The value of \( \displaystyle \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx \) is:

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

If \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - 3\hat{k} \), find the magnitude of \( \vec{a} \times \vec{b} \).

• (A) \( \sqrt{75} \)
• (B) \( \sqrt{86} \)
• (C) \( \sqrt{110} \)
• (D) \( \sqrt{102} \)

Find the truth value of the statement: "If 2 is even, then 5 is prime."

• (A) True
• (B) False
• (C) Cannot be determined
• (D) None of these

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

• (A) \( y = e^{-x}(x + C) \)
• (B) \( y = xe^{-x} + C \)
• (C) \( y = e^{x}(x + C) \)
• (D) \( y = e^{-x} + C \)

Determine the distance of the point \( (1,2,3) \) from the plane \( 2x + 3y - z = 7 \).

• (A) \( \frac{2}{\sqrt{14}} \)
• (B) \( \frac{4}{\sqrt{14}} \)
• (C) \( \frac{1}{\sqrt{14}} \)
• (D) \( \frac{3}{\sqrt{14}} \)

Calculate the moment of inertia of a uniform circular disc of mass \(M\) and radius \(R\) about its diameter.

• (A) \( \frac{1}{2}MR^2 \)
• (B) \( \frac{1}{4}MR^2 \)
• (C) \( MR^2 \)
• (D) \( \frac{3}{2}MR^2 \)

A wire of resistance \(R\) is stretched to triple its original length. Find the new resistance.

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

Find the energy of a photon with a wavelength of \(4000\,\AA\).
(Take \( h = 6.63 \times 10^{-34}\,Js \), \( c = 3 \times 10^8\,m/s \)).

• (A) \(4.97 \times 10^{-19}\,J\)
• (B) \(3.31 \times 10^{-19}\,J\)
• (C) \(6.63 \times 10^{-19}\,J\)
• (D) \(2.48 \times 10^{-19}\,J\)

Determine the centripetal force acting on a \(1000\,kg\) car moving at \(20\,m/s\) around a curve of radius \(50\,m\).

• (A) \(4000\,N\)
• (B) \(8000\,N\)
• (C) \(10000\,N\)
• (D) \(20000\,N\)

What is the change in internal energy of a system if \(500\,J\) of heat is added and \(200\,J\) of work is done by the system?

• (A) \(300\,J\)
• (B) \(700\,J\)
• (C) \(200\,J\)
• (D) \(500\,J\)

What is the oxidation number of Manganese (Mn) in the compound \( KMnO_4 \)?

• (A) \(+4\)
• (B) \(+6\)
• (C) \(+7\)
• (D) \(+2\)

Identify the product formed when Ethyl alcohol is heated with concentrated \( H_2SO_4 \) at \(443\,K\).

• (A) Ethane
• (B) Ethene
• (C) Diethyl ether
• (D) Acetaldehyde

Which reagent is used in the Reimer--Tiemann reaction to convert phenol to salicylaldehyde?

• (A) \(CHCl_3\) and \(NaOH\)
• (B) \(HNO_3\)
• (C) \(KMnO_4\)
• (D) \(HCl\)

Calculate the pH of a \(0.001\,M\) \(NaOH\) solution at \(298\,K\).

• (A) \(11\)
• (B) \(10\)
• (C) \(3\)
• (D) \(7\)

Identify the type of crystal defect found in \(NaCl\) crystals.

• (A) Frenkel defect
• (B) Schottky defect
• (C) Metal excess defect
• (D) Interstitial defect