Find the value of \( k \) if the function \( f(x) = \dfrac{k\sin x + 2\cos x}{\sin x + \cos x} \) is increasing for all \( x \).

• (A) \( k < 2 \)
• (B) \( k = 2 \)
• (C) \( k \geq 2 \)
• (D) \( k \leq 2 \)

Calculate the area of the region bounded by the curve \( y^2 = 4x \) and the line \( x = 3 \).

• (A) \(6\sqrt{3}\)
• (B) \(8\sqrt{3}\)
• (C) \(12\sqrt{3}\)
• (D) \(4\sqrt{3}\)

Find the general solution of the differential equation \( \dfrac{dy}{dx} + \dfrac{y}{x} = x^2 \).

• (A) \( y = \dfrac{x^3}{4} + \dfrac{C}{x} \)
• (B) \( y = \dfrac{x^3}{3} + \dfrac{C}{x} \)
• (C) \( y = \dfrac{x^3}{2} + \dfrac{C}{x} \)
• (D) \( y = x^3 + \dfrac{C}{x} \)

Determine the shortest distance between the lines \( \dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{4} \) and \( \dfrac{x-2}{3} = \dfrac{y-4}{4} = \dfrac{z-5}{5} \).

• (A) \( \dfrac{2}{\sqrt{3}} \)
• (B) \( \dfrac{2}{\sqrt{5}} \)
• (C) \( \dfrac{2}{\sqrt{2}} \)
• (D) \( \dfrac{2}{\sqrt{29}} \)

If \( \vec{a} \) and \( \vec{b} \) are unit vectors and the angle between them is \(60^\circ\), find \( |\vec{a}-\vec{b}| \).

• (A) \(1\)
• (B) \(\sqrt{2}\)
• (C) \(\sqrt{3}\)
• (D) \(2\)

A particle performs SHM with an amplitude of 5 cm. Find its acceleration when it is 3 cm from the mean position (assume \( \omega = 2 \, rad/s \)).

• (A) \(3\ cm/s^2\)
• (B) \(6\ cm/s^2\)
• (C) \(12\ cm/s^2\)
• (D) \(24\ cm/s^2\)

Calculate the de-Broglie wavelength of an electron accelerated through a potential difference of \(100 V\).

• (A) \(1.227\ Å\)
• (B) \(0.1227\ nm\)
• (C) \(12.27\ nm\)
• (D) \(0.01227\ nm\)

Find the equivalent resistance between two opposite corners of a cube made of wires, each having resistance \(R\).

• (A) \( \dfrac{R}{6} \)
• (B) \( \dfrac{5R}{6} \)
• (C) \( \dfrac{R}{3} \)
• (D) \( \dfrac{2R}{3} \)

Determine the ratio of the speed of sound in Oxygen to that in Hydrogen at the same temperature.

• (A) \(1:4\)
• (B) \(1:2\)
• (C) \(2:1\)
• (D) \(4:1\)

What is the oxidation state of Phosphorus in \( H_3PO_4 \)?

• (A) \(+3\)
• (B) \(+5\)
• (C) \(+1\)
• (D) \(+7\)

Which of the following polymers is a copolymer: Buna-S, Polythene, or PVC?

• (A) Polythene
• (B) PVC
• (C) Buna-S
• (D) All of these

Calculate the pH of a \(0.001\, M\) \(NaOH\) solution at \(25^\circ C\).

• (A) \(3\)
• (B) \(7\)
• (C) \(11\)
• (D) \(13\)

Name the reagent used in the conversion of benzene to nitrobenzene.

• (A) Concentrated \(HNO_3\) only
• (B) Concentrated \(H_2SO_4\) only
• (C) Mixture of concentrated \(HNO_3\) and \(H_2SO_4\)
• (D) Dilute \(HNO_3\)