Digital SAT Practice Questions

Let the quadratic function \( f(x) = ax^2 + bx + c \) have a vertex at the coordinate \( (4, -2) \) and intersect the x-axis at \( x = 2 \) and \( x = 6 \). The...

What is the sum of all real values of \( x \) that satisfy the rational equation below? \[ \frac{x}{x-4} + \frac{4}{x+4} = \frac{32}{x^2 - 16} \]...

In a circle with center \( O \), points \( P \), \( Q \), and \( R \) lie on the circumference such that the inscribed angle \( \angle PQR = \frac{\pi}{4} \) ra...

Let \( P(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are real constants. When \( P(x) \) is divided by \( x - 2 \), the remainder is 5....

The exponential equation \( 2^{2x+1} - 17(2^x) + 8 = 0 \) has two real solutions, \( m \) and \( n \). What is the value of \( m + n \)?...

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

A circle in the \( xy \)-plane has the equation \( x^2 - 4x + y^2 + 6y = k \), where \( k \) is a constant. A line with the equation \( y = 2x - 3 \) intersects...

The rational function \( f(x) = \frac{ax^2 + bx + c}{3x^2 + dx - 6} \), where \( a, b, c, \) and \( d \) are constants, has a horizontal asymptote at \( y = 4 \...

A plane parallel to the circular base of a right circular cone intersects the cone, separating it into a smaller cone and a frustum. The volume of the smaller c...

A dataset consists of 100 positive integers. The mean of the dataset is 54, and the median is 60. The sum of the 25 largest integers in the dataset is exactly 2...

In the \( xy \)-plane, a circle has the equation \( (x - 6)^2 + y^2 = 9 \). A line passing through the origin is tangent to the circle in the first quadrant. Wh...

The quadratic function \( p(x) \) has its vertex at the point \( (h, k) \) and intersects the \( x \)-axis at exactly two points: \( (2, 0) \) and \( (12, 0) \)...

A regular hexagon is inscribed in a circle of radius 6. If a point is chosen at random from the interior of the circle, what is the probability that the point l...

A manufacturing plant produces a liquid mixture of Chemical A and Chemical B. Initially, a 100-gram batch of the mixture contains exactly 30% Chemical A by weig...

Given the equation \( 2^{x+3} - 2^{x+1} = 3^{y+2} - 3^y \), where \( x \) and \( y \) are integers, what is the value of \( x + y \)?...

In the \(xy\)-plane, the parabola given by the equation \( y = -2x^2 + 4x + c \) intersects the line given by the equation \( y = mx + 10 \) at exactly one poin...

The function \( f \) is defined by \( f(x) = \frac{x^3 + ax^2 + bx + c}{x^2 - 4x + 3} \), where \( a, b \), and \( c \) are constants. The graph of \( y = f(x) ...

In the \(xy\)-plane, a circle with center \( O \) has radius \( r \). Points \( A \) and \( B \) lie on the circle such that the measure of minor arc \( AB \) i...

For positive real numbers \( x \) and \( y \), the equations \( x^y = y^x \) and \( y = 4x \) are satisfied simultaneously. What is the exact value of \( x + y ...

An urn contains \( r \) red marbles and \( b \) blue marbles, where \( r \) and \( b \) are positive integers. If two marbles are drawn at random from the urn w...

The system of equations consists of a parabola \( y = px^2 - 4x + 5 \) and a line \( y = 2x + r \), where \( p \) and \( r \) are non-zero real constants. If th...

An isosceles trapezoid has bases of length \( 18 \) and \( 32 \). A circle is inscribed within the trapezoid such that it is tangent to all four of its sides. W...

The polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) has real roots at \( x = 1 \), \( x = -2 \), and a double root at \( x = k \). If the \( y \)-intercept o...

The equation \( 2^{2x+3} - 17(2^x) + 2 = 0 \) has two real solutions, \( x_1 \) and \( x_2 \). What is the value of \( x_1 + x_2 \)?...