Digital SAT Practice Questions

In the \( xy \)-plane, the graph of the equation \( x^2 + y^2 - 10x + 6y + c = 0 \) is a circle, where \( c \) is a constant. If the line \( y = x - 1 \) is tan...

In the \( xy \)-plane, the graph of \( y = x^4 - 2x^3 + kx^2 + mx - 8 \) intersects the \( x \)-axis at \( (2, 0) \). If the remainder when the polynomial is di...

In the \( xy \)-plane, a circle with center \( O \) is given by the equation \[ x^2 + y^2 - 10x + 6y + 9 = 0 \] Line \( L \) is tangent to the circle at point \...

In the \( xy \)-plane, the parabola \( y = ax^2 + bx + c \) has its vertex at \( (2, 1) \) and passes through the point \( (4, 9) \). A line with the equation \...

A farmer wants to construct a rectangular enclosure adjacent to a straight river. The side along the river requires no fencing. The enclosure is divided into 4 ...

In the complex plane, a sequence of complex numbers is defined by \( z_1 = 1 + i \) and \( z_{n+1} = (1 - i)z_n \) for all integers \( n \geq 1 \). What is the ...

The polynomial \( P(x) = x^4 - 2x^3 + kx^2 - 4x + 8 \), where \( k \) is a constant, is perfectly divisible by the polynomial \( x^2 + 2 \). What is the value o...

A rhombus is drawn in the \( xy \)-plane. The lengths of its two diagonals are 10 and 24. A circle is inscribed perfectly within the rhombus, such that it is ta...

How many distinct real solutions does the following equation have? \[ \frac{x}{x-4} - \frac{4}{x+4} = \frac{32}{x^2-16} \]...

A dataset consists of 11 distinct positive integers. The median of the dataset is 20, and the arithmetic mean is 25. What is the maximum possible value of the l...

The function \( f(x) = a \sin(bx) + c \), where \( a \), \( b \), and \( c \) are positive constants, models a physical phenomenon. The maximum value of \( f(x)...

The system of equations is given by \( y = 2x^2 - kx + 5 \) and \( y = 3x - 7 \), where \( k \) is a constant. For which of the following values of \( k \) does...

The polynomial \( p(x) = ax^3 + bx^2 + cx + d \) has distinct zeros at \( x = -2 \), \( x = 1 \), and \( x = 3 \). The polynomial \( q(x) \) is defined as \( q(...

If \( 8^{x+y} = 16^{x-y} \) and \( 9^{x} = 27^{y+2} \), what is the value of \( x + y \)?...

In a certain high school, the ratio of the number of juniors to the number of seniors is \( 3:2 \). The average (arithmetic mean) score on a standardized test f...

In the \( xy \)-plane, the terminal side of an angle in standard position with measure \( \theta \) radians intersects the unit circle at the point \( (a, b) \)...

In the \( xy \)-plane, the graphs of the equations \( y = cx^2 - 4x + 5 \) and \( y = 2x + b \) intersect at exactly one point. Which of the following equations...

A circle with center \( O \) has a radius of \( 12 \). Points \( A \) and \( B \) lie on the circle such that the length of the minor arc \( AB \) is \( 10\pi \...

The polynomial \( p(x) = x^3 + kx^2 + mx + 24 \) is divisible by \( (x - 2) \). When \( p(x) \) is divided by \( (x - 1) \), the remainder is \( 30 \). What is ...

If \( x > 1 \) and the equation \[ \frac{\sqrt[3]{x^a}}{\sqrt[4]{x^b}} = x^2 \] is true, and it is also given that \( 2a + 3b = 30 \), what is the value of \( a...

In a certain school district, all high schools are classified as either Category A or Category B. The Category A schools have an average of \( 400 \) students p...

The equation \[ \frac{2x}{x-3} + \frac{1}{x+a} = \frac{4x^2 - 5x + 3}{x^2 + (a-3)x - 3a} \] has exactly one valid real solution for \( x \). If \( a \) is a pos...

A circle in the \( xy \)-plane has the equation \[ x^2 + y^2 - 14x + 6y + k = 0 \] where \( k \) is a constant. If the circle is tangent to the line \( 3x - 4y ...

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...