If \( \sin x \cos x = \frac{1}{4} \), then the general solution is:

• (A) \( \frac{n\pi}{2} + (-1)^n \frac{\pi}{12} \)
• (B) \( \frac{n\pi}{2} + \frac{\pi}{4} \)
• (C) \( n\pi \pm \frac{\pi}{6} \)
• (D) \( \frac{n\pi}{2} + (-1)^n \frac{\pi}{6} \)

If \( n \in \mathbb{Z} \), then the expression \( \frac{2^n}{(1 - i)^{2n}} + \frac{(1 + i)^{2n}}{2^n} \) is equal to:

• (A) \( 2(-1)^n \)
• (B) \( 0 \)
• (C) \( 2 \)
• (D) \( 2^n \)

The value of \( \int \frac{x^2 - 1}{(x^4 + 3x^2 + 1)\tan^{-1}(x + \frac{1}{x})} \, dx \) is:

• (A) \( \log|\tan^{-1}(x + \frac{1}{x})| + C \)
• (B) \( \tan^{-1}(x + \frac{1}{x}) + C \)
• (C) \( -\log|\tan^{-1}(x + \frac{1}{x})| + C \)
• (D) \( \frac{1}{2} \log |x + \frac{1}{x}| + C \)

If \( f(x) = \int \frac{x^2}{(1 - x^2)(1 + \sqrt{1 - x^2})} \, dx \) and \( f(0) = 2 \), then \( f\left(\frac{1}{2}\right) \) is:

• (A) \( 2 + \frac{1}{2} \log 3 - \frac{\sqrt{3}}{2} \)
• (B) \( 2 + \log 3 - \sqrt{3} \)
• (C) \( 2 + \sqrt{3} - \log 3 \)
• (D) \( 2 + \frac{\sqrt{3}}{2} - \frac{1}{2} \log 3 \)

If \( A = \begin{bmatrix} \sin\alpha & -\cos\alpha
\cos\alpha & \sin\alpha \end{bmatrix} \) and \( \alpha \in \left( \frac{\pi}{2}, \frac{3\pi}{2} \right) \). If \( A + A^T = I \), then \( \alpha = \)

• (A) \( \frac{2\pi}{3} \)
• (B) \( \frac{5\pi}{6} \)
• (C) \( \frac{\pi}{3} \)
• (D) \( \frac{4\pi}{3} \)

The range of the function \( y = \log(\sin x) \) where \( \sin x > 0 \) is:

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

Let \( f(x) \) be defined by \( f(x) = \begin{cases} \int_x^6 (|t-2| + 3) dt, & x > 4
2x + 8, & x \leq 4 \end{cases} \). Then at \( x = 4 \), \( f(x) \) is:

• (A) Continuous but not differentiable
• (B) Differentiable
• (C) Discontinuous
• (D) None of these

If \( y = (x-1)(x-2)(x-3)\dots(x-100) \), and the value of \( \frac{dy}{dx} \) at \( x = 0 \) is equal to \( \lambda \left( \frac{100!}{100C_5} \right) \), then \( \lambda \) is:

• (A) \( \frac{1}{120} \)
• (B) \( 120 \)
• (C) \( 1 \)
• (D) \( \frac{1}{24} \)

If a vector \( \vec{c} \) is coplanar with \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} - \hat{j} + 2\hat{k} \) such that \( \vec{c} \cdot \vec{a} = 1 \) and \( \vec{c} \cdot \vec{b} = 2 \), then \( \vec{c} \) is:

• (A) \( \frac{1}{3} (-\hat{i} + 5\hat{j} + 3\hat{k}) \)
• (B) \( \frac{1}{3} (5\hat{i} - \hat{j} + 3\hat{k}) \)
• (C) \( \hat{i} - \hat{j} + \hat{k} \)
• (D) \( \frac{1}{2} (\hat{j} + \hat{k}) \)

If \( n \) is an odd natural number and \( I_n = \int_0^1 e^x (x-1)^n dx \), then \( I_n + nI_{n-1} \) is equal to:

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