Digital SAT Practice Questions

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...

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...