Digital SAT Practice Questions

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...

A circle in the xy-plane has the equation \( x^2 + y^2 - 6x + 8y = c \), where \( c \) is a constant. If the circle intersects the y-axis at exactly one point, ...

The population of a certain bacteria colony is currently \( P \). The population is projected to decrease by \( x\% \) each day for the next 5 days, and then in...

In the xy-plane, an angle with measure \( \theta \) radians is in standard position, and its terminal side passes through the point \( (-\sqrt{3}, 1) \). The te...

Consider the system of equations below: \[ y = ax^2 + bx + c \] \[ y = mx + k \] In the system of equations above, \( a, b, c, m, \) and \( k \) are constants, ...

The system of equations is given by \[ y = a x^2 + b \] \[ x^2 + y^2 = r^2 \] Let \( a \), \( b \), and \( r \) be positive constants such that \( b = r \). How...

The equation below is true for all \( x eq \frac{3}{a} \): \[ \frac{2x^2 + bx + c}{ax - 3} = -4x + 5 + \frac{R}{ax - 3} \] where \( a \), \( b \), \( c \), and ...

A circle in the \( xy \)-plane has the equation \( x^2 + y^2 - 2hx + 4ky + 100 = 0 \), where \( h \) and \( k \) are positive constants such that \( h = 3k \). ...

Solution X contains \( p \% \) acid by volume, and Solution Y contains \( q \% \) acid by volume. If \( x \) gallons of Solution X are mixed with \( y \) gallon...

The function \( f \) is defined by \( f(x) = c \cdot d^x \), where \( c \) and \( d \) are positive constants. The graph of \( y = f(x) \) in the \( xy \)-plane...

The equations \[ y = 3x^2 - kx + 7 \] and \[ y = mx - 5 \] are graphed in the \( xy \)-plane, where \( k \) and \( m \) are positive constants. The line interse...