Content
Part
School Book Exams
Algebra exam 1 : Page 2
1
Part
Algebra exam 2 : Page 3
Calculus exam 1 : Page 4
Calculus exam 2 : Page 5
Practice Exams
Practice exam 1 : Page 6
2
Practice exam 2 : Page 9
Practice exam 3 : Page 12
Practice exam 4 : Page 15
Practice exam 5 : Page 18
Practice exam 6 : Page 21
Practice exam 7 : Page 24
Practice exam 8 : Page 27
Practice exam 9 : Page 30
Practice exam 10 : Page 33
Algebra – School book Exam 1
ANSWER THE FOLLOWING QUESTIONS:
(1) Choose the correct answer: 1) If 𝑛 + 1 = 30 𝑛 − 1, then the value of 𝑛 is ……………
a) 5
b) 6
c) 29
d) 30
2
2) The value of the series: ∑15
𝑟=1(𝑟 + 𝑟 + 1) is …………..
a) 1375
b) 3720
c) 14400
d) 2232000
3) The number of the terms of the arithmetic sequence (7,11,15, … ,271) is …………
a) 34
b) 67
c) 169
d) 9313
4) If 𝑥 > 0, then the common ratio of the geometric sequence (4, 𝑥 − 3,2 𝑥 + 6, … ) is …………
a) 1
b) 5
c) 3
d) 24
(2) (a) if 5𝑃𝑟 = 2 ×6𝑃𝑟−1 , find the value of: 𝑟
(b) find the order of the first negative term of the terms of the sequence( 152 − 9 𝑛), then find
the greatest sum can be got from the terms of this sequence.
(3) (a) How many different three – digit even numbers can be formed from the set of the numbers
{2,3,4,5,7}?
(b) Find the geometric sequence whose terms are positive, the sum of the first three terms
equals 14 and its first term is greater than its second term by 4, then find the sum of infinite
number of its terms starting from its first term
(4) (a) find the geometric sequence whose sum of its first three terms equals
27
171
32
and its second
term equals 16, then find its tenth term.
(b) A 25 – row theatre; the first row contains 20 seats, the second row contains 22 seats and the
third row contains 24 seats and so on … Find the number of seats in all the theatre rows.
(5) (a) if 2 n+m 𝐶2 = 190, n-2m 𝑃3 = 60,find the value of: 𝑛 and 𝑚.
(b) An arithmetic sequence whose sum of its first and last terms is 26 and the sum of its terms
is 468, find the number of its terms. If its tenth term equals 47, find this sequence.
Algebra – School book Exam 2
ANSWER THE FOLLOWING QUESTIONS:
(1) choose the correct answer:
1) the number of ordered pairs (𝑎, 𝑏) which can be formed from the elements of the set {1,2,3}
where 𝑎 ≠ 𝑏 is …………..
a) 2
b) 3
c) 6
d) 9
8
2) The nth term of the sequence (2,2, 3 , 4, … ) is ………….
a) 𝑛 − 1
b) 2𝑛 − 1
c) 2𝑛−1
3) The sum of the first 25 terms of the sequence (3 − 2 𝑛) is ………………
a) 650
b) 600
c) – 575
4) If (𝑥, 𝑦, 𝑧, … ) is a geometric sequence, then ………..
a) 2𝑦 < 𝑥 + 𝑧
b) 𝑦 2 > 𝑥 𝑧
c) 𝑦 = 𝑥 𝑧
d)
2𝑛
𝑛
d) – 600
d) √𝑦 = 𝑥 𝑧
2) (a) if 25𝐶2𝑟+1 =25𝐶3𝑟−1 , find the value of: r
(b) Find the number of terms that must be taken from the terms of the sequence
(−43, −36, −29, … ) starting from its first term to get a sum of 221.
3) (a) A university student learns different eight subjects and he cannot join the next grade till
he succeeds in six subjects at least. How many ways can the student join the next grade?
(b) A geometric sequence in which the sum of an infinite number of its terms starting from
its first term equals 108 and its first term is greater than its second term by 12 Find the sequence
and the sum of its first seven terms
4) (a) Find the sum of the odd ordered terms of the arithmetic sequence (2 ,5 , 8 . . . , 110)
(b) An agricultural crops storing company has seven warehouses to store the wheat so that
the first warehouse holds 270 tons and each warehouse after that can hold two third of the
amount of the directly previous warehouse. Can the company store 800 tons of wheat? What is
the greatest amount of wheat the company can store in its warehouses to the nearest ton?
5) (a) if 3𝑛 − 7 = 120, find the value of n𝐶𝑛−1
(b) Insert 28 arithmetic means between 4 and 91, then find the sum of the terms of the
arithmetic sequence resulted.
Calculus – School book Exam 1
ANSWER THE FOLLOWING QUESTIONS:
(1) choose the correct answer:
𝑑𝑦
1) If 𝑦 = sin 2𝑥, then 𝑑 𝑥 when 𝑥 =
a) 2
𝜋
6
equals ………….
1
b) 1
c) 2
d) √3
2
2) If cos θ= 3 , then cos 2 𝜃 = ⋯
4
a) 9
3
b) 2
3) ∫(2 𝑥 + 3)4 d 𝑥 = ⋯
1
a) 5 (2 𝑥 + 3)5 + 𝐶
c)
1
b) 10 (2 𝑥 + 3)5 + 𝐶
−1
d) √3
9
1
c) 10 (2 𝑥 + 3)3 + 𝐶
d) 10(2 𝑥 + 3)3 + 𝐶
4) The average rate of change of the function 𝑓 where 𝑓(𝑥) = 𝑥 2 when 𝑥 varies from 3 to 3.1
equals ……
a) 0.61
b) 6.1
c) 9
d) 9.61
(2) (a) Find the first derivative if: 𝑦 = 𝑥 2 𝑠𝑖𝑛 2 𝑥
𝑠𝑖𝑛 2 𝑥
(b) prove that: 1+𝑐𝑜𝑠 2 𝑥 = 𝑡𝑎𝑛 𝑥
(3) (a) find the slope of the tangent to the curve of the function: 𝑓: 𝑓(𝑥) =
(b) find (1) ∫(𝑥 2 + 2 𝑥)𝑑 𝑥
𝑥 2 +3
𝑥−2
,= 1
(2) ∫(𝑠𝑖𝑛 𝑥 − 𝑐𝑜𝑠 𝑥)2 𝑑 𝑥
1
4) (a) Find the point(s) lying on the curve of the function: 𝑦 = 𝑥−3 at which the tangent is parallel
to the straight line 𝑥 + 𝑦 = zero
(b) From a house top of 25 meters high, the measurement of the angle of elevation of a tower
top was 70° and the measurement of the angle of depression of the lower base was 30°, find the
height of the tower known that the bases of the house and tower are at the same horizontal level.
(5) (a) if the function 𝑓 where 𝑓(𝑥) = {
𝑥 2 − 2 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑥 ≤ 2
is differentiable when 𝑥 = 2,
2 𝑎 𝑥 − 3 𝑏 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑥 > 2
find the values of 𝑎 and 𝑏.
d𝑦
(b) Find d 𝑥 if: 𝑦 = (z3 − z2 ), z = 2 𝑥 + 1 , when 𝑥 = −1
Calculus – School book Exam 2
ANSWER THE FOLLOWING QUESTIONS:
(1) choose the correct answer:
1) The slope of the tangent to the curve of the function 𝑓 where 𝑓(𝑥) = 3 𝑥 2 + 2 𝑥 − 1 when 𝑥 = 2
equals …..
a) 4
b) 8
c) 17
d) 14
2) sin A cos B − cos A sin B=…
a) sin (A+B)
b) cos (A+B)
3) ∫
𝑥 2 +3 𝑥
𝑥
d
d𝑥
d) cos (A-B)
𝑑𝑥=⋯
1
b) 2 𝑥 2 + 3 𝑥 + 𝐶
a) 𝑥 + 3
4)
c) sin (A-B)
c) 𝑥 2 + 3 𝑥 + 𝐶
d)
𝑥 3 +3 𝑥 2
𝑥2
(sin 𝑥 cos 𝑥) = ⋯
a) sin 𝑥
b) cos 𝑥
1
c) 2 cos 2 𝑥
d) cos 2 𝑥
2) (a) if 𝑦 = 𝑓(𝑥) where 𝑦 = 𝑥 2 − 𝑎 𝑥, find the slope of the tangent to the curve of the function 𝑓
at the point (3,0) lying on it.
5
(b) if sec A = 4 and csc B =
13
5
, where 𝐴 and 𝐵 are the measurements of two acute angles, find
𝑠𝑒𝑐 (𝐴 − 𝐵)
3) (a) discuss the differentiability of the function 𝑓 where: 𝑓(𝑥) = {
𝑥2
,𝑥 > 2
4 𝑥 − 1, 𝑥 ≤ 2
when 𝑥 = 2
(b) find ∫(1 − 𝑐𝑜𝑠 𝑥)2 𝑑 𝑥
4) (a) A ship sailed from a certain point in the direction of 60° North of the west at velocity 26
km./hr and at the same time and place, another ship sailed in the direction of the east at velocity
15 km./hr. Find the distance between two ships after 3 hours.
