GRE Math Practice Test

If \( a, b, \) and \( c \) are the angles of a triangle, and \( a = 2b \) and \( b = 2c \), what is the measure of angle \( c \)?

What is the derivative of \( f(x) = e^{2x} \sin(x) \)?

If \( f(x) = \ln(x) \), what is \( f''(e) \)?

Evaluate the integral: \( \int (3x^2 - 2x + 1) dx \).

Determine the convergence or divergence of the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \).

What is the area enclosed by the curve \( y = x^2 \) and the lines \( x = -1, x = 2, \) and \( y = 0 \)?

What is the volume of the solid obtained by rotating the region bounded by \( y = x^3 \), \( y = 0 \), and \( x = 2 \) around the y-axis?

Find the area under the curve \( y = e^{2x} \) from \( x = 0 \) to \( x = 1 \).

Evaluate the definite integral \( \int_{0}^{2} (3x^2 - 2x) dx \).

Calculate the derivative of \( f(x) = \sqrt{x}(x^2 + 1) \).

Determine the limit: \( \lim_{x \to \infty} \frac{4x^3 - x}{2x^3 + 5} \).

Find the integral of \( \int \frac{1}{x \ln(x)} dx \).

Find the derivative of \( g(t) = t^4 \ln(t) \).

Evaluate: \( \lim_{x \to 0} \frac{\sin(3x)}{x} \).

Calculate the derivative of \( h(x) = \frac{x^2 + 1}{x^2 - 1} \).

What is the second derivative of \( f(x) = \cos(x^2) \)?

Find the equation of the tangent line to the curve \( y = x^3 - 3x \) at \( x = 2 \).

What is the derivative of \( f(x) = x^5 \sin(x^2) \)?

Find the volume of the solid formed by revolving the region bounded by \( y = x^2 \) and \( y = 0 \) around the x-axis between \( x = 0 \) and \( x = 1 \).

Evaluate the limit: \( \lim_{x \to 0} \frac{e^{x} - 1}{x} \).

Find the indefinite integral: \( \int x \cos(x^2) dx \).

Calculate the derivative of \( f(x) = \frac{1}{x^2 + x + 1} \).

If \( f(x) = x^3 - 3x + 2 \), find \( f^{-1}(1) \).

Compute the area between the curves \( y = x^2 \) and \( y = x \) for \( 0 \leq x \leq 1 \).

Evaluate the integral: \( \int \frac{dx}{x\sqrt{x^2 - 1}} \).

Determine the convergence or divergence of the series \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n^3} \).

Compute the limit: \( \lim_{x \to \infty} \frac{\sqrt{x^2 + x} - x}{x} \).

What is the maximum value of the function \( f(x) = -x^2 + 4x - 3 \) on the interval [0, 3]?

Find the derivative of \( g(t) = t^4 \ln(t) \).

Evaluate the integral: \( \int \frac{dx}{x\sqrt{x^2 - 1}} \).

Find the derivative of \( g(y) = \arctan(\sqrt{y}) \).

If \( (2x - 1)(x + 3) = 0 \), what are the possible values of \( x \)?

If the sum of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) is equal to its product of roots, which of the following is true?

What is the solution to the system of equations \( 2x + 3y = 10 \) and \( 4x - y = -2 \)?

If \( 3^{2x-1} = 27 \), what is the value of \( x \)?

The expression \( \frac{x^2 - 4}{x^2 - 2x - 3} \) can be simplified to:

Given \( f(x) = x^2 - 4x + 4 \), for what value of \( x \) is \( f(x) = 0 \)?

If \( 5^{2x-1} = 125 \), what is the value of \( x \)?

Solve the inequality \( 2(x - 3) > 3(x + 1) \).

If \( \frac{1}{x-1} + \frac{1}{x+1} = \frac{2}{x} \), what is the value of \( x \)?

Given a quadratic equation \( x^2 + kx + 16 = 0 \), where \( k \) is a constant. If the equation has two distinct real roots, which of the following must be true?

What are the solutions to the equation \( x^2 - 5x + 6 = 0 \)?

If \( x \) and \( y \) satisfy the equation \( 3x - 2y = 6 \), which of the following represents \( y \) in terms of \( x \)?

What is the coefficient of \( x \) in the expansion of \( (2x - 3)^4 \)?

For what values of \( x \) is the inequality \( \frac{x + 1}{x - 2} < 0 \) true?

The sum of the interior angles of a polygon is 1260 degrees. How many sides does the polygon have?

If \( 2x + 3y = 12 \) and \( 3x + 2y = 12 \), what is the value of \( x - y \)?

What is the hundredth term of the arithmetic sequence 3, 6, 9, ... ?

In a certain sequence, the term \( a_n = \frac{1}{n(n+1)} \) for \( n \geq 1 \). What is the sum of the first 10 terms of this sequence?

A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls is 9?

What is the value of \( n \) in the permutation equation \( P(n, 3) = 60 \)?

If \( x \) and \( y \) are integers such that \( x^2 + y^2 = 25 \), what is the maximum possible value of \( x \)?

If a set contains all numbers \( x \) such that \( 1 < x^2 < 9 \), what is the range of \( x \)?

A circle has a circumference of \( 8\pi \). What is the area of the circle?

If \( 2x + 3y = 12 \) and \( 4x - 5y = 1 \), what is the value of \( x \)?

A rectangle is inscribed in a circle of radius \( r \). If the perimeter of the rectangle is \( 20 \), what is the maximum possible area of the rectangle?

What is the remainder when \( 2^{100} \) is divided by 7?

If \( \log_2 x + \log_2 y = 5 \) and \( \log_2 x - \log_2 y = 1 \), what is the value of \( x \)?

If \( \sin \theta = \frac{3}{5} \) and \( 0 < \theta < \frac{\pi}{2} \), what is \( \cos \theta \)?

What is the equation of the line perpendicular to \( y = 2x + 3 \) and passing through the point (1, -2)?