ACT Math - Full-Length Practice Test 2
Calculator is allowed on all questions. Figures are not necessarily drawn to scale.
Question 1
If \(5x-9=31\), what is \(x\)?
Show solution
\(5x=40\), so \(x=8\).
Answer: A
Question 2
A store increases a \(50\) price by \(20\%\). What is the new price?
Show solution
\(50(1.20)=60\).
Answer: G
Question 3
The rectangle shown has side lengths 12 and 5. What is the length of its diagonal \(d\)?
Show solution
\(d=\sqrt{12^2+5^2}=\sqrt{169}=13\).
Answer: C
Question 4
What is the median of \(4,7,9,11,18\)?
Show solution
The middle value is 9.
Answer: J
Question 5
Which expression is equivalent to \(3(4y+2)-5y\)?
Show solution
\(12y+6-5y=7y+6\).
Answer: A
Question 6
If \(f(x)=2x^2-1\), what is \(f(3)\)?
Show solution
\(f(3)=2(9)-1=17\).
Answer: G
Question 7
A bag contains 3 red, 5 blue, and 2 green marbles. What is the probability of selecting a green marble?
Show solution
Use \(P(\text{event})=\frac{\text{favorable outcomes}}{\text{total outcomes}}\).
There are 10 marbles and 2 are green, so \(P(\text{green})=\frac{2}{10}=\frac15\).
Answer: C
Question 8
What is \(2^3\cdot2^4\)?
Show solution
\(2^3\cdot2^4=2^7=128\).
Answer: J
Question 9
The line shown passes through \((0,3)\) and \((3,0)\). What is its slope?
Show solution
Use the slope formula \(m=\frac{y_2-y_1}{x_2-x_1}\).
Substitute the two points: \(m=\frac{0-3}{3-0}=-1\).
Answer: A
Question 10
A circle has radius 9. What is its circumference?
Show solution
\(C=2\pi r=18\pi\).
Answer: G
Question 11
An arithmetic sequence has first term 5 and common difference 6. What is the fourth term?
Show solution
For an arithmetic sequence, use \(a_n=a_1+(n-1)d\).
Here \(a_1=5\) and \(d=6\).
The fourth term is \(a_4=5+(4-1)(6)=5+18=23\).
Answer: C
Question 12
Which equation has solution \(x=-4\)?
Show solution
\(-4+6=2\), so \(x+6=2\) is satisfied.
Answer: J
Question 13
The scatterplot shown has which type of association?
Show solution
As \(x\) increases, the plotted values generally decrease, indicating a negative association.
Answer: A
Question 14
If \(g(x)=x+7\), which value of \(x\) gives \(g(x)=12\)?
Show solution
\(x+7=12\), so \(x=5\).
Answer: G
Question 15
What is the area of a triangle with base 14 and height 6?
Show solution
\(A=\frac12(14)(6)=42\).
Answer: C
Question 16
Solve the system \(2x+y=11\) and \(x-y=1\). What is \(x\)?
Show solution
Adding gives \(3x=12\), so \(x=4\).
Answer: J
Question 17
If \(x^2-2x-15=0\), what is the positive solution?
Show solution
\((x-5)(x+3)=0\), so the positive solution is 5.
Answer: A
Question 18
The parabola shown has \(x\)-intercepts \(-2\) and \(4\). What is the equation of its axis of symmetry?
Show solution
The axis lies midway between the roots: \(x=\frac{-2+4}{2}=1\).
Answer: G
Question 19
A data set has mean 24. If 5 is added to every value, what is the new mean?
Show solution
Adding 5 to every observation adds 5 to the mean, giving 29.
Answer: C
Question 20
For \(x\ne3\), which expression is equivalent to \(\frac{x^2-9}{x-3}\)?
Show solution
\(x^2-9=(x-3)(x+3)\).
The expression simplifies to \(x+3\).
Answer: J
Question 21
The graph of \(y=|x|-4\) is shifted how from the graph of \(y=|x|\)?
Show solution
Subtracting 4 shifts the graph vertically downward by 4 units.
Answer: A
Question 22
If \(\sin\alpha=\frac35\) for an acute angle \(\alpha\), what is \(\cos\alpha\)?
Show solution
A \(3\)-\(4\)-\(5\) right triangle gives \(\cos\alpha=\frac45\).
Answer: G
Question 23
A \(90^\circ\) sector of a circle has radius 10. What is the area of the sector?
Show solution
Use the circle area formula \(A=\pi r^2\).
The full circle area is \(\pi(10)^2=100\pi\).
The sector is one fourth of the circle, so its area is \(\frac14(100\pi)=25\pi\).
Answer: C
Question 24
If \(P(A)=0.6\), \(P(B)=0.5\), and \(P(A\cap B)=0.3\), what is \(P(A\cup B)\)?
Show solution
\(P(A\cup B)=0.6+0.5-0.3=0.8\).