(b) if
d𝑦
𝑦 = 𝑧 5 + 3 , 𝑧 = (𝑥 − 1)3 , find the value of: d 𝑥 when 𝑥 = 2
𝑥 2 +1
5
d𝑦
5) (a) if 𝑦 = ( 𝑥−3 ) ,find: d 𝑥 when 𝑥 = 1
(b) find the tangent equation to the curve of: 𝑦 = 2 𝑥 sin 𝑥 cos 𝑥 when 𝑥 = 𝜋
Practice Exam 1
1) How many three different digit numbers could be formed from the set of digits {1 , 3 , 6 , 7} ?
a) 9
b) 12
c) 64
d) 24
1
2) The tangent equation of the curve: 𝑦 = 𝑥 2 at the point which lies on the curve and its
𝑥 −coordinate = −1 is …….
a) 𝑦 = 2𝑥 − 3
b) 𝑦 = 2𝑥 + 3
c) 𝑦 = 3𝑥 + 2
3) 1 − 2 sin2 35° = …………
a) sin 70°
b) cos 70°
4)
17 𝑃𝑟
17 𝑃 𝑟−1
d) 𝑥 = 2𝑦 + 3
c) cos 35°
d) otherwise
= …….
a) 𝑟
b) 𝑟 − 1
c) 7 − 𝑟
d) 8 − 𝑟
5) If (0 , 1 , 2 , 3 , … … . ) Is an arithmetic sequence where Sm is the sum of the first 𝑚 of the terms
with odd orders and 𝑆𝑛 is the same of the first 𝑛 of the terms with even orders then
a)
m2 −1
b)
n2
m2 −m
n2
c)
m(m+1)
n2 −1
d)
m2 −m
Sm
Sn
= ……..
n2 +n
6) A university students studies eight different subjects and cannot be promoted to the next grade
till he succeeds in six subjects at least by how many ways can the student promoted to the next
grade ?
a) 17 𝐶3
b) 37
c) 28
d) 8
7) The number of the different ways can 4 students sit on 4 seats in form of a row equals …….
a) 4 + 4
b) 4 × 4
c) 4 × 3 × 2 × 1
d) 1
8)
d
d𝑥
(3𝑥 2 + 5𝑥) = …… at 𝑥 = −1
a) −1
b) −3
c) −6
d) 3
9) If the geometric mean of the two numbers 𝑎 , 𝑏 equals 4 and the arithmetic mean of the two
1
1
1
numbers 𝑎 , 𝑏 equals 4 , then 𝑎 + 𝑏 = …….
a) 32.5
b) 8
c) 10
d) 16
10) From a point on the ground , a person observed the top of a tower to be at an angle of elevation
of measure 32° , then this person walked 50 𝑚. away from the tower base. He found that the
top of the tower had an angle of elevation 27° , then the height of the tower ≃ ……𝑚
a) 260
b) 220
c) 180
d) 138
11) In the arithmetic sequence (𝑇𝑛 ) = (32 , 28 , 24 , … . ) Find the least number of terms which makes
the sum is as great as possible starting from the first term ?
a) 8
b) 9
c) 10
d) 11
12) ∫ cos(2𝜋) 𝑑𝑥 = ……
a) sin(2𝜋) + 𝑐
b) cos(2𝜋) + 𝑐
c) (sin 2𝜋)𝑥 + 𝑐
d) 𝑥 + 𝑐
3
5
13) If sin A = 5 , where : 90° < 𝐴 < 180° , sin 𝐵 = 13 where 𝐵 is an acute angle , find
cos(A − B) , tan 2B
33
120
63
a) − 65 , 119
120
14
b) 65 , 119
65
c) 65 , −2
d) 14 , 2
𝑑𝑦
14) If 𝑦 = (z + 1)3 , z = 𝑥 5 − 3 , then 𝑑𝑥 = ……. at 𝑥 = 1
a) 15
b) −15
c) 60
d) −60
1
15) The rate of change of the function 𝑓 ∶ 𝑓(𝑥) = √𝑥 at 𝑥 = 4 equals …….
a) 2
1
b) 1
1
c) 2
d) 4
16) The sum of infinite terms of a geometric sequence (Tn ) = (3)3−𝑛 equals …….
a) 27
b)
27
c) 3
4
d)
27
2
17) Number of solutions of the equation n𝑃2 = 2 in ℤ is ……
a) zero
b) 1
c) 2
d) infinite number
9
18) The next term in the geometric sequence (8 , 6 , 2 ,
a)
11
8
19) If 𝑓(𝑥) = {
a) 2
27
27
8
9
b) 16
, … . . ) is …….
81
c) 4
d) 32
𝑥2 + 2 , 𝑥 ≥ 1
, then 𝑓 ′ (1+ ) = ……
𝑥2 − 2 , 𝑥 < 1
b) −2
c) zero
d) not exist
20) The number of terms of a geometric sequence its first term = 243 and its
last term = 1 and the sum of its terms = 364 equals ……
a) 6
b) 7
c) 4
d) 8
21) sin 75° cos 15° + cos 75° sin 15° = ………
a) zero
1
b) 2
c) 1
d)
√3
2
3
22) The tangent to the curve of the function 𝑦 = √𝑥 at 𝑥 = 0 is …….
a) the 𝑥 −axis
b) the 𝑦 −axis
c) the straight line 𝑦 = 𝑥
d) the straight line 𝑥 + 𝑦 = 0
23) (1 + tan2 𝑥)(1 − sin2 𝑥)(sin 𝑥 cos 𝑥) = …….
a) 1
b) sin 2𝑥
1
c) 2 sin 2𝑥
d) 2 sin 2𝑥
24) If (𝑎1 , 𝑎2 , 𝑎3 , … ) is an arithmetic sequence with common difference (𝑑1 ) and (𝑏1 , 𝑏2 , 𝑏3 , … ) is
an arithmetic sequence with common difference (𝑑2 ) , then (𝑎1 + 𝑏2 , 𝑎2 + 𝑏2 , 𝑎3 + 𝑏3 , … )
a) is an arithmetic sequence whose base (𝑑1 + 𝑑2 )
b) is an arithmetic sequence whose base (𝑑1 𝑑2 )
c) is an arithmetic sequence whose base (𝑑1 − 𝑑2 )
d) in not an arithmetic sequence
25) If 𝑥 ∈ [0 , 𝜋[ and
𝜋
a) 9
tan 𝑥−cot 55°
1+tan 𝑥 cot 55°
4𝜋
b)
= 1 , then 𝑥 = ……..
c)
9
5𝜋
d)
9
4𝜋
9
or
13𝜋
9
26) If 𝑓 is an even function and diiferentiable on ℝ and 𝑓 ′ (2) = 3 , then 𝑓 ′ (−2) = …..
a) 3
1
b) −3
1
c) 3
d) − 3
27) If 𝑎 , 𝑏 , 𝑐 form a geometric sequence , 𝑟 is its common ratio , then all the following statements
are true except ……
𝑏
𝑏2
𝑐
a) 𝑟 = 𝑎
b) 𝑟 = 𝑏
𝑏+𝑐
c) 𝑟 = 𝑎𝑐
d) 𝑟 = 𝑎+𝑏
28) ∫ sin2 𝑥 𝑑𝑥 + ∫ cos 2 𝑥 𝑑𝑥 = ……. +𝑐
a) sin 𝑥 + cos 𝑥
b) 𝑥
1
1
c) 3 sin3 𝑥 + 3 cos3 𝑥
d) sin 𝑥 − cos 𝑥
29) The series : 1 + 4 + 9 + 16 is written in summation notation as ………
2
a) ∑16
b) ∑4𝑟=1(𝑟 2 )
c) ∑4𝑟=1 𝑟
𝑟=1(𝑟 )
30) If 𝑓 ∶ 𝑓(𝑥) = {
𝑘 𝑥2 , 𝑥 > 2
is differentiable at 𝑥 = 2 , find the value of 𝑎
4𝑥 − 𝑎 , 𝑥 ≤ 2
31) First negative term in the sequence (96 , 93 , 90 , … ) is …..
a) T33
b) T34
c) T35
d) T36
32) Which of the following is a geometric mean for the two quantities 𝑎4 , 𝑏16 ?
a) 𝑎4 𝑏16
b) 𝑎2 𝑏 8
c) 𝑎2 𝑏 4
d) 𝑎 𝑏 4
d) ∑16
𝑟=1 𝑟
Practice Exam 2
1) ∫
𝑥 2 +3𝑥
𝑥
d𝑥 = ……….
1
b) 2 𝑥 2 + 3𝑥 + 𝑐
a) 𝑥 + 3
c) 𝑥 2 + 3𝑥 + 𝑐
2) The solution set of the equation 11𝐶𝑟 = 11𝐶𝑟+2 is ……
a) 3
b) −3
c) ± 3
d)
𝑥 3 +3𝑥 2
𝑥2
d) 6
3) The sum of the first term and fourth term in a decreasing geometric sequence = 70 The sum
of the second and third terms = 60 , find the sum of infinite terms starting form its first
term.
4) If 𝑎 , 𝑏 , 𝑐 , 𝑑, 𝑒 are positive numbers forming a geometric sequence , then the geometric
mean of these terms is ……
a) 𝑐
b) √𝑎 𝑏 𝑐 𝑑 𝑒
c) −𝑐
d) −√𝑎 𝑏 𝑐 𝑑 𝑒
5) The area of the triangle whose side lengths are 5 , 6 , 7 cm. equals ….. 𝑐𝑚2 .
a) 3√6
b) 6√6
c) 15
d) 105
6) If (𝑥 , 7 , 𝑦) form an arithmetic sequence and (𝑥 + 2 , 5 , 𝑦 − 6) form a geometric sequence ,
then 𝑦 − 𝑥 = ……
a) 3
b) 8
c) 11
d) 14
7) sin 75° sin 75° − cos 75° cos 75° = …….
