Digital SAT Practice Questions

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...

The polynomial \( p(x) = ax^4 - 3x^3 + bx^2 + 4x - 6 \) is divided by \( (x - 2) \), resulting in a remainder of 10. When \( p(x) \) is divided by \( (x + 1) \)...

Consider the following system of equations: \[ 8^{x} \cdot 4^{y} = 32 \] \[ 9^{x} \cdot 27^{y-1} = 81 \] If \( (x, y) \) is the solution to the system of equati...

In the quadratic equation \( 2x^2 - px + q = 0 \), \( p \) and \( q \) are real constants. If one of the solutions to the equation is \( 3 - 4i \), where \( i =...

In the \( xy \)-plane, a circle has the equation \( x^2 + y^2 - 10x + 6y + c = 0 \), where \( c \) is a constant. If the area of the circle is \( 16\pi \), what...

The system of equations is given by: \[ y = x^2 - 6x + k \] \[ y = mx - 4 \] where \( k \) and \( m \) are constants. The equations intersect at exactly one poi...

Let \( P(x) = x^3 + ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are real constants. The polynomial \( P(x) \) has a root at \( x = 3 + 2i \), where \(...

In a circle with center \( O \) and radius \( r \), points \( J \) and \( K \) lie on the circle. The area of sector \( JOK \) is \( \frac{5}{12} \) the area of...

A radioactive isotope decays such that its mass, \( M(t) \), in grams, after \( t \) days is modeled by the function \( M(t) = M_0 (0.5)^{\frac{t}{h}} \), where...

Consider the equation: \[ \frac{2x^2}{x^2 - 9} - \frac{5}{x + 3} = \frac{x}{x - 3} \] What is the sum of all valid solutions to the equation above?...

The system of equations is given by: \[ y = mx - 4 \] \[ x^2 + y^2 - 6x + 8y + 21 = 0 \] For what values of the constant \( m \) does the system have exactly on...