Digital SAT Practice Questions

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) ...

The value of a rare antique automobile is projected to increase by \( r\% \) every 5 years. If the value of the automobile increases by exactly \( 72.8\% \) eve...

In the \( xy \)-plane, the terminal side of an angle of measure \( \alpha \) in standard position intersects the unit circle at point \( (x, y) \). If \( y = -\...

The system of equations below has no solutions. \[ a x + 3 y = 12 \] \[ -5 x + b y = 8 \] If \( a \) and \( b \) are integers, which of the following is the max...

The equation \( y = cx^2 - 4x + 5 \) represents a parabola in the \( xy \)-plane, where \( c \) is a constant. The line \( y = kx - 3 \) intersects the parabola...

Let \( P(x) = ax^4 + bx^3 + cx^2 + dx + e \), where \( a, b, c, d, \) and \( e \) are constants. When \( P(x) \) is divided by \( (x - 2) \), the remainder is 1...

A circle has radius \( r \) and center \( O \). Point \( P \) lies outside the circle such that the distance from \( P \) to \( O \) is \( d \). Two distinct li...

In a large population of registered voters, 60% are homeowners and 50% are over the age of 50. If the probability of randomly selecting a voter who is both a ho...

A scientist is observing two bacterial cultures. Culture X has an initial population of \( P_0 \) and doubles every \( a \) hours. Culture Y has an initial popu...

The system of equations is given by: \[ y = 2x^2 - 4x + c \] \[ y = kx - 5 \] where \( c \) and \( k \) are constants. The system has exactly one real solution ...

The function \( f \) is defined by \( f(x) = \frac{ax^2 + bx - 120}{2x^2 - 14x + 20} \), where \( a \) and \( b \) are constants. The graph of \( y = f(x) \) in...

In the xy-plane, a circle is defined by the equation \( x^2 + y^2 - 10x + 6y = c \), where \( c \) is a constant. A line with the equation \( y = \frac{4}{3}x -...

The absolute value inequality \( |kx - 12| \le c \) has a solution set of \( -2 \le x \le 8 \), where \( k \) and \( c \) are positive constants. What is the va...

A solid glass right circular cone has a height of \( H \) centimeters. A horizontal cut parallel to the base separates the solid cone into a smaller top cone an...

In the \( xy \)-plane, the system of equations is given by: \[ x^2 + y^2 - 8x + 6y + c = 0 \] \[ 3x - 4y = 12 \] where \( c \) is a constant. For what value of ...