1
a) 2
b)
√3
2
c) 1
d) zero
5
1
8) If 𝐴 and 𝐵 are two acute angles and tan A = 6 , tan B = 11 , then A + B = ……..
a) 30°
b) 60°
c) 45°
d) 75°
9) If Sn is the sum of the first 𝑛 terms form an arithmetic sequence and S2n = 3 Sn , then S3n ∶
Sn = …….
a) 4
b) 6
c) 8
d) 10
10) The number of ways that 5 students can sit on 7 seats in one row equals …….
a) 7
b) 5
c) 7𝑃5
d) 7𝐶5
11) The arithmetic sequence whose sixth term = 20 and the ratio between the fourth term and
tenth term equals 4 ∶ 7 , then the sum of the first fifteen terms started from its third term =
…….
a) 360
b) 380
c) 400
d) 420
12) If sin 𝑥 + cos 𝑥 = √2 , then sin 2𝑥 = …….
a) 1
1
b) 2
c)
√3
2
d) zero
13) If the average rate of change in 𝑓 equals 2.4 when 𝑥 changes from 4 to 4.2 , then the
variation in 𝑓 = …….
a) 0.32
b) 0.48
c) 3.6
d) 7.2
14) The geometric sequence whose first term is 𝑎 and its common ratio 𝑟 is decreasing if ………
a) 𝑎 > 0 , −1 < 𝑟 < 0 b) 𝑎 > 0 , 0 < 𝑟 < 1
c) 𝑎 < 0 , −1 < 𝑟 < 0 d) 𝑎 < 0 , 0 < 𝑟 < 1
d𝑦
15) If 𝑦 = tan 𝑥 , then : d𝑥 = …….
a) 1 + 𝑦
c) 1 + 𝑦 2
b) 1 − 𝑦
d) 1 − 𝑦 2
𝑥 2 + 2𝑥 , 𝑥 ≤ 1
is ……. At 𝑥 = 1
4𝑥 − 1 , 𝑥 > 1
a) continuous but not differentiable
b) continuous and differentiable
c) not continuous and not differentiable
d) not continuous but differentiable
16) The function 𝑓 ∶ 𝑓(𝑥) = {
17) In the opposite figure :
𝐴𝐵𝐶 is a right-angled triangle at 𝐵 ,
prove that : 𝑥 + 𝑦 = 45°
18) If 𝑛 + 1 = 30 𝑛 − 1 , then : 𝑛 = ……
a) 5
b) 6
c) 29
d) 30
19) If 𝑛 + 1𝐶 𝑛−1 = 36 and 𝑥𝑃 3 = 120 , find 3𝑛 − 4𝑥
dy
20) If 𝑥 2 + 𝑦 2 = 9 + 2𝑥𝑦 , then : dx = …….
a) −1
𝑥+𝑦
b) 1
c) 𝑦−𝑥
𝑥−𝑦
d) 𝑥+𝑦
21) ∫(2𝑥 − 5)6 𝑑𝑥 = ……… +𝑐
a) (𝑥 2 − 5𝑥)6
1
b) 14 (2𝑥 − 5)7
1
c) 14 (𝑥 2 − 5𝑥)7
22) Which of the following functions is differentiable at = 2 ?
23) If 𝑓(3 − 2𝑥) = 3𝑥 2 + 1 , then : 𝑓 ′ (7) = …….
a) −12
b) −2
c) 6
d) 42
1
d) 2 (2𝑥 − 5)7
56
16
24) If sin(A + B) = 65 , sin(A − B) = − 65 , then sin A cos B = ……..
5
4
a) 13
7
b) 13
5
c) 13
d) − 13
25) If 5 geometric means are inserted between 𝑎 and 𝑏 , then the third mean is ……
1
4
a) 𝑎5 𝑏 5
b)
𝑎𝑏
4
c) √𝑎𝑏
2
1
d) 𝑎5 𝑏 5
26) The 𝑛𝑡ℎ term of an arithmetic sequence = 𝑚2 and the 𝑚𝑡ℎ term = 𝑛2 , then the common
difference of the sequence 𝑑 = …….
a) 𝑚2 + 𝑛2 − 2
b) 𝑚 + 𝑛
c) −𝑚 − 𝑛
d) −𝑚 + 𝑛
27) The number of terms of the geometric sequence (5 , 10 , 20 , … . , 1280) equals …….. terms.
a) 8
b) 7
c) 10
d) 9
28) If 𝑛 = 𝑎 , then 𝑛 − 1 = …….
a) 𝑎 − 1
b) 𝑛 𝑎
𝑎
c) 𝑛 + 𝑎
d) 𝑛
29) ∫ cos(3𝑥 + 1) 𝑑𝑥 = 𝑎 sin(3𝑥 + 1) + 𝑐 , then 𝑎 = ……
a) 3
1
b) 3
c) 1
1
d) 9
30) The first term of a geometric sequence equals the sum of the next infinite terms , then the
common ratio of this sequence equals …….
a) 0.5
b) 0.333
c) 0.25
d) 0.666
1
31) Find the equation of the tangent to the curve of the function 𝑓 ∶ 𝑓(𝑥) = 𝑥+1 at the point
(0 , 1) which lies on it.
32) The first term of an arithmetic sequence = 5 , Tn+1 = Tn + 3 , then the fifth term = …….
a) 12
b) 20
c) 17
d) 19
Practice Exam 3
1) If n𝐶10 = n𝐶14 , then :
a) 24
25𝐶
𝑛
= ……
b) 25
c) 1
d) 49
2) The first term of a geometric sequence is (𝑎) and its last term (𝑙) and its number of terms
(𝑛) , then the product of all its terms of 𝑎 , 𝑙 , 𝑛 is …….
𝑛
a)
𝑎 2
(𝑙 )
𝑛
b) (𝑎 𝑙)𝑛
𝑛
c) (𝑎 𝑙) 3
d) (𝑎 𝑙) 2
3) How many even numbers consists of 3 different digits could be formed from the set of
digits {2 , 3 , 4 , 5 , 7}
4) If 𝑓(𝑥) = 𝑥 2 + 3 , then the rate of change of the function 𝑓 at 𝑥 = 5 equals ……
a) 2
b) 5
c) 10
d) 20
5) In the arithmetic sequence (Tn ) , Tn − Tm = ……..
a) 𝑑
b) (𝑛 − 𝑚)
c) 𝑑(𝑛 − 𝑚)
6) The value of the series ∑7𝑟=3(3𝑟 − 1) equals ……..
a) 62
b) 70
c) 75
𝜋
d) (𝑚 + 𝑛)
d) 77
3
7) If 𝜃 ∈ ]0 , 2 [ , sin 𝜃 = 5 , then tan 2𝜃 = ……..
a)
15
24
b) 17
8
3
c) 4
d)
24
7
8) If 𝑎 , 𝑏 , 𝑐 are positive real numbers , prove that : (𝑎 + 𝑏)(𝑏 + 𝑐)(𝑐 + 𝑎) ≥ 8 𝑎 𝑏 𝑐
9) The sum of infinite terms of the sequence (32 , 16 , 8 , … ) equals ………
a) 72
b) 64
c) 48
d) 24
10) As a person approaches to the base of a tower on the horizontal line passes through the
base of the tower then the measure of the elevation angle of the top of the lower will …..
a) decrease
b) remain constant
c) increase
d) vanish
11)
1+cos 2𝑥
sin 2𝑥
a) tan 𝑥
= …….
b) cos 𝑥
12) If n𝐶𝑟 = n𝑃𝑟 , then : 𝑟 = ……….
a) zero
b) 1
c) sin 𝑥
c) zero or 1
d) cot 𝑥
d) 2 or zero
13) In the opposite figure :
𝐴𝐵𝐶 is a right-angled triangle ,
then : tan 𝜃 = ……….
2
3
a) 11
b) 11
c) 2
d) 4
1
3
14) Water is poured into a tank at rate that the quantity of water each day is twice the quantity
of water was poured the day before if 12 liters of water poured on the first day , then the
day that 1536 liters of water were poured is the …… day.
a) 6𝑡ℎ
b) 7𝑡ℎ
c) 8𝑡ℎ
d) 10𝑡ℎ
15) Find each of the following :
1) ∫ (2𝑥 + 3)4 𝑑𝑥
2) ∫ sin5 4𝑥 sin 8𝑥 𝑑𝑥
𝑑𝑦
16) If 𝑦 = sin 2𝑥 , then : 𝑑𝑥 = ……. at 𝑥 =
a) 2
b) 1
𝜋
6
1
c) 2
d) √3
17) An arithmetic sequence consists of 15 terms and its middle term is 23 , then the sum of
terms of this sequence = ………
a) 345
b) 225
c) 450
d) 690
1
18) Find the solution set of the equation : cos 𝑥 − 2 sin2 2 𝑥 = 0 where 0 < 𝑥 < 360°
1
19) If ∫ 𝑥 𝐾 𝑑𝑥 = 3 𝑥 3 + 𝑐 , then : 𝐾 = …….
a) −1
b) 1
c) 2
d) 3
20) 7 arithmetic means are inserted between the two numbers : −24 , 16 , then the fourth mean
= …….
a) −14
b) −9
c) −4
d) 1
21) If 4 friends shake hands each other. How many hand shakes are done between them ?
a) 16
b) 8
c) 6
d) 4
22) The slope of the tangent to the curve 𝑦 = 3𝑥 2 + 2𝑥 + 1 at 𝑥 = 2 equals ……
a) 5
b) 8
c) 14
d) 17
23) If 𝑓 is a function and : 𝑓(1) = 5 , 𝑓 ′ (1) = 4 , then : lim 𝑓(𝑥) =……..
a) 5
b) 4
c) 9
𝑥→1
d) does not exist
1
24) ∫ 𝑥 2 (4𝑥 − 𝑥 2 ) 𝑑𝑥 = ……. +𝑐
a) 4𝑥 3 − 1
b) 𝑥 4 − 𝑥
1
1
c) 2𝑥 2 − 𝑥 3
25) The solution set in ℝ for the equation (𝑥)! = 1 is …….
a) {1}
b) {zero}
c) {0 , 1}
d) 𝑥 4 − 𝑥 3
d) {1 , −1}
26) The point lies on the curve of the function 𝑦 = (𝑥 − 3)2 − 1 at which the tangent parallel to
the straight line 2𝑥 + 𝑦 − 3 = 0 is ……
a) (3 , 1)
b) (1 , 3)
c) (2 , 0)
d) (3 , 0) or (0 , 4)
27) If (𝑎 , 𝑏 , 𝑐 , … . )is an arithmetic sequence , then :
(𝑎 + 2𝑏 − 𝑐)(2𝑏 + 𝑐 − 𝑎)(𝑎 + 2𝑏 + 𝑐) = ……..
a) 3 𝑎𝑏𝑐
b) 4 𝑎𝑏𝑐
c) 8 𝑎𝑏𝑐
d) 16 𝑎𝑏𝑐
𝑑𝑦
28) If 𝑦 = (𝑧 + 1)3 , 𝑧 = 𝑥 5 − 1 , then 𝑑𝑥 = …….
a) 𝑥15
b) 𝑥 8
c) 15 𝑥14
d) 8 𝑥 7
1
1
1
29) If ( 𝑎 , 𝑏 , 𝑐 , … . ) is a geometric sequence , its common ratio is (𝑟) , then (𝑎 , 𝑏 , 𝑐 , … . )
Represents a geometric sequence , its common ratio equals ……
1
a) 𝑟
1
c) 𝑟 2
b) 𝑟
d) 𝑟 2
30) The measure of the positive angle that the tangent to the curve of the function 𝑓 where
𝑥+2
𝑓(𝑥) = 𝑥−2 makes with the positive direction of 𝑥 − 𝑎𝑥𝑖𝑠 at the point (0 , −1) equals ……….
a) 45°
1
b) 67 2 °
c) 135°
d) 150°
𝜋
𝜋
31) The value of 𝑥 which makes the expression :cos 𝑥 cos 6 + sin 𝑥 sin 6 has minimum value is
…….
a)
𝜋
3
b)
𝜋
2
c) 𝜋
d)
7𝜋
6
32) The sum of 𝑛 terms of a sequence is given by the relation Sn = 𝑛2 − 2𝑛 , then its fifth term =
…….
a) 35
b) 15
c) 10
d) 7
Practice Exam 4
1) The sum of the series : 89 + 85 + 81 + ⋯ + 33 equals …….
a) 1830
2) If 𝑓(𝑥) =
b) 1630
1
𝑥 2 +1
c) 915
d)
915
2
and : 𝑓(𝑎) = 𝑓′(𝑎) , then : 𝑓(𝑎) = ……
a) 3
1
1
b) 2
c) 4 d) 4
3) ∫ cos(2𝑥 + 3) 𝑑𝑥 = …….
