Pre Calculus - Spring Semester
Name _____________________________
Final Exam - Review 1
Per/Sec. _____ _ Date ________
1.
If sin B > 0 and cos B < 0 then B lies in
quadrant(s) _ _.
8.
Use trigonometric identities to simplify
cos (; + x)
2.
If cos B < 0 and tan B > 0 then B lies in
quadrants(s) _ _.
9.
Simplify: cos (3;
10.
3.
If tan a > 0 and csc a < 0 then a lies in
quadrant(s) _ _.
Given that sin a =
and cos {3 = - ~, a and (3
are in quadrant III, then sin( a - (3) = __.
11.
4.
Express sin 45 ° cos 15 ° - cos 45 ° sin 15 ° as a
trigonometric function of a single angle and
simplify.
If sinA =~, cosB =~, LA is in quadrant II,
and LB is in quadrant IV , evaluate cos(A+B).
(Answer in radical form and put everything over
a common denominator.)
5.
Express cos35°cos25 ° + sin 35°sin25° as a
trigonometric function of a single angle.
12.
If a and {3 are second-quadrant angles and
sin a = and sin (3 = ~, find the exact value of
cos(a - (3). .
13.
If tan a =
14.
If csc B = with L B is in quadrant III. What
is the value of sin(2B)?
6.
7.
Express cos 35° cos 25° + sin 35° sin 25° as a
trigonometric function of a single angle.
Use trigonometric identities to simplify
sin
x)
e; -
+ B)
-!?
l
f2
and cos a < 0, find cos 2a.
¥
Page 2
15.
i
If csc A = - with L A is in quadrant III. What
is the value of cos(2A)?
23.
(cote+1)2-2cotO=csc 2 0
24.
Express 8 sin x cos x in terms of a single
trigonometric function.
Simplify.
16.
sin 2 0+ cos 2 a+ tan 2 a
Simplify.
17.
sec 2 a- tan 2 a+ cot 2 a
18.
cos asec a-
cosO
--a
sec
25.
cosOsecO -
cosO
--a
sec
Verify each identity.
26.
a a
cos a+ sin a cos a- sin a _
+
.
csc sec
sm
cos A
a
cscO
- - - - - = cos
tanO + cot a
27.
Simplify:
2cosx
sin 2x
1 + tanO
. a = csc a+ sec a
sm
28.
Simplify:
sin a
1 + cosO
cot 0- 1
a
---=cot
1 - tan a
29.
Simplify:
sin,B + tan,B
1 + cos,B
a
Verify each identity.
19.
20.
21.
22.
(sin 0+ cose)2 - 1 = 2sin ecoso
-
Page 3
Verify each identity.
30.
csc e + cot e =
Solve.
sine
e
1 - cos
Solve.
31.
2 cos 2 e+ 7 cos e = 4
32.
2 sin 2 e = 9 sin e + 5
.33.
2 sin 2 e = sin e + 1
34.
2sine·
35.
2 cos 2 e + sin e = 2
36.
Find all values of x that satisfy sin 2x = 1 for
o :S x < 21[".
cose =
sine
37.
3 sin 2 e = cos 2 e
38.
cos 2 e- 3sine = 3
39.
sin 2 e- 3 cos e = 3
Acces format version 3.49F
EducAide Software
Licensed for use by Gabrielino High School
© 1997-2001
Pre Calculus - Spring Semester
Final Exam - Review 1
6/3/2008
Answer List
3.
m
0
6.
cos 10 0
8.
- sinx
9.
sine
12.
2VW + 2
9
2.
m
5.
cos 10
7.
II
1
2
-cosx
10.
-85"
II.
-
13.
119
169
14.
120
169
15.
25
16. sec 2 e
17.
csc 2 e
18.
sin 2 e
19.
20.
21.
23.
24.
4 sin 2x
27.
cscx
I.
4.
13
22.
25.
sin 2 e
28.
csce - cot e
3I.
34.
37.
,,­
26.
5,,­
3' 3
o, 3'
,­
,,­
5,,­
3\1'5 ­ 2,;7
12
5,,­
3'
7r
7,,­
1111"
6' 6' 6' """"6
7
29.
tanjJ
30.
32.
7,,­
6' """"6
33.
35.
0,
38.
2"
3,,­
39.
2.
