1. The figure below shows a circle divided into 6 equal parts.
8
How many parts should be shaded to illustrate ¾¾¾ of the circle?
12
A. 2 B. 3 C. 4 D. 6 E. 5
Answer is C.
Solution: 8/12 = 4/6 Thus, shade in 4 parts of the circle.
2. Pat's mom gave him 3/4 of an apple pie.
He ate 2/3 of what was given to him.
How much of the whole pie did he eat?
A. 1/2 B. 5/7 C. 5/12 D. 17/12 E. 8/9
Answer is A.
Solution: 2/3 x 3/4 = 1/2
3. -2 - 5·(-1 + 3) =
A. -14 B. 20 C. -20 D. 14 E. -12
Answer is E.
Solution: -2 - 5·(-1 + 3) = -2 - 5·(2)
= -2 - (10)
= -12
4. If 3/4 ounce of a substance costs $52 ,
then how many ounces can you buy with $78?
A. 1 B. 1.25 C. 1.5 D. 1.125 E. 2
Answer is D.
Solution: .75 ounce x ounces
¾¾¾¾¾¾ = ¾¾¾¾¾¾
$52 $78
52x = 78(.75)
52x = 58.5
x = 1.125
5. Fran is now making $2356.25 per month thanks to a big 45% raise.
How much was Fran making per month before the raise?
A. $1295.94 B. $1060.31 C. $1625 D. $2311.25 E. $2211.25
Answer is C.
Solution: $145 after raise $2356.25 after raise
¾¾¾¾¾¾¾¾¾¾¾¾ = ¾¾¾¾¾¾¾¾¾¾¾¾
$100 before raise $x before raise
145x = 235625
x = $1625
6. A field is in the shape of a square. If the area of the field is
48400 square yards, find the number of yards of fence needed to enclose
the field.
A. 193600 yards B. 220 yards C. 96800 yards
D. 880 yards E. 440 yards
Answer is D.
Solution: Let x = a side of the square field.
x·x = the area of the field = 48400
x2 = 48400
x = 220 yards
Perimeter = 220(4) = 880 yards
Part B
7. If x = -2, then -x2 + 3x - 5 =
A. -7 B. -15 C. 50 D. -50 E. 10
Answer is B.
Solution: -1(-2)2 + 3(-2) - 5 = -1(4) + 3(-2) - 5
= -4 + (-6) - 5
= -15
8. If 2x - 3y = 12 and x = 9, then find y.
A. 2 B. -10 C. 10 D. -2 E. -3
Answer is A.
Solution: 2(9) - 3y = 12
18 - 3y = 12
-3y = -6
y = 2
9. Given 5 - 2(-3x + 4) - (-x + 2) = 3 + (2x - 1), then x =
A. 5/7 B. .6 C. .8 D. 1.4 E. 13/14
Answer is D.
Solution: 5 - 2(-3x + 4) - (-x + 2) = 3 + (2x - 1)
5 + 6x - 8 + x - 2 = 3 + 2x - 1
7x - 5 = 2x + 2
5x = 7
x = 1.4
-5x 2
10. Given ¾¾¾ - 1.25 = .75x - ¾, then x =
2 5
17 ______ ______ ______ ______
A. - ¾ B. .2615384 C. .5076923 D. -.5076923 E. -.4857142
65
Answer is A.
Solution:
-5x 2
¾¾¾ - 1.25 = .75x - ¾
2 5
Multiply each term by 100.
-250x - 125 = 75x - 40
-325x = 85
x = -17/65
(-3x2y3)3
11. Simplify: ¾¾¾¾¾¾ =
81xy20
-x5 -x6 -x4 -x5 x3
A. ¾¾¾ B. ¾¾¾ C. ¾¾¾ D. ¾¾¾ E. ¾¾¾¾¾
3y11 3y11 3y14 9y11 -19683y51
Answer is A.
