Math 96 - Final Exam Sample Packet A
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve.
1) Find the volume of the circular cylinder below whose diameter is 12 m and height is 8 m.
1)
12 m
8m
A) 288π m3
B) 108π m 3
C) 648π m 3
D) 54π m 3
2) The sum of two numbers is -2. Three times the first number equals 4 times the second number.
Find the two numbers.
A)
8
6
and
7
7
B) -2 and 12
C) -8 and -6
D) -
8
6
and 7
7
3) A boat moves 5 kilometers upstream in the same amount of time it moves 20 kilometers
downstream. If the rate of the current is 9 kilometers per hour, find the rate of the boat in still
water.
2
A) 6 km/hr
B) 9 km/hr
C) 3 km/hr
D) 15 km/hr
3
4)
3
4x - 2 =
3
x +6
A) 2
2)
3)
4)
8
B)
3
4
C)
3
4
D)
5
5) The amount of water in a leaky bucket is given by the linear function f(t) = 125 - 7t, where f(t) is
in ounces and t is in minutes. Find the amount of water in the bucket after 4 minutes.
A) 28 oz
B) 118 oz
C) 153 oz
D) 97 oz
5)
6) x4 + 36 = 37x2
A) -1, 1, -6, 6
6)
7)
4x + 6 - 3 = 0
3
A)
4
B) 1, 36
C) -i, i, -6i, 6i
D) -1, 1, -6i, 6i
7)
B)
4
3
C) 12
Appendix A - 1
D) ∅
Graph the piecewise defined function.
8)
-4
if x ≤ -1
f(x) =
-2 + x if x > -1
8)
A)
B)
y
-4
y
4
4
2
2
-2
2
4
6x
-4
-2
2
-2
-2
-4
-4
C)
6x
4
D)
y
y
6
4
4
2
-4
-2
2
2
4
6x
-6
-4
-2
2
4
6
x
-2
-2
-4
-4
-6
Solve the equation.
9) (4x + 2)2 = 36
A) 10, -10
9)
B) 0, 1
C) 1, 2
D) 1, -2
C) (-2, -19)
D) (2, -61)
Find the vertex of the graph of the quadratic function.
10) f(x) = -7x2 - 14x - 5
A) (1, -26)
Solve for x.
11) log
3/5
A)
B) (-1, 2)
10)
x=2
9
25
11)
B) 23/5
C)
9
5
Appendix A - 2
D)
3
5
Graph the function. Find the vertex.
12) f(x) = -5x2
12)
A) vertex: (0, 0)
B) vertex: (0, 0)
y
-10
y
10
10
5
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
C) vertex: (0, 0)
10 x
5
10 x
D) vertex: (0, -5)
y
-10
5
y
10
10
5
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. Then factor the
trinomial.
13) x2 + 1 x + _______
13)
3
1 2
A) x2 + 1 x + 1 = x +
B) x2 + 1 x + 36 = (x +6)2
3
3
9
3
1 2
1 2
C) x2 + 1 x + 1 = x +
D) x2 + 1 x + 1 = x +
6
3
3
36
3
6
Appendix A - 3
Graph the exponential function.
14) f(x) = 4x + 1
14)
A)
B)
y
-10
y
10
10
5
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
C)
5
10 x
5
10 x
D)
y
-10
y
10
10
5
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
Find the power of i.
15
15) i
A) 1
15)
B) -i
C) i
D) -1
B) 7, -5
C) 7, 5
D) -7, 5
Solve the equation.
16) x2 + 2x - 35 = 0
A) -7, 1
17)
11 = 6 - 1
x
x
A)
18)
16)
11
6
17)
B) 2
C) 1
D)
3
5
1 + 2 =
-3
x + 6 x + 3 x2 + 9x + 18
A) -6
18)
B) 0
C) 3
Appendix A - 4
D) no solution
( 3x - 5 )
19) 4
= 256
19)
A) 128
B) 3
20) log (x + 2) - log x = 2
9
9
1
A)
40
C)
1
64
D) -3
20)
B) 40
C) 9
D)
2
81
Solve the proportion.
4 = 5
21)
y-6 y+6
21)
A) 54
B) 12
D) -
C) - 54
4
3
Graph the function.
