ACT Math Diagnostic Quiz

Calculator is allowed on all questions. Figures are not necessarily drawn to scale.

Question 1

What is \(\frac{3}{4}+\frac{1}{8}\)?

A) \(\frac{1}{2}\)
B) \(\frac{5}{8}\)
C) \(\frac{7}{8}\)
D) \(1\)

Show solution

Use a common denominator: \(\frac{3}{4}=\frac{6}{8}\). Then \(\frac{6}{8}+\frac{1}{8}=\frac{7}{8}\).

Answer: C

Question 2

What value of \(x\) satisfies \(5x-7=18\)?

F) 4
G) 5
H) 6
J) 7

Show solution

Add 7 to both sides: \(5x=25\). Divide by 5 to get \(x=5\).

Answer: G

Question 3

If \(f(x)=2x^2-3\), what is \(f(2)\)?

A) \(-1\)
B) 1
C) 3
D) 5

Show solution

Substitute \(x=2\): \(f(2)=2(2^2)-3=8-3=5\).

Answer: D

Question 4

The triangle shown has base 12 units and height 7 units. What is its area?

F) 36
G) \(42\)
H) \(72\)
J) \(84\)

Show solution

The area of a triangle is \(\frac12 bh\). Thus \(\frac12(12)(7)=42\).

Answer: G

Question 5

The five values in a data set are 6, 8, 9, 12, and 15. What is the median?

A) 6
B) 7
C) 8
D) 9

Show solution

The values are already ordered. The middle of the 5 values is 9.

Answer: D

Question 6

What is the slope of the line shown?

F) \(-3\)
G) \(-\frac13\)
H) \(\frac13\)
J) \(3\)

Show solution

Using the plotted points \((0,-2)\) and \((2,4)\), the slope is \(\frac{4-(-2)}{2-0}=\frac62=3\).

Answer: J

Question 7

Which matrix is equal to \(3\begin{bmatrix}2&-1\\0&4\end{bmatrix}\)?

A) \(\begin{bmatrix}6&-3\\0&12\end{bmatrix}\)
B) \(\begin{bmatrix}5&2\\3&7\end{bmatrix}\)
C) \(\begin{bmatrix}6&-1\\0&12\end{bmatrix}\)
D) \(\begin{bmatrix}2&-3\\0&4\end{bmatrix}\)

Show solution

Multiply every entry by 3: \(3(2)=6\), \(3(-1)=-3\), \(3(0)=0\), and \(3(4)=12\).

Answer: A

Question 8

For the system \(2x+y=11\) and \(x-y=1\), what is the value of \(x\)?

F) 3
G) 4
H) 5
J) 6

Show solution

Add the equations: \(3x=12\), so \(x=4\).

Answer: G

Question 9

The graph shown has horizontal asymptote \(y=1\). Which of the following could be its equation?

A) \(y=2^x+1\)
B) \(y=2^x-1\)
C) \(y=x^2+1\)
D) \(y=2x+1\)

Show solution

The graph shows exponential growth shifted up 1 unit, so its horizontal asymptote is \(y=1\). Thus \(y=2^x+1\).

Answer: A

Question 10

A circle has radius 8. The central angle shown measures \(90^\circ\). What is the area of the sector?

F) \(4\pi\)
G) \(8\pi\)
H) \(12\pi\)
J) \(16\pi\)

Show solution

A \(90^\circ\) sector is one-fourth of the circle. Its area is \(\frac14\pi(8^2)=16\pi\).

Answer: J

Question 11

The table summarizes 70 students. If one of the students is selected at random, what is the probability that the student is in Grade 11 and plays an instrument?

	Plays an instrument	Does not play an instrument	Total
Grade 10	18	12	30
Grade 11	14	26	40
Total	32	38	70

A) \(\frac{1}{5}\)
B) \(\frac{7}{20}\)
C) \(\frac{2}{5}\)
D) \(\frac{4}{7}\)

Show solution

There are 14 Grade 11 students who play an instrument out of 70 students total. The probability is \(\frac{14}{70}=\frac15\).

Answer: A

Question 12

For \(x\ne3\), which value of \(x\) satisfies \(\frac{2}{x-3}+1=\frac{5}{x-3}\)?

F) 4
G) 5
H) 6
J) 8

Show solution

Subtract 1 and multiply by \(x-3\): \(2+(x-3)=5\). Thus \(x-1=5\), so \(x=6\).

Answer: H

Question 13

The function \(f\) is defined by \(f(x)=\begin{cases}2x+1,&x<3\\x^2-4,&x\ge3.\end{cases}\) What is \(f(3)-f(2)\)?

A) \(0\)
B) \(2\)
C) \(4\)
D) \(6\)

Show solution

Since \(3\ge3\), \(f(3)=3^2-4=5\). Since \(2<3\), \(f(2)=2(2)+1=5\). Therefore \(f(3)-f(2)=0\).

Answer: A

Question 14

In the figure, \(PT\) is tangent to the circle at \(T\), \(OT=5\), and \(OP=13\). What is \(PT\)?

F) 10
G) 12
H) 13
J) 18

Show solution

A radius to a tangent point is perpendicular to the tangent, so \(\triangle OTP\) is right. Thus \(PT=\sqrt{13^2-5^2}=\sqrt{144}=12\).

Answer: G

Question 15

A committee of 2 students is chosen at random from 5 seniors and 3 juniors. What is the probability that both students chosen are juniors?

A) \(\frac{1}{28}\)
B) \(\frac{1}{14}\)
C) \(\frac{3}{28}\)
D) \(\frac{3}{14}\)

Show solution

There are \(\binom{8}{2}=28\) possible committees and \(\binom{3}{2}=3\) all-junior committees. The probability is \(\frac{3}{28}\).

Answer: C