Digital SAT Practice Questions

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 \( x_1 + x_2 \)?...

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...