Digital SAT Practice Questions

In the \( xy \)-plane, the graph of the equation \( x^2 + y^2 - 12x + ky + 45 = 0 \) is a circle. If the circle is tangent to the y-axis, and \( k > 0 \), what ...

The equation \( \frac{6x^3 - 13x^2 + 21x - k}{3x - 2} = 2x^2 - 3x + 5 + \frac{7}{3x - 2} \) is true for all values of \( x eq \frac{2}{3} \), where \( k \) is a...

What is the sum of all real solutions to the equation \( x - 4\sqrt{x} = 21 \)?...

A radioactive isotope, Isotope A, decays over time according to the model \( A(t) = A_0 (0.5)^{\frac{t}{h}} \), where \( A_0 \) is the initial mass, \( A(t) \) ...

Text 1 Historians have traditionally attributed the Late Bronze Age collapse (c. 1200 BCE) to a sudden, catastrophic series of environmental cataclysms, primari...

In a dense literary critique, a scholar examines the evolution of a renowned 19th-century novelist’s prose style: "Though her early manuscripts were characteriz...

In standard macroeconomic models, the velocity of money—the rate at which currency changes hands in an economy—is typically assumed to be relatively stable, all...

In her analysis of Charlotte Brontë's use of psychological space, literary critic Dr. Aris Thorne argues that the motif of silence is deliberately subverted in ...

Despite its barren, desolate appearance, the Atacama Desert is home to a startling diversity of extremophile microbes. Astrobiologists are particularly interest...

For how many real values of \( k \) does the equation \[ \frac{x^2 - kx + 20}{x - 4} = x + 2 \] have no solution for \( x \)?...

In the \( xy \)-plane, the circle given by the equation \[ x^2 + y^2 - 6x + 8y = 0 \] intersects the positive x-axis at point \( A \). A line tangent to the cir...

The graph of the quadratic function \( f(x) = ax^2 + bx + c \) has its vertex at \( (4, -10) \) and intersects the x-axis at \( x = 2 \) and \( x = 6 \). A new ...

In the \( xy \)-plane, the terminal ray of angle \( \theta \) in standard position intersects the unit circle at point \( P(a, b) \), where \( a 0 \). If \( \ta...

A dataset consists of 99 positive integers \( a_1, a_2, \ldots, a_{99} \) such that \( a_1 \le a_2 \le \ldots \le a_{99} \). The median of the dataset is 50, th...

Investigating the "hygiene hypothesis," epidemiologist Maria Strachan analyzed historical demographic data regarding autoimmune disorders. The hypothesis posits...

The development of quantum electrodynamics (QED) is universally regarded as a triumph of twentieth-century physics, yielding theoretical predictions that match ...

Despite her reputation as a staunch defender of classical economic principles, the senator’s proposed trade legislation was far from orthodox. Critics noted tha...

The intricate social hierarchies of the spotted hyena are entirely matrilineal, meaning that females invariably outrank males in the clan's structure. In fact, ...

In 1856, the young chemist William Perkin was attempting to synthesize quinine, a valuable anti-malarial drug, from coal tar. His experiment was a spectacular f...

In the \(xy\)-plane, the graph of the equation \( x^2 + y^2 - 10x + 6y + 9 = 0 \) is a circle. For what value of \(b\) does the line given by the equation \( y ...

The polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) has strictly real coefficients. It is given that \( P(1 + i) = 0 \) and \( P(2) = 0 \). When \( P(x) \) i...

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

A circle with center \( O \) has a radius of \( 6 \). A secant line from a point \( P \) located outside the circle intersects the circle at points \( A \) and ...

The quadratic equation \( 3x^2 - kx + 12 = 0 \) has two distinct real roots, \( r_1 \) and \( r_2 \). If it is given that \( \frac{r_1}{r_2} + \frac{r_2}{r_1} =...