1
a) 2 sin(2𝑥 + 3) + 𝑐
b) 2 sin(2𝑥) + 𝑐
1
c) 2 sin(2𝑥 + 3) + 𝑐
d) sin(2𝑥 + 3) + 𝑐
4) cos 2 𝑐 − cos 2𝑐 = …….
a) sin 𝑐
c) sin2 𝑐
b) cos 𝑐
d) cos 2 𝑐
5) If 𝑛 3𝑛 − 1 = 240 , and8𝑃𝑟+1 = 336 find 2n𝐶𝑟
6) If 𝑦 =
1
z−1
z+1
d𝑦
, 𝑧 = √𝑥 2 + 3 , then d𝑥 = ……. at 𝑥 = 1
2
a) 2
7) If
1
b) 9
tan 𝑥
1−tan2 𝑥
c) 9
1
d) 3
= 3 , then tan 2𝑥 = …….
a) 3
b) −3
c) 6
𝑑) − 6
21
8) ∑10
𝑟=1(2𝑟 + 1) + ∑𝑟=11(2𝑟 + 1) =…….
a) ∑21
𝑟=1(2𝑟 + 1)
2
b) ∑21
𝑟=1(2𝑟 + 1)
c) ∑21
𝑟=1(4𝑟 + 2)
d) ∑21
𝑟=1(2𝑟 + 2)
𝑎𝑥 2 + 1 , 𝑥 ≥ 2
is differentiable at 𝑥 = 2 , then 𝑎 = …….
4𝑥 − 3 , 𝑥 < 2
c) 3
d) 4
9) If the function 𝑓 ∶ 𝑓(𝑥) = {
a) 1
b) 2
𝑑𝑦
10) If 𝑦 = √𝑥 + √𝑥 + √𝑥 + √… , then :
= ……
𝑑𝑥
a) 1
1
b) 𝑥𝑦
1
c) 2𝑦+1
1
d) 2𝑦−1
11) If (29 , 𝑥 , … . , 3𝑥 . 95) is an arithmetic sequence, then 𝑥 = ……
a) 21
b) 31
c) 95
d) 124
12) If (Tn ) = (3 × 2−𝑛 ) is a geometric sequence then the sum of infinite terms starting from its
first term = ………
a) 2
b) 3
c) 4
d) 9
13) If 𝑛 is divisible by 7 and 13 , then ………
a) 𝑛 ≤ 7
b) 𝑛 = 10
c) 7 ≤ 𝑛 ≤ 13
d) 𝑛 ≥ 13
14) The number of two different digit numbers can be formed from the digits {3 , 4 , 0 , 7} equals
………..
a) 6
15) If sin A =
b) 8
c) 9
d) 12
4
12
where0 < 𝐴 < 90° , cos 𝐵 = − 13 where ° < 𝐵 < 180° , find csc(𝐴 − 𝐵).
5
16) On the ground 50 metrers away from the base of a tower , the top of the tower has an
elevation angle of measure 30° , then the height of the tower = …….. 𝑚.
a) 50 sin 30°
b) 50 cos 30°
c) 50 tan 30°
d) 50√3
17) The sum of the second mean and fourth mean from an arithmetic sequence equals 12 and
the seventh mean is more than the third mean by 4 , then the sequence is …….
a) (3 , 4 , 5 , … )
b) (3 , 5 , 7 , … . )
c) (5 , 4 , 3 , … )
d) (3 , 7 , 11 , … )
18) The slope of the tangent to the curve 𝑦 = sin 2𝑥 at 𝑥 =
a) 1
b) 𝑧𝑒𝑟𝑜
c) −1
𝜋
equals ……..
2
d) −2
19) The average rate of change of the function 𝑓 ∶ 𝑓(𝑥) = 𝑥 2 + 1 when 𝑥 changes from 2 to 2.5
equals ……..
a) 4.5
b) 5.4
c) 0.54
d) 0.45
20) If the geometric mean of the two numbers 9 and 𝑦 is 15 , then 𝑦 = …….
a) 135
b) 10
c) 25
d) ± 25
21) The number of ways of arranging 7 kids in a circle equals ……..
a) 1
b) 7
c) 720
d) 5040
22) If 𝑛 ∈ ℤ+ and 𝑓(𝑥) = 𝑛 𝑥 𝑛 , 𝑓 ′ (1) = 9 , then 𝑛 = …….
a) 2
b) 3
c) 4
d) 5
23) In the given figure :
tan(∠𝐵𝐶𝐷) =……..
3
31
a) 5
b)
c)
d) 4
31
8
24) ∑𝑛
𝑟=0 𝑛𝐶 𝑟 = ……
a) 2𝑟
3
3
b) 2𝑛
c) 𝑛
d) 𝑟
25) If a clock chimes once at one o’clock and twice at two and so on , then find the number of
chimes of this clock on one day.
26) If (𝑥 , 𝑦, 𝑧) are different positive numbers from an arithmetic sequence and 𝑎 is the
geometric mean between 𝑥 and 𝑦 and 𝑏 is the geometric mean between 𝑦 and 𝑧 , then …….
𝑎
𝑎
a) 𝑦 > 𝑎𝑏
b) 𝑦 2 > 𝑎𝑏
c) 𝑦 2 < 𝑏
d) 𝑦 < 𝑏
27) If sin 𝑥 cos 3 𝑥 − cos 𝑥 sin3 𝑥 =
1
1
a) 2
b) 4
1
8
, then : sin 4𝑥 = ……..
1
c) 1
d) 8
28) 𝑛 arithmetic means are inserted between 3 and 51 , then the sum of the formed arithmetic
sequence equals ………
a) 27(𝑛 − 2)
b) 27(𝑛 − 1)
1
29) The sequence (
√3
,
2
√3
c) 27(𝑛 + 1)
d) 27(𝑛 + 2)
, √3 , … ) is ………
a) a geometric sequence with common ratio 2
b) a geometric sequence with common ration
√3
3
c) an arithmetic sequence with common difference 1
d) an arithmetic sequence with common difference
30)
𝑑
𝑑𝑥
√3
3
(𝑓(𝑥). 𝑔(𝑥)) = …….
a) 𝑓 ′ (𝑥) . 𝑔′ (𝑥)
b) 𝑓(𝑥). 𝑔′ (𝑥)
c) 𝑓 ′ (𝑥) . 𝑔(𝑥)
d) 𝑓(𝑥) . 𝑔′ (𝑥) + 𝑓 ′ (𝑥). 𝑔(𝑥)
31) For any geometric sequence ,𝑇1 × 𝑇5 = ……..
a) (𝑇3 )2
b) (𝑇1 )2
c) (𝑇5 )2
d) (𝑇2 )2
32) Find the equation of the tangent to the curve 𝑥 4 + 𝑦 4 = 17 at the point (1 , 2)
Practice Exam 5
1) The geometric mean of the numbers: 3,9 and 1 is ………….
a) ±3√3
b) ±3
c) 3
d) 9
2) If 𝑓(𝑥) = 5 g(𝑥) + 20 , g̀ (𝑥) = ⋯
a)𝑓̀ (𝑥)
3)
d
d𝑥
b) 𝑓̀(𝑥) − 20
1̀
d) 5 𝑓 (𝑥)
c) 5 𝑓̀(𝑥)
𝜋
(sin 6 ) = ⋯
a) 𝑐𝑜𝑠
𝜋
1
b) 2
6
4) If the function 𝑓: 𝑓(𝑥) = {
a) – 2
𝜋
c) zero
d) 6
𝑥2 + 𝑎 , 𝑥 ≤ 2
is differentiable at 𝑥 = 2, then 𝑏 − 𝑎 = ⋯
𝑎 𝑥 + 𝑏, 𝑥 > 2
b) – 4
c) 4
d) zero
5) From the top of a hill, a man observed the angles of depression of the top and the base of a
tower and their measures were 22° 𝑎𝑛𝑑 30°respectively where the two bases of the hill and
the tower are on the same horizontal level, if the height of the tower 50 𝑚, find the height of
hill to the nearest meter.
