Digital SAT Practice Questions

In the \( xy \)-plane, the equations for a parabola and a line are given by the system below: \[ y = x^2 - 4x + c \] \[ y = mx + 2 \] The system of equations ab...

In right triangle \( XYZ \) with the right angle at \( Y \), \( \sin(X) = a \) and \( \cos(X) = b \). If \( a - b = \frac{1}{5} \), what is the value of \( a \t...

For all \( x > 0 \), the expression \( \sqrt[3]{x^a \cdot \sqrt{x^b}} \) is mathematically equivalent to \( x^{\frac{11}{6}} \). If \( a \) and \( b \) are posi...

The graph of \( y = f(x) \) in the \( xy \)-plane is a parabola with vertex \( (-3, 5) \). The function \( g \) is defined by \( g(x) = a \cdot f(x - h) + k \),...

A data set consists of 15 distinct positive integers. The median of the data set is 40, and the range of the data set is 74. What is the maximum possible value ...

In the \( xy \)-plane, the graph of the equation \( x^2 + y^2 - 6x + 8y = 0 \) intersects the graph of the linear equation \( y = 2x - 10 \) at points \( A \) a...

The polynomial \( p(x) = 3x^3 - kx^2 + 17x - 10 \), where \( k \) is a constant, has a factor of \( (x - 2) \). If \( p(x) \) can be rewritten in the factored f...

A local coffee shop sells two coffee blends: Blend A and Blend B. Blend A consists of 20% Arabica beans and 80% Robusta beans by weight. Blend B consists of 60%...

If \( \frac{9^{2x-1}}{\sqrt{27^{x+2}}} = 3^{x-4} \), what is the value of \( x \)?...

In the \( xy \)-plane, a circle with center \( O \) passes through the origin \( (0, 0) \) and has a radius of \( 4\sqrt{2} \). A line \( \ell \) is tangent to ...

The polynomial function \( p \) is defined by \( p(x) = x^4 - 2x^3 + cx^2 + dx - 15 \), where \( c \) and \( d \) are constants. When \( p(x) \) is divided by \...

The quadratic equation \( 2x^2 - 5x - 4 = 0 \) has distinct roots \( u \) and \( v \). What is the value of \( \frac{u}{v} + \frac{v}{u} \)?...

A circle in the \( xy \)-plane has the equation \( x^2 + y^2 - 6x + 8y + c = 0 \), where \( c \) is a constant. If the line \( y = x - 1 \) is tangent to the ci...

A list of \( 7 \) distinct positive integers has a median of \( 15 \) and a range of \( 20 \). What is the maximum possible arithmetic mean of this list of numb...

For a certain quadratic function \( f \), it is known that \( f(2x) - 2f(x) = 2x^2 - 5 \) for all real values of \( x \). If the minimum value of \( f(x) \) in ...

The quadratic equation \( 2x^2 - 2kx + (k^2 - 5k + 6) = 0 \), where \( k \) is a constant, has exactly one real solution for \( x \). What is the sum of all pos...

The circle in the \( xy \)-plane has the equation \( x^2 + y^2 - 10x + 6y + c = 0 \), where \( c \) is a constant. If the area of the circle is \( 16\pi \), wha...

If \( x > 1 \) and \( \frac{\sqrt[3]{x^a \cdot x^b}}{x^{a/2}} = x^4 \), what is the value of \( 2b - a \)?...

The function \( f(x) \) is defined as \( f(x) = |x - 3| - 2 \). A new function \( g(x) \) is defined as \( g(x) = -f(2x) + 4 \). What is the area of the polygon...

The polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) has roots at \( x = 1 \), \( x = -2 \), and \( x = 3 \). When \( P(x) \) is divided by \( x + 1 \), the r...

The equation \( 2x^2 - kx + 3 = 0 \) has distinct roots \( r_1 \) and \( r_2 \). If \( r_1^2 + r_2^2 = 10 \), what is the positive value of the constant \( k \)...

In the xy-plane, the circle given by the equation \( x^2 + y^2 - 12x + 8y + c = 0 \) is tangent to the line \( y = x \). What is the value of the constant \( c ...

The function \( f \) is defined by \( f(x) = \frac{ax + b}{cx + d} \), where \( a \), \( b \), \( c \), and \( d \) are non-zero constants. The graph of \( y = ...

Alloy A consists of 40% copper and 60% zinc by mass. Alloy B consists of 70% copper and 30% zinc by mass. A manufacturer combines \( x \) grams of Alloy A with ...