PreCalculus Final
Cumulative Semester Final
Name
3 
2

1) Determine the exact value of cos(   ) : sin   ,     ; cos   , 0    .
5 2
5
2
a)
8  3 21
25
b)
8  3 21
25
c)
6  4 21
25
d)
6  4 21
25
2) Complete the following identity:
sin 
sin 

?
1  sin  1  sin 
a) sin tan
b) 1  cot 
c) sec csc
d) 2 tan 2 
3) Solve the equation on the interval 0    2 : 2cos  2   3  0
a)
b)
c)
d)

6
5 7 11
,
,
6
6
6
,

11
12 12

6

,
11
6
,
11 13 23
,
,
12 12
12
12
,
4) Solve the equation on the interval 0    2 : 1  7sin x  6cos2 x
a)
b)
c)
d)

6

3
,
5
6
,
2
3
7 11
,
6
6

6
,
7
6
5) Determine the angle between v = -5i + 7j and w = -6i - 4j . Round your answer to one decimal place.
a) 110.80
b) 88.20
c) 20.70
d) 90.90
6) Determine the cross product v  w given v = -5i + 6j - 4k and w = -3i + 4j - 4k .
a) -8i – 8j – 18k
b) -40i – 32j – 38k
c) -8i – 8j – 2k
d) -2i + 8j + 8k
 
5
5
 i sin
7) Write the expression in the standard form a + bi:  3  cos
6
6
 
9 9 3
i
a)  
2
2
b) 
9 3 9
 i
2
2
9 9 3
i
c)  
2
2
d) 
9 3 9
 i
2
2
4

 .

8) Determine an equation for the parabola satisfying the stated conditions: Vertex at (7, -1) and Focus at (2, -1).
a) ( x  7)2  4( y 1)
b) ( x  7)2  4( y 1)
c) ( y  1)2  20( x  7)
d) ( y  1)2  20( x  7)
9) Determine the vertices of the hyperbola: 12 x2  9 y 2  48x 126 y  501  0
a) (5, -7), (-1, -7)
b) (2, -4), (2, -10)
c) (-2, 10), (-2, 4)
d) (1, 7), (-5, 7)
10) Write an equation for the following graph:
( x  1) 2 ( y  2) 2

1
a)
1
4
1
-2
b)
-1
1
( x  1)
( y  2)

1
1
2
2
2
4
-2
-3
-4
( x  1) 2 ( y  2)2

1
1
2
-5
2 x  y  z  3
x yz 4
3x  2 y  5 z  1
11) Use Cramer’s Rule to solve for z:
 3 1 1
2
1  3
4 1 1
1 2 5
a) z 
2 1 1
1
1
1 1 1
3 2 5
3
-1
( x  1) 2 ( y  2)2

1
c)
1
4
d)
2
b) z 
3 2
4
1
2
1 1
1
1
1
3 2 5
2 1 1
1 1 1
3 2 5
c) z 
2 1 3
1 1
3 2
4
1
3 3 1
4 4 1
1 1 5
d) z 
2 1 3
1 1
3 2
4
1
12) Write the partial fraction decomposition of the rational expression:
a)
b)
c)
d)
2x  3

2 x3  3 x 2
( x 2  5)2
.
10 x  15
x 2  5 ( x 2  5)2
2x  3
x 5
2
2x  3


10 x  15
( x 2  5)2
10 x  15
x  5 ( x 2  5)2
2
2x  3
x 5
2

10 x  15
( x 2  5)2
x  0
y  0

13) Maximize z  2 x  5 y subject to 
.
x

2
y

6

9 x  3 y  27
a) 13.8
10
9
8
7
6
b) 12
5
4
c) 14.6
3
d) 15
2
1
1
2
3
4
5
6
7
8
9 10
14) Perform each row operation in order, (1) followed by (2) followed by (3), on the given augmented matrix.
1
2
4 5 
(1) R2  3R1  R2


(2) R3  4 R1  R3
 3 5 13 18 
 4 10 13 19 
(3) R3  2 R2  R3
1 2 4 5


a) 0 1 1 3
0 0 1 5
1 2 4 5 


b) 0 1 1 3
0 0 1 5
1 2 4 5 


c)  0 1 1 3 
 0 0 1 5 
1 2 4 5 


d) 0 1 1 3
0 0 1 5 
15) Determine the 201st term of the sequence: 79,  76,  73,  70, ...
a) 603
b) 521
c) 119
d) 524
16) After being struck with a hammer, a gong vibrates 22 vibrations in the first second and in each second
2
thereafter makes
as many vibrations as in the previous second. Find how many vibrations the gong makes
3
before it stops vibrating.
a) 76 vibrations
b) 56 vibrations
c) 33 vibrations
d) 66 vibrations
17) Express the expression using the Binomial Theorem:  5 x  2 y  .
3
a) 125x3  450 x2 y  60 xy 2  8 y3
b) 125x3 150 x2 y  60 xy 2  8 y3
c) 125x3  50 x2 y  20 xy 2  8 y3
d) 5x3  60 x2 y  30 xy 2  2 y 2
18) Determine the sum of the following sequence:
a) 3276.6
b) 3276.4
c) 6553.4
d) 6553.2
1 2 4 8
8192
     
5 5 5 5
5