6) The number of terms of the arithmetic sequence (5,9,13, … ,205) is ……….
a) 35
b) 45
c) 51
d) 50
7) If the number of ways of choosing 3 elements together from n elements is 10, then 𝑛 = ⋯
a) 30
b) 10
c) 6
d) 5
8)
𝑐𝑜𝑠 40° 𝑐𝑜𝑠 20°−𝑠𝑖𝑛 40° 𝑠𝑖𝑛 20°
𝑠𝑖𝑛 15° 𝑐𝑜𝑠 15°
=⋯
1
a) 1
b) 2
9) In the opposite figure:tan 𝑥 = ⋯
7
5
a) 17
b) 13
c) 12
d) 35
5
12
c) 2
d) 3
10) ∑6𝑟=2(𝑟 2 + 𝑟 + 1) = ⋯
a) 1115
b) 115
c) 1015
d) 5115
11) If the arithmetic mean of two positive numbers is 7.5 and their geometric mean is 6, then
the difference between the two numbers = ………….
a) 3
b) 5
c) 7
d) 9
12) ∫(2 𝑥 + 1)−4 𝑑𝑥 = ⋯
a) (2 𝑥 + 1)−3 + 𝑐
b) (2 𝑥 + 1)5 + 𝑐
c)
−1
6
(2 𝑥 + 1)−3 + 𝑐
d) −8𝑥 2
𝑑𝑦
13) If 𝑦 = (𝑧 + 1)5 , 𝑧 = 𝑥 2 − 𝑥 + 1 , 𝑑𝑥 = ⋯ 𝑎𝑡 𝑥 = 1
a) 80
14) If cos B =
a) zero
b) 100
1
3
c) 120
d) 160
, cos 2 B = ⋯
b)
−2
3
15) If 3𝑛 − 7 = 120 , then n𝐶𝑛−1 = ⋯
a) zero
b) 1
c)
−7
9
c) 4
2
d) 3
d) 64
16) If (Tn ) is an arithmetic sequence in which T1 +T5 +T10 =64, then the sum of the first 15 terms =
…..
a) 120
b) 180
c) 240
d) 360
17) If n𝐶𝑚 = 1, 𝑚 = ⋯where n , 𝑚 ∈ 𝑁 , 𝑚 ≤ 𝑛
a) 1
b) zero
c) 1 or n
d) zero or n
18) The slope of the tangent to the curve of the function 𝑦 = sin 𝑥 cos 𝑥 = ⋯
a) cos 𝑥 − sin 𝑥
b) sin2 x cos2 𝑥
c) cos 2 5
d) 1
19) In the opposite figure:
the area of ∆ 𝐴𝐵𝐶 = ⋯ 𝑐𝑚2
a) 6√6
b) 9√6
c) 9√3
d) 2√6
20) Find the number of terms necessary to be taken from the sequences (25,23,21, … ) starting
from the first term to make the sum equal to 120.
21) An arithmetic sequence in which S5 -S4 =20, S8 -S7 =29, T51 =…
a) 49
b) 98
c) 155
1
22) If 𝑓(𝑥) = ∫ 𝑥 𝑑 𝑥 , 𝑓̀ (2) = ⋯
1
c) 2
b) 13
c) 13
a) does not exist
b) 𝑥 + 𝑐
d) 158
1
d) 2
23) If ∑25
𝑟=1(3 + 𝑘 𝑟) = 100 , 𝑘 = ⋯
a) 25
1
1
d) 25
24) An infinite geometric sequence in which the first and the second terms are two positive
integers, and their sum is 3, then S∞ =…
a) 4
b) 8
c) 64
d) 1023
25) The license plates of cars in a governorate start with three letters followed by three digits
except zero. How many plates can be got assuming that there is no repetition for any letter
or digit in the license plates?
26) If 𝑎 and 𝑏 are two arithmetic means between 𝑥 and 𝑦 , 𝑙, 𝑚 are two geometric means
between 𝑥 and 𝑦, then
𝑥+𝑦
a) 2 𝑥 𝑦
𝑎+𝑏
𝑙𝑚
=⋯
b)
2𝑥𝑦
c)
𝑥+𝑦
−1 1 −1
27) The nth term of the geometric sequence ( 2 , 4 ,
−1 𝑛−1
a)( 2 )
1 𝑛−1
8
b) (2)
𝑥+𝑦
𝑥𝑦
d)
𝑥𝑦
𝑥+ 𝑦
, … ) is …………
1 𝑛
c) (2)
−1 𝑛
d) ( 2 )
28) Find the average rate of change in the volume of the cube when its edge length varies from
5 cm to 7 cm
a) 125
b) 343
c) 218
d) 109
29) ∫(sin2 3 𝑥 + cos2 3 𝑥 + tan2 3 𝑥)𝑑 𝑥 = ⋯ + 𝑐
1
a) 3 𝑡𝑎𝑛 3 𝑥
1
b) 𝑠𝑒𝑐 2 3 𝑥
c) 3 𝑠𝑒𝑐 2 3 𝑥
d) 𝑡𝑎𝑛 3 𝑥
30) The number of diagonals of octagon = …………
a) 8𝐶2
b)6 ×8𝐶2
c) 8𝐶2 − 8
1
d) 2 8𝐶2
31) If the tangent to the curve: 𝑦 = 𝑥 3 − 3 𝑥 2 makes an obtuse angle with the positive direction
of 𝑥 − 𝑎𝑥𝑖𝑠, then 𝑥 ∈ ⋯
a) [0,2]
b) ]0,2[
c) 𝑅 − [0,2]
d) 𝑅 − ]0,2[
32) csc 2A+cot 2A=…
a) tan A
b) cot A
c) sec A
d) csc A
Practice Exam 6
1) If g(𝑥) = 3 𝑥 + 5 , 𝑓(𝑥) = {
a) 66
𝑥,
𝑥<0
, then (𝑓 ⃘g)̀(2) = ⋯
2
𝑥 , 𝑥≥0
b) 54
c) 12
d) 3
2) Find the equation of the tangent to the curve of the function: f:f(x)=2 tan x-cos2 x at the
point (0, −1)
7
𝜋
3) If cos 2 A= 25 , then 𝑠𝑖𝑛 𝐴 = ⋯ where 𝐴 ∈ ]0, 2 [
16
3
a) 25
4
b) 5
9
4) If 𝑓(𝑥) = 𝑥 + 𝑥 , 𝑓̀ (𝑥) = 0 when 𝑥 = ⋯
a) 3
b) – 3
5
c) 5
d) 4
c) ±3
d) ±9
5) If (8, 𝑎, … , 𝑏, 68) is an arithmetic sequence, the number of its terms 16 , then 𝑏 − 𝑎 = ⋯
a) 64
b) 76
c) 52
d) 60
6) If 𝑛 − 2 = 24, then 8𝐶𝑛 = ⋯
a) 24
b) 26
c) 28
d) 32
7) A ship sailed from a certain point in the direction of 60° north of the west with velocity 26
km/hr at the same time and place another ship moves in the direction of east with velocity
15 km/hr, find the distance between the two ships after 3 hours.
8) The sum of the sequence (3,6,12, … ,384) equals …………
a) 756
b) 567
c) 657
d) 765
9) If n𝑃𝑟−1 = 6720, n𝐶𝑟−1 = 56, then find the value of each r and n.
10) The number of ways of choosing a book and a magazine from a set of 6 books and 7
magazines is …
a) 42
b) 13
c) 1
d) 7
7
11) If sin A + cos A = where 𝐴 is an acute angle, then 𝑐𝑜𝑠 2 𝐴 = ⋯
24
a) 25
5
b) ±
24
25
7
c) 25
7
d) ± 25
12) ∫(2 𝑥 + 1)5 𝑑 𝑥 = ⋯
a) (2 𝑥 + 1)6 + 𝑐
13) Is any arithmetic sequence (Tn ),
a) 5
1
b) (2 𝑥 + 1)6
T45 +T51
T48
b) 4
c) 2 (2 𝑥 + 1)6 + 𝑐
1
d) 12 (2 𝑥 + 1)6 + 𝑐
=…
c) 3
d) 2
14) A geometric sequence in which the sum of an infinite number of its terms starting from its
first term equals 108 and its first term is greater than its second term by 12, then find the
sequence.
15) The number if arrangements formed from 3 elements that can be formed from 6 elements
equals …
a) 3
b) 6 𝑃3
c) 6 𝐶3
d) 6 × 3
16) If some arithmetic means are inserted between 8 and 62 and the sum of the second and
sixth means equals 40, then the number of this means = ………..
a) 13
b) 15
c) 17
d) 19
…
17) tan A-tan B= cos A cos B
a) sin (A+B)
b) sin (A-B)
c) cos (A-B)
18) ∑9𝑘=1(cos (10 𝑘) − sin(10 𝑘)) = ⋯
a) 2
b) ∑9𝑘=1 = 𝑘
c) – 1
d) sin A-sin B
d) ∑9𝑘=1(sin(10 𝑘) − cos(10 𝑘))
𝑑𝑦
19) If 𝑦 = 𝑧 3 − 5 , 𝑧 = 2 𝑥 2 − 3 𝑥 , 𝑑 𝑥 = ⋯ at 𝑥 = 1
a) 1
b) 3
c) 5
d) 7
20) If the equation of the normal to the curve 𝑓(𝑥) at the point (2, −1) is 𝑥 − 2 𝑦 = 4 , ̀𝑓 (2) = ⋯
a) 2
b) – 2
c) 1
d) – 1
21) In any geometric sequence (Tn ),
a) 1
𝑇7 ×𝑇11
=…
(𝑇9 )2
b) ±1
𝜋
c)
77
9
d) 2
𝜋
22) If 𝑓(𝑥) = 𝑡𝑎𝑛 𝑥 , 𝑓 (𝑥 + 4 ) × 𝑓 (𝑥 − 4 ) = ⋯
a) – 1
b) 1
c) 𝑡𝑎𝑛 2 𝑥
d) −𝑡𝑎𝑛 2 𝑥
23) If 𝑎, 𝑏, 𝑐 are three positive consecutive terms of a non-constant geometric sequence,
then……..
a)
𝑎+𝑐
2
b)
>𝑏
𝑎+𝑐
2
c)
<𝑏
𝑎+𝑐
d) 𝑏 2 = 𝑎 + 𝑐
=𝑏
2
24) The average rate of change of 𝑓 where 𝑓(𝑥) = 𝑥 2 + 3 𝑥 + 5 when 𝑥 varies from 1 to 3 equals
…..
a) 1
b) 3
c) 7
d) 9
25) The solution set of the equation:
a) {5}
26) ∫
b) {6}
2 cos2 𝑥+1
cos2 𝑥
𝑥
10
= x-1𝑃𝑥−3
is ……………..