5.
8.
II.
14.
17.
20.
23.
26.
29.
32.
35.
TRI NH 26
CM1 II 27
CM1 II 13
CM1 II 59
CM1 IJ 48
TRI QA 52
TRI QC 67
TRI QC 41
TRI QC 93
CM1 ill 54
TRI QE 15
3. TRI NH 30
6.
CM1 II 34
CM1 II 11
9.
12. CM1 II 56
15. CM1 IJ 46
18. TRI QA 57
21. TRI QC 71
24. CM1 IJ 38
27. CM1 IH 34
30. TRI QC 78
33. TRI QE 13
36. CM1 IK 24
39. TRI QE 30
1111"
lL
5,,­
7r
6 '
36.
,,­
7,,­
11,,­
2' 6' """"6
,,­
5,,­
4' 4"
7r
Catalog List
1.
4.
7.
10.
13.
16.
19.
22.
25.
28.
3I.
34.
37.
TRI NH 25
CM1 II 29
CM1 II 18
CM1 II 48
CM1 IJ 54
TRI QA 49
TRI QC 70
TRI QC 42
TRI QA 57
CM1 IH 50
TRI QE 16
TRI QE 52
TRI QE 5
TRI QE 27
38. TRI QE 29
Pre Calculus - Spring Semester
Name _________________________
Final Exam - Review 2
Per/Sec. ________ Date ________
Graph each function.
1.
2.
3.
4.
5.
6.
y = sin
(!x)
-5...}150 - ...}294 + 4.)486
11.
-5V28 + 6V63 -
12.
3V20 + 4V2 -
13.
8m 2 + 20m - 48
6m 2 - m -12
14.
n 2 + 2n - 8 n 2 - 5n + 4
16 - n 2
n 2 + 3n - 10
y=2sin(3x)
2VI75
y=sin(-~x)
2V45
y = cos (~x)
9m 2
18m2
16
+ 3m - 36
y = cos(30)
y = - 2 + cos (0
Simplify.
7.
8.
10.
(-ab2 )4
(ab 5 ) (ab)3
+ 7r)
15.
2c2 + c - 10
c3
2c2 -
-
9c
C -
15
-
Page 2
Solve.
Write the equation of the line.
2
16.
(3x-3)3=9
17.
(m 2
18.
19.
20.
21.
-
25. contains (-5,0) and is perpendicular to the line
through (10,-3) and (6,-11)
1)~ = 16
26 .
contains (-7,8) and is perpendicular to the line
through (3,-7) and (13,-15)
27.
contains (2,4) and is perpendicular to the line
through (8, -4) and (5, -8)
28.
What is the domain of f(x) = x2 _ 5x
29.
Find the domain of f( x) =
.y2a + 6 = -2
Solve: /3x
+ 2/ =
x
+8
2
+6 ?
Solve: /3x - 71 = 2x + 3
2
1
.
x -4x-2
Write an equation that can be used to solve
15x - 71 = x - 3.
30. Find the domain of f(x) = J2x
Solve.
22.
14 - 4w 1< 16
23.
15t -1012: 15
24.
15k - 41 ~ 6
+ 3.
Page 3
Graph.
Find the inverse. Is it a function?
X
31.
f(x) =
{
3,
32.
33.
h(x)
={
f(x)=
{
8
2x-+ 2',
-x,
-3,
if x < -3
if -3 ::; x ::; 3
if x > 3
2x + 3,
x 2,
1,
if x > -3
'f
I x::; - 3
if x < 0
ifO:Sx < 2
if x ~ 2
Given f(x) = 2x + 5, g(x)
find the following.
34 .
g(h(x))
35.
go f(9)
36 .
f
0
g(7)
=x2 -
10 and h(x)
= 3x -
8,
= 10 + 3x 2
37.
f(x)
38.
f(x) = 4x 3
+8
Acces format version 3.49F
© 1997-2001 EducAide Software
Licensed for use by Gabrielino High School
Pre Calculus - Spring Semester
Final Exam - Review 2
6/3/2008
Answer List
I.
2.
3.
4.
5.
6.
7.
-25n
-6­
a 5b
8.
1
9.
10. 4V6
II.
-2y7
12. 4'1'2
13.
14.
-(n-I)
n+5
15.
C
±vf65
18.
-19
2I.
