Digital SAT Practice Questions

Suppose a polynomial \( P(x) = x^2 - ax + b \) has distinct roots \( r_1 \) and \( r_2 \). A second polynomial \( Q(x) = x^2 - cx + d \) has roots \( r_1^3 \) a...

In right triangle \( ABC \), angle \( B \) is a right angle. The lengths of the sides are \( AB = c \), \( BC = a \), and \( AC = b \). Point \( D \) lies on th...

A specific radioactive isotope decays exponentially according to the function \( N(t) = N_0 (0.5)^{\frac{t}{h}} \), where \( N_0 \) is the initial amount of the...

In the equation below, \( a \) and \( b \) are real numbers, and \( i = \sqrt{-1} \). \[ \frac{3 + ai}{b - 2i} = 1 - i \] What is the value of \( a + b \)?...

In biological taxonomy, the concept of a "species" is notoriously porous. The traditional Biological Species Concept (BSC) defines a species as a population who...

Literary critic Julian Vance argues that in her 1922 poetry collection *The Glass Orchard*, fictional poet Elara Vance decisively departs from the Romantic trad...

Despite the newly elected committee’s superficial gestures toward structural reform, their actual policy proposals remained entirely _____, stubbornly adhering ...

The newly discovered exoplanet Kepler-186f is located in the habitable zone of its host star, but astrophysicist Dr. Elena Vargas cautions against premature opt...

Developing a successful quantum computer requires maintaining qubits in a state of superposition. This delicate quantum state is notoriously fragile, as even th...

The graph of the degree-4 polynomial \( f(x) = x^4 + bx^3 + cx^2 + dx + e \) intersects the graph of the linear function \( g(x) = mx + k \) at exactly three po...

If \( (\sqrt{3})^{x+2y} = 27^{x-y} \) and \( 25^{x+y} = \left(\frac{1}{125}\right)^{y-2} \), what is the value of \( x \)?...

In the \( xy \)-plane, a circle with radius \( r \) is centered at the origin. A line intersects the circle at points \( P \) and \( Q \). The coordinates of \(...

At a large corporation, the engineering department has \( 50\% \) more employees than the marketing department, and the sales department has \( 50\% \) fewer em...

A rectangular garden is to be fenced on exactly three sides, with a straight river forming the natural boundary for the fourth side. The total length of the fen...

Biologist Elena K. argues that the severe decline of *Myotis lucifugus* (little brown bat) populations in the Northeast is driven by a critical behavioral misma...

Text 1 The advent of the mechanized loom and the broader tide of industrialization stand as the ultimate emancipators of the working class. By replacing the ago...

The decipherment of Linear B in 1952 provided incontrovertible evidence that the Mycenaeans spoke an early form of Greek. The architect of this monumental disco...

Early 20th-century sociologist Émile Durkheim conceptualized 'anomie' as a state of normlessness arising from rapid societal shifts, leading to profound individ...

The 'Laffer Curve' in macroeconomics suggests that as tax rates increase from 0%, total tax revenue initially increases. However, after a certain optimal rate, ...

The quadratic equation \( 2x^2 - cx + d = 0 \) has roots \( m \) and \( n \). A second quadratic equation, \( x^2 + cx - 2d = 0 \), has roots \( p \) and \( q \...

In the \( xy \)-plane, a circle with center \( O \) at the origin has a radius of \( 8 \). The points \( A \) and \( B \) lie on the circle such that the measur...

The circle given by the equation \( x^2 - 10x + y^2 + 6y = -18 \) is tangent to the line \( y = mx - \frac{7}{4} \) at exactly one point in the \( xy \)-plane. ...

If \( x \) and \( y \) are positive real numbers such that \( x^{3/2} y^{1/2} = 24 \) and \( x^{1/2} y^{3/2} = 54 \), what is the value of \( x + y \)?...

In a certain population of trees, the height of a tree \( H \), in meters, is modeled by the function \( H(t) = C \cdot (1.04)^t \), where \( t \) is the age of...