Digital SAT Practice Questions

The function \( f(x) = a \sin(bx) + c \), where \( a \), \( b \), and \( c \) are positive constants, models a physical phenomenon. The maximum value of \( f(x)...

The system of equations is given by \( y = 2x^2 - kx + 5 \) and \( y = 3x - 7 \), where \( k \) is a constant. For which of the following values of \( k \) does...

The polynomial \( p(x) = ax^3 + bx^2 + cx + d \) has distinct zeros at \( x = -2 \), \( x = 1 \), and \( x = 3 \). The polynomial \( q(x) \) is defined as \( q(...

If \( 8^{x+y} = 16^{x-y} \) and \( 9^{x} = 27^{y+2} \), what is the value of \( x + y \)?...

In a certain high school, the ratio of the number of juniors to the number of seniors is \( 3:2 \). The average (arithmetic mean) score on a standardized test f...

In the \( xy \)-plane, the terminal side of an angle in standard position with measure \( \theta \) radians intersects the unit circle at the point \( (a, b) \)...

In the \( xy \)-plane, the graphs of the equations \( y = cx^2 - 4x + 5 \) and \( y = 2x + b \) intersect at exactly one point. Which of the following equations...

A circle with center \( O \) has a radius of \( 12 \). Points \( A \) and \( B \) lie on the circle such that the length of the minor arc \( AB \) is \( 10\pi \...

The polynomial \( p(x) = x^3 + kx^2 + mx + 24 \) is divisible by \( (x - 2) \). When \( p(x) \) is divided by \( (x - 1) \), the remainder is \( 30 \). What is ...

If \( x > 1 \) and the equation \[ \frac{\sqrt[3]{x^a}}{\sqrt[4]{x^b}} = x^2 \] is true, and it is also given that \( 2a + 3b = 30 \), what is the value of \( a...

In a certain school district, all high schools are classified as either Category A or Category B. The Category A schools have an average of \( 400 \) students p...

The equation \[ \frac{2x}{x-3} + \frac{1}{x+a} = \frac{4x^2 - 5x + 3}{x^2 + (a-3)x - 3a} \] has exactly one valid real solution for \( x \). If \( a \) is a pos...

A circle in the \( xy \)-plane has the equation \[ x^2 + y^2 - 14x + 6y + k = 0 \] where \( k \) is a constant. If the circle is tangent to the line \( 3x - 4y ...

The polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) has roots at \( x = 1 \), \( x = -2 \), and \( x = 3 \). When \( P(x) \) is divided by \( (x+1) \), the r...

The function \( P(t) = P_0 (a)^{t} \) models the population of a certain bacteria culture after \( t \) hours. The population triples every 45 minutes. If the f...

In a recent survey of a certain population, the ratio of people who drink coffee to those who do not is \( 3:1 \), and the ratio of people who drink tea to thos...

In the \( xy \)-plane, the graph of the equation \( y = 2x^2 + bx + 8 \) intersects the line \( y = cx - 10 \) at exactly one point. If \( b \) and \( c \) are ...

A regular hexagon is inscribed in a circle with center \( O \). The area of the circle is \( 144\pi \). A sector of the circle is formed by two adjacent vertice...

Pump A can fill a tank in \( x \) hours, and Pump B can fill the same tank in \( y \) hours, where \( x \) and \( y \) are positive constants. When Pump A and P...

Let \( P(x) = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants. When \( P(x) \) is divided by \( (x - 2) \), the remainder is 15. When \(...

The equation \[ \frac{x^2 - kx + 16}{x - 4} = x + 5 \] has no solution for \( x \). If \( k > 0 \), what is the value of the constant \( k \)?...

In the \(xy\)-plane, the graphs of the quadratic functions \( y = ax^2 + bx + c \) and \( y = dx^2 + ex + f \) intersect the \(x\)-axis at the exact same two di...

Suppose a polynomial \( P(x) = x^2 - ax + b \) has distinct roots \( r_1 \) and \( r_2 \). A second polynomial \( Q(x) = x^2 - cx + d \) has roots \( r_1^3 \) a...

In right triangle \( ABC \), angle \( B \) is a right angle. The lengths of the sides are \( AB = c \), \( BC = a \), and \( AC = b \). Point \( D \) lies on th...