-27x6y9 -x5
Solution: ¾¾¾¾ = ¾¾¾¾
81xy20 3y11
12. Multiply(2x - 1)(x + 1)2 =
A. 4x4 + 4x3 - 3x2 - 2x + 1 B. 2x3 - 1 C. 2x3 - x2 + 2x - 1
D. 2x3 + 3x2 - 1 E. 4x4 + 1
Answer is D.
Solution:
(2x - 1)(x + 1)2 = (2x - 1)(x + 1)(x + 1)
= (2x - 1)(x2 + 2x + 1)
= 2x3 + 3x2 - 1
13. Consider the picture below.
How far is it, by boat, across the lake from the campgrounds to the store?
A. 8 miles B. 4 miles C. 15 miles
__
D. 34 miles E. Ö34 miles
Answer is E.
Solution: Use the Pythagorean theorem. 52 + 32 = c2
25 + 9 = c2
34 = c2
__
Ö34 = c
Part C
14. Find the coordinates of the x intercept for 5x - 3y = -15
A. (0, 5) B. (-3, 0) C. (0, 0)
D. (3, 10) E. (5, -3)
Answer is B.
Solution: We need to determine the coordinates of the point where
the graph crosses the x axis.
Let y = 0.
5x - 3(0) = -15
5x = -15
x = -3
Thus,(-3, 0).
15. Determine the slope of this line from the graph below:
-1 1
A. ¾¾¾ B. ¾¾¾ C. -3 D. 3 E. 6
3 3
Answer is C.
Solution: The slope of the line is negative.
Using the two given points, the change in y is 6 units
while the change in x is 2 units.
Slope = -(change in y / change in x)
6
Slope = - ¾¾ = -3
2
2
16. The graph below that looks most like the graph of y = - ¾¾ x + 1 is
3
A. B.
C. D.
E.
Answer is B.
Solution: Let x = 0, then y = 1.
Let x = -3, then y = 3.
Thus,(0, 1)and(-3, 3)are two points on the line.
17. Simplify:
A. X2 B. X4 C. X-2 D. X-3 E. X-5
Answer is B.
Solution: Multiply each exponent by -2.
X3
You will have ¾¾ which is X3-(-1) = X4
X-1
18. Solve the system
3x - 4y = 18
2x - 5y = 19
Add your answers as x + y.
A. -1 B. 1 C. 5 D. -5 E. 0
Answer is A.
Solution: Multiply the top row by 2 and the bottom row by -3.
6x - 8y = 36
-6x + 15y = -57
¾¾¾¾¾¾¾¾¾¾
7y = -21
y = -3
Now substitute into the original top row and solve for x. 3x - 4(-3) = 18
3x + 12 = 18
3x = 6
x = 2
Thus, x + y = -1.
19. If y = ½ 5x - 10 ½, we know that y ³ 0.
One branch of the graph is y = 5x - 10.
The other is y =
A. 5x + 10 B. -5x - 10 C. 10x + 5 D. 10x - 5 E. -5x + 10
Answer is E.
Solution:
The graph is a VEE shape with vertex at(2,0), found by setting 5x - 10 = 0.
The right branch of the VEE is y = 5x - 10 and
the left branch is y = -5x + 10.
Just change the signs.
20. Solve ½ -5x + 10 ½ < 25.
A. x < 7 B. x > -3 C. -3 < x < 7 D. x < -3 E. -7 < x < 0
Answer is C.
Solution: -25 < -5x + 10 < 25
-35 < -5x < 15
7 > x > -3
You could write: -5x + 10 < 25 and 5x - 10 < 25
These are the result of the 2 branches of the VEE graph and y = 25.
Solving we obtain: x > -3 and x < 7
21. Solve the following equation for all x and ,then,
add your answers together:
-16x + 3 -2(-x + 4) -13
¾¾¾¾¾¾¾¾¾ - ¾¾¾¾¾¾¾¾¾ = ¾¾¾¾¾¾¾¾¾
x2 + x - 6 x - 2 x + 3
A. 3/2 B. 1/2 C. -3/2 D. 0 E. -1/2
Answer is E.
