Rhombus
SOLUTION:
D. Rhomboid
A trapezoid is a 4-sided flat shape with straight sides that has a pair of
opposite sides parallel.
A rhombus looks like a diamond. All sides have equal length. Opposite sides
are parallel, and opposite angles are equal.
A rhomboid is a parallelogram in which adjacent sides are of unequal lengths
and angles are non-right angled.
A trapezium is defined by the properties it does not have. It has no parallel sides.
Any quadrilateral drawn at random would probably be a trapezium.
88. Find the equation of the line tangent to the curve y = x3 – 6x2 + 5x + 2 at its
point of inflection.
A. 7x – y
B. -7x + y = 0
C. 7x +y = 10
D. -7x – y = 10
SOLUTION:
y = x³ − 6x 2 + 5x + 2
y = −4
y ′ = 3x² − 12x + 5
3(2)2 − 12(2) + 5 = −7 → m
; x=0
y" = 6x − 12 = 0
y − y1 −= m(x − x1
y = (2)3 − 6(2)2 + 5(2) +
y + 4 = −7(x − 2)
2
y + 4 = −7x + 14
P. O. I. (2, −4)
y ′ = 3x 2 − 12x + 5 = m
7x + y = 10
89. Find the area of the polygon with vertices at 2 + 3i, 3 + i, -2 – 4i, -1 + 2i.
A. 47/5
B. 47/2
C. 45/2
D.45/4
SOLUTION:
1
2
1
2
1
2
1
(3.16)(3.61)sin(37.28) + (3.61)(2.24)sin(60.26) +
2
1
(2.24)(4.12)sin(77.47) + (4.12)(4.47)sin(49.39) +
2
(4.47)(3.16)sin(116.5718.43) =
𝟒𝟕
𝟐
𝐨𝐫 𝟐𝟑. 𝟓𝟎 𝐬𝐪 𝐮𝐧𝐢𝐭𝐬
90. Find the radius of curvature of y = x3 at x =1.
A. 5.27
B. 4.27
C. 6.27
SOLUTION:
R =? y = x3 @ x = 1
R=
[1+(y′)²]3/2
y"
y ′ = 3x² = 3(1)2 = 3
y" = 6x = 6(1) = 6
D. 7.27
[1 + (3)²]3/2
R=
= 5.27
6
91. Determine the probability of throwing a total of 8 in a single throw with two
dice, each of whose faces is numbered from 1 to 6.
A. 1/3
B. 1/18
C. 5/36
D. 2/9
SOLUTION:
Let E5 = event of getting a sum of 8. The number which is a sum of 8 will be
E5 = [(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)] = 5
Therefore, probability of getting a sum of 8
Number of favorable outcomes
P=Total number of possible outcome
=
𝟓
𝟑𝟔
92. Find the distance between the point (3, 2, -1) and the plane 7x – 6y + 6z +
8 = 0.
A. 1
B. 2
C. 3
D. 4
SOLUTION:
d=
A(x)+B(y)+C(z)+D
√x²+y²+z²
= 1
7(3)−6(2)+6(−1)+8
d=
1√7²+6²+6²
= 1
93. How many parallelograms are formed by a set of 4 parallel lines intersecting
another set of 7 parallel lines?
A. 123
B. 124
C. 125
D. 126
SOLUTION:
𝐦(𝐦−𝟏)𝐧(𝐧−𝟏)
𝟒
[𝟕(𝟕−𝟏)(𝟒)(𝟒−𝟏)]
𝟒
= 𝟏𝟐𝟔
94. The graphical representation of the cumulative frequency distribution in a
set of statistical data is called:
A. Ogive
B. Histogram C. Frequency polyhedron D. mass diagram
SOLUTION:
Cumulative Frequency is an important tool in Statistics to tabulate data in an
organized manner. A curve that represents the cumulative frequency
distribution of grouped data on a graph is called a Cumulative Frequency Curve
or an Ogive. Representing cumulative frequency data on a graph is the most
efficient way to understand the data and derive results.
95. Find the area bounded by the curve defined by the equation x2 = 8y and its
latus rectum.
A. 11/3
SOLUTION:
B. 32/3
x² = 8y
8
a = 4 = 2, LR = 8
4
x²
A = ∫−4 (2 − 8 ) dx
A=
𝟑𝟐
𝟑
𝐬𝐪. 𝐮𝐧𝐢𝐭𝐬
C. 16/3
D. 22/3
96.
Evaluate lim (i z 4 + 3z² − 10i)
z→2i
A. -12 +6i
SOLUTION:
B. 12 - 6i
C. 12 +6i
D. -12 – 6i
C. 10
D. 2.71828
lim (i z 4 + 3z² − 10i)
z→2i
= i(2i)4 + 3(2i)2 − 10i
= i(24 i4 ) + 3(22 i2 ) − 10i
= 16i − 12 − 10i = −𝟏𝟐 + 𝟏𝟔𝐢
97. Naperian logarithm have a base of
A. 3.1416
B. 2.171828
SOLUTION:
The number e frequently occurs in mathematics (especially calculus) and is an
irrational constant (like π). Its value is e = 2.718 281 828 ...
𝐞 = 𝟐. 𝟕𝟏𝟖𝟐𝟖
98. If an aviator flies around the world at a distance 2km above the equator,
how many more km will he travel than a person who travels along the equator?
A. 12.6 km
B. 16.2 km
C. 15.8 km
D. 18.5 km
SOLUTION:
1 rev = 2π
(2km)(2π) = 4π = 𝟏𝟐. 𝟓𝟔𝟔 𝐨𝐫 𝟏𝟐. 𝟔 𝐤𝐦
99. Find the volume of a spherical whose central angle is pi/5 radians on a
sphere of radius 6 cm.
A. 90.48 cu. cm B. 86.40 cu. cm C. 78.46 cu. cm
D. 62.48 cu. cm
SOLUTION:
θ=
V=
π
5
rad , r = 6 cm
πr³θ
270
Vwedge =
π 180
)
5 π
π(6)3 ∙( x
270
= 𝟗𝟎. 𝟒𝟖 𝐜𝐮. 𝐜𝐦
100. What is the coefficient of the (x -1)3 term in the Taylor series expansion of
f(x) = lnx expanded about x = 1?
A. 1/6
B. 1/4
C. 1/3
D. 1/2
SOLUTION:
T(x) = ∑∞
𝑘=0
𝑓 (𝑘) (1)
𝑘!
(𝑥 − 1)𝑘

f(1) = 0

𝑓 ′ (1) = 1
1

𝑓 ′′ (1) = -1
2

𝑓 ′′′ (1) = 2
f(x) = lnx
1
𝑓 ′ (x) = 𝑥
𝑓 ′′ (x) =- 𝑥 2
𝑓 ′′′ (x) = 𝑥 3
1
1
(𝑥 − 1) − (𝑥 − 1)2 + (𝑥 − 1)3
2
3
𝟏
Ans. 𝟑