Digital ACT Practice Questions

For what value of $\theta$, where $0 \le \theta \le \pi$, does $\sin(2\theta) = \cos(\theta)$ have the greatest number of solutions?...

If $x$ and $y$ are positive integers such that $x^2 - y^2 = 31$, what is the value of $x$?...

What is the matrix product $\begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 \\ -2 \end{bmatrix}$?...

In a geometric sequence of positive real numbers, the third term is $18$ and the sixth term is $486$. What is the first term of this sequence?...

If $\tan(\alpha) = \frac{4}{3}$ and $\pi < \alpha < \frac{3\pi}{2}$, what is the value of $\sin(2\alpha)$?...

How many integers between $100$ and $999$, inclusive, are divisible by both $6$ and $9$?...

What is the area, in square units, of the triangle in the standard $(x,y)$-coordinate plane with vertices at $(0, 0)$, $(6, 2)$, and $(4, 8)$?...

If the polynomial $f(x) = x^3 - 3x^2 + 4x - 12$ has a real root at $x = 3$, what are the other two roots of $f(x)$?...

If log_3(x - 2) + log_3(x + 6) = 2, what is the value of x?...

What is the sum of the infinite geometric series 18 - 12 + 8 - 16/3 + ...?...

An ellipse in the standard xy-coordinate plane has the equation (x - 2)^2 / 25 + (y + 1)^2 / 9 = 1. What is the distance between the two foci of this ellipse?...

What are the equations of the asymptotes of the hyperbola with equation x^2 / 16 - y^2 / 25 = 1?...

Which of the following complex numbers is equivalent to the expression (3 + 4i) / (1 - 2i)?...

Two similar solid cones have surface areas of 36pi square centimeters and 81pi square centimeters, respectively. If the volume of the smaller cone is 48 cubic c...

In the expansion of (2x - 3)^5, what is the coefficient of the x^3 term?...

If sin(theta) = 3/5 and theta is an acute angle in Quadrant I, what is the value of sin(2*theta)?...

Which of the following datasets has the greatest standard deviation?...

What is the equation of the tangent line to the circle x^2 + y^2 = 25 at the point (3, 4)?...

What is the value of z in the solution to the system: x + y + z = 6, 2x - y + z = 3, and x + 2y - z = 2?...

In triangle ABC, side a = 5, side b = 12, and angle A = 30°. How many distinct triangles can be formed with these measurements?...

A committee of 3 people is to be selected from a group of 7 candidates: Alice, Bob, and 5 others. If Alice and Bob refuse to serve on the committee together, ho...

What is the determinant of the 2x2 matrix \(\begin{pmatrix} 5 & -3 \\ 4 & 2 \end{pmatrix}\)?...

If \(\cos \theta = -\frac{5}{13}\) and \(\pi < \theta < \frac{3\pi}{2}\), what is the value of \(\tan \theta\)?...

In \(\triangle PQR\), the measure of \(\angle P = 45^\circ\), the measure of \(\angle Q = 30^\circ\), and side \(p = 10\sqrt{2}\). What is the length of side \(...