PAX Domain 3: Mathematics

If \( \frac{5}{x} = \frac{10}{x + 2} \), what is the value of \( x \)?

A cylindrical tank with a radius of 4 meters and a height of 10 meters is filled with water. If the water is pumped out to fill cubic containers with side lengths of 2 meters, how many containers can be filled?

Solve the equation for \( x \): \( 2\log(x) - \log(5) = \log(40) \).

If the sequence is defined by \( a_n = 3^n + n^2 \), what is the fifth term?

A researcher records the growth of a bacterial culture every hour. The number of bacteria triples every hour. If the culture started with 500 bacteria, how many bacteria will there be after 4 hours?

If \( \sin(x) = \frac{1}{2} \) and \( x \) is in the first quadrant, what is \( \cos(x) \)?

Evaluate the limit: \( \lim_{{x \to 3}} \frac{x^2 - 9}{x - 3} \).

The function \( f(x) = x^3 - 6x^2 + 11x - 6 \) has a root at \( x = 1 \). What are the other two roots?

What is the sum of the infinite series \( 6 + 2 + \frac{2}{3} + \frac{2}{9} + \ldots \)?

A set of data has a mean of 50 and a standard deviation of 5. If the data is transformed using the function \( y = 2x + 3 \), what is the new standard deviation?

A sequence is defined as \( a_n = (-1)^n \cdot n \). What is the sum of the first 5 terms of this sequence?

A train travels 60 miles in 2 hours and then 90 miles in 3 hours. What is the average speed for the entire trip?

What is the inverse function of \( f(x) = 2x - 3 \)?

The sides of a triangle are in the ratio 3:4:5. If the perimeter of the triangle is 36 cm, what is the length of the longest side?

If the function \( f(x) = x^2 + bx + c \) has a maximum point at \( x = -3 \), what is the value of \( b \)?

A box contains 6 red, 4 blue, and 5 green balls. If three balls are picked at random, what is the probability that they are all red?

Solve the inequality: \( 3(2x - 4) > 6(x - 1) \).

If \( f(x) = \sqrt{x} \) and \( g(x) = x^2 \), what is \( f(g(4)) \)?

Calculate the determinant of the matrix \(\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}\

What is the limit as \(x\) approaches 0 of \(\frac{\sin(x)}{x}\

If the series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) converges to \(\frac{\pi^2}{6}\), what is the sum of its first four term

Solve for \(x\) in the equation \( \log_2(x) + \log_2(8) = 4 \

What is the solution set for the inequality \( 3x - 4 < 2(x - 1) \

A fair coin is flipped 5 times. What is the probability of getting exactly 3 head

Find the equation of the line perpendicular to \( y = 2x + 3 \) that passes through the point (1, -2

If a matrix \(A\) has an inverse \(A^{-1}\), what is the determinant of \(A^{-1}\) when \(\det(A) = 5\",,,,

Evaluate the integral \( \int 3x^2 dx \

Solve the system of equations: \( \begin{cases} 3x + 2y = 5 \\ x - y = 3 \end{cases} \

Find the range of the function \( f(x) = 2x^2 - 3x + 1 \

What is the next number in the sequence: 2, 6, 18, 54, ..

A particle moves along the x-axis so that its position at time \( t \) is given by \( x(t) = t^3 - 6t^2 + 9t \). At what time \( t \) is the velocity zer

Find the area of a sector of a circle with radius 6 cm and a central angle of \( 30^{\circ} \

If \( \sin(x) = \frac{1}{2} \) and \( x \) is in the first quadrant, what is \( \cos(x) \)?

The sides of a triangle are in the ratio 3:4:5. If the perimeter of the triangle is 36 cm, what is the length of the longest side?

A box contains 6 red, 4 blue, and 5 green balls. If three balls are picked at random, what is the probability that they are all red?

A cylindrical tank with a radius of 4 meters and a height of 10 meters is filled with water. If the water is pumped out to fill cubic containers with side lengths of 2 meters, how many containers can be filled?

Evaluate the limit: \( \lim_{{x \to 3}} \frac{x^2 - 9}{x - 3} \).