Digital SAT Practice Questions

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...

The rational function \( f \) is defined by \( f(x) = \frac{ax + b}{cx + d} \), where \( a, b, c, \) and \( d \) are constants. In the \( xy \)-plane, the graph...

In the \( xy \)-plane, the circle with equation \( x^2 + y^2 - 10x + 6y + c = 0 \) is tangent to the line \( y = \frac{3}{4}x + 2 \). What is the value of the c...

A radioactive isotope decays such that its mass, \( M(t) \), in grams, after \( t \) days is modeled by the function \( M(t) = M_0 (0.5)^{\frac{t}{h}} \), where...

Consider the system of linear equations below: \[ 3x - 4y = 12 \] \[ ax + by = c \] If the system has infinitely many solutions, and \( a, b, \) and \( c \) are...

The polynomial \( P(x) = x^3 + ax^2 + bx + c \) has roots at \( x = 2 \) and \( x = -3 \). When \( P(x) \) is divided by \( (x - 1) \), the remainder is \( -12 ...

In the \( xy \)-plane, the parabola given by the equation \( y = cx^2 - 4x + c \) intersects the line \( y = 2x - 5 \) at exactly one point. If \( c > 0 \), wha...

The graph of the equation \( y = ax^2 + bx + c \) in the \( xy \)-plane, where \( a \), \( b \), and \( c \) are constants and \( a > 0 \), has its vertex at \(...

The polynomial \( P(x) = x^4 - 5x^3 + cx^2 - 15x + d \), where \( c \) and \( d \) are constants, is perfectly divisible by \( (x - 3)^2 \). What is the value o...

In the \( xy \)-plane, a circle is given by the equation \[ x^2 + y^2 - 10x + 8y + 16 = 0 \] A line passing through the origin with equation \( y = mx \), where...

If \( (\sqrt{3})^{x} \cdot 9^{y} = 27\sqrt{3} \) and \( 4^{x} \cdot \left(\frac{1}{2}\right)^{y} = 32 \), what is the value of \( x - y \)?...

In right triangle \( \triangle ABC \), the right angle is at \( B \). The lengths of sides \( AB \) and \( AC \) are \( 3 \) and \( 5 \), respectively. Point \(...

In the \( xy \)-plane, the parabola given by the equation \( y = -x^2 + 8x - 14 \) and the line given by the equation \( y = 2x + k \), where \( k \) is a const...

In the \( xy \)-plane, the graph of the equation \( x^2 + y^2 - 12x + ky + 45 = 0 \) is a circle. If the circle is tangent to the y-axis, and \( k > 0 \), what ...

The equation \( \frac{6x^3 - 13x^2 + 21x - k}{3x - 2} = 2x^2 - 3x + 5 + \frac{7}{3x - 2} \) is true for all values of \( x eq \frac{2}{3} \), where \( k \) is a...