Solution: Multiply out -2(-x + 4) to get 2x - 8
Factor
x2 + x - 6 = (x - 2)(x + 3)
Multiply each fraction by(x - 2)(x + 3)
You will get:
-16x + 3 - (2x - 8)(x + 3) = -13(x - 2)
-16x + 3 - 2x2 + 2x + 24 = -13x + 26
-2x2 - x + 1 = 0 Multiply through each side by -1.
(You could use the quadratic formula.)
2x2 + x - 1 = 0
Now factor: (2x - 1)(x + 1) = 0
Write 2x - 1 = 0 or x + 1 = 0.
Thus, x = 1/2 or x = -1. Thus 1/2 + (-1)= -1/2.
22. Given y = -x2 - 10x + 15, find the x coordinate of the vertex.
A. 5 B. 0 C. -5 D. -10 E. 15
Answer is C.
Solution:
Draw a graph as follows:
Write
y = -1[x2 + 10x + 25 - 15 - 25]
y = -1[(x + 5)2 - 40]
y = -1(x + 5)2 + 40
Vertex:(-5, 40)
-b
You could use the vertex formula x = ¾¾¾
2a
23. Find the radius of the circle with equation
x2 - 6x + y2 + 10y + 18 = 0
__
A. Ö18 B. 4 C. -6 D. 10 E. 18
Answer is B.
Solution: Complete the square in x and y.
x2 - 6x + 9 + y2 + 10y + 25 + 18 - 9 - 25 = 0
(x - 3)2 + (y + 5)2 = 16
__
The radius is Ö16 = 4. The center is at(3, -5).
24. Solve for all x: 12x3 + 2x2 = 30x. Choose the best answer.
A. 0 B. 3/2 C. -5/3 D. 30 E. A, B, and C
Answer is E.
Solution: Write 12x3 + 2x2 - 30x = 0
Factor 2x(6x2 + x - 15) = 0
2x(2x - 3)(3x + 5) = 0
2x = 0 or 2x - 3 = 0 or 3x + 5 = 0
Thus, x = 0 or x = 3/2 or x = -5/3
Part D
25. If f(x) = 3x - 1 and x = 2, then find f(f(x)).
A. 14 B. 5 C. 18 D. 15 E. .5
Answer is A.
Solution: f(2) = 3(2) - 1
= 5
We want f(f(2)). f(5) = 14.
______
26. Find the range of f(x) = 3 + Ö4x - 8 .
A. x ³ 2 B. y ³ 3 C. y > 3 D. y > 5 E. x < 2
Answer is B.
Solution:
________ _____
Write 3 + Ö4(x - 2) = 3 + 2Öx - 2
_
The graph is that of 2Öx translated 2 units in x and 3 units in y.
The range is y ³ 3.
Draw a graph as follows:
27. Simplify ln (ex)
A. e B. ex C. x D. ln e E. ln x
Answer is C.
Solution: Let f(x) = ln x. Now f-1(x) = ex .
Thus, f(f-1(x)) = x .
28. If y = f(x) = x2 - 4 with x ³ 0,
find the inverse function where y = f-1(x).
_
A. y = 2x B. y = 2x - 4 C. y = Öx
_____
D. y = Öx + 4 E. y = x
Answer is D.
Solution: f(x) is one to one. Switch the x and y.
Write x = y2 - 4.
_____
Thus, solving for y we get y = Öx + 4 .
_____
Note that we would not want y = -Öx + 4 .
29. If sin x = -.6 and cos x < 0, then find tan x .
A. .8 B. .6 C. -.75 D. .75 E. -.8
Answer is D.
Solution:
30. If cos x = .5 and cot x < 0, then find x.
A. 240° B. 210° C. 30° D. 300° E. 330°
Answer is D.
Solution:
Recalling that the cos 60° = .5 and that cos 300° = .5
and that cot 300° < 0,we observe that x = 300°.