c) {7}
d) {8}
𝑑 𝑥 = ⋯+𝑐
a) 2 𝑥 + tan 𝑥
b) 2 sin 𝑥 + tan 𝑥
c) 2 tan2 𝑥 + 1
d) sin 𝑥 + cos 𝑥
27) The tenth term in the sequence (1,1,2,3,5,8,13, … ) is ………..
a) 29
b) 34
c) 55
d) 89
3
28) If tan 2 𝑥 = 4 , tan 𝑥 = ⋯
1
a) 3 or − 3
1
b) − 3 or − 3
1
1
c) −3 or 3
1
d) 3 or 3
1
29) The sum of series (1 + 𝑥 + 𝑥 2 + ⋯ ) equals ……….. where 𝑥 > 1
1
a) 𝑥−1
𝑑
b)
𝑥
1−𝑥
𝑥
c)
𝑥−1
𝑥
d) 𝑥 2 −1
𝟑𝟑
30) 𝑑 𝑥 ( √𝑥 2 ) = …
𝟑
a) √2𝑥
1𝟑
b) √2𝑥
3
c)
2
𝟑
3 √𝑥
2𝟑
d) √𝑥
3
31) The general term of the sequence ((2 × 3), (3 × 4), (4 × 5), (5 × 6), … ) is Tn =…
a) (𝑛 − 1)(𝑛 + 1)
b) 𝑛(𝑛 + 1)
c) 2 𝑛(𝑛 + 1)
d) (𝑛 + 1)(𝑛 + 2)
32) If 𝑓(𝑥) = (2𝑥 + 1) × ℎ(𝑥) and 𝑓(2) = 15 , ℎ̀(2) = 4, 𝑓̀(2) = ⋯
a) 26
b) 28
c) 30
d) 32
Practice Exam 7
1) The first derivative for the function :𝑦 = (6𝑥 3 + 3𝑥 + 10)10 at 𝑥 = −1 equals ……..
[a] −150 [b] −50
[c] 50
[d] 210
2) The average rate of change of the function 𝑓: 𝑓(𝑥) = 𝑥 2 when 𝑥 changes from 5 to 5.2 is
……
[a] 0.1
[b] 0.2
[c] 10.2
[d] 2.04
2
3) If sin 𝑥 = 3, then cos 2𝑥 = ………..
[a]
1
[b] −
9
1
[c]
9
8
[d]
9
7
9
4) In ∆𝐴𝐵𝐶, 𝑎 = 18cm, 𝑏 = 30 cm., 𝑐 = 24 cm., then the area of the circle inscribed in the
triangle = ………cm2
[a] 6𝜋
[b] 216
[c] 36𝜋
[d] 9𝜋
1
5) ∫ 2 sin(6 𝜋) 𝑑𝑥 = …………. +𝑐
1
[a] 2 cos(3 𝜋)
1
[b] −2 cos(3 𝜋)
𝑑𝑦
[c] 𝑥
6) If 𝑦 = 2𝑥 sin 𝑥 cos 𝑥, then 𝑑𝑥 = ……….. at 𝑥 =
[a] −𝜋
[b] 𝜋
[d] 2
𝜋
2
𝜋
𝜋
[c] 2
[d] − 2
7) If 𝑔(𝑥) = 𝑥 2 − 3, 𝑓(𝑥) = 𝑥 2 + 2, then (𝑓 ∘ 𝑔)′ (𝑥) = ………..
[a] 4𝑥(𝑥 2 − 3)
[b] 4𝑥
[c] 2𝑥(2𝑥 2 − 1)
1 𝑟−1
8) The value of the series ∑∞
𝑟=1 20 × (2)
[a] 40
[b] 80
[d] 2𝑥(𝑥 2 − 4)
equals ………..
[c] 100
[d] 400
9) The sum of the integers between 2 and 100 which divisible by 3 equals…………
[a] 1632
[b] 1683
[c] 2466
[d] 3366
10) If
1
9
+
1
10
=
𝑥
11
, then :𝑥 = ……………..
[a] 1
[b] 11
[c] 121
[d] 132
11) The number of ways of answering only 4 questions in an exam consists of 6 questions =
…….
[a] 30
[b] 15
[c] 24
[d] 10
𝑑𝑦
12) If 𝑦 4 = 𝑥 3 , then 𝑑𝑥 equals ……………. At 𝑥 = 1
4
[a] ± 3
[b] 1
3
[c] ± 4
[d] zero
13) In the opposite figure:
̅̅̅̅ )
The distance between two houses is 50 m., the top of the house (𝐶𝐷
has an elevation angle of measure 35° from the top of the house
̅̅̅̅ ), the height of the house (𝐴𝐵
̅̅̅̅ ) = 20 m. and the bases of the two
(𝐴𝐵
̅̅̅̅ ) to the
houses on the same horizontal plane, find the height of (𝐶𝐷
nearest meter.
14) tan(135° + A) = …………..
[a] −1 + tan A
sin A−cos A
[b] sin A+cos 𝐴
sin A+cos A
[c] sin 5−cos A
[d] 1 + tan A
15) From the set of letters {a,b,c,d,e,f} , the number of ways of selecting two different letters
taking order in consideration equals ………….
[a] 6P2
[b]6C2
[c] (6)2
[d] (2)6
16) If the straight line : 𝑦 + 𝑥 − 1 = 0 touches the curve of the function 𝑓: 𝑓(𝑥) = 𝑥 2 − 3𝑥 + 𝑎,
then 𝑎 = ……
[a] 1
[b] 2
[c] 3
[d] 4
17) The terms of an arithmetic sequence are positive T7 = 2 T4 − 6 and the first, second and
fifth term form a geometric sequence, then the common difference of the arithmetic
sequence could be …………
[a] 6
[b] 12
[c] 15
[d] 18
18) The geometric mean of two positive numbers equals 8 and their arithmetic mean is more
than their geometric mean by 2, then the difference between the two numbers = ……………
[a] 4
[b] 8
[c] 12
[d] 16
1
1
19) If Sn is the sum of the first 𝑛 terms from the geometric sequence ( 1 , 2 , 4 , … ), 𝑆𝑛′ is the sum
1 1
of the first 𝑛 terms from the geometric sequence (1, − 2 , 4 , … ) where 𝑛 is an even, then 𝑆𝑛 =
………….
[a] 2S'n
[b] 3S'n
3
[c] 2 S'n
20) The tenth term from the sequence (13,16,19, … ,100) equals ………..
[a] 27
[b] 32
[c] 35
2
[d] 3 S'n
[d] 40
21) If 2𝑛 = 24, mCn =6C2𝑛+1 , find the value of 𝑚
1
22) If B − 2A = 180° and tan 𝐴 = 2, then tan B = …………..
1
[a] 3
2
[b] 3
4
[c] 3
3
[d] 4
23) The first five terms in the sequence in which 𝑇1 = 1, 𝑇2 = 2 , 𝑇𝑛 = 𝑇𝑛−1 + 𝑇𝑛−2 forever 𝑛 > 2
is ………….
[a] (1,2,3,4,5)
[b] (1,2,4,8,16)
[c] (1,2,3,5,8)
[d] (1,2,3,6,12)
24) If ∑𝑛𝑟=1(3) = 12 and 𝑓(𝑥) = 3 − 2𝑥 + 𝑥 2 , find ∑𝑛𝑟=1(𝑓(𝑟)).
25) In an arithmetic sequence, T17 = 73, T73 = 17, then the order of the term whose value equals
zero is …………
[a] 36
[b] 89
[c] 90
[d] 91
𝑑𝑦
26) If 𝑦 = 𝑧 7 , z = 𝑥 3 + 2𝑥 2 − 4, find 𝑑𝑥 at 𝑥 = 1
27) If 𝑥 + 𝑦 =
5𝜋
6
, then (sin 𝑥 − cos 𝑦)2 + (cos 𝑥 − sin 𝑦)2 = ……………
3
[a] 1
[b] 2
[c] 2
[d] 3
28) If (a,b,c,20,k,e,f) form an arithmetic sequence, then a + b + c + k + e + f = …………
[a] 60
[b] 80
[c] 100
[d] 120
29) lim
𝜋
4
ℎ→0
sin ℎ
[a]
𝜋
4
sin( +ℎ)−sin( )
ℎ
ℎ
= …………….
[b] sin ℎ
[c] cos ℎ
𝜋
[d] cos 4
30) If (Tn ),(T'n ) are two geometric sequences which of the following form a geometric sequence?
[a] (Tn )k
[b] (k T'n )
[c] (Tn T'n )
[d] all the previous
31) ∫
𝑥−
1
2
√2𝑥−1
1
d𝑥 = ………. +𝑐
[a] 6 √2𝑥 − 1
1
[b] 6 √(2𝑥 − 1)3
32) If 7 =7P𝑥 , then 𝑥 = …………..
[a] 6 or 7
[b] 7
1
[c] 6√2𝑥−1
[c] 1 or zero
1
[d] 2 √(2𝑥 − 1)3
[d] 5040
Practice Exam 8
1) If 𝑛 − 5 = 1, then 𝑛 ∈ ……..
[a] {6}
[b] {5,6}
2) If 𝑓: 𝑓(𝑥) = {
3 𝑎𝑥 + 5
𝑏 𝑥 2 + 4𝑎
[a] 2
[c] {1}
[d] {5}
,𝑥 ≤ 1
is differentiable at 𝑥 = 1, 𝑓(1) = 11, then 𝑎 + 𝑏 = ……….
,𝑥 > 1
[b] 3
[c] 4
[d] 5
3) A 100 meter tower is built on a rock from a point on the ground on the horizontal plane
passing through the base of the rock, the measure of the elevation angles of the top and the
base of the tower were 76°, 46° respectively, find the height of the rock to the nearest meter.
4) The perimeter of a triangle = 12 cm. and its area = 6 cm2, then the radius length of the circle
touches its sides internally = ……….. cm
1
a. [a] 1
[b] 2
[c] 2
[d] 5
5) The order of the term whose value equals zero in the arithmetic sequence (22,20,18,….) is
………..
[a] 8
[b] 10
[c] 12
[d] 14
6) By how many ways a man and two women can be elected to form a committee from 5 men
and 14 women?