22) f(x) = -|x + 2| + 6
22)
y
10
-10
10
x
-10
A)
B)
y
y
10
-10
10
10
x
-10
-10
10
-10
Appendix A - 5
x
C)
D)
y
y
10
10
-10
10
x
-10
10
-10
23) f(x) =
A)
x
-10
x- 4
23)
B)
y
-10
y
10
10
5
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
C)
5
10 x
5
10 x
D)
y
-10
y
10
10
5
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
Rationalize the denominator and simplify.
24)
33+
4
4
A) -7 - 2
24)
12
B) 7 + 2
12
C) 2
Appendix A - 6
12 - 7
D) 7 - 2
12
Write the series with summation notation.
25) 14 + 20 + 26 + 32
4
4
A) ∑ (6i + 2)
B) ∑ (6i + 8)
i=1
25)
C)
i=1
4
∑
(i + 13)
i=1
D)
4
∑
14i
i=1
Use the quadratic formula to solve the equation.
26) 4x2 + 1 = 3x
A)
-3 - i
8
C)
3-i
8
7 , -3 + i
8
7 , -3 + i
8
26)
7
7
B)
-3 - i
8
D)
3-i
8
7 , 3 +i 7
8
7 , 3 +i 7
8
Evaluate.
27)
27)
7 0 0
6 3 9
6 2 8
A) -42
B) 294
C) 47
D) 42
Write the first five terms of the sequence whose general term is given.
28) an = n2 - n
A) 2, 6, 12, 20, 30
B) 1, 4, 9, 16, 25
C) 0, 2, 6, 12, 20
28)
D) 0, 3, 8, 15, 24
Find an equation of the line.
29) Through (-2, 4); perpendicular to y = 2x - 2
A) y= -2x + 3
B) y = 2x + 3
29)
C) y =
1
x+ 3
2
Write the function in the form y = a(x - h)2 + k.
30) f(x) = -2x2 + 4x - 9
A) y = -2(x + 1)2 - 15
C) y = -2(x - 1)2 - 7
D) y= -
1
x+ 3
2
30)
B) y = -2(x + 2)2 - 25
D) y = -2(x - 2)2 - 13
Appendix A - 7
!
Math 96 - Final Exam Sample Packet B
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the equation.
1) log (x + 2) - log x = 2
5
5
A) 12
B)
2) x2 + 9x - 36 = 0
A) 12, -3
3)
1)
2
25
C) 5
D)
1
12
2)
B) -12, 3
C) -12, 1
D) 12, 3
1 + 4 =
-2
x + 7 x + 5 x2 + 12x + 35
A) 5
3x - 5
4) 27
3)
B) no solution
C) 0
D) -7
5x
=9
1
A)
15
4)
B) -
1
15
C) 15
D) - 15
Solve the proportion.
3 = 2
5)
y +2 y - 2
A) 4
5)
B)
4
5
C) 10
D) - 10
Solve.
6)
3x - 2 - 2 = 0
2
A)
3
6)
B) 2
C) 4
D) ∅
7) One number is four more than a second number. Two times the first number is 12 more than four
times the second number. Find the two numbers.
A) - 10 and - 14
B) 2 and - 2
C) 1 and - 3
D) 3 and - 1
7)
8) How many real solutions are possible for a system of equations whose graphs are a parabola and
a hyperbola?
A) 1, 2, 3, or 4 real solutions
B) 0, 1, 2, or 3 real solutions
C) 1, 2, or 3 real solutions
D) 0, 1, 2, 3, or 4 real solutions
8)
Appendix B - 1
9) Find the volume of the circular cylinder below. Leave your answer in terms of π.
9)
10 m
17 m
A) 220π m 3
10) x4 - 7x2 - 144 = 0
A) -4, 4, -3, 3
B) 2420π m 3
C) 110π m 3
D) 425π m3
10)
B) -16, 16, -9i, 9i
C) -3, 3, -4i, 4i
D) -4, 4, -3i, 3i
11) A painter can finish painting a house in 6 hours. Her assistant takes 8 hours to finish the same
job. How long would it take for them to complete the job if they were working together?
