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.	Which of the following is equal to ? A. 2 B. 2 C. 4 D. 10 E. 10

2.	In the figure below, X is on , XYZ measures 45^o, and AXZ measures 130^o. What is the measure of XZY ? F. 45^o G. 60^o H. 85^o J. 95^o K. 105^o

3.	What is the slope of the line 5y = -3x + 10 ? A. -3 B. - C. D. 2 E. 10

4.	What is the least value of x that satisfies the equation x² - 7x + 6 = 6 ? F. -7 G. -6 H. -4 J. 0 K. 1

5.	A box of cereal contains 18 cups of cereal. At most, how many persons can you serve from this box of cereal if each serving must be at least cup? A. 14 B. 18 C. 19 D. 24 E. 25

6.	What is the largest possible product for 2 even integers whose sum is 34 ? F. 64 G. 68 H. 120 J. 240 K. 288

7.	In the graphs below, all axes have the same scale. One of the graphs shows y = 3x + 1. Which is that graph? A. D. B. E. C.

8.	The measures of the angles of a triangle are in the ratio of 2x:3x:5x as illustrated below. What is the measure of the smallest angle in the triangle? F. 18^o G. 20^o H. 30^o J. 36^o K. 40^o

9.	There are n students in a class. If, among those students, p% play at least 1 musical instrument, which of the following general expressions represents the number of students who play NO musical instrument? A. np B. .01np C. D. E. 100(1 - p)n

10.	If x + 2y = 1, and 2x + y = 5, then x + y = ? F. 1 G. 2 H. 3 J. 4 K. 5

11.	What integer most nearly approximates ? A. 20 B. 40 C. 63 D. 200 E. 2,000

12.	Over all real numbers x, what is the maximum value of 4 sin 3x ? F. 1 G. H. 3 J. 4 K. 12

Set 3: Mathematics

1. The correct answer is A.

B. = =
C. = =
D.
E. = =

2. The correct answer is H.

AXZ is an exterior angle of XZY.
By the exterior angle theorem, 45^o + m(XZY) = 130^o, so
m(XZY) = 130^o - 45^o = 85^o.
Or, by definition of supplementary angles, m(YXZ) + 130^o = 180^o, so m(YXZ) = 180^o - 130^o = 50^o.
Then for XZY, the angle sum is 45^o + 50^o + m(XZY),
and 180^o = 95^o + m(XZY) and m(XZY) = 180^o - 95^o = 85^o.

F. If m(XZY) were 45^o, then XZY would have angle sum
50^o + 45^o + 45^o = 140^o 180^o so m(XZY) 45^o.

G. If m(XZY) were 60^o, then XZY would have angle sum
50^o + 45^o + 60^o = 155^o 180^o so m(XZY) 60^o.

J. If m(XZY) were 95^o, then XZY would have angle sum
50^o + 45^o + 95^o = 190^o 180^o so m(XZY) 95^o.

K. If m(XZY) were 105^o, then XZY would have angle sum
50^o + 45^o + 105^o = 200^o 180^o so m(XZY) 105^o.

3. The correct answer is B.

5y = -3x + 10

(5y) = (-3x + 10)
y = -x + 2
This is the slope-intercept form, y = mx + b, where m is the slope, so the slope is -.

A. (0,2) is on the line. If the slope were -3, then (1,-1) would be on the line

since = -3, but 5(-1) -3(1) + 10 since -5 7. So (1,-1) isn't on the line, and the slope isn't -3.

C. (0,2) is on the line. If the slope were , then (3,7) would be on the line since = = , but 5(7) -3(3) + 10 since 35 1. So (3,7) isn't on the line, and the slope isn't .

D. (0,2) is on the line. If the slope were 2, then (1,4) would be on the line since = = 2, but 5(4) -3(1) + 10 since 20 7. So (1,4) isn't on the line, and the slope isn't 2.

E. (0,2) is on the line. If the slope were 10, then (1,12) would be on the line since = = 10, but 5(12) -3(1) + 10
since 60 7. So (1,12) isn't on the line, and the slope isn't 10.

4. The correct answer is J.

x² - 7x + 6 = 6, so x² - 7x = 0, x(x - 7) = 0, so x = 0 or x = 7.
To check, if x = 0, 0 - 0 + 6 = 6. If x = 7, 7² - 7(7) + 6 = 6.
So 0 and 7 both satisfy x² - 7x + 6 = 6 and the lesser of the two values is 0.

