Find the value of \( k \) if the function \( f(x) = \frac{k \cos x}{\pi - 2x} \) is continuous at \( x = \frac{\pi}{2} \).

• (A) \( 1 \)
• (B) \( 2 \)
• (C) \( \frac{\pi}{2} \)
• (D) \( \pi \)

If \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - 3\hat{k} \), find the unit vector perpendicular to both \( \vec{a} \) and \( \vec{b} \).

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

Evaluate the integral \( \displaystyle \int_{0}^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} \, dx \).

• (A) \(0\)
• (B) \( \dfrac{\pi}{4} \)
• (C) \( \dfrac{\pi}{2} \)
• (D) \( \pi \)

Find the area of the region bounded by the parabola \( y^2 = 4ax \) and its latus rectum.

• (A) \( \dfrac{4a^2}{3} \)
• (B) \( \dfrac{8a^2}{3} \)
• (C) \( \dfrac{16a^2}{3} \)
• (D) \( \dfrac{32a^2}{3} \)

Identify the truth value of the statement pattern \( (p \land \sim q) \rightarrow r \).

• (A) Tautology
• (B) Contradiction
• (C) Contingency
• (D) None of these

Calculate the moment of inertia of a uniform ring of mass \( M \) and radius \( R \) about a tangent in its own plane.

• (A) \( MR^2 \)
• (B) \( 2MR^2 \)
• (C) \( \dfrac{3}{2}MR^2 \)
• (D) \( \dfrac{5}{2}MR^2 \)

A capillary tube of radius \( r \) is dipped in water; find the height \( h \) to which water rises if the surface tension is \( T \).

• (A) \( \dfrac{T}{\rho g r} \)
• (B) \( \dfrac{2T}{\rho g r} \)
• (C) \( \dfrac{4T}{\rho g r} \)
• (D) \( \dfrac{T}{2\rho g r} \)

Find the ratio of the de Broglie wavelengths of a proton and an alpha particle accelerated through the same potential difference.

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

Determine the energy of a photon in eV if its wavelength is \( 4000\ \AA \).

• (A) \( 3.1 \, eV \)
• (B) \( 2.5 \, eV \)
• (C) \( 1.55 \, eV \)
• (D) \( 4.2 \, eV \)

In a Young's Double Slit Experiment, find the fringe width if the distance between slits is doubled and the screen distance is halved.

• (A) Fringe width becomes half
• (B) Fringe width becomes one-fourth
• (C) Fringe width remains unchanged
• (D) Fringe width becomes double

What is the IUPAC name for the compound \( CH_3 - CH(OH) - CH_2 - CHO \)?

• (A) 3-hydroxybutanal
• (B) 2-hydroxybutanal
• (C) 4-hydroxybutanal
• (D) 1-hydroxybutanal

Calculate the spin-only magnetic moment of a \( Mn^{2+} \) ion (Atomic number = 25).

• (A) \( 3.87 \, BM \)
• (B) \( 4.90 \, BM \)
• (C) \( 5.92 \, BM \)
• (D) \( 6.93 \, BM \)

Which of the following polymers is prepared by condensation polymerization: Nylon-6,6 or Polythene?

• (A) Polythene
• (B) Nylon-6,6
• (C) Both
• (D) None of these

Find the oxidation state of Phosphorus in \( H_3PO_4 \) and \( H_3PO_3 \).

• (A) \(+5\) and \(+3\)
• (B) \(+3\) and \(+5\)
• (C) \(+4\) and \(+3\)
• (D) \(+5\) and \(+4\)

Identify the major product formed when phenol reacts with bromine water.

• (A) Bromobenzene
• (B) 2,4,6-tribromophenol
• (C) o-bromophenol
• (D) p-bromophenol