Digital ACT Practice Questions

The polynomial \( P(x) = x^4 - 2x^3 - 11x^2 + 12x + 36 \) has a known root of \( x = 3 \) with multiplicity 2. What are the other two roots of the polynomial?...

A circle in the \( xy \)-plane has its center at \( (3, -2) \) and is tangent to the line \( 4x - 3y = 25 \). What is the area of the circle?...

For \( 0 \le \theta < 2\pi \), how many distinct values of \( \theta \) satisfy the equation \( \sin(2\theta) + \cos(\theta) = 0 \)?...

A hyperbola in the standard \( xy \)-coordinate plane is defined by the equation: \[ \frac{(x - 2)^2}{25} - \frac{(y + 3)^2}{36} = 1 \] What is the slope of the...

A high school academic challenge team consists of 5 seniors and 4 juniors. If a 3-person travel squad is chosen entirely at random, what is the probability that...

Given that \( \sin(\theta) = \frac{3}{5} \) and \( \theta \) lies in Quadrant II \( \left(\frac{\pi}{2} < \theta < \pi\right) \), what is the exact value of \( ...

The first term of an infinite geometric series is 18, and the sum of the series is 54. What is the common ratio \( r \) of the series?...

Matrices \( A \) and \( B \) are given by: \[ A = \begin{pmatrix} 4 & -3 \\ 2 & k \end{pmatrix}, \quad B = \begin{pmatrix} 6 & 1 \\ -2 & 3 \end{pmatrix} \] If t...

In triangle \( ABC \), side \( AB \) has length 7 meters, side \( BC \) has length 8 meters, and \( \cos(\angle B) = \frac{1}{4} \). What is the exact length, i...

For the imaginary unit \( i = \sqrt{-1} \), which of the following expressions is equivalent to: \[ \frac{5 + 2i}{3 - 4i} \]?...

Matrix \( A \) is defined as \( \begin{pmatrix} 2k & 4 \\ 3 & k-1 \end{pmatrix} \). For which of the following values of \( k \) will Matrix \( A \) have no mat...

In triangle \( \triangle PQR \), the length of \( PQ = 7 \), \( QR = 10 \), and the measure of \( \angle PQR = 120^\circ \). What is the exact length of side \(...

A quadratic function \( f(x) = ax^2 + bx + c \) has a minimum value of \( -4 \) at \( x = 1 \). If \( f(0) = -3 \), what is the value of \( a+b+c \)?...

A small, specialized library contains a collection of 6 distinct historical fiction novels and 4 distinct non-fiction biographies. A student, unaware of the spe...

In $\triangle ABC$, the length of side $a$ is $8$ cm, the length of side $b$ is $11$ cm, and the measure of angle $C$ between them is $60^\circ$. What is the ex...

If $i = \sqrt{-1}$, what is the value of the complex expression $\frac{5 - 3i}{2 + 4i}$ in standard form $a + bi$?...

An ellipse centered at the origin in the standard $(x,y)$-coordinate plane has a major horizontal axis of length $14$ and a minor vertical axis of length $8$. W...

A committee of $4$ students is to be randomly selected from a pool of $7$ seniors and $5$ juniors. What is the probability that the committee contains exactly $...

If $\log_2(x) + \log_2(x - 6) = 4$, what is the real solution for $x$?...

Vectors $\mathbf{u} = \langle 3, -4 \rangle$ and $\mathbf{v} = \langle k, 6 \rangle$ are perpendicular in the standard coordinate plane. What is the value of th...

Given the $2 \times 2$ matrix $M = \begin{bmatrix} x & 3 \\ 4 & x - 1 \end{bmatrix}$, for what positive value of $x$ is the determinant of $M$ equal to $8$?...

In an arithmetic sequence, the $4^{\text{th}}$ term is $19$ and the $10^{\text{th}}$ term is $49$. What is the $25^{\text{th}}$ term of this sequence?...

What is the remainder when the polynomial $P(x) = 2x^4 - 5x^3 + 3x - 7$ is divided by $(x - 2)$?...

If $\sin(\theta) = -\frac{5}{13}$ and $\tan(\theta) > 0$, what is the exact value of $\cos(\theta)$?...