Digital SAT Practice Questions

A scientist is studying a radioactive isotope in a laboratory. The mass of the isotope decreases by \( 20\% \) every \( 6 \) days. If the initial mass of the is...

Which of the following represents all solutions to the rational equation below? \[ \frac{x}{x - 4} - \frac{4}{x + 4} = \frac{32}{x^2 - 16} \]...

The equation of a circle in the \( xy \)-plane is given below: \[ x^2 + y^2 - 10x + 8y - 8 = 0 \] What is the radius of the circle?...

The system of linear equations below has infinitely many solutions. \[ 5x - 7y = 15 \] \[ kx - \frac{21}{2}y = m \] If \( k \) and \( m \) are constants, what i...

The equation \( y = -3x^2 + bx + c \) represents a parabola in the \(xy\)-plane, where \( b \) and \( c \) are constants. If the vertex of the parabola is at th...

A circle in the \(xy\)-plane has its center at the origin. The total area of the circle is \( 144\pi \). A sector of the circle is bounded by radii \( OA \) and...

The population of a certain species of fish in a newly conserved lake is modeled by the function \( P(t) = 1200(0.82)^{\frac{t}{3}} \), where \( t \) is the num...

The polynomial \( p(x) = 2x^3 - 5x^2 + ax + b \), where \( a \) and \( b \) are constants, is perfectly divisible by \( (x - 2) \). When \( p(x) \) is divided b...

The system of equations is given by: \[ y = ax^2 + 4x - 5 \] \[ y = -2x + b \] where \( a \) and \( b \) are constants, and \( a > 0 \). If the system of equati...

What is the sum of all valid solutions to the given equation? \[ \frac{x^2 - 3x}{x + 2} = \frac{10}{x + 2} + 2 \]...

A circle in the \( xy \)-plane has the equation \( x^2 + y^2 - 12x + ky = 13 \), where \( k \) is a positive constant. If the area of the circle is \( 85\pi \),...

The population of a specific bacterial culture decreases by \( p\% \) every \( h \) hours, where \( p \) and \( h \) are positive constants. If the initial popu...

The function \( f \) is defined by \( f(x) = c \cdot 3^{kx} \), where \( c \) and \( k \) are positive constants. If \( f(x + 2) = 16 \cdot f(x) \) for all real...

The system of equations below consists of a parabola and a line: \[ y = x^2 - 6x + k \] \[ y = 2x - 5 \] For what value of the constant \( k \) does the system ...

Let \( P(x) = ax^3 + bx^2 - 14x + 8 \), where \( a \) and \( b \) are constants. If \( P(x) \) is divisible by both \( (x - 2) \) and \( (x + 1) \), what is the...

The equation \( 2x^2 + 2y^2 - 12x + 16y = 10 \) represents a circle in the \( xy \)-plane. If points \( A \) and \( B \) lie on the circle such that the line se...

A radioactive isotope decays such that its mass decreases by \( 18\% \) every 4 hours. If the initial mass of the isotope is \( M_0 \) grams, which of the follo...

What is the sum of all valid solutions to the following equation? \[ \frac{x}{x - 3} - \frac{2}{x + 3} = \frac{18}{x^2 - 9} \]...

The system of equations below is given by: \[ y = a x^2 + b x + c \] \[ y = m x + k \] where \( a \), \( b \), \( c \), \( m \), and \( k \) are constants, and ...

The rational expression \( \frac{3x^3 + 4x^2 + x - 15}{x + 2} \) is equivalent to the expression \( A x^2 + B x + C + \frac{D}{x + 2} \) for all \( x eq -2 \), ...

If \( x \) and \( y \) are the solutions to the system of equations below, what is the value of the product \( xy \)? \[ 8^x \cdot 16^y = 2^{24} \] \[ \frac{9^x...

A data set consists of 15 distinct integers. The median of the data set is 40, and the range is 50. If the highest and lowest integers are both removed to form ...

In the \( xy \)-plane, an angle \( \theta \) in standard position has its terminal ray intersecting the unit circle at point \( P \). The x-coordinate of point ...

Let \( P(x) = x^3 + ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. When \( P(x) \) is divided by \( x - 2 \), the remainder is \( 14 \). W...