5x -7 = -x
24.
-~5 -< k <
- 2
4;::r
16.
10, -8
17.
19.
-~, 3
20.
t, 10
22.
-3
23.
t ~ 5 or
25. Y=-2I x -25
26.
y-8= ~(x+7)
27. y-4=-~(x-2)
28.
29.
all reals except 2 ± V6
30.
32.
33 .
35.
519
36.
83
no
38.
{jx:t
39.
{jxf; yes
TRl PA 74
TRI PA 79
TRl AA 62
TRl BE 19
TRI AE 95
TRI BJ 174
CM1 LA 48
ALG OE 67
TRI ED 131
APC BB 4
TRl HF 40
TRI HB 34
TRI HC 26
2.
5.
8.
II.
14.
17.
20.
23.
26.
29.
32.
35.
38.
TRl PA 105
TRl PA 70
TRl AA 76
TRI BE 18
TRI AE 103
TRl BJ 175
CM1 LA 47
ALG OE 66
TRl ED 132
APC BB 5
TRl HF 49
TRI HB 58
TRI HC 49
<w <5
x¥-2 and x¥-3
3I.
2
34.
9x
37.
±JX 310;
-
48x
+ 54
8
;
t $-1
yes
[-~,oo)
Catalog List
I.
4.
7.
10.
13.
16.
19.
22.
25.
28.
3I.
34.
37.
3.
6.
9.
12.
15.
18.
2I.
24 .
27.
30.
33.
36.
39 .
TRI PA 84
TRl PC 60
TRI AA 77
TRl BE 28
ALG LF 84
TRI BJ 176
ALG OE 78
TRI ED 130
APC BB 7
TRl HF 48
TRI HB 53
TRI HC 51
+3
Pre Calculus - Spring Semester
Name ____________________________
Final Exam - Review 3
Per/Sec. __________ Date _________
1.
Y varies inversely as X and if X = 18 then
Y = 35. Find Y when X = 21.
8.
Write in factored form an equation for the grapb
shown?
y =P(x)
2.
3.
If y varies directly as x and x
find x when y = 98.
= 3 when y = 7,
x
4
If y varies jointly as x and z, and y = 12 when
x = 3 and z = 2, what is the value of y when
x = 5 and z = 6?
9.
Write the equation for the graph.
Graph.
= 2x2 -
4.
y
5.
y = 3x 2
-
24x + 69
l8x
x
+ 28
Use long division.
6.
y = 4x 2 + 24x + 44
7.
Find the real zeros of the function and write an
equation for the graph in factored form ..
+ 5p3 + p2 + 20p -
10.
(p4
12.
(3k4
12) --;- (p2
+ 4)
y
,
x
f\J
+ 2k 3 + 4k2 + 6k -
15) --;- (k 2 + 3)
Page 2
13.
14.
15.
Factor the polynomial P(x) =
completely.
X3
+ 2x2 -
llx - 12
State all horizontal and vertical asymptotes of the
function.
22.
x2
f(x) = x2 _
23.
x-3
q(x)=x 2 -x_20
25.
Evaluate: logg r;v;
v27
26.
Evaluate: log327)3
27.
Evaluate: log3..yg
-
4
X _
6
Given x = 2 is a root of 6x 3 - 13x 2 + 4 = 0,
factor the polynomial P(x) = 6x 3 - 13x 2 + 4.
One factor of x 3 + 3x 2 - 16x - 48 is x
are the remaining two factors?
+ 4.
What
Find all real solutions.
16.
17.
18.
x3
-
2x2 - 3x + 6
=0
1
x 3 + 3x 2 - 5x - 15 = 0
x3 + x 2
2x - 2
-
+ x3 -
19.
x4
21.
x4 - 3x 3
=0
26x 2 + 24x = 0
-
25x 2 - 21x
=0
Page 3
Write as the sum or difference of logarithms with no
exponents.
28.
Vab
log­
e
{f;
=
log(5y + 4)
37.
log(2y - 1)
38.
log(d)
39.
Solve for x: log 81 = 4 log x
4
29.
loge
3
32
y z
{[s
6
30.
loga
58
y z
Solve.
31.
8 x = 164x ­
33.
