Digital SAT Practice Questions

A circle in the \( xy \)-plane has the equation \[ x^2 + y^2 - 14x + 6y + k = 0 \] where \( k \) is a constant. If the circle is tangent to the line \( 3x - 4y ...

The polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) has roots at \( x = 1 \), \( x = -2 \), and \( x = 3 \). When \( P(x) \) is divided by \( (x+1) \), the r...

The function \( P(t) = P_0 (a)^{t} \) models the population of a certain bacteria culture after \( t \) hours. The population triples every 45 minutes. If the f...

In a recent survey of a certain population, the ratio of people who drink coffee to those who do not is \( 3:1 \), and the ratio of people who drink tea to thos...

While nineteenth-century geologist Charles Lyell was a champion of uniformitarianism—the idea that the Earth was shaped by slow-moving geological forces still i...

Some evolutionary biologists suggest that the rapid speciation of cichlid fishes in Lake Victoria is driven primarily by sexual selection—specifically, female m...

Text 1 Advocates of the nineteenth-century gold standard argued that tying a nation’s currency to a physical commodity prevented reckless government spending an...

The newly discovered exoplanet Kepler-1649c orbits its red dwarf host star within the habitable zone, a region where liquid water could theoretically exist. How...

Archaeological consensus has long held that the transition from foraging to agriculture in the ancient Levant was driven strictly by necessity: a changing clima...

In the \( xy \)-plane, the graph of the equation \( y = 2x^2 + bx + 8 \) intersects the line \( y = cx - 10 \) at exactly one point. If \( b \) and \( c \) are ...

A regular hexagon is inscribed in a circle with center \( O \). The area of the circle is \( 144\pi \). A sector of the circle is formed by two adjacent vertice...

Pump A can fill a tank in \( x \) hours, and Pump B can fill the same tank in \( y \) hours, where \( x \) and \( y \) are positive constants. When Pump A and P...

Let \( P(x) = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants. When \( P(x) \) is divided by \( (x - 2) \), the remainder is 15. When \(...

The equation \[ \frac{x^2 - kx + 16}{x - 4} = x + 5 \] has no solution for \( x \). If \( k > 0 \), what is the value of the constant \( k \)?...

Historian Evelyn Ward argues that the rapid industrialization of the 19th-century British textile industry was primarily driven by the sudden availability of ch...

Text 1 Classical utilitarianism dictates that moral actions are strictly those that maximize overall happiness and minimize suffering for the greatest number of...

Jupiter’s moon Europa is encased in a massive crust of ice, beneath which planetary scientists strongly suspect lies a vast, global ocean of liquid water. The i...

Hidden beneath the dense, impenetrable canopy of the remote Guatemalan jungle for over a millennium, ______ Which choice completes the text so that it conforms ...

Despite the diplomatic committee's exhaustive, months-long efforts to mediate the territorial dispute, the two factions remained firmly entrenched in their resp...

In the \(xy\)-plane, the graphs of the quadratic functions \( y = ax^2 + bx + c \) and \( y = dx^2 + ex + f \) intersect the \(x\)-axis at the exact same two di...

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 \( a + b \)?...