Digital SAT Practice Questions

The population of a certain bacteria colony triples every \( k \) hours. If the population increases by exactly \( 400\% \) in \( 14 \) hours, what is the value...

Let \( p(x) = x^3 + ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. When \( p(x) \) is divided by \( x - 2 \), the remainder is \( 15 \). W...

A circle with center \( O \) has a radius of \( 5 \). A chord \( AB \) has a length of \( 8 \). What is the area of the largest triangle that can be inscribed i...

How many distinct real solutions does the equation \[ (x^2 - 4x + 3)^2 - 5(x^2 - 4x + 3) - 6 = 0 \] have?...

In the \(xy\)-plane, a parabola with equation \( y = ax^2 + bx + c \) passes through the points \( (-1, 0) \), \( (0, -6) \), and \( (3, 0) \). A second parabol...

The polynomial \( P(x) = x^3 + cx^2 + dx - 12 \), where \( c \) and \( d \) are constants, is perfectly divisible by \( (x - 2)^2 \). What is the value of \( c ...

The circle with equation \( x^2 + y^2 - 10x + 6y + c = 0 \) has a radius of \( r \). Points \( P \) and \( Q \) lie on the circle such that the length of the mi...

A high-security digital lock requires a 5-digit PIN using the digits 0 through 9. A valid PIN must contain the digit '7' exactly twice, and no other digit in th...

The atmospheric pressure \( P \), in kilopascals (kPa), at an altitude of \( h \) kilometers above Earth's surface can be modeled by the equation \( P(h) = 101 ...

\[ y = 2x^2 - 4x + 7 \] \[ y = kx - 1 \] In the given system of equations, \( k \) is a constant. If the system has exactly one real solution \( (x, y) \), what...

\[ x^2 + y^2 - 10x + 24y + 153 = 0 \] What is the minimum possible distance from the origin \( (0,0) \) to a point \( (x, y) \) on the graph of the given equati...

\[ f(x) = x^{15} - 2x^{14} + ax^3 + bx - 6 \] In the polynomial function \( f \) defined above, \( a \) and \( b \) are constants. When \( f(x) \) is divided by...

\[ 8^{2x - 1} = 16^{\frac{y}{2}} \] \[ 9^{x + y} = 27^{x + 1} \] If \( (x, y) \) is the solution to the system of equations above, what is the value of \( x + y...

In right triangle \( ABC \), the right angle is at vertex \( B \). Point \( D \) lies on the hypotenuse \( AC \) such that line segment \( BD \) is perpendicula...

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...