25- 2x
34.
log2(x + 1)
1
= 56x-3
+ log2(x -
1) = 3
+ log(2d + 1) = log(7d)
Acces format version 3.49F
© 1997-2001
EducAide Software
Licensed for use by Gabrielino High School
Pre Calculus - Spring Semester
Final Exam - Review 3
6/3/2008
Answer List
1.
Y=30
2.
42
4.
(6, -3), (6, -2~), Y = -3i
5.
(3,1), (3, 1b), Y
7.
y=(x+1)(x+~)(x-2)
8.
y=(x-2)2(x+1)
10.
p2
3
11.
x2 + x
13.
(x - 3)(x + l)(x + 4)
14.
(x - 2)(3x - 2)(2x + 1)
16.
2,
±V3
17.
-3,
19.
0, I, 4, -6
20.
0, 3, -2, 5
22.
x
23.
x
26.
7
"2
29.
! loge x -
32.
+ 5p -
= 3, y = 1
3
25 . -4
28.
~ log a + ~ log b - log c
3I.
X
34.
x=3
= rt
37. Y --
- §.3
3.
60
6.
9.
(-3,8), (-3,8fB), y
y = x2 - 4
12.
3k 2 + 2k - 5
15.
(x
18.
-1,
21.
0, -1, 7, -3
24.
x
27.
2
3
30.
3 log"
x=2
33.
X -
35.
x=4
36.
X
38.
d=3
39.
3
2.
5.
8.
II.
14.
17.
20.
23.
26.
29.
32.
35.
38.
CM1 FA 22
TR1 JA 42
3.
6.
9.
12.
15.
18.
2I.
24.
27.
30.
33.
36.
39.
CM1 FA 35
TR1 JA 44
=
=H
+8
±v's
-4, x
= 5,
y
=0
loge Y - ~ loge Z
+ 3)
and (x - 4)
±J2
= -2 , x = 3, y = 0
X -
~ log a Y - 4 loga Z
3
-10
83
= 15
Catalog List
I.
4.
7.
10.
13.
16.
19.
22.
25 .
28.
3I.
34.
37.
CM1 FA 24
TR1 JA 41
ALG EI45
CM1 PC 5
TR1 GH 28
TR1 GH 81
TR1 ID 57
CM1 OA 59
TR1 KC 82
TRI KF 14
TRI KF 147
TR1 KF 152
ALG EI46
CM1 PC 7
TR1 GH 27
TR1 GH 82
TR1 ID 30
CM1 OA 57
TR1 KC 85
TR1 KF 13
TR1 KF 138
TR1 KF 159
= 7~
ALG EI 47
CM1 PC 9
TR1 GH 26
TR1 GH 83
TR1 ID 31
CM1 OA 62
TRl KC 86
TRI KF 15
TR1 KF 148
CM1 OD 20
Pre Calculus - Spring Semester
Name ____________________________
Final Exam - Review 4
Per/Sec. _________ Date ________
Solve.
1.
Solve.
y=
X
2
+1
7.
- 3x - 2y + z
= -3
2x + 3y + 2z = 7
x+y=3
x+y+z=O
2.
=1- x
x + 2y = 1
y2
8.
5x - y - 2z
=1
- 3x + 2y + 3z
x - 2y - z
3.
y - 2x - 3
x2 - Y = 0
= -10
=0
9.
4.
=2
At a student bake sale cakes sold for $4 each
and pies sold for $5 each. The students sold a
total of 75 cakes and pies and made $340. Write
a system of equations that describes the number
of each ticket sold.
+ 2z = 7
- 2x + 2y + 3z = -2
2x + 3y + 2 = -12
5x + y
Graph the intersection.
10.
y 2: 2x - 1
yS;-(x-l)2+3
5.
6.
The entrance fee to a club was $10 for
non-members and $2 for members. If 500 tickets
were sold and the total amount of money taken
in was $2600, how many non-members bought
tickets?
Jennie purchased 3 packages of the cheaper pop
and 4 packages of the more expensive pop for
a total of $57. Rob purchased 7 packages of
the cheaper pop and 11 packages of the more
expensive pop for a total of $148. How much
was the cheaper package of pop?
11.
x 2 + y2 S; 9
y + x 2 2: -1
12.
3y - 2x
<6
y> (x - 2)2 - 1
Page 2
Find the sum, if it exists.
13.
100 + 50 + 25 + ...
