Digital SAT Practice Questions

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...

What is the sum of all real solutions to the equation \( x - 4\sqrt{x} = 21 \)?...

A radioactive isotope, Isotope A, decays over time according to the model \( A(t) = A_0 (0.5)^{\frac{t}{h}} \), where \( A_0 \) is the initial mass, \( A(t) \) ...

For how many real values of \( k \) does the equation \[ \frac{x^2 - kx + 20}{x - 4} = x + 2 \] have no solution for \( x \)?...

In the \( xy \)-plane, the circle given by the equation \[ x^2 + y^2 - 6x + 8y = 0 \] intersects the positive x-axis at point \( A \). A line tangent to the cir...

The graph of the quadratic function \( f(x) = ax^2 + bx + c \) has its vertex at \( (4, -10) \) and intersects the x-axis at \( x = 2 \) and \( x = 6 \). A new ...

In the \( xy \)-plane, the terminal ray of angle \( \theta \) in standard position intersects the unit circle at point \( P(a, b) \), where \( a 0 \). If \( \ta...

A dataset consists of 99 positive integers \( a_1, a_2, \ldots, a_{99} \) such that \( a_1 \le a_2 \le \ldots \le a_{99} \). The median of the dataset is 50, th...

In the \(xy\)-plane, the graph of the equation \( x^2 + y^2 - 10x + 6y + 9 = 0 \) is a circle. For what value of \(b\) does the line given by the equation \( y ...

The polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) has strictly real coefficients. It is given that \( P(1 + i) = 0 \) and \( P(2) = 0 \). When \( P(x) \) i...

A radioactive isotope decays exponentially such that its mass decreases by \( p \\% \) every \( h \) hours. If the mass of the isotope decreases by \( q \\% \) ...