[a] 5C1 ×14C2
[b] 19C3
[c] 6P1 ×14P2
[d] 19P3
7) The sum of infinite terms of geometric sequence (𝑇𝑛 ), if T1 = 1, Tn = 2 Tn+1 equals
…………..
[a] ∞
8) If
[b] 2
2𝑛× 𝑛−1
2𝑛−1× 𝑛+1
[a] 3
[c] 3
2
[d] 2
1
[c] 7
[d] 9
1
= , then 𝑛 = ……………
3
[b] 5
dy
9) If 𝑦 = (𝑥 2 + 2)(5 − 𝑥), then dx = ………… at 𝑥 = 1
[a] −15
[b] 5
[c] −5
10) If 𝑓 is a function and 𝑓 ′ (1) = 2𝑓(1) = 4, then lim
ℎ→0
4
[a] zero
𝑓(1+ℎ)−2
[b] 3
3ℎ
[d] 2
= …………
1
[c] 12
[d] 4
11) tan 𝑥 − tan 25° = 1 + tan 𝑥 tan 25°, then 𝑥 = ……….
[a] 20°
[b] 60°
[c] 70°
[d] 110°
𝑥
12) ∫ sec 2 2 𝑑𝑥 = ……….. +𝑐
1
𝑥
[a] 2 tan 2
𝑥
[b] 2 tan 2
1
[c] 2 tan 𝑥
[d] 2 tan 𝑥
13) If 𝑦 = sin 𝑥, then 𝑦 ′ = …………
[a] 𝑦 cos 𝑥
[b] 𝑦 tan 𝑥
[c] 𝑦 cot 𝑥
[d] 𝑦 sin 𝑥
14) In the opposite figure:
ABCD is a rectangle, then sin 𝜃 =……………
[a]
24
5
24
4
[b] 25
12
[c] 5
[d] 25
1
3
𝑥
7
15) If 𝑥1 , 𝑥2 are two roots of the equation 𝑥 2 + (𝑚 − 1)𝑥 + 𝑛 = zero and ∑2𝑟=1 𝑥𝑟 = 3, ∑2𝑟=1 = ,
then 𝑛 − 𝑚 = ….
[a] 7
[b] −2
[c] 9
[d] 16
16) All terms of a geometric sequences are positive and 𝑇1 = 4𝑇3 , 𝑇2 + 𝑇5 = 36, then the sum of
its first seven terms = ……..
[a] 49
[b] 127
[c] 189
[d] 215
17) If the sum of 𝑛 terms from an arithmetic sequence defined by the rule 𝑆𝑛 = 2𝑛(7 − 𝑛), find
the number of terms should be taken from the first term to get sum equals −240.
18) How many terms should be taken from the sequence (35,30,25,….) starting from the first
term to get sum 135?
[a] 6 or 9
[b] 6 or 15
[c] 9 or 15
[d] 15 or 18
19) If 2n+1C2n-1 − 2 ×n+2Cn = 46, find the value of 𝑛.
20) The number of boys in a class is twice the number of girls, if the number of ways of
choosing a boy and a girl is 72, then the number of boys = …………
[a] 4
[b] 6
[c] 12
[d] 18
d
21) If dx (𝑎𝑥)3 = 24 at 𝑥 = 1, then 𝑎 = …………..
1
[a] 4
1
[b] 1 2
[c] 1
[d] 2
22) If tan 𝑥 − cot 𝑥 = 3, then tan 2𝑥 =……………..
[a] 6
2
[b] 3
2
[c] − 3
3
[d] 2
1
1
1
23) If the arithmetic mean between 𝑎, 𝑏 equals 9, the arithmetic mean between 𝑎 , 𝑏 equals 4,
then the positive geometric mean between 𝑎, 𝑏 equals …………..
[a] 6.5
[b] 6
[c] 3
[d] 2
24) The radius length of a circle is 𝑟, then the average rate of change in its area when 𝑟 changes
from (𝑟1) to (𝑟1 + ℎ) is ………..
[a] 𝜋 𝑟12
[b] 2 𝜋 𝑟1
[c] 𝜋(2𝑟1 + ℎ)
[d] 𝜋𝑟1 + ℎ
25) If (x,y,z,…..) form a geometric sequence, then ……….
[a] 2𝑦 < 𝑥 + 𝑧
[b] 𝑦 2 > 𝑥𝑧
[c] 𝑦 = 𝑥𝑧
[d] √𝑦 = 𝑥𝑧
3
26) Find the equation of the normal to the curve 𝑦 = tan(𝜋 − 4 𝑥)at the point (𝜋, 1)
25
27) If cos2 𝐴 = 169 where ∈]𝜋,
120
3𝜋
2
A
[ , then cos 2 = …………..
4
[a] 169
5
[b] 13
2
[c] − 13
[d] −
[c] 15 𝑥14
[d] 8 𝑥 7
√13
d𝑦
28) If 𝑦 = (𝑧 + 1)3 , 𝑧 = 𝑥 5 − 1, then d𝑥 = …………
[a] 𝑥15
[b] 𝑥 8
29) If the sum of the first 𝑛 terms from a geometric sequence is given by the relation
𝑆𝑛 = 128 − 27−𝑛 , then the common ratio of the sequence = ………….
[a] 2
1
[b] −2
1
[c] 2
[d] − 2
30) If ∫ 𝑥√𝑥 2 − 15d𝑥 = 𝑎√(𝑥 2 − 15)3 + 𝑐, then 𝑎 = ……………..
1
[a] 3
3
[b] 3
1
[c] 2
[d] 2
31) In an arithmetic sequence, the fifth mean is the ……….. term
[a] fifth
[b] fourth
[c] sixth
[d] tenth
32) Which of the following geometric sequence have the same common ratio:
[1] (𝑎, 𝑏,
2
b
a
,…)
[a] only 1 , 2
1
[2] (ab ,
[b] only 1 , 3
1
2
b
,
a
3
b
,…)
1
1
b
[3] ( − b , − a , − a2 ,….)
[c] only 2 , 3
[d] 1 , 2 , 3
Practice Exam 9
1) The sum of odd ordered terms from the arithmetic sequence (2,5,8,….,110) equals …………
[a] 1064
[b] 1008
[c] 1640
[d] 2072
2) ∫ 𝑓(𝑥). 𝑓 ′ (𝑥)𝑑𝑥 = …………….
1
[a] 𝑓(𝑥) + 𝑐
[b] [𝑓(𝑥)]2 + 𝑐
1
[c] 2 𝑓(𝑥) + 𝑐
2
3
3) If tan A = 4 , tan 𝐵 =
33
12
5
where A and B are two acute angles, then sin(𝐴 − 𝐵) = …………..
33
[a] 65
[d] [𝑓(𝑥)]2 + 𝑐
[b] − 65
56
56
[c] 65
[d] − 65
4) The slope of the curve to the function 𝑦 = (2𝑥 − 3)5 at 𝑥 = 2 equals ………….
1
[a] 1
[b] 12
cos2 𝑥
[c] 5
[d] 10
dy
5) If = 1+sin 𝑥 , then dx = …………..
[a] sin 𝑥
[b] cos 𝑥
[c] − sin 𝑥
[d] − cos 𝑥
1 1 1
6) In the geometric sequence (8 , 4 , 2 , 1, … ), then the order of the term whose value = 1024 is
………
[a] 10
[b] 12
[c] 14
[d] 16
7) If 𝑥 > 0 , then the common ratio of the geometric sequence (4, 𝑥 − 3,2𝑥 + 6, … ) equals
………
[a] 1
[b] 5
[c] 3
[d] 24
8) If lim
𝑓(1+2ℎ)−𝑓(1−3ℎ)
ℎ→0
ℎ
[a] 35
= 35 , then 𝑓 ′ (1) = …………
[b] 7
[c] 5
[d] 1
9) If 𝑓(𝑥) = 3𝑥 2 + 2, then the rate of change of the function 𝑓 at 𝑥 = 2 equals ………….
[a] 6
[b] 8
[c] 10
[d] 12
10) If
tan 𝑥
1−tan2 𝑥
= 3, then tan 2𝑥 = ……………..
[a] 3
[b] −3
[c] 6
[d] −6
11) From the top of a minaret, a ship has a depression angle of measure 38°, if the distance
between the ship and the base of the minaret is 220 meters, then find the height of the
minaret from the sea level to the nearest meter.
12) If sin 𝑥 + cos 𝑥 =
2
[a] 9
√2
,
3
then sin 2𝑥 = ………….
2
[b] 3
7
[c] 9
7
[d] − 9
13) In the opposite figure:
tan 𝜃 = …………….
[a]
4
√65
[b]
16
[c] 1
5
14) If 𝑦 = cos 𝑥, then 𝑦 − 𝑦 ′ = ……….
[a] sin 𝑥 − cos 𝑥
[b] cos 𝑥 − sin 𝑥
[d]
16
37
[c] sin 𝑥 + cos 𝑥
[d] 0
15) If (Tn ) is an arithmetic sequence where Tn = 3n + 2, then the arithmetic mean between
T5 , T7 equals ……….
[a] 6
[b] 12
[c] 20
[d] 24
16) If nPr = 60, nCr = 10, then n + r = ……………..
[a] 3
[b] 5
[c] 8
[d] 13
17) If n arithmetic means are inserted between the two numbers 𝑎 and 𝑏, then the sum of these
means equals ……..
[a]
𝑎+𝑏
2
[b] 𝑛(
𝑎+𝑏
2
)
[c] 𝑛(𝑏 − 𝑎)
𝑛
[d] 2 (𝑏 − 𝑎)
18) The geometric mean of two positive number = 20 and their arithmetic mean is more than
their geometric mean by 5, then the difference between the two numbers …………
[a] 20
[b] 30
[c] 40
[d] 50
19) The opposite figure represents the curve of the function 𝑓, then 𝑓 ′ (2)
is …………
[a] positive
[b] negative
[c] zero
[d] undefined
20) If log 2 , log(2𝑛 − 1) , log(2𝑛 + 3) from an arithmetic sequence, find the value of 𝑛.
21) Football teams compete in a league each pair of teams play once and the number of
matches in the league is 153 matches then the number of competing teams = ………..