14.
~+
16.
17.
18.
19.
is +
1~5 + ...
22.
In a geometric progression, the first term is 243
and the common ratio is ~. Find the 8th term.
23.
In a geometric progression, the first term is 100
and the common ratio is
Find the 12th term.
24.
The first term of a geometric sequence with
common ratio V7 is 4. What is the 41st term?
25.
In now many ways can 12 people be divided into
hockey teams of 6 players each?
26.
How many ways can 3 pencils be chosen from a
box of 12?
27.
Out of 20 softball players on a team, 2 are
selected at random to be co-captains. How many
different outcomes are possible?
28.
Seth is to select a center and guard for his
basketball team from a group of 7 people. Find
the number of possible outcomes.
!.
Find the sum of the series 3 + 5 + 7 + 9 + ... + 57.
Find the sum of the series 8 + 2 - 4 - 10· .. - 106.
Find the sum of the series 6+9+ 12+ 15+··· +60.
In a geometric progression, the first term is 256
and the common ratio is ~. Find the 7th term.
20.
In a geometric sequence, the first term is 3J2
and the 7th term is 24J2. Find the common
ratio .
21.
In a geometric sequence, the first term is 2 and
the the 7th term is 250. Find the common ratio.
Page 3
29.
There are 20 girls in a beauty pageant. A queen,
a first runner-up and a second runner-up are to
be chosen. How many different outcomes are
possible?
30.
In a track meet, 7 runners compete for first,
second and third place. How many different
ways can the runners place if there are no ties?
31.
There are 5 nickels, 7 dimes, and 9 pennies in a
coin purse. Suppose two coins are to be selected,
without replacing the first one. What is the
probability of selecting a penny and then a
dime?
32.
There are 6 plates, 5 saucers, and 5 cups on the
counter. Andrew accidentally knocks off two and
breaks them . What is the probability that he
broke a cup and a saucer, in that order?
33.
If you roll a die and pick a marble from a
bowl containing 5 white, 3 yellow, and 6 black
marbles, what is the probability that you will
roll a 2 on the die and a yellow marble?
Acces format version 3.49F
EducAide Software
Licensed for use by Gabrielino High School
© 1997-2001
Pre Calculus - Spring Semester
Final Exam - Review 4
6/3/2008
Answer List
l.
(-2,5), (1,2)
2.
(1,0), (-3,2)
3.
(3,9), (-1,1)
4.
x +y = 75
5.
200
6.
$7.00
4x
+ 5y =
340
7.
(-3,5, -2)
8.
(0,7, -4)
9.
(1,-6,4)
10.
graph
11.
graph
12.
13. 200
14.
3
4"
15.
graph
1
-6
16. 840
17.
-980
18.
627
20.
±v'2
2l.
±v'5
22.
16
128
"""9
23.
24.
4(7)20
25.
924
26.
25
512
220
28.
42
29.
6840
3l.
3
32.
48
33.
2.
5.
8.
TRI JH 8
CM1 DE 32
ALG QD 26
3.
TRI JH 2
6.
CM1 DE 45
9.
ALG QD 28
12.
15. TRl LK 32
18. TRI LG 2
2l. TRI LH 31
24. CM1 QD 62
27. SAT EB 26
30.
33. MMA EE 66
19.
729
20
27.
190
210
30.
5
1
28
Catalog List
l.
4.
7.
10.
13.
16.
19.
22.
25.
28.
3l.
TRl JH 7
ALG QD 27
ll.
TRl LK 13
TRI LG 1
TRl LH 28
TRI LH 26
MMA EE 26
MMA EE 22
MMA EE 53
14.
17.
20.
23.
26.
29.
32.
TRl LK
TRI LG
TRl LH
TRI LH
PRE PI
38
4
29
25
34
MMA EE 56
Pre Calculus - Spring Semester
Name _ _ _ _ _ _ _ _ _ _ _ _ ___
Final Exam - Review 5
Per/Sec. _ _ _ _ _ Date _ _ _ __
1.
2.
Simplify: sin
(3;
+
e)
Solve.
8.
2sin 2 e -Hsine - 6 = 0
9.
Solve the following for x, where 0
Answer in terms 7r.
¥
If tan e = and e lies in quadrant II, then
what is the value of cos 2e ?
cos x - 2sinxcosx = 0
Simplify.
