Digital SAT Practice Questions

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...

A specific radioactive isotope decays exponentially according to the function \( N(t) = N_0 (0.5)^{\frac{t}{h}} \), where \( N_0 \) is the initial amount of the...

In the equation below, \( a \) and \( b \) are real numbers, and \( i = \sqrt{-1} \). \[ \frac{3 + ai}{b - 2i} = 1 - i \] What is the value of \( a + b \)?...

The graph of the degree-4 polynomial \( f(x) = x^4 + bx^3 + cx^2 + dx + e \) intersects the graph of the linear function \( g(x) = mx + k \) at exactly three po...

If \( (\sqrt{3})^{x+2y} = 27^{x-y} \) and \( 25^{x+y} = \left(\frac{1}{125}\right)^{y-2} \), what is the value of \( x \)?...

In the \( xy \)-plane, a circle with radius \( r \) is centered at the origin. A line intersects the circle at points \( P \) and \( Q \). The coordinates of \(...

At a large corporation, the engineering department has \( 50\% \) more employees than the marketing department, and the sales department has \( 50\% \) fewer em...

A rectangular garden is to be fenced on exactly three sides, with a straight river forming the natural boundary for the fourth side. The total length of the fen...

The quadratic equation \( 2x^2 - cx + d = 0 \) has roots \( m \) and \( n \). A second quadratic equation, \( x^2 + cx - 2d = 0 \), has roots \( p \) and \( q \...

In the \( xy \)-plane, a circle with center \( O \) at the origin has a radius of \( 8 \). The points \( A \) and \( B \) lie on the circle such that the measur...

The circle given by the equation \( x^2 - 10x + y^2 + 6y = -18 \) is tangent to the line \( y = mx - \frac{7}{4} \) at exactly one point in the \( xy \)-plane. ...

If \( x \) and \( y \) are positive real numbers such that \( x^{3/2} y^{1/2} = 24 \) and \( x^{1/2} y^{3/2} = 54 \), what is the value of \( x + y \)?...

In a certain population of trees, the height of a tree \( H \), in meters, is modeled by the function \( H(t) = C \cdot (1.04)^t \), where \( t \) is the age of...

The system of equations is given by: \[ y = 3x^2 - kx + 12 \] \[ y = 2x - 4 \] where \( k \) is a constant. For which of the following values of \( k \) does th...

Which of the following represents all valid real solutions to the equation? \[ \frac{x^2 - 2x - 8}{x - 4} = \sqrt{2x + 5} \]...