Digital SAT Practice Questions

The function \( f \) is defined by \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In the \( xy \)-plane, the vertex of the parab...

A right rectangular prism has a total surface area of \( 144 \) and a space diagonal of length \( \sqrt{52} \). What is the sum of the lengths of all 12 edges o...

In a certain animal population, \( 40\% \) of the individuals exhibit Trait A, and \( 30\% \) of the individuals exhibit Trait B. The event of an individual exh...

In the \( xy \)-plane, the terminal side of an angle of measure \( \theta \) radians in standard position intersects the unit circle at point \( P \). The termi...

The polynomial \( P(x) = x^4 + c x^3 + d x^2 + c x + 1 \), where \( c \) and \( d \) are real constants, has \( x = 1 + i \) as a root (where \( i = \sqrt{-1} \...

A circle in the \( xy \)-plane is given by the equation \[ x^2 + y^2 - 14x - 6y + 33 = 0 \] Line \( L \) is tangent to the circle at the point \( (10, 7) \). Li...

A radioactive isotope decays such that its mass decreases by \( p\\% \) every \( h \) hours. If the mass of the isotope decreases by \( q\\% \) every \( 3h \) h...

How many integers \( x \) satisfy the inequality \[ |x^2 - 12x + 20| \le -x^2 + 12x - 20 \]?...

The circle \( C \) in the \( xy \)-plane is given by the equation \[ x^2 + y^2 = 36 \] A triangle is formed by the origin \( O(0,0) \), the point \( A(6, 0) \),...

In the \( xy \)-plane, the graph of the line \( y = 2x + 5 \) intersects the graph of the parabola \( y = x^2 - 4x + c \) at exactly one point. What is the \( y...

A regular hexagon is inscribed in a circle whose equation in the \( xy \)-plane is \( x^2 + 2\sqrt{3}x + y^2 - 6y = 26 \). What is the area of the hexagon?...

If \( \frac{3x^3 - kx^2 + 13x - 4}{x - 4} = ax^2 + bx + c \) is true for all values of \( x eq 4 \), what is the value of \( a - b + c \)?...

The population of a certain species of fish in a lake is modeled by the function \( P(t) = 800(1.69)^{\frac{t}{a}} \), where \( t \) is the time in months. It i...

In the \( xy \)-plane, the terminal ray of an angle in standard position with measure \( \alpha \) radians intersects the circle with equation \( x^2 + y^2 = 36...

In the \( xy \)-plane, the graph of the exponential function \( f(x) = a(b)^x \), where \( a \) and \( b \) are positive constants, passes through the points \(...

The equation \( 2x^2 + 2y^2 - kx + 16y + 14 = 0 \), where \( k \) is a positive constant, represents a circle in the \( xy \)-plane. If the circle is tangent to...

A system of two linear equations is shown below: \[ \frac{3}{2}x - \frac{k}{4}y = 12 \] \[ cx - 5y = 2k \] In the system above, \( c \) and \( k \) are positive...

The function \( g \) is defined by \( g(x) = x^2 - 6x + 13 \). For all real numbers \( x \), what is the minimum value of \( g(g(x)) \)?...

Solution A contains \( p\% \) of a chemical acid by volume, and Solution B contains \( 3p\% \) of the same acid by volume. When \( x \) liters of Solution A are...

Let \( P(x) = x^3 + ax^2 + bx + c \) be a polynomial where \( a \), \( b \), and \( c \) are rational numbers. It is given that \( x = 3 + \sqrt{2} \) is a root...

In the \( xy \)-plane, the circle with equation \( x^2 + y^2 - 10x + 6y + c = 0 \) is exactly tangent to the line \( y = 2x - 1 \). What is the value of the con...

A specific financial asset routinely loses \( 87.5\% \) of its current value every 12 years. The value of the asset, in dollars, \( t \) years after its initial...

In triangle \( ABC \), point \( D \) lies on side \( AB \) and point \( E \) lies on side \( AC \) such that the line segment \( DE \) is parallel to side \( BC...

The system of equations below has exactly two distinct real solutions for \( (x,y) \). \[ y = -x^2 + 8x - 12 \] \[ y = k \] If the distance between the two poin...