3.
sec ecos e + sin ecsc e
Solve.
Verify each identity.
4.
5.
6.
10.
2 cos 2 e + sin e = 2
11.
Solve: (x+1)2=64
12.
Solve: 14x-121=7x+3
cos e
sece + tan e
- - - - - = 1- sine
3
Simplify: (sin e + cos e)2 + 2 sin ecos e
Simplify:
1 + cos 2e
2
Graph.
13.
Verify each identity.
7.
.
- -sece
- - - = sIn
(]
Ll
tane + cot e
g(x)=
{
<0
X
if x
-'2,
x2,
if x ~ 1
ifO~x < l
~
x < 27r.
Page 2
Given f(x) = 2x + 5, g(x) = x 2 - 10 and h(x) = 3x - 8,
find the following.
14.
Write as the sum or difference of logarithms with no
exponents.
f(g(x))
22.
15.
Given y varies jointly as x and the positive
square root of z, and inversely as w. Also, y = 3
when x = 2, Z = 4, and w = 16. Find y when
x = 15, Z = 36, and w = 5.
Use long division.
16.
17.
(2c 4
-
6c3
25c2
-
+ 48c + 72) -;- (~ -
log (
iffj)
Solve.
= 27 2x - 1
23.
9 1 - 3x
24.
log x
25.
Solve for x: log 64
+ log(x -
3)
=1
8)
= 2 log x
Given x = -2 is a root of 6x 3 + 11x2 - 4x - 4 = 0,
factor the polynomial P(x)=6x 3 +11x 2 -4x-4.
Solve.
26.
Find all roots.
18.
x
x 3 - x 2 - 3x
+3=
Find all real solutions.
x4 - 8x 3
-
11x2
+ 18x =
0
State all horizontal and vertical asymptotes of the
function.
20.
21.
\
g ()
x
x2
+ 4x -
12
x
.
= ----­
+ 5y = 11
0
27.
19.
y2 = 5 - x
+ 3y+z = 3
2x + 5y - 2z = -4
x + 6y + 2z = 0
x
Find the sum, if it exists.
28.
29.
Find the sum of the series 14 + 11 + 8 + 5· .. - 82.
30.
If you flip a coin and pick a card from a
standard 52-card deck, what is the probability
you will get a head and a heart?
Evaluate: log 1 27
3
Acces format version 3.49F
EducAide Software
Licensed for use by Gabrielino High School
© 1997-2001
Pre Calculus - Spring Semester
Final Exam - Review 5
6/12/2008
Answer List
1.
- cos ()
4.
7.
10.
o,
'If
5'1f
6' 6'
7r
13.
2.
119
-169
5.
8.
7'1f
6'
II.
15
14.
2x2 ­
3.
2
1 + 2 sin 2()
6.
cos 2 ()
1l1f
-6
9.
6' 2' 6 ' 2
12.
IT
15.
216
18.
I,
15
'If
'If
16.
17. (x
19.
0, 9, -2, 1
20.
x=O
21.
-3
23.
_
x - 12
24.
x=5
~ log d
2)(2x
5
+ 1)
3'1f
9
2c2 - 6c ­ 9
+ 2)(3x ­
5'1f
±v'3
22.
~ log b + ~ log c ­
25.
8
26.
(1,2), (-4,3)
27.
(6,-2,3)
28.
7
29.
-1122
30.
8"
2.
5.
8.
II.
14.
17.
20.
23.
26.
29.
CM1 IJ 50
CM1 IH 61
TRl QE 28
CM1 CG 31
TRI HB 33
CM1 PC 8
TRl ID 34
TRl KF 16
TRI JH 5
TRI LG 5
3.
6.
9.
12.
15.
18.
2I.
24.
27.
30.
TRl QA 55
CM1 IJ 20
CM1 IK 30
CM1 LA 46
CM1 FA 28
TRI GH 25
CM1 OA 64
TRI KF 143
ALG QD 25
MMA EE 68
1
Catalog List
I.
4.
7.
10.
13.
16.
19.
22.
25.
28.
CM1 II 12
TRI QC 76
TRI QC 69
TRl QE 27
TRI HF 46
ALG EI 48
TRl GH 84
TRI KC 84
CM1 OD 18
TRI LK 39