[a] 9
[b] 13
[c] 18
[d] 19
22) How many three different digit number could be formed from the digits {2,4,5,7} such that
it is smaller than 500?
[a] 6
[b] 12
[c] 𝐶34
[d] 𝑃34
23) If 4Cr+2 =4C2−r , then 𝑟 ∈ ………….
[a] {0}
[b] {−1,4}
[c] {−2, −1,0,1,2}
[d] {0,3}
24) The first term of an infinite geometric sequence = 1 and its common ratio equals (𝑦) then
the sum of the squares of its terms is ………..
1
[a] 1−𝑦
1
[b] 1−𝑦 2
1
[d] 𝑦 2
1
[d] 𝑦
1
25) 2 (tan 𝜃 + cot 𝜃) = ………..
[a] cos 2𝜃
[b] sin 2𝜃
[c] csc 2𝜃
[d] sec 2𝜃
26) Find the value of each of the following:
[1] ∫(sin 𝑥 + cos 𝑥)
2
[2] ∫ cot 𝑥 sin 𝑥 𝑑𝑥
𝑑𝑥
27) The first term of a geometric sequence (𝑎) = 2 and its common ratio 𝑟 = 1, then the sum of
the first 10 terms = ….
[a] 20
[b] 2
[c] 10
[d] 1024
28) The derivative of 𝑥 6 with respect to 𝑥 3 is …………
[a] 𝑥 3
[b] 2𝑥 3
[c] 3𝑥 2
[d] 6𝑥 6
1
29) If the side lengths of a triangle are 2 𝑛! , (𝑛 − 2)! , (2 − 𝑛)! cm., then the numerical value of
the area of the triangle = …….. cm2
[a]
√3
2
[b]
√3
4
[c]
1
√3
8
√3
[d] 16
𝑑
30) If 𝑓 ′ (𝑥) × 𝑔(𝑥) + 𝑔′ (𝑥) × 𝑓(𝑥) = 𝑥 + 𝑥, then 𝑑𝑥 [𝑓(𝑥) × 𝑔(𝑥)] = …………. At 𝑥 = 2
3
[a] 4
[b] 1
31) 2 + 4 + 6 + 8 + ⋯ + 30 = ……………
[a] ∑30
[b] ∑15
𝑟=1 𝑟
𝑟=1 𝑟
5
[c] 2
[d] 3
[c] ∑15
𝑟=1 2𝑟
[d] ∑30
𝑟=1 2𝑟
32) The number of terms of an arithmetic sequence is (𝑛), then the term with order (𝑘) from the
end is the term with order = ……… from the beginning.
[a] 𝑘
[b] 𝑛 − 𝑘
[c] 𝑛 − 𝑘 + 1
[d] 𝑛 − 𝑘 + 2
Practice Exam 10
1) A rubber ball is released from 10 m. high on the ground. Each time the ball rebounds to half
the height which it falls from, find the total distance the ball covered till stop.
2)
𝑑
𝑑𝑥
𝜋
(tan 4 ) = …………
𝜋
[b] sec 2 4
[a] 1
[c] zero
[d] 2
3) If 𝑃 is half the perimeter of the triangle 𝐴𝐵𝐶, 𝑃 + 𝑎 = 35 cm., 𝑃 + 𝑏 = 34 cm., 𝑐 = 15 cm.,
then the area of ∆𝐴𝐵𝐶 = ............... cm2
[a] 64
[b] 72
[c] 84
[d] 96
4) 1 + cos 4A = …………..
[a] 2 cos2 4A
[b] cos2 2A
[c] cos2 4A
1
3
7
[d] 2 cos 2 2A
15
5) The sum of the first 𝑛 terms of the series: 2 + 4 + 8 + 16 + …………….
[a] 2𝑛 − 𝑛 − 1
[b] 1 − 2𝑛
𝑥2 − 4
𝑎𝑥 − 8
1
[b] 3 2
6) If the function 𝑓: 𝑓(𝑥) = {
[a] 2
[c] 2𝑛 − 1
[d] 𝑛 + 2−𝑛 − 1
,𝑥 > 2
is differentiable at 𝑥 = 2, then 𝑎 = ……………..
,𝑥 ≤ 2
[c] 3
[d] 4
7) If n-mP3 = 210, n+mC4 = 715, then find the value of each of 𝑚 and 𝑛
8) If (n2 + 9n + 20) =
[a] n + 3
𝑛+5
𝑥
, then 𝑥 = ………………
[b] 𝑛 + 3
3
𝜋
[c] n + 4
5
[d] n+5C2
𝜋
9) In ∆𝐴𝐵𝐶: tan 𝐴 = 4 , 𝐴 ∈]0, 2 [ , cos 𝐵 = 13 , 𝐵 ∈]0, 2 [, find tan(𝐴 + 𝐵)
10) ∫(𝑥 + 2)(𝑥 − 2)𝑑𝑥 = ………….
[a] 𝑥 + 4 + 𝑐
1
[b] 3 𝑥 3 − 4𝑥 + 𝑐
[c] 𝑥 2 − 4𝑥 + 𝑐
[d] (𝑥 2 − 4)2 + 𝑐
11) From a balcony of a house 20 m. high, a man observed that the top and the base of a tower
in front of him have elevation angle and depression angle of measures = 𝜃, giving that the
house the tower on the same horizontal plane, then the height of the tower = ………… m
[a] 20 tan 𝜃
20
[b] tan 𝜃
𝑑𝑦
[c] 40 csc 𝜃
𝑑𝑧
12) If 𝑦 = 𝑧 2 + 𝑧, 𝑧 = 2𝑥 2 , then 𝑑𝑥 − 𝑑𝑥 = …………..
[a] 4𝑥 4 + 2𝑥 2 [b] 16𝑥 2
[d] 40
[c] 16𝑥 3
[d] 8𝑥 6
1
13) The nth term of the sequence Tn = n − 1 where 𝑛 ∈ ℤ+ represents ……….. sequence.
[a] an increasing
[b] a decreasing
[c] a constant
[d] an oscillate
1
14) The number of horizontal tangents to the curve of the function 𝑓(𝑥) = 3 𝑥 3 − 𝑥 2 + 3 is
……….
[a] zero
[b] 1
[c] 2
[d] 3
15) If (4, 𝑏, 𝑐) form an arithmetic sequence and (2, 𝑏 + 3, 5𝑐) form a geometric sequence, the 𝑐 −
𝑏 = ………..
[a] −3
[b] 17
[c] 10
[d] 3
1−tan2 𝑥
16) 1+tan2 𝑥 = …………..
[a] cos 2𝑥
[b] sin 2𝑥
[c] cos 𝑥
[d] sin 𝑥
17) The number of ways cam prime number consists of three different digits be formed from
the set of digits 3,4,5 is …
[a] 6
[b] 3
[c] 1
[d] zero
18) If nC10 > nC9 , then 𝑛 ……………
[a] = 19
[b] > 19
[c] < 19
[d] ≤ 19
19) The greatest number of terms should be taking from the sequence (25,21,17, … ) starting
from the first term to keep the sum positive = ………….
[a] 12
[b] 13
[c] 14
[d] 15
20) Find each of the following:
tan 𝑥
3
[1] ∫ √9𝑥 − 1𝑑𝑥
[2] ∫ (cot 𝑥 + 1) 𝑑𝑥
21) If the last term of an arithmetic sequence is 10 times its first term, and the term before the
T
last one equals the sum of the fourth and the fifth terms then :T6 = ………….
[a] 3𝑑
[b] 2
3
[c] 2𝑑
[d] 𝑎 + 𝑑
22) By how many ways can a president and vice president be selected from a 12 member
committee?
[a] 2
[b] 23
[c] 66
[d] 132
23) The average rate of change of the volume of a cube when its edge changes from 3 cm. to 5
cm. equals ……..
[a] 98
[b] 49
[c] 125
[d] 9
T
2
T
24) If (𝑇𝑛 ) is an arithmetic sequence in which T4 = 3, then T6 = …………
2
[a] 3
4
[b] 5
7
6
[c] 7
8
3
[d] 4
25) The equation of the perpendicular to the curve 𝑓(𝑥) at the point (2, −1) is 𝑥 − 2𝑦 = 4, then
𝑓 ′ (2) = ……….
[a] 2
[b] −2
[c] 1
[d] −1
26) If (𝑎, 𝑏, 𝑐, … ) is a geometric sequence, its common ratio 2 , then ………..
𝑐
𝑐
[a] 𝑎 = 2
𝑏2
𝑎
[b] 𝑏 = 2
[c] 𝑏 = 2
[d] 𝑎𝑐 = 2
27) The first and second terms of an infinite geometric sequence are positive integers and their
sum = 3, then 𝑆∞ = ….
[a] 4
[b] 8
[c] 65
[d] 1024
28) The equation of the tangent to the curve 𝑦 =
[a] 𝑦 = 3𝑥 − 2
[b] 𝑦 = 2𝑥 − 3
1−2𝑥
𝑥−2
at the point (1,1) equals …………..
[c] 𝑦 = −3𝑥 + 2
[d] 𝑦 = 2𝑥 + 3
29) If (𝑎, 𝑏, 𝑐) is an arithmetic sequence its common difference is (𝑚), then (3𝑎 , 3𝑏 , 3𝑐 ) is
………….
[a] an arithmetic sequence, its common difference = 3
[b] a geometric sequence, its common ratio = 3
[c] an arithmetic sequence, its common difference = 3𝑚
[d] a geometric sequence, its common ratio = 3𝑚
30) If sin 32° = 𝑥, then sin 4° cos 4° cos 8° cos 16° = ………….
𝑥
𝑥
𝑥
[a] 2
[b] 4
[c] 8
𝑥
[d] 16
31) If 𝑓(𝑥) = √𝑥 2 + 9, then 𝑓 ′ (−4) = …………..
4
[a] − 5
[b] 5
1
1
[c] 10
[d] − 10
32) The number of terms in a geometric sequence is (2n) and its common ratio (𝑟), then the
ratio between the sum of its odd ordered terms to the sum of its even ordered terms equals
…………
1
[a] r
1
[b] r2
[c] r2
[d]
n
r