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

In the given equation, \( (3x + p)(5x^2 - 45)(3x^2 - 16x + 6p) = 0 \), where \( p \) is a positive constant. The sum of the solutions to the equation is \( \frac{20}{3} \). What is the value of \( p \)?

What percentage of \(700\) is \(350\)?

One gallon of paint will cover 200 square feet of a surface. A room has a total wall area of \(w\) square feet. Which equation represents the total amount of paint \(P\), in gallons, needed to paint the walls of the room twice?

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

\( y = \frac{1}{2}x + 5 \)
\( y = -\frac{1}{2}x + 5 \)

For the function q, the value of q(x) decreases by 45% for every increase in the value of x by 1. If q(0) = 14, which equation defines q?

Poll results:
Candidate P - 725 votes
Candidate Q - 275 votes

According to the poll of 1,000 voters, Candidate P received 725 votes, and Candidate Q received 275 votes. If 8,000 people vote in the election, how many more votes would Candidate P receive compared to Candidate Q?

In the given equation, \( (5x + p)(4x^2 - 32)(2x^2 - 10x + 2p) = 0 \), where \( p \) is a positive constant. The sum of the solutions to the equation is \( 15 \). What is the value of \( p \)?

Starting with 5,000 bacteria, a biologist records 40,000 bacteria after five hours. If the growth follows \( P = C(2)^{rt} \), where \( P \) is the bacterial count and \( t \) is time in hours, what is the value of \( r \)?

The function \( f \) is defined by \( f(x) = 300(0.2)^x \). What is the value of \( f(0) \)?

In the equation \( y = x^2 - 16x + 40 \), which relates \( x \) and \( y \), for what value of \( x \) does \( y \) reach its minimum?

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?

The function \( f(x) = \frac{1}{6}(x - 10)^2 + 4 \) describes the height of a basketball above the court \( f(x) \), in feet, \( x \) seconds after it was thrown, where \(\) 0 < x < 20 [/latex]. Which of the following is the best interpretation of the vertex of the graph of [latex] y = f(x) [/latex] in the [latex] xy [/latex]-plane?

The measure of angle P is \( \frac{5\pi}{6} \) radians. The measure of angle Q is \( \frac{\pi}{4} \) radians greater than the measure of angle P. What is the measure of angle Q, in degrees?

A baker used dough to make loaves of bread. The function g(x) = -3x + 50 approximates the amount of dough, in pounds, the baker had remaining after making x loaves of bread. Which statement is the best interpretation of the y-intercept of the graph of y=g(x) in the xy-plane in this context?

Value: 10, 15, 20, 25, 30

Data set A frequency: 3, 5, 7, 9, 11

Data set B frequency: 11, 9, 7, 5, 3

Data set A and Data set B each contain 35 values. The table shows the frequencies of the values for each data set. Which of the following statements best compares the means of the two data sets?

Given the system of equations:

\( 20x + 3y = 150 \)
\( 4x + y = 30 \)

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

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?

The function \( f \) is defined by \( f(x) = 500(0.05)^x \). What is the value of \( f(0) \)?

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

\( y = 3x + 7 \)
\( y = 3x - 4 \)

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 \( r \)?

One gallon of paint will cover 250 square feet of a surface. A room has a total wall area of \(w\) square feet. Which equation represents the total amount of paint \(P\), in gallons, needed to paint the walls of the room twice?

\(x(px - 90) = -49.\) In the given equation, p is an integer constant. If the equation has no real solution, what is the least possible value of p?

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?

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 \( p \)?

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?

A scientist observes an initial population of 1,500 cells. Twelve hours later, the population grows to 24,000. Using the exponential growth formula \( P = C(2)^{rt} \), where \( P \) is the cell count at \( t \) hours, determine the value of \( r \).

h(t) = 200 - 5t
The function h models the amount of water, in gallons, in a container t hours after it begins to leak. According to the model, what is the predicted amount of water, in pints, leaking from the container each day?

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?

Triangles \( \triangle PQR \) and \( \triangle STU \) are congruent, where \( P \) corresponds to \( S \), and \( Q \) and \( T \) are right angles. The measure of angle \( P \) is 25°. What is the measure of angle \( U \)?

What percentage of \(400\) is \(120\)?

\(x^2 - 4x - 5 = 0.\) One solution to the given equation can be written as \(2 + \sqrt{k}\), where \(k\) is a constant. What is the value of \(k\)?

\( 3y - 9 = 3(y - 3) \). How many solutions does the given equation have?

The population of a certain bacteria colony is initially 2,000. The population doubles every 30 minutes. Which equation represents the population \( p \) after \( x \) hours?

Which ordered pair is a solution to the given equations:

\( y = (x + 4)(x - 2) \)
\( y = 5x - 4 \)

A national park has an area of 10,240,000 square yards. What is the area, in square miles, of this park? (1 mile = 1760 yards)

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

| x | y |
|----|----|
| \(q\) | -10 |
| \(q - 9\) | -40 |

The y-intercept of the line is at \((q + 5, b)\), where \(q\) and \(b\) are constants. What is the value of \(b\)?

Value: 8, 16, 24, 32, 40

Data set A frequency: 5, 5, 10, 5, 5

Data set B frequency: 6, 8, 6, 8, 6

Data set A and Data set B each contain 30 values. The table shows the frequencies of the values for each data set. Which of the following statements best compares the means of the two data sets?

Square X has side lengths that are 50 times the side lengths of square Y. The area of square X is \( k \) times the area of square Y. What is the value of \( k \)?

For \(x > 0\), the function \(p\) is defined as follows: \(p(x)\) equals 75% of \(x\). Which of the following could describe this function?

The half-life of a radioactive substance is 3 years. The initial quantity of the substance is 5 grams. Which equation best represents the quantity \( Q \) of the substance remaining after \( x \) minutes?

A line in the xy-plane has a slope of \( -2 \) and passes through the point \( (3, 7) \). Which of the following equations represents this line?

Value: 5, 10, 15, 20, 25

Data set A frequency: 6, 8, 10, 8, 6

Data set B frequency: 7, 8, 9, 8, 7

Data set A and Data set B each contain 38 values. The table shows the frequencies of the values for each data set. Which of the following statements best compares the means of the two data sets?

One of the factors of \(5x^3 + 35x^2 + 60x\) is \(x + b\), where \(b\) is a positive constant. What is the smallest possible value of \(b\)?

Given the system of equations:

\( 18x + y = 81 \)
\( 3x + y = 27 \)

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