Digital SAT Practice Questions

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 \( q \\% \) ...

A circle with center \( O \) has a radius of \( 6 \). A secant line from a point \( P \) located outside the circle intersects the circle at points \( A \) and ...

The quadratic equation \( 3x^2 - kx + 12 = 0 \) has two distinct real roots, \( r_1 \) and \( r_2 \). If it is given that \( \frac{r_1}{r_2} + \frac{r_2}{r_1} =...

Let \( P(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are real constants. When the polynomial \( P(x) \) is divided by \( x^2 + 1 \), th...

In the \( xy \)-plane, the circle with equation \[ x^2 + (y - 3)^2 = 10 \] intersects the parabola with equation \[ y = x^2 + k \] at exactly three distinct poi...

The solution to the equation \[ 2^{x+2} \cdot 3^{x-1} = 5^{2x} \] is \( x \). Which of the following expressions is equivalent to \( x \)?...

Acute triangle \( ABC \) is inscribed in a circle with a radius of \( 12 \). If \( \sin(A) = \frac{3}{4} \) and the measure of angle \( B \) is \( 30^\circ \), ...

A merchant sells two types of gems: rubies and sapphires. A ruby costs $120 and a sapphire costs $85. The merchant sold a total of \( N \) gems for exactly $4,1...

The system of equations is given by \[ y = cx^2 - 4x + 7 \] \[ y = 2x + k \] where \( c \) and \( k \) are constants. If the system has exactly one real solutio...

In the \( xy \)-plane, a circle with center \( O \) passes through the origin and has a radius of \( 4 \). Points \( A \) and \( B \) lie on the circle such tha...

What is the sum of all valid solutions to the equation below? \[ \frac{x}{x - 3} - \frac{2}{x + 3} = \frac{18}{x^2 - 9} \]...

A microbiologist is observing two bacterial cultures, Culture A and Culture B. Culture A initially contains \( 10,000 \) bacteria and grows by \( p\% \) every \...

The polynomial \( P(x) = ax^3 + bx^2 + cx + d \) has a remainder of \( 5 \) when divided by \( (x - 2) \), and a remainder of \( -3 \) when divided by \( (x + 1...

In the \( xy \)-plane, the graph of the equation \( y = 3x^2 - 4x + c \), where \( c \) is a constant, intersects the graph of the line \( y = 2x - 5 \) at exac...

The equation of a circle in the \( xy \)-plane is given by \( x^2 - 12x + y^2 + ky = -20 \), where \( k \) is a positive constant. If the radius of the circle i...