Digital SAT Practice Questions

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? \[ \frac{x^2 - 2x - 8}{x - 4} = \sqrt{2x + 5} \]...

A circle in the \( xy \)-plane has the equation \( x^2 + y^2 - 10x + 6y + 18 = 0 \). A straight line passing through the center of the circle intersects the cir...

In a certain school district, the average (arithmetic mean) test score for High School A is \( 84 \), and the average test score for High School B is \( 72 \). ...

The polynomial \( p(x) = ax^3 + bx^2 + cx + d \) has roots at \( x = -2 \), \( x = 3 \), and \( x = 5 \). If \( p(x) \) is divided by \( x - 4 \), the remainder...

A radioactive isotope decays such that its mass, \( M(t) \), in grams, after \( t \) years is given by the function \( M(t) = M_0 (0.5)^{\frac{t}{h}} \), where ...

A circle in the \( xy \)-plane has the equation \[ x^2 + y^2 - 6x + 8y = 75 \] A line with the equation \( y = mx + b \) is tangent to the circle at the point \...

The quadratic equation \[ 3x^2 + px + q = 0 \] where \( p \) and \( q \) are constants, has roots \( x = \frac{-2 + \sqrt{10}}{3} \) and \( x = \frac{-2 - \sqrt...

A data set consists of 50 positive integers. The mean of this data set is \( m \), and the standard deviation is \( s \). A new data set is formed by multiplyin...

The function \( f(x) = a \sin(bx + c) + d \) has a maximum value of \( 12 \) and a minimum value of \( 2 \) in the \( xy \)-plane. The period of the function is...

The polynomial \( P(x) = x^4 - 3x^3 + kx^2 - 12x + 28 \) is exactly divisible by \( x^2 + 4 \). What is the value of the constant \( k \)?...

A circle in the \( xy \)-plane passes through the origin \( (0,0) \) and has its center at \( (h, k) \). The circle intersects the positive x-axis at \( (a, 0) ...