Mathematical Mastery Challenge

What is the limit of the sequence \( \frac{1}{n} \) as \( n \) approaches infinity?

• 0

• 1

• Infinity

• Undefined

If \( f(x) = e^x \), what is the derivative of \( f(x) \)?

• \( e^x \)

• \( x \cdot e^x \)

• 1

• \( e^{x^2} \)

Evaluate the integral \( \int \sin(x) \, dx \).

• \( -\cos(x) \)

• \( \cos(x) \)

• \( x \sin(x) \)

• \( -x \cos(x) \)

What is the value of the determinant of the following matrix? \[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \]

• -2

• 2

• 10

• -10

Solve for \( x \) in the equation \( \log_2(x) = 5 \).

• 10

• 25

• 32

• 50

What is the sum of the infinite geometric series \( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \)?

• 1

• 2

• \( \frac{1}{2} \)

• \( \frac{3}{2} \)

If \( A \) and \( B \) are two events with \( P(A) = 0.5 \), \( P(B) = 0.4 \), and \( P(A \cap B) = 0.2 \), what is \( P(A \cup B) \)?

• 0.7

• 0.9

• 0.5

• 0.2

What is the solution to the differential equation \( \frac{dy}{dx} = 3y \)?

• \( y = Ce^{3x} \)

• \( y = 3Ce^x \)

• \( y = Ce^{x^3} \)

• \( y = 3x + C \)

Find the eigenvalues of the matrix \( \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix} \).

• 2 and 5

• 3 and 4

• 1 and 6

• -1 and 8

What is the Maclaurin series expansion of \( \cos(x) \)?

• \( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \)

• \( 1 + \frac{x^2}{2!} - \frac{x^4}{4!} + \cdots \)

• \( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \)

• \( x + \frac{x^3}{3!} - \frac{x^5}{5!} + \cdots \)