Given \( f(x) = x - 1 \), \( h(1) = 4 \), and \( h'(1) = -2 \). If \( g(x) = (f[4(h(x)) + 3])^{2} \), find the value of \( g'(1) \).

• (A) \(-288\)
• (B) \(288\)
• (C) \(-144\)
• (D) \(144\)

The area in sq. units bounded by the curve \( y = 2\sqrt{1 - x^{2}} \) and the x-axis is :

• (A) \( \pi \)
• (B) \( 2\pi \)
• (C) \( \pi/2 \)
• (D) \( 4\pi \)

The objective function is \( Z = 3x + 5y \). The difference between the maximum and minimum values of \( Z \) is \( 3a \), subject to constraints: \( x + y \geq 1, 5x + 10y \leq 50, x, y \geq 0 \). Find the value of \( a \).

• (A) \( 3 \)
• (B) \( 6 \)
• (C) \( 9 \)
• (D) \( 27 \)

The surrounding temperature is \( 20^{\circ}C \). A body cools from \( 100^{\circ}C \) to \( 60^{\circ}C \) in 20 minutes. Find the time taken for the body to cool down to \( 30^{\circ}C \).

• (A) \( 40 \) minutes
• (B) \( 50 \) minutes
• (C) \( 60 \) minutes
• (D) \( 80 \) minutes

For the parabola \( y^{2} = 16x \), find the distance between the focus and the directrix.

• (A) \( 2 \)
• (B) \( 4 \)
• (C) \( 8 \)
• (D) \( 16 \)

The lines \( \vec{r} \times \vec{a} = \vec{b} \times \vec{a} \) and \( \vec{r} \times \vec{b} = \vec{a} \times \vec{b} \) intersect at a point, where \( \vec{a} = \hat{i} + \hat{j} \) and \( \vec{b} = \hat{i} - \hat{k} \). Find the point of intersection.

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

The statement \( \neg(p \leftrightarrow q) \) is logically equivalent to:

• (A) \( \neg p \leftrightarrow \neg q \)
• (B) \( \neg p \rightarrow q \)
• (C) \( \neg(p \rightarrow \neg q) \)
• (D) \( p \leftrightarrow \neg q \)

If \( A = \begin{bmatrix} \sec\theta & -\tan\theta \\
-\tan\theta & \sec\theta \end{bmatrix} \) and \( A + adj A = 4I \), then find the value of \( \theta \).

• (A) \( \pi/6 \)
• (B) \( \pi/4 \)
• (C) \( 0 \)
• (D) \( \pi/3 \)

Let \( f(x) = \sqrt{x^{2} + 1} \), \( g(x) = \frac{x+1}{x^{2}+1} \), \( h(x) = 2x - 3 \). Then, \( f'(h'(g'(x))) = ? \)

• (A) \( 2/\sqrt{5} \)
• (B) \( -2/\sqrt{5} \)
• (C) \( 0 \)
• (D) \( 1/\sqrt{5} \)

Evaluate: \( \int \frac{x + 1}{x(1 + x e^{x})^{2}} d x \)

• (A) \( \ln \left| \frac{x e^{x}}{1 + x e^{x}} \right| + \frac{1}{1 + x e^{x}} + C \)
• (B) \( \ln|x e^{x}| - \ln|1 + x e^{x}| - \frac{1}{1 + x e^{x}} + C \)
• (C) \( \ln \left| \frac{1 + x e^{x}}{x e^{x}} \right| + \frac{1}{1 + x e^{x}} + C \)
• (D) \( \frac{1}{1 + x e^{x}} + C \)