Digital SAT Practice Questions

A circle with center \( O \) has a radius of \( 6 \). A secant line from a point \( P \) located outside the circle intersects the circle at points \( A \) and ...

The quadratic equation \( 3x^2 - kx + 12 = 0 \) has two distinct real roots, \( r_1 \) and \( r_2 \). If it is given that \( \frac{r_1}{r_2} + \frac{r_2}{r_1} =...

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

In the \( xy \)-plane, the circle with equation \[ x^2 + (y - 3)^2 = 10 \] intersects the parabola with equation \[ y = x^2 + k \] at exactly three distinct poi...

The solution to the equation \[ 2^{x+2} \cdot 3^{x-1} = 5^{2x} \] is \( x \). Which of the following expressions is equivalent to \( x \)?...

Acute triangle \( ABC \) is inscribed in a circle with a radius of \( 12 \). If \( \sin(A) = \frac{3}{4} \) and the measure of angle \( B \) is \( 30^\circ \), ...

A merchant sells two types of gems: rubies and sapphires. A ruby costs $120 and a sapphire costs $85. The merchant sold a total of \( N \) gems for exactly $4,1...

The system of equations is given by \[ y = cx^2 - 4x + 7 \] \[ y = 2x + k \] where \( c \) and \( k \) are constants. If the system has exactly one real solutio...

In the \( xy \)-plane, a circle with center \( O \) passes through the origin and has a radius of \( 4 \). Points \( A \) and \( B \) lie on the circle such tha...

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

A microbiologist is observing two bacterial cultures, Culture A and Culture B. Culture A initially contains \( 10,000 \) bacteria and grows by \( p\% \) every \...

The polynomial \( P(x) = ax^3 + bx^2 + cx + d \) has a remainder of \( 5 \) when divided by \( (x - 2) \), and a remainder of \( -3 \) when divided by \( (x + 1...

In the \( xy \)-plane, the graph of the equation \( y = 3x^2 - 4x + c \), where \( c \) is a constant, intersects the graph of the line \( y = 2x - 5 \) at exac...

The equation of a circle in the \( xy \)-plane is given by \( x^2 - 12x + y^2 + ky = -20 \), where \( k \) is a positive constant. If the radius of the circle i...

The polynomial \( P(x) = x^3 + ax^2 + bx - 24 \), where \( a \) and \( b \) are constants, has roots at \( x = 2 \) and \( x = -3 \). If \( P(x) \) is divided b...

The function \( f \) is defined by \( f(x) = 2^{3x - 1} \). The graph of the function \( g \) in the \( xy \)-plane is the result of translating the graph of \(...

For all \( x eq \frac{2}{3} \) and \( x eq -1 \), the equation below is true: \[ \frac{Ax + B}{3x - 2} - \frac{4}{x + 1} = \frac{5x^2 - Cx + 10}{(3x - 2)(x + 1)...

The system of equations is given by: \[ y = 3x^2 - 12x + k \] \[ y = mx - 5 \] where \( k \) and \( m \) are constants. The graphs of the equations intersect at...

In the \( xy \)-plane, Circle A has the equation \( x^2 - 10x + y^2 + 8y = -5 \). Circle B is formed by translating Circle A to the left by \( 4 \) units and up...

What is the sum of all distinct real solutions to the equation below? \[ \frac{2x^2 - 24x + 64}{x^2 - 16} = x - 5 \]...

A marketing survey categorizes respondents by age ("Under 30" or "30 & Older") and by their preferred subscription tier ("Basic" or "Premium"). The following re...

The mass of a radioactive isotope is modeled by the decay function \( M(d) = M_0 \left(\frac{1}{4\sqrt{2}}\right)^{\frac{d}{k}} \), where \( M_0 \) is the initi...

The parabola defined by the equation \( y = -2x^2 + bx + c \) is graphed in the \( xy \)-plane, where \( b \) and \( c \) are constants. The line defined by the...

The polynomial \( P \) is defined by \( P(x) = x^3 + ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The equation \( P(x) = 0 \) has roots ...