F, G, H, and K cannot be correct because x² - 7x + 6 = 6 is a quadratic equation and quadratic equations have at most two real solutions. Both 0 and 7 have been shown to be solutions, so no other numbers can be solutions.

5. The correct answer is E.

If x is the number of people, then is the size of the serving, so if each serving is at least cup.

So 4 3x, 75 3x, 25 x, so x 25, and any number of people 25 or fewer can be served with a serving size of at least cup.

If x > 25, x > (25), x > 18 .

So for more than 25 people to be served the minimum amount of cup, the box would have to hold more than 18 cups. Therefore, the maximum number of people who could be served at least cup is 25.

6. The correct answer is K.

If n is one integer, then 34 - n is the other. To maximize n(34 - n), consider y = x(34 - x) = 34x - x² =
-(x² - 34x) =
-(x² - 34x + 289) + 289 = -(x - 17)² + 289.

The graph is a parabola that turns downward and has maximum point (17,289). Applying this to the task of maximizing n(34 - n) for even integers n, the closest even value to 17 is 16 (or 18), so n = 16 and 34 - n = 18 (or n = 18 and 34 - n = 16). The product is 288. So the even integers are 16 and 18. Their sum is 34, and their product is maximum.

F-J are not correct because each is less than 288.

7. The correct answer is C.

The graph must have slope 3 and y-intercept 1. This graph has a positive slope and a positive y-intercept.

A.

The line goes through (0,0), so it can't have y-intercept 1.

B.

The line crosses the y-axis at a point below the x-axis, so the y-intercept can't be 1.

D.

The line has a negative slope, so the slope can't be 3.

E.

The line has a negative slope, so the slope can't be 3.

8. The correct answer is J.

2x + 3x + 5x = 180, 10x = 180, x = 18, so the smallest angle has measure 2(18) = 36^o.

F. If the smallest angle is 18^o, the others are 27^o and 45^o, but then the angle sum of the triangle is 18^o + 27^o + 45^o = 90^o, not 180^o.
G. If the smallest angle is 20^o, the others are 30^o and 50^o, but then the angle sum of the triangle is 20^o + 30^o + 50^o = 100^o, not 180^o.
H. If the smallest angle is 30^o, the others are 45^o and 75^o, but then the angle sum of the triangle is 30^o + 45^o + 75^o = 150^o, not 180^o.
K. If the smallest angle is 40^o, the others are 60^o and 100^o, but then the angle sum of the triangle is 40^o + 60^o + 100^o = 200^o, not 180^o.

9. The correct answer is C.

If p% play a musical instrument, then (100 - p)% don't; (100 - p)% is ,
since "%" means "out of 100." The number of students who do not play an instrument is n or .
If n = 100 and p = 30, then 30 play an instrument and 70 don't.

A. np = 100(30) = 3,000, not 70

B. .01 np = .01(100)(30) = 30, not 70

D. = = -290,000, not 70
E. 100(1 - p)n = 100(1 - 30)(100) = -290,000, not 70

10. The correct answer is G.

If x + 2y = 1, then x = 1 - 2y.
Substituting into 2x + y = 5, 2(1 - 2y) + y = 5,
2 - 4y + y = 5, 2 - 3y = 5, -3y = 3, y = -1.
Then x = 1 - 2(-1) = 1 + 2 = 3.

Check: If x = 3 and y = -1, 3 + 2(-1) = 3 - 2 = 1 and 2(3) + (-1) = 6 - 1 = 5, so the solution for the system is x = 3 and y = -1 and then x + y = 2.

F, H, J, K. x + 2y = 1 and 2x + y = 5 are different nonparallel lines because the slope of the first line is - and the slope of the second line is -2. Different nonparallel lines intersect in at most one point and that point is (3,-1). The sum of any two numbers is unique, so none of the other answers can be correct.

11. The correct answer is C.

or 60; 63 60.

Or, is about 7, is about 9, so
is about 63. So 63 is the best approximation.

12. The correct answer is J.

The maximum value for sin X, where X is any function of x, is 1 so sin 3x 1 for all x. Then multiplying by 4, 4 sin 3x 4, so the maximum value of 4 sin 3x is 4.

F, G, H. When x = , 4 sin 3x = 4 sin = 4 sin = 4 1 = 4. Since the values in F-H are less than 4, none can be the maximum value of 4 sin 3x.

K. If 12 were a value of 4 sin 3x, then 12 = 4 sin 3x for at least one value of x. But then 3 = sin 3x for at least one value of x. But sin 3x 1 for all x, so 12 can't be a value of 4 sin 3x.