Act Mathematics Test 1 Answers

Sixty Maths questions, with answers and detailed solutions, similar to the questions in the ACT maths test. The answers are at the bottom of the page.

4 x 10^-5 =
A) - 40,000
B) - 200
C) 0.0004
D) 0.00004
E) 0.20

Which of the following is equivalent to (x)(x)(x)(x³), for all x?
A) 4x
B) 6x
C) x⁶
D) 4x⁶
E) 4x⁴
There are 15 balls in a box: 8 balls are green, 4 are blue and 3 are white. Then 1 green and 1 blue balls are taken from the box and put away. What is the probability that a blue ball is selected at random from the box?
A) 3/13
B) 4/15
C) 3/15
D) 4/13
E) 2/13

The length of a rectangle is 3 times its width. If the width of the rectangle is 5 inches, what is the rectangle's area, in square inches?
A) 15
B) 20
C) 30
D) 75
E) 40
For all x >2, (2x² + 2x - 12) / (x - 2) simplifies to
A) 2(x - 2)
B) x + 3
C) 2(x + 3)(x - 2)
D) x - 2
E) 2(x + 3)

If the hypotenuse of a right triangle is 10 inches long and one of its legs is 5 inches long, how long is the other leg?
A) 5
B) 5√3
C) 5√5
D) 75
E) 10 - √5
In the standard (x,y) coordinate plane, the graph of (x + 3)² + (y + 5)² = 16 is a circle. What is the circumference of the circle, expressed in coordinate units?
A) 4π
B) 5π
C) 3π
D) 8π
E) 25π
How many solutions are there to the equation x² - 7 = 0?
A) 1
B) 2
C) 4
D) 7
E) 14

A circle with center (4,-5) is tangent to the y-axis in the standard (x,y) coordinate plane. What is the radius of this circle?
A) 4
B) 5
C) √(41)
D) 16
E) 25
Angle A is an acute angle and sin(A) = 11/14. What is the value of cos(A)?
A) 3/14
B) √3 / 14
C) 5√3 / 14
D) √(3/14)
E) 5/14
What are the values of a and b, if any, where - a|b + 4| > 0?
A) a > 0 and b ≠ -4
B) a > 0 and b ≠ 4
C) a < 0 and b ≥ -4
D) a < 0 and b ≠ -4
E) a < 0 and b ≤ -4
In a shipment of televisions, 1/50 of the televisions are defective. What is the ratio of defective to non defective televisions?
A) 1/200
B) 1/50
C) 1/49
D) 49/1
E) 50/1
Which of the following is divisible (with no remainder) by 4?
A) 214133
B) 510056
C) 322569
D) 952217
E) 214395
A particle travels 1 x 10⁶ meters per second in a straight line for 5 x 10^-6 seconds. How many meters has it traveled?
A) 2 x 10¹¹
B) 5 x 10¹²
C) 5 x 10^-12
D) 5
E) 5 x 10^-36
The length of sides AB and AC in the triangle below are equal. What is the measure of angle ∠ A?
.

A) 70^o
B) 55^o
C) 40^o
D) 110^o
E) 20^o
In the figure below ABC is an equilateral triangle. AH is perpendicular to BC and has a length of 2√3 inches. What is the area, in square inches, of triangle Δ ABC.
.

