대교 ACT

DIRECTIONS: Solve each problem, choose the correct answer, and then fill in the corresponding oval on your answer document.

Do not linger over problems that take too much time. Solve as many as you can; then return to the others in the time you have left for this test.

You are permitted to use a calculator on this test. You may use your calculator for any problems you choose,

but some of the problems may best be done without using a calculator.

Note: Unless otherwise stated, all of the following should be assumed.

1 Illustrative figures are NOT necessarily drawn to scale.
2. Geometric figures lie in a plane.
3. The word line indicates a straight line.
4. The word average indicates arithmetic mean.

1.	If 5a = 25, then a² = ? A. 5 B. 20 C. 25 D. 125 E. 625

2.	A rock group gets 30% of the money from sales of their newest compact disc. That 30% is split equally among the 5 group members. If the disc generates $1,000,000 in sales, how much does one group member receive? F. $ 30,000 G. $ 50,000 H. $ 60,000 J. $200,000 K. $300,000

3.	If the expression x² – kx – 12 is equal to zero when x = 4, what is the value of k ? A. 7 B. 1 C. –1 D. –3 E. –7

4.	If x = 4, then = ? F. G. H. J. 6 K. 8

5.	If = , then u = ? A. –3 B. –1 C. 3 D. 4 E. 5

6.	What positive value of k would make the lines below parallel in the standard ( x, y) coordinate plane? kx + 2y = 8 18x + ky = 12 F. 0 G. 2 H. 6 J. 8 K. 12

7.	A circular coin has a radius of inch. When lying flat, how much area does the coin cover, in square inches? A. B. C. D. E.

8.	If the angles X and Y each measure between 0° and 90°, and if sin X = cos Y, what is the sum of the measures of the angles X and Y ? F. 30° G. 45° H. 60° J. 90° K. 135°

9.	Lengths for the triangle below are given in feet. What is the value of x ? A. 2 B. C. 4 D. 5 E.

10.	Which of the following is the graph of the solution set of –2x > 4 ? F. G. H. J. K.

11.	Which of the following is an equation of a line that passes through the origin and is parallel to the line 2x – y = 5 ? A. –2x + y = 5 B. 2x + y = 5 C. x – y = 0 D. 2x – y = 0 E. 2x + y = 0

Set 4: Mathematics

1. The correct answer is C.

5a = 25 so a = = 5; a² = 5 · 5 = 25

A. If a² = 5, then a = and 5a = 5 = 25 so a² 5.

B. If a² = 20, then a = and 5a = 5 = 25 so a² 20.

D. If a² = 125, then a = and 5a = 5 = 25 so a² 125.

E. If a² = 625, then a = = 25 and 5a = 5(25) = 125 25 so a² 625.

2. The correct answer is H.

30% of $1,000,000 is 0.3($1,000,000) = $300,000.
When split equally among 5 group members, each gets ($300,000) = $60,000.

F. If each got $30,000, the group's share would be 5($30,000) = $150,000, but that's only 15% of $1,000,000.

G. If each got $50,000, the group's share would be 5($50,000) = $250,000, but that's only 25% of $1,000,000.

J. If each got $200,000, the group's share would be 5($200,000) = $1,000,000, but that's 100% of $1,000,000.

K. If each got $300,000, the group's share would be 5($300,000) = $1,500,000, and that's more than the amount of sales generated.

3. The correct answer is B.

4² – k(4) – 12 = 0, so 4 – 4k = 0, k = 1 when x = 4, x² – x – 12 = 4² – 4 – 12 = 0.

A. If k = 7, then x² – 7x – 12 should be 0 when x = 4. But 4² – 7(4) – 12 = –24 0.

C. If k = –1, then x² + x – 12 should be 0 when x = 4. But 4² + 4 – 12 = 8 0.

D. If k = –3, then x² + 3x – 12 should be 0 when x = 4. But 4² + 3(4) – 12 = 16 0.

E. If k = –7, then x² + 7x – 12 should be 0 when x = 4. But 4² + 7(4) – 12 = 32 0.

4. The correct answer is H.

= =

F. Incorrect because = only when x = , since = =

G. Incorrect because = only when x = , since = = = =

J. Incorrect because = 6 only when x = –3, since = = 6

K. Incorrect because = 8 only when x = – , since = = = 8

5. The correct answer is E.

= , so 5(u + 4) = (u – 2)15, 5u + 20 = 15u – 30, 10u = 50, u = 5.

Check: = and = =

A. If u = –3, = = –1 and = = 15 and –1 15, so –3 doesn't satisfy the equation.

B. If u = –1, = = – and = = = 5 and – 5, so –1 doesn't satisfy the equation.

C. If u = 3, = = 5 and = = and 5 , so 3 doesn't satisfy the equation.

D. If u = 4, = = and = = and , so 4 doesn't satisfy the equation.

6. The correct answer is H.

Parallel lines have equal slopes; kx + 2y = 8,
so 2y = –kx + 8, y = x + 4.
18x + ky = 12, so ky = –18x + 12, y = x + . The slopes must be equal, so
= , –k² = –36, k = ±6. A positive value is required, so k = 6.

