If \( \cos 4x = \cos 3x \), find the general solution for \( x \).

• (A) \( x = 2n\pi \) or \( x = \frac{2n\pi}{7} \), where \( n \in \mathbb{Z} \)
• (B) \( x = n\pi \) or \( x = \frac{n\pi}{7} \), where \( n \in \mathbb{Z} \)
• (C) \( x = \frac{2n\pi}{7} \) only
• (D) \( x = 2n\pi \) only

Evaluate the integral: \( \int \frac{\sin x}{\sin 4x} \, dx \)

• (A) \( -\frac{1}{8} \log \left| \frac{1+\sin x}{1-\sin x} \right| + \frac{1}{4\sqrt{2}} \log \left| \frac{1+\sqrt{2}\sin x}{1-\sqrt{2}\sin x} \right| + C \)
• (B) \( \frac{1}{8} \log \left| \frac{1+\sin x}{1-\sin x} \right| - \frac{1}{4\sqrt{2}} \log \left| \frac{1+\sqrt{2}\sin x}{1-\sqrt{2}\sin x} \right| + C \)
• (C) \( -\frac{1}{8} \log \left| \frac{1+\sin x}{1-\sin x} \right| + \frac{1}{4\sqrt{2}} \log \left| \frac{1-\sqrt{2}\sin x}{1+\sqrt{2}\sin x} \right| + C \)
• (D) None of these

Find the eigenvalues of the matrix \( A = \begin{pmatrix} 0 & 0 & -1
0 & -1 & 0
0 & 0 & -1 \end{pmatrix} \).

• (A) \( 0, 1, 1 \)
• (B) \( 0, -1, -1 \)
• (C) \( 1, -1, 0 \)
• (D) \( -1, -1, -1 \)

Consider the following logical statements:

R: If \( p \rightarrow q \) is false, then \( p \lor q \) is false.

S: If \( p \leftrightarrow q \) is false, then \( p \lor q \) is false.

Evaluate the truth values of R and S.

• (A) Both R and S are true
• (B) R is true and S is false
• (C) R is false and S is true
• (D) Both R and S are false

Evaluate the integral: \( \int \frac{2x}{x^2-5x+4} \, dx \)

• (A) \( \frac{8}{3} \log |x - 4| - \frac{2}{3} \log |x - 1| + C \)
• (B) \( \frac{2}{3} \log |x - 4| - \frac{8}{3} \log |x - 1| + C \)
• (C) \( \frac{8}{3} \log |x - 1| - \frac{2}{3} \log |x - 4| + C \)
• (D) \( \frac{2}{3} \log |x - 1| - \frac{8}{3} \log |x - 4| + C \)