A) 12
B) 2√3
C) 4√3
D) 8√3
E) cannot be determined from the given information
If 8y = 3x - 11, then x =
A) (88/3) y
B) (8/3) y + 11
C) (8/3) y - 11
D) (8y - 11)/3
E) (8y + 11)/3
Which of the statements describes the solution set for -2(x + 8) = -2x + 20?
A) x = - 2 only
B) x = 0 only
C) x = 20 only
D) There are no solution for this equation.
E) All real numbers are solutions of this equation.
When graphed in the (x,y) coordinate plane, at what point do the lines 2x + 3y = 5 and x = -2 intersect?
A) (-2,0)
B) (-2,5)
C) (0,5/3)
D) (0,5)
E) (-2,3)
The area of a trapezoid is 0.5 h (b1 + b2), where h is the altitude, and b1 and b2 are the lengths of the parallel bases. If a trapezoid has an altitude of 15 inches, an area of 105 square inches, and one of the bases 22 inches, what is the perimeter, in inches, of the trapezoid?
A) 8
B) 45
C) 60
D) 30
E) This trapezoid does not exist.
If you drove at average speed of 66 miles per hour, what distance, in miles, did you drive in 99 minutes?
A) 1.5
B) 0.7
C) 65.34
D) 108.9
E) 150
If x and y are any real numbers such that 0 < x < 2 < y , which of these must be true?
A) x < (xy)/2 < y
B) 0 < xy < 2x
C) x < xy < 2
D) 0 < xy < 2
E) xy < y
In the right triangle ABC below, what is the cosine of angle A?
.
A) 3/4
B) 4/3
C) 5/4
D) 3/5
E) 4/5
The graph of y = 2/(x - 3) is shown below
.
Among the following, which is the best possible graphical representation of y = -2/|x - 3|
.
What is the slope of the line
4x = -3y + 8
A) 4
B) -3/4
C) -4/3
D) 2
E) 8
In the figure below, B is on AD, ∠ BAC and ∠ ACB measure 26� and 131� respectively. What is the measure of ∠CBD?
.
A) 26�
B) 157�
C) 23�
D) 154�
E) 49�
Which of the following is equal to √45
A) 15
B) 5√3
C) 9√5
D) 3√5
E) 3
What is the smallest value of x that satisfies the equation
x(x + 4) = -3
A) -1
B) 0
C) 1
D) 3
E) -3
A group of 7 friends are having lunch together. Each person eats at least 3/4 of a pizza. What is the smallest number of whole pizzas needed for lunch?
A) 7
B) 5
C) 6
D) 28
E) 21
All axes in the graphs below have the same scales. Which one of the graphs shows the graph of y = -2x + 2?
.
The measures of angles A, B and C of a triangle are in the ratio 3:4:5. What is the measure, in degrees, of the largest angle?
A) 75�
B) 15�
C) 12�
D) 90�
E) 60�
There are n students in a school. If r% among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?
A) nr/100
B) n(1 - r)
C) 100(1 - r)n
D) n(1 - r) / 100
E) n(100 - r) / 100
If x + 4y = 5 and 5x + 6y = 7, then 3x + 5y = ?
A) 12
B) 6
C) 4
D) 2
E) 1
Which integer is nearest to √2100 / √7?
A) 3
B) 30
C) 43
D) 150
E) 17
For all real numbers x, the minimum value of 1 + 2cos(4x) is
A) 0
B) 1
C) -1
D) -2
E) -4
What is the largest possible product for 2 odd integers whose sum is equal to 32?
A) 34
B) 64
C) 255
D) 256
E) 1024
If (a + b)² = 25 and (a - b)² = 45, then a² + b² = ?
A) 35
B) 70
C) 625
D) 2025
E) 4900
If a = 3, then 2 / (1/7 + 1/a) = ?
A) 5
B) 21/10
C) 20
D) 10
E) 21/5
A company makes a profit equal to 25% of its sales. The profit is shared equally among the 4 owners of the company. If the company generates sales of $5,000,000, how much money does each one of the owners get?
A) 12,500
B) 312,500
C) 500,000
D) 1,250,000
E) 12,500,000
If the expression x³ + 2hx - 2 is equal to 6 when x = -2, what is the value of h?
A) 0
B) -2
C) -4
D) 4
E) 6
If -3/(a - 3) = 3/(a + 2), then a = ?
A) -3
B) -2
C) 1/2
D) 2
E) 3
For what negative value of k, the system of equations below has no solutions?
2x + ky = 5
kx + (k + 4)y = 7
A) -2
B) -5
C) -4
D) -7
E) -1
The two legs of a right triangle measure 6 and 8 inches respectively. What is the area of the circle that contains all 3 vertices of the triangle?
A) 24Pi
B) 25Pi
C) 36Pi
D) 64Pi
E) 48Pi
X and Y are acute angles such that tanY = cotX. What is the sum, in degrees, of the measures of the angles X and Y?
A) 90�
B) 45�
C) 60�
D) 30�
E) 120�
The measures of the sides of the right triangle below are in inches. What is the value of x?
.

A) 1/√3
B) √2
C) √3
D) 6/√3
E) 2/√3
Which of the graphs below is the solution set of -3x ≤ 12
.
Which of the lines below is not parallel to the line 6x - 2y = 10?
A) 3x - y = 7
B) -6x + 2y = 20
C) 3x + y = 7
D) 6x - 2y = 5
E) x - y/3 = 9
For what value of k the equation below has no solutions?
2x + 3 = x - 2kx - 5
A) 1
B) 2
C) -1
D) 0.5
E) -0.5
|5 - 2| - |6 - 9| = ?
A) 6
B) 0
C) -8
D) 22
E) -6
What is the average of 7/8 and 3/4?
A) 13/8
B) 5/6
C) 5/3
D) 5/12
E) 13/16
A solution is made of water and pure acid. If 75% of the solution is water, how many litters of pure acid are in 20 litters of this solution?
A) 10
B) 15
C) 25
D) 5
E) 20
The total surface area of all six sides of the rectangular box below is equal to 128 square inches. What is x in inches?
.

A) 6
B) 4
C) 16
D) 8
E) 12
The diagonal of a square has a measure of 12 inches. What is the perimeter, in inches, of this square.
A) 6√2
B) 72
C) 24√2
D) 144
E) 48
The lengths in the rectangle below are in inches. What is the shaded area in square inches?
.

A) 70
B) 34
C) 21
D) 8
E) 38
In the right triangle ABC, C is a right angle and the measure of angle B is 60�. If BC is 20 inches long, then how long is AC?
.

A) 20√3
B) 20
C) √3
D) 40
E) 20/√3
In the figure below ABC is a right triangle. ABDE is a square of area 200 square inches and BCGF is a square of 100 square inches. What is the length, in inches, of AC?
.

A) 10√3
B) 10√2
C) 300
D) 10
E) 15
If x = 2.0001, which of the following expressions has the largest value?
A) 2 / (x + 2)
B) 2 / (x - 2)
C) (x + 2) / 2
D) 2 / x
E) x / 2
In the rectangle below, the measure of the length AD is 3 times the measure of the width AB. What is the slope of the line segment BD?
.

A) 3
B) 1/3
C) -1/3
D) -3
E) -1/4
What is the product of the two real solutions of the equation

2x = 3 - x²
A) 3
B) 2
C) -2
D) 6
E) -3

Which of the following equations corresponds the data in the table?

x	y
-20	-1/8
-10	-1/3
0	1/2
5	2/9
20	1/12

A) y = 1 / ( x + 2)
B) y = 2 / ( x + 4)
C) y = (x - 1) / ( x + 2)
D) y = 2 / ( x - 2)
E) y = 1 / ( 2x + 2)

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