SAT Randomized Questions - 1 Full Math Practice Test - Answers and Detailed Explanations at the END

For \(f(x) = -3x^2 + 12x - 5\), let \(g(x) = f(x - 6)\). For what value of \(x\) does \(g(x)\) reach its maximum?

A researcher initially measures 8,000 units of a certain substance. Six hours later, the substance's quantity has increased to 64,000 units. Assuming exponential growth, the formula \( P = C(2)^{rt} \) represents the amount of substance, where \( C \) is a constant and \( P \) is the quantity after \( t \) hours. What is the value of \( r \)?

Square P has side lengths that are 20 times the side lengths of square Q. The area of square P is \( k \) times the area of square Q. What is the value of \( k \)?

The given equation describes the relationship between the number of cats, \( x \), and the number of dogs, \( y \), that can be cared for at a pet shelter on a given day. If the shelter cares for 24 dogs on a given day, how many cats can it care for on this day?

\( 3.5x + 7y = 140 \)

Given the system of equations:

\( 10x + 6y = 120 \)
\( 2x + y = 15 \)

The solution to the system is \( (x, y) \). What is the value of \( y \)?

3x + 5y = 24

6x = 10y - b

In the given system of equations, b is a constant. If the system has no solution, what is the value of b?

If \(\frac{c}{d} = 2\) and \(\frac{40c}{qd} = 2\), what is the value of \(q\)?

For \(x > 0\), the function \(h\) is defined as follows: \(h(x)\) equals 80% of \(x\). Which of the following could describe this function?

In a lab experiment, a cell culture begins with 3,000 cells. Eight hours later, the cell count increases to 24,000. Using the formula \( P = C(2)^{rt} \), where \( C \) and \( r \) are constants, and \( P \) is the number of cells \( t \) hours after the initial count, find the value of \( r \).

If \( 39 \) is \( p \% \) of \( 65 \), what is \( p \% \) of \( 39 \)?

In the xy-plane, the equation \( 9x^2 + 144px + 9y^2 - 96py = -576p^2 \) represents a circle. The length of the radius of the circle is np, where n and p are positive constants. What is the value of n?

The measure of angle X is \( \frac{\pi}{6} \) radians. The measure of angle Y is \( \frac{\pi}{3} \) radians greater than the measure of angle X. What is the measure of angle Y, in degrees?

\(m(x) = 7500(0.70)^{x/12}\)

The function \(m\) gives the value, in dollars, of a piece of laboratory equipment after \(x\) months of use. If the equipment's value decreases each year by \(q\)% of its value from the preceding year, what is the value of \(q\)?

One of the factors of \(4x^3 + 32x^2 + 96x\) is \(x + b\), where \(b\) is a positive constant. What is the smallest possible value of \(b\)?

A runner used water from a bottle during a marathon. The function j(x) = -0.5x + 10 approximates the volume, in liters, of water the runner had remaining after x kilometers of the marathon. Which statement is the best interpretation of the y-intercept of the graph of y=j(x) in the xy-plane in this context?

Square R has side lengths that are 12 times the side lengths of square S. The area of square R is \( k \) times the area of square S. What is the value of \( k \)?

In the xy-plane, the equation \( 16x^2 + 192px + 16y^2 - 128py = -1024p^2 \)represents a circle. The length of the radius of the circle is np, where n and p are positive constants. What is the value of n?

In a recent referendum, 4 times as many people voted against a measure as those who voted in favor of it. A survey indicated that 12,000 more people voted against it than in favor. How many people voted in favor of the measure?

f(x) = 7(2)^x. The function f is defined by the given equation. If g(x) = f(x + 4), which of the following equations defines the function g?

The function \( f \) is defined by \( f(x) = 400(0.5)^x \). What is the value of \( f(0) \)?

The table below gives the coordinates of two points on a line in the xy-plane:

| x | y |
|----|----|
| \(p\) | -7 |
| \(p + 8\) | 21 |

The y-intercept of the line is at \((p - 6, b)\), where \(p\) and \(b\) are constants. What is the value of \(b\)?

\(x^2 - 6x + 3 = 0.\) One solution to the given equation can be written as \(3 + \sqrt{k}\), where \(k\) is a constant. What is the value of \(k\)?

