Digital SAT Practice Questions

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 ...

In the xy-plane, an angle with measure \( \theta \) radians is in standard position, and its terminal side intersects the unit circle at point \( P \). The x-co...

The quadratic function \( f \) is defined by \( f(x) = a(x - h)^2 + k \), where \( a \), \( h \), and \( k \) are constants. The graph of \( y = f(x) \) in the ...

In the \(xy\)-plane, a circle with center \( O \) has the equation \( x^2 + y^2 - 12x + 8y = 12 \). Point \( A \) lies on the circle, and a line is drawn tangen...

Let \( P(x) = x^4 + ax^3 - 7x^2 + bx - 12 \), where \( a \) and \( b \) are constants. When \( P(x) \) is divided by \( (x - 2) \), the remainder is \( -18 \). ...

In a certain school district, the ratio of the number of elementary school teachers to high school teachers is \( 5 : 3 \). The average (arithmetic mean) annual...

For positive constants \( m \) and \( n \), the following system of equations is true: \[ m^k \cdot n^{3k} = 10^{14} \] \[ 2\log_{10}(m) + 6\log_{10}(n) = 16 \]...

The function \( f \) is defined by \( f(x) = x^2 - 6x + c \), and the function \( g \) is defined by \( g(x) = |x - 3| + d \), where \( c \) and \( d \) are con...

A circle with center \( O \) has a radius of \( 6 \). A triangle \( ABC \) is inscribed in the circle such that \( AB \) is a diameter of the circle. A second, ...

A company operates entirely out of a North office and a South office. Currently, the average (arithmetic mean) salary of employees in the North office is \( \$8...

The population of City A grows by \( 50\% \) every 5 years, and the population of City B grows by \( q\% \) every 10 years, where \( q \) is a positive constant...