3
7
A) 5 hr
B) 3 hr
C) 7 hr
D)
hr
7
24
11)
12) A boat moves 5 kilometers upstream in the same amount of time it moves 18 kilometers
downstream. If the rate of the current is 8 kilometers per hour, find the rate of the boat in still
water.
12
1
2
A) 6
km/hr
B) 3
km/hr
C) 14
km/hr
D) 8 km/hr
13
13
13
12)
13) An arrow is fired into the air with an initial velocity of 160 feet per second. The height in feet of
the arrow t seconds after it was shot into the air is given by the function h(t) = -16t2 + 160t. Find
13)
the maximum height of the arrow.
A) 720 ft
B) 80 ft
Evaluate the expression.
5
14) ∑ (2i - 2)
i=2
A) 12
C) 1200 ft
D) 400 ft
14)
B) 18
C) 16
Solve the nonlinear system of equations for real solutions.
2
2
15) 4x - 2y = 6
-x2 + y 2 = 1
A) ( 5, 2), ( 5, -2), (- 5, 2), (- 5, -2)
C) (2, 5), (2, - 5), (-2, 5), (-2, - 5)
D) 20
15)
B) (2,
D) (2,
5), (2, - 5)
5), (-2, - 5)
Write in terms of i.
16) -1600
A) ±40
16)
B) -40i
C) i
Appendix B - 2
40
D) 40i
Graph the function.
17) y = log x
17)
2
A)
B)
y
-6
-4
y
6
6
4
4
2
2
-2
2
4
6
x
-6
-4
-2
-2
-2
-4
-4
-6
-6
C)
2
4
6 x
2
4
6 x
D)
y
y
6
-6
-4
4
4
2
2
-2
2
4
6 x
-6
-4
-2
-2
-2
-4
-4
-6
Appendix B - 3
18) f(x) = -|x| - 5
18)
A)
B)
y
-10
y
10
10
5
5
-5
10 x
5
-10
-5
-5
-5
-10
-10
C)
5
10 x
5
10 x
D)
y
-10
y
10
10
5
5
-5
10 x
5
-10
-5
-5
-5
-10
-10
Solve the equation. Give an exact solution.
19) e
(x + 6)
=3
A) -3
19)
B) e
3
C) ln 3 - 6
3
D) e - 6
Find the value of the logarithmic expression.
1
20) log
5 125
A) 3
B) -3
20)
C) - 1
25
D)
1
3
Find the indicated term of the sequence.
21) The eighth term of the arithmetic sequence whose first term is -3 and whose common difference
is -3
A) -27
B) 18
C) -24
D) 21
Appendix B - 4
21)
Sketch the graph of the quadratic function. Give the vertex and axis of symmetry.
1
22) f(x) = (x - 3)2 - 1
5
A) vertex (1, -3); axis x = 1
B) vertex (-1, 3); axis x = -1
y
-10
y
10
10
5
5
-5
10 x
5
-10
-5
5
-5
-5
-10
-10
C) vertex (3, -1); axis x = 3
y
10
10
5
5
-5
10 x
5
-10
-5
5
-5
-5
-10
-10
Write an equation of the circle with the given center and radius.
23) (1, 2);
5
A) (x + 3)2 + (y - 9)2 = 5
B) (x + 1)2 + (y + 2)2 = 5
C) (x - 1)2 + (y - 2)2 = 5
D) (x + 2)2 + (y + 1)2 = 25
24)
6
5
11
10 x
23)
Use the Pythagorean theorem to find the unknown side of the right triangle.
24)
A)
10 x
D) vertex (-3, -1); axis x = -3
y
-10
22)
B)
31
C)
Appendix B - 5
41
D) 31
Find the power of i.
25) i
13
25)
A) -1
B) i
C) 1
D) -i
Solve the equation for x. Give an exact solution.
26) ln 3x = 4
3
A)
e
4
B) e
26)
4
3
4
C)
e
3
D) 12
Simplify the radical expression. Assume that all variables represent positive real numbers.
27) 50k 7q 8
A) 5k 7q 8
2k
B) 5k 3q 4
C) 5k 3q 4
2
2k
D) 5q 4
27)
2k 7
Solve the equation for the indicated variable.
