Digital SAT Practice Questions

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...

The equation of a circle in the \( xy \)-plane is given by: \[ x^2 + y^2 + 6x - 8y - 75 = 0 \] Line \( \ell \) is tangent to this circle at the point \( P(-3, 1...

The equation of a circle in the \( xy \)-plane is given by: \[ x^2 + y^2 - 12x - 16y = 0 \] Line \( \ell \) is tangent to this circle at the point \( P(6, 18) \...

The equation of a circle in the \( xy \)-plane is given by: \[ x^2 + y^2 + 4x + 10y - 35 = 0 \] Line \( \ell \) is tangent to this circle at the point \( P(-2, ...

The equation of a circle in the \( xy \)-plane is given by: \[ x^2 + y^2 - 14x + 2y - 14 = 0 \] Line \( \ell \) is tangent to this circle at the point \( P(7, 7...

A marine bacterial culture has an initial population of 250 and grows according to the model: \[ P(t) = 250 \cdot (1.44)^{\frac{t}{4}} \] where \( t \) is the t...

A marine bacterial culture has an initial population of 500 and grows according to the model: \[ P(t) = 500 \cdot (1.69)^{\frac{t}{6}} \] where \( t \) is the t...

A marine bacterial culture has an initial population of 100 and grows according to the model: \[ P(t) = 100 \cdot (1.96)^{\frac{t}{8}} \] where \( t \) is the t...

A marine bacterial culture has an initial population of 800 and grows according to the model: \[ P(t) = 800 \cdot (2.25)^{\frac{t}{10}} \] where \( t \) is the ...

A marine bacterial culture has an initial population of 1200 and grows according to the model: \[ P(t) = 1200 \cdot (2.56)^{\frac{t}{12}} \] where \( t \) is th...

A marine bacterial culture has an initial population of 400 and grows according to the model: \[ P(t) = 400 \cdot (1.21)^{\frac{t}{2}} \] where \( t \) is the t...