Digital SAT Practice Questions

Consider the equation \[ \frac{2x^2 + kx - 15}{x - 3} = 2x + 7 \] where \( k \) is a constant. What is the sum of all possible values of \( k \) for which the e...

In the \( xy \)-plane, a circle with center \( O \) has a radius of \( 6\sqrt{2} \). Triangle \( ABC \) is inscribed in the circle such that the length of arc \...

A biologist models the growth of a bacteria colony. The colony's population doubles every 15 hours. The function \( P(m) \) represents the population \( m \) mi...

The quadratic function \( f \) is defined by \( f(x) = -2x^2 + 12x - 14 \). The function \( g \) is defined by \( g(x) = f(x - c) + d \), where \( c \) and \( d...

Let \( f(x) = ax^7 + bx^5 + cx^3 + dx + 5 \), where \( a \), \( b \), \( c \), and \( d \) are constants. If \( f(-3) = 12 \), what is the value of \( f(3) \)?...

In the xy-plane, the parabola \( y = -2x^2 + bx + c \) has its vertex at \( (h, k) \) and intersects the x-axis at \( x = -1 \) and \( x = 5 \). A line with equ...

The equation \( x^2 + y^2 - 2ax + 4by + c = 0 \) represents a circle in the xy-plane, where \( a \), \( b \), and \( c \) are positive constants. The circle is ...

In the xy-plane, the line \( y = mx + d \) intersects the parabola \( y = 3x^2 + bx + c \) at exactly one point. If this unique point of intersection is exactly...

Set A consists of 15 integers with a mean of 40 and a standard deviation of 4. Set B consists of 25 integers with a mean of 60 and a standard deviation of 6. Se...

The equation \( x^2 - kx + p = 0 \) has roots \( r_1 \) and \( r_2 \). The equation \( x^2 - px + k = 0 \) has roots \( r_1 + 2 \) and \( r_2 + 2 \). If \( k \)...

In the \( xy \)-plane, a circle has the equation \( x^2 + y^2 - 2ax - 2by + c = 0 \). The circle intersects the \( x \)-axis at points \( A \) and \( B \), wher...

The function \( f(x) = c \cdot d^x \) is defined such that \( c \) and \( d \) are positive constants. If \( f(x+3) = 64 \cdot f(x) \) for all real values of \(...

Let \( P(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are real constants. If two of the roots of the equation \( P(x) = 0 \) are \( 3 - ...

The function \( g(x) = \frac{ax^2 + bx + c}{2x^2 - 8} \), where \( a, b, \) and \( c \) are constants, has a horizontal asymptote at \( y = 3 \) and vertical as...

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 ...