Answer: J
Question 25
Which equation represents a line perpendicular to \(y=\frac12x+4\)?
Show solution
A perpendicular slope is the negative reciprocal of \(\frac12\), which is \(-2\).
Answer: A
Question 26
If \(h(x)=3(2^x)\), what is \(h(4)\)?
Show solution
\(h(4)=3(16)=48\).
Answer: G
Question 27
A sample of 200 voters contains 118 who favor a proposal. What proportion of the sample favors it?
Show solution
\(118/200=0.59\).
Answer: C
Question 28
Two rays form a \(45^\circ\) angle as shown. What is the measure of its supplementary angle?
Show solution
Supplementary angles total \(180^\circ\).
The measure is \(180-45=135^\circ\).
Answer: J
Question 29
If \(f(x)=x^2+1\) and \(g(x)=x-2\), what is \(f(g(4))\)?
Show solution
\(g(4)=2\), then \(f(2)=5\).
Answer: A
Question 30
What is the solution of \(|2x-5|=7\) with the greater value of \(x\)?
Show solution
\(2x-5=7\) gives \(x=6\); the other solution is \(x=-1\).
The greater is 6.
Answer: G
Question 31
If \(z=3-4i\), what is \(z\overline z\)?
Show solution
\(z\overline z=(3-4i)(3+4i)=3^2+4^2=25\).
Answer: C
Question 32
If \(\log_3(x-1)=2\), what is \(x\)?
Show solution
\(x-1=3^2=9\), so \(x=10\).
Answer: J
Question 33
Let \(A=\begin{bmatrix}1&2\\-3&4\end{bmatrix}\) and \(B=\begin{bmatrix}5&-1\\2&0\end{bmatrix}\). What is the bottom-left entry of \(A+B\)?
Show solution
The bottom-left entry is \(-3+2=-1\).
Answer: A
Question 34
A sequence is defined by \(a_1=2\) and \(a_n=2a_{n-1}+1\). What is \(a_4\)?
Show solution
Use the recurrence to build each term from the previous one.
Starting with \(a_1=2\), \(a_2=2(2)+1=5\).
Then \(a_3=2(5)+1=11\) and \(a_4=2(11)+1=23\).
Answer: G
Question 35
For \(f(x)=\frac{2x+1}{x-4}\), which value is excluded from the domain?
Show solution
The denominator cannot be zero, so \(x\ne4\).
Answer: C
Question 36
The vector shown has terminal point \((-3,3)\). What is its magnitude?
Show solution
The magnitude is \(\sqrt{(-3)^2+3^2}=\sqrt{18}=3\sqrt2\).
Answer: J
Question 37
If \(p(x)=x^3-4x^2-x+4\), which is a factor of \(p(x)\)?
Show solution
\(p(1)=1-4-1+4=0\): \(x-1\) is a factor.
Answer: A
Question 38
In the right triangle shown, what is \(\tan\theta\)?
Show solution
Relative to \(\theta\), opposite is 7 and adjacent is 24: \(\tan\theta=\frac7{24}\).
Answer: G
Question 39
A random variable \(X\) takes values 0, 1, and 4 with probabilities 0.2, 0.5, and 0.3, respectively. What is \(E(X)\)?
Show solution
\(E(X)=0(0.2)+1(0.5)+4(0.3)=1.7\).
Answer: C
Question 40
If \(f(x)=\begin{cases}x^2,&x<2\\3x-1,&x\ge2\end{cases}\), what is \(f(f(1))\)?
Show solution
\(f(1)=1\), and since 1 is still less than 2, \(f(1)=1\) again.
Answer: J
Question 41
A circle has equation \((x-2)^2+(y+3)^2=49\). Which point is its center?
Show solution
In \((x-h)^2+(y-k)^2=r^2\), the center is \((h,k)=(2,-3)\).
Answer: A
Question 42
If \(1\le a\le4\) and \(2\le b\le6\), what is the greatest possible value of \(\frac{3b}{a}\)?
Show solution
Maximize the numerator and minimize the denominator: \(a=1\), \(b=6\), giving \(18\).
Answer: G
Question 43
A quadratic function has zeros \(-5\) and 3 and satisfies \(f(0)=-30\). Which equation could define \(f\)?
Show solution
\(f(x)=a(x+5)(x-3)\).
Since \(f(0)=-15a=-30\), \(a=2\).
Answer: C
Question 44
In a normal-shaped distribution, a score is 2 standard deviations above the mean. If the mean is 70 and the standard deviation is 6, what is the score?
Show solution
\(70+2(6)=82\).
Answer: J
Question 45
A rectangle has perimeter 40. If its length is \(x\), its area is \(A(x)=x(20-x)\). What is the maximum possible area?
Show solution
\(A(x)=-x^2+20x\) has vertex at \(x=10\), giving \(A=10(10)=100\).
Answer: A