Digital SAT Practice Questions

A circle in the xy-plane has the equation \( x^2 + y^2 - 6x + 8y = c \), where \( c \) is a constant. If the circle intersects the y-axis at exactly one point, ...

The population of a certain bacteria colony is currently \( P \). The population is projected to decrease by \( x\% \) each day for the next 5 days, and then in...

In the xy-plane, an angle with measure \( \theta \) radians is in standard position, and its terminal side passes through the point \( (-\sqrt{3}, 1) \). The te...

Consider the system of equations below: \[ y = ax^2 + bx + c \] \[ y = mx + k \] In the system of equations above, \( a, b, c, m, \) and \( k \) are constants, ...

The system of equations is given by \[ y = a x^2 + b \] \[ x^2 + y^2 = r^2 \] Let \( a \), \( b \), and \( r \) be positive constants such that \( b = r \). How...

The equation below is true for all \( x eq \frac{3}{a} \): \[ \frac{2x^2 + bx + c}{ax - 3} = -4x + 5 + \frac{R}{ax - 3} \] where \( a \), \( b \), \( c \), and ...

A circle in the \( xy \)-plane has the equation \( x^2 + y^2 - 2hx + 4ky + 100 = 0 \), where \( h \) and \( k \) are positive constants such that \( h = 3k \). ...

Solution X contains \( p \% \) acid by volume, and Solution Y contains \( q \% \) acid by volume. If \( x \) gallons of Solution X are mixed with \( y \) gallon...

The function \( f \) is defined by \( f(x) = c \cdot d^x \), where \( c \) and \( d \) are positive constants. The graph of \( y = f(x) \) in the \( xy \)-plane...

The equations \[ y = 3x^2 - kx + 7 \] and \[ y = mx - 5 \] are graphed in the \( xy \)-plane, where \( k \) and \( m \) are positive constants. The line interse...

Consider the equation \[ \frac{2x^2 + kx - 15}{x - 3} = 2x + 7 \] where \( k \) is a constant. What is the sum of all possible values of \( k \) for which the e...

In the \( xy \)-plane, a circle with center \( O \) has a radius of \( 6\sqrt{2} \). Triangle \( ABC \) is inscribed in the circle such that the length of arc \...

A biologist models the growth of a bacteria colony. The colony's population doubles every 15 hours. The function \( P(m) \) represents the population \( m \) mi...

The quadratic function \( f \) is defined by \( f(x) = -2x^2 + 12x - 14 \). The function \( g \) is defined by \( g(x) = f(x - c) + d \), where \( c \) and \( d...

Let \( f(x) = ax^7 + bx^5 + cx^3 + dx + 5 \), where \( a \), \( b \), \( c \), and \( d \) are constants. If \( f(-3) = 12 \), what is the value of \( f(3) \)?...

In the xy-plane, the parabola \( y = -2x^2 + bx + c \) has its vertex at \( (h, k) \) and intersects the x-axis at \( x = -1 \) and \( x = 5 \). A line with equ...

The equation \( x^2 + y^2 - 2ax + 4by + c = 0 \) represents a circle in the xy-plane, where \( a \), \( b \), and \( c \) are positive constants. The circle is ...

In the xy-plane, the line \( y = mx + d \) intersects the parabola \( y = 3x^2 + bx + c \) at exactly one point. If this unique point of intersection is exactly...

Set A consists of 15 integers with a mean of 40 and a standard deviation of 4. Set B consists of 25 integers with a mean of 60 and a standard deviation of 6. Se...

The equation \( x^2 - kx + p = 0 \) has roots \( r_1 \) and \( r_2 \). The equation \( x^2 - px + k = 0 \) has roots \( r_1 + 2 \) and \( r_2 + 2 \). If \( k \)...

In the \( xy \)-plane, a circle has the equation \( x^2 + y^2 - 2ax - 2by + c = 0 \). The circle intersects the \( x \)-axis at points \( A \) and \( B \), wher...

The function \( f(x) = c \cdot d^x \) is defined such that \( c \) and \( d \) are positive constants. If \( f(x+3) = 64 \cdot f(x) \) for all real values of \(...

Let \( P(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are real constants. If two of the roots of the equation \( P(x) = 0 \) are \( 3 - ...

The function \( g(x) = \frac{ax^2 + bx + c}{2x^2 - 8} \), where \( a, b, \) and \( c \) are constants, has a horizontal asymptote at \( y = 3 \) and vertical as...