28) P = A
for t
1 + rt
A) t = P - 1
Ar
28)
B) t = A - P
Pr
C) t = P - Ar
D) t = P - A
1 +r
Find the inverse of the one-to-one function.
29) f(x) =
3
x- 7
A) f-1(x) = x + 7
29)
B) f-1(x) = x3 + 49
C) f-1(x) = x3 + 7
D) f-1(x) =
1
3
x +7
Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
5
30)
x +6
A)
- 30 - 5 x
x - 36
B)
- 30 + 5 x
x - 36
C)
- 30 + 5 x
x2 - 36
Appendix B - 6
D)
- 30 - 5 x
x + 36
30)
Math 96 - Final Exam Sample Packet C
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Given the matrix in echelon form, find the solution for the system.
1)
1 1 4 2
0 1 -2 6
0 0 1 -3
A) (2, 6, -3)
B) (14, 0, -3)
C) (-3, 0, 14)
Complete the indicated row operation.
2) Replace R 2 in -5 8 1 with -4R 1 + R 2.
-3 0 6
A)
-5 8 1
-8 8 7
C)
17 -32 2
-3
0 6
1)
D) (2, 12, -3)
2)
B)
-5
8 1
17 -32 2
D)
-5
-23
8 1
32 10
Find the sum of the terms of the infinite geometric sequence.
2 2
3) 2, - , , ...
3 9
A)
4
3
B) 2
3)
2
3
C) -
D)
3
2
Graph the function.
4) f(x) = -3x2
A)
4)
B)
y
y
4
4
2
2
-4
-2
2
4
-4
x
-2
2
4
x
-2
-2
-4
-4
C)
D)
y
y
4
4
2
2
-4
-2
2
4
-4
x
-2
2
-2
-2
-4
-4
Appendix C - 1
4
x
5) f(x) = -|x| - 4
5)
A)
B)
y
-10
y
10
10
5
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
C)
5
10 x
5
10 x
D)
y
-10
y
10
10
5
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
Appendix C - 2
6) y = log
4
x
6)
A)
B)
y
y
6
4
4
2
2
-6
-4
-2
2
4
6 x
-6
-4
-2
2
-2
4
6 x
-2
-4
-4
-6
C)
D)
y
-6
-4
y
6
6
4
4
2
2
-2
2
4
6 x
-6
-4
-2
2
-2
-2
-4
-4
-6
-6
4
6
x
Solve the equation.
7) x2 + 6x + 5 = 0
A) -1, -5
7)
B)
2, -1
C) 10, -5
D) 1, 5
Solve.
8) A boat moves 7 kilometers upstream in the same amount of time it moves 15 kilometers
downstream. If the rate of the current is 9 kilometers per hour, find the rate of the boat in still
water.
1
7
3
A) 13 km/hr
B) 7 km/hr
C) 9 km/hr
D) 24 km/hr
8
8
4
8)
9) An arrow is fired into the air with an initial velocity of 96 feet per second. The height in feet of
the arrow t seconds after it was shot into the air is given by the function h(t) = -16t2 + 96t. Find
9)
the maximum height of the arrow.
A) 240 ft
B) 144 ft
C) 432 ft
D) 48 ft
10) How many real solutions are possible for a system of equations whose graphs are an ellipse and
a circle?
A) 0, 1, 2, or 3 real solutions
B) 0, 1, 2, 3, or 4 real solutions
C) 1, 2, or 3 real solutions
D) 1, 2, 3, or 4 real solutions
Appendix C - 3
10)
11) One number is four more than a second number. Two times the first number is 6 more than four
times the second number.
A) - 7 and - 11
B) 6 and 2
C) 5 and 1
D) 4 and 0
11)
12) x4 - 21x2 - 100 = 0
A) -5, 5, -2i, 2i
12)
13)
B) -2, 2, -5i, 5i
C) -5, 5, -2, 2
B) 16
12
C)
5
D) -25, 25, -4i, 4i
5x - 4 - 4 = 0
A) 4
13)
D) ∅
Use the Pythagorean theorem to find the unknown side of the right triangle.