F. If k = 0, the lines are 2y = 8 or y = 4, which is horizontal, and 18x = 12 or x = , which is vertical. A horizontal line cannot be parallel to a vertical line.

G. If k = 2, the lines are 2x + 2y = 8 or y = 4 – x, which has slope –1, and 18x + 2y = 12 or
y = 6 – 9x, which has slope –9. –1 –9, so the lines cannot be parallel.

J. If k = 8, the lines are 8x + 2y = 8 or y = 4 – 4x, which has slope –4, and 18x + 8y = 12 or y = – x, which has slope – . –4 – , so the lines cannot be parallel.

K. If k = 12, the lines are 12x + 2y = 8 or y = 4 – 6x, which has slope –6, and 18x + 12y = 12 or y = 1 – x, which has slope – . –6 – , so the lines cannot be parallel.

7. The correct answer is E.

A = r² so A = ² =

A. In order for to be the area, the radius would have to be , since ²= , but the radius is inch, not inch.

B. In order for to be the area, the radius would have to be , since ²= , but the radius is inch, not inch.

C. In order for to be the area, the radius would have to be , since ² = , but the radius is inch, not inch.

D. In order for to be the area, the radius would have to be , since ² = , but the radius is inch, not inch.

8. The correct answer is J.

Since sin x = cos(90° – x) for all x, if sin x = cos y, then Y = 90° – X and X + Y = X + (90° – X) = 90°. Conversely, if X + Y = 90°, Y = 90° – X, cos Y = cos(90° – X) = cos 90° cos X + sin 90° sin X = 0 · cos X + 1 · sin X = sin X.

F. If X + Y = 30°, then Y = 30° – X and
sin X = cos(30° – X)
sin X = cos 30° cos X + sin 30° sin X
sin X = cos X + sin X
sin X = cos X
=
tan X = , so X = 60°, Y = –30°.
But both X and Y must be between 0° and 90°, so
X + Y 30°.

G. If X + Y = 45°, then Y = 45° – X and
sin X = cos(45° – X)
sin X = cos 45° cos X + sin 45° sin X
sin X = cos X + sin X
1 – sin X = cos X
=
tan X = + 1, so X = 67.5°, Y = –22.5°.
But both X and Y must be between 0° and 90°,
so X + Y 45°.

H. If X + Y = 60°, then Y = 60° – X and
sin X = cos(60° – X)
sin X = cos 60° cos X + sin 60° sin X
sin X = cos X + sin X
1 – sin X = cos X
=
tan X = 2 + , so X = 75°, Y = –15°.
But both X and Y must be between 0° and 90°, so
X + Y 60°.

K. If X + Y = 135°, then Y = 135° – X and
sin X = cos(135° – X)
sin X = cos 135° cos X + sin 135° sin X
sin X = cos X + sin X
1 – sin X = cos X
=
tan X = –( + 1), so X = 112.5°, Y = 22.5°.
But both X and Y must be between 0° and 90°, so
X + Y 135°.

9. The correct answer is A.

The triangle is a 30°-60°-90° triangle so the sides are in the ratio 1: :2.
So = , 2 = x, x = 2.

B. If x were , then, according to the ratios for 30°-60°-90° triangles, the hypotenuse would be 2 feet long. But this is the length of a leg, and the length of the hypotenuse must be greater than the lengths of the legs.

C–E. Since these values all exceed 2 , none can be correct because each would imply that 60° is less than 30°, since the smaller angle must be opposite the smaller side.

10. The correct answer is F.

–2x > 4 so – (–2x) < – (4), x < –2 and the graph is all points to the left of –2.

G, H, and K are not correct because each includes x = 0; –2(0) = 0 and 0 4.

J is not correct because J includes x = 3; –2(3) = –6 and –6 4.

11. The correct answer is D.

Lines parallel to 2x – y = 5 have the same slope as 2x – y = 5. The slope-intercept form for 2x – y = 5 is y = 2x – 5, so the slope is 2. The desired line has slope 2 and passes through (0,0). Its equation is then y = 2x or 2x – y = 0.

A. The line with equation –2x + y = 5 does not go through (0,0), since –2(0) + 0 5.
B. The line with equation 2x + y = 5 does not go through (0,0), since 2(0) + 0 5.
C. The slope of the line with equation x – y = 0 is 1, since the slope-intercept form is y = x. So the line with equation x – y = 0 is not parallel to the line 2x – y = 5.
E. The slope of the line with equation 2x + y = 0 is –2, since the slope-intercept form is
y = –2x. So the line with equation 2x + y = 0 is not parallel to the line 2x – y = 5.