Digital SAT Practice Questions

The polynomial \( P(x) = x^3 + ax^2 + bx - 24 \), where \( a \) and \( b \) are constants, has roots at \( x = 2 \) and \( x = -3 \). If \( P(x) \) is divided b...

The function \( f \) is defined by \( f(x) = 2^{3x - 1} \). The graph of the function \( g \) in the \( xy \)-plane is the result of translating the graph of \(...

For all \( x eq \frac{2}{3} \) and \( x eq -1 \), the equation below is true: \[ \frac{Ax + B}{3x - 2} - \frac{4}{x + 1} = \frac{5x^2 - Cx + 10}{(3x - 2)(x + 1)...

The system of equations is given by: \[ y = 3x^2 - 12x + k \] \[ y = mx - 5 \] where \( k \) and \( m \) are constants. The graphs of the equations intersect at...

In the \( xy \)-plane, Circle A has the equation \( x^2 - 10x + y^2 + 8y = -5 \). Circle B is formed by translating Circle A to the left by \( 4 \) units and up...

What is the sum of all distinct real solutions to the equation below? \[ \frac{2x^2 - 24x + 64}{x^2 - 16} = x - 5 \]...

A marketing survey categorizes respondents by age ("Under 30" or "30 & Older") and by their preferred subscription tier ("Basic" or "Premium"). The following re...

The mass of a radioactive isotope is modeled by the decay function \( M(d) = M_0 \left(\frac{1}{4\sqrt{2}}\right)^{\frac{d}{k}} \), where \( M_0 \) is the initi...

The parabola defined by the equation \( y = -2x^2 + bx + c \) is graphed in the \( xy \)-plane, where \( b \) and \( c \) are constants. The line defined by the...

The polynomial \( P \) is defined by \( P(x) = x^3 + ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The equation \( P(x) = 0 \) has roots ...

The expression \( \frac{x^2 + kx + 16}{x - 3} \) can be rewritten in the equivalent form \( x + p + \frac{r}{x - 3} \), where \( k \), \( p \), and \( r \) are ...

A circle in the \( xy \)-plane has the equation \( x^2 + y^2 - 12x + 8y + 27 = 0 \). Points \( A \) and \( B \) lie on the circle such that the length of minor ...

A data set consists of 9 strictly distinct positive integers. The median of the data set is 20, and the range is 17. What is the maximum possible arithmetic mea...

The polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants, has roots at \( x = 1 \), \( x = -2 \), and ...

In the \( xy \)-plane, the circle defined by the equation \( x^2 + y^2 - 6x + 8y = c \) has an area of \( 16\pi \). A line defined by the equation \( y = mx + b...

The rational equation \[ \frac{kx^2 - 18}{2x^2 - mx + n} = 4 \] has infinitely many solutions for \( x \), where \( k \), \( m \), and \( n \) are constants. Wh...

A specific radioactive isotope decays such that its mass decreases by 19% every 45 years. The mass \( M(t) \), in grams, of the isotope \( t \) decades after an...

In the \( xy \)-plane, a circle with radius \( r \) is completely inscribed within a rhombus. The lengths of the diagonals of the rhombus are 10 and 24. What is...

Let \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) be a polynomial, where \( a \), \( b \), \( c \), and \( d \) are constants. The polynomial \( P(x) \) is perfectly ...

In triangle \( PQR \), the measure of angle \( Q \) is \( 90^\circ \). The length of the hypotenuse \( PR \) is \( \sqrt{104} \). If \( \sin(P) + \cos(P) = \fra...

The equation \( \frac{x^2 - cx + 12}{x - 4} = 2 \) has exactly one real solution for \( x \). If \( c \) is an integer, what is the value of \( c \)?...

A data set consists of 15 distinct positive integers. The median of the data set is 25, and the range of the data set is 40. What is the maximum possible sum of...

A line \( L \) is tangent to the parabola \( y = x^2 \) at the point \( (2, 4) \). A circle is tangent to line \( L \) at the point \( (2, 4) \) and is also tan...

In the \( xy \)-plane, the quadratic equation \( y = -2x^2 + bx + c \) intersects the line passing through the origin at exactly one point, \( (3, 5) \). What i...