14)
14)
5
3
A) 4
B) 2
7
Find the distance between the pair of points.
15) (7, -7) and (3, -5)
A) 12 3 units
B) 6 units
Solve the equation.
16) log (x + 2) - log x = 2
5
5
1
A)
12
4x - 3
17) 27
A)
18)
C) 22
D)
22
15)
C) 2
5 units
D) 12 units
16)
B) 5
C)
2
25
D) 12
2x
=9
9
8
17)
B)
8
9
9
8
C) -
D) -
8
9
1 + 3 =
-3
x + 7 x + 4 x2 + 11x + 28
A) 4
18)
B) 0
C) -7
Appendix C - 4
D) no solution
Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
4
19)
x +7
A)
- 28 + 4 x
x2 - 49
B)
- 28 + 4 x
x - 49
C)
- 28 - 4 x
x + 49
D)
- 28 - 4 x
x - 49
Use the quadratic formula to solve the equation.
20) 2x2 + 12x = - 7
22 , -12 +
2
-12 2
C)
-6 - 22 , -6 + 22
4
4
22
A)
19)
20)
2 , -6 +
2
2
B)
-6 2
D)
-6 - 22 , -6 + 22
2
2
Find the indicated term of the sequence.
21) The eighth term of the arithmetic sequence whose first term is 6 and whose common difference is
-4
A) -26
B) 34
C) 38
D) -22
21)
Find the indicated term for the sequence whose general term is given.
n
22) an = (-1) ; a
18
n +4
A) -
9
11
22)
B)
1
72
1
22
C) -
Appendix C - 5
D)
1
22
Graph the piecewise defined function.
23)
-x + 4 if x < 0
f(x) =
3x - 4 if x ≥ 0
23)
A)
B)
y
-6
-4
y
6
6
4
4
2
2
-2
2
4
6
x
-6
-4
-2
-2
-2
-4
-4
-6
-6
C)
2
4
6
x
2
4
6
x
D)
y
-6
-4
y
6
6
4
4
2
2
-2
2
4
6
x
-6
-4
-2
-2
-2
-4
-4
-6
-6
Appendix C - 6
Sketch the graph of the equation. Find its vertex.
24) x = y 2 + 2
24)
A) vertex (0, 2)
B) vertex (2, 0)
y
-10
y
10
10
5
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
C) vertex (0, -2)
10 x
5
10 x
D) vertex (-2, 0)
y
-10
5
y
10
10
5
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
Write as a logarithmic equation.
-2
25) 4 = 1
16
A) log
C) log
25)
1 = -2
B) log
1 =4
D) log
4 16
-2 16
4
-2 = 1
16
1/16
4 = -2
For the given functions f and g, find the composition.
26) f(x) = x3 + 2x; g(x) = 3x
Find (g ∘ f)(x).
A) 27x3 + 6x
B) 3x3 + 2x
26)
C) 3x3 + 6x
Appendix C - 7
D) 27x2 + 6x
Find the surface area of the figure.
27)
27)
2m
6m
4m
A) 88 m2
B) 48 m2
C) 96 m2
D) 44 m2
Solve the equation. Give an exact solution.
28) e
(x + 6)
=3
A) -3
28)
B) ln 3 - 6
3
C) e - 6
D) e
-3
Use the given right triangle to find the trigonometric function.
29) sin A
A)
3
5
B)
5
4
C)
29)
4
5
D)
3
4
Solve the equation for x. Give an exact solution.
30) ln 11x = -2
11
A)
e
-2
30)
-2
B)
e
11
C) e
- 2
11
Appendix C - 8
D) -22
Final Exam Sample Packet Answer Keys
1.
2.
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4.
5.
6.
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12.
13.
14.
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30.
Form A
A
D
D
B
D
A
A
A
D
B
A
C
C
C
B
D
B
D
B
A
A
A
C
C
B
D
D
C
D
C
Form B
D
B
B
D
C
B
B
D
D
D
B
C
D
D
C
D
D
B
C
B
C
C
C
B
B
C
C
B
C
B
1
Form C
B
B
D
A
D
B
A
D
B
B
C
A
A
D
C
A
A
D
B
D
D
D
A
B
A
C
A
B
C
B