In the xy-plane, the equation \( 4x^2 + 64px + 4y^2 - 32py = -256p^2 \) represents a circle. The length of the radius of the circle is np, where n and p are positive constants. What is the value of n?

The function \( h(t) = 45,000 \cdot (1.04)^{t/200} \) represents the number of organisms in a culture \( t \) minutes after starting. How much time, in hours, is needed for the culture's population to double?

A right triangle has legs with lengths of \( 8 , \text{cm} \) and \( 15 , \text{cm} \). If the length of the hypotenuse, in cm, can be written in the form \( \sqrt{d} \), where \( d \) is an integer, what is the value of \( d \)?

Let \(f(x) = 5x^2 - 20x + 95\) and define \(g(x) = f(x + 3)\). For what value of \(x\) does \(g(x)\) reach its minimum?

5x + 7y = 20 + n

14y = 10x - 35

In the given system of equations, n is a constant. If the system has no solution, what is the value of n?

At how many points do the graphs of the given equations intersect in the xy-plane?

\( 5x + 4y = 16 \) and \( -10x - 8y = -32 \)

For \(x > 0\), the function \(q\) is defined as follows: \(q(x)\) equals 130% of \(x\). Which of the following could describe this function?

Given \(f(x) = 3x^2 + 24x + 59\), the function \(g\) is defined by \(g(x) = f(x - 4)\). For what value of \(x\) does \(g(x)\) reach its minimum?

A construction worker used concrete to build foundations. The function k(x) = -8x + 64 approximates the amount of concrete, in cubic feet, the worker had remaining after building x foundations. Which statement is the best interpretation of the y-intercept of the graph of y=k(x) in the xy-plane in this context?

\(g(x) = 8000(0.75)^{x/6}\)

The function \(g\) gives the value, in dollars, of a certain piece of equipment after \(x\) months of use. If the value of the equipment decreases each year by \(q\)% of its value from the preceding year, what is the value of \(q\)?

If \(\frac{r}{s} = 6\) and \(\frac{48r}{ks} = 6\), what is the value of \(k\)?

The measure of angle S is \( \frac{2\pi}{3} \) radians. The measure of angle T is \( \frac{\pi}{4} \) radians greater than the measure of angle S. What is the measure of angle T, in degrees?

The exponential function \( g \) is defined by \( g(x) = 25 \cdot b^x \), where \( b \) is a positive constant. If \( g(2) = 625 \), what is \( g(\frac{1}{2}) \)?

\(x(nx - 72)\) = -36. In the given equation, n is an integer constant. If the equation has no real solution, what is the least possible value of n?

At how many points do the graphs of the given equations intersect in the xy-plane?

\( y = -4x + 3 \)
\( y = -4x + 3 \)

The equation describes the relationship between the number of parrots, \( p \), and the number of snakes, \( s \), that can be cared for in a wildlife rehabilitation center. If the center cares for 10 snakes, how many parrots can it care for?

\( 4p + 2s = 100 \)

In the given equation, \( (7x + p)(5x^2 - 25)(4x^2 - 14x + 5p) = 0 \), where \( p \) is a positive constant. The sum of the solutions to the equation is \( 10 \). What is the value of \( p \)?

\(t = \frac{n}{5x + 6y}\). The given equation relates the distinct positive numbers \(t, n, x,\) and \(y\). Which equation correctly expresses \(5x + 6y\) in terms of \(t\) and \(n\)?

A vote on a school funding bill showed that 3 times as many voters voted against the bill as those who voted in favor. It was also reported that 18,000 more voters voted against it than in favor. How many voters voted against the bill?

7x - 4y = 8

14y = kx + 16

In the given system of equations, k is a constant. If the system has no solution, what is the value of k?

The table below gives the coordinates of two points on a line in the xy-plane:

| x | y |
|----|----|
| \(r\) | 8 |
| \(r + 4\) | -24 |

The y-intercept of the line is at \((r - 2, b)\), where \(r\) and \(b\) are constants. What is the value of \(b\)?

\(x^2 - 10x + 2 = 0.\) One solution to the given equation can be written as \(5 + \sqrt{k}\), where \(k\) is a constant. What is the value of \(k\)?