If \( I = \int \frac{\sin x+\sin^{3} x}{\cos 2x} d x = P \cos x+Q \log \left|\frac{\sqrt{2} \cos x-1}{\sqrt{2} \cos x+1}\right|+C \), then the values of \( P \) and \( Q \) are respectively:

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

In a triangle \( \Delta ABC \), if \( a, b \), and \( c \) are in arithmetic progression, then \( \cos A + 2 \cos B + \cos C = \)

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

If the area of triangle \( ABC \) is \( b^2-(c-a)^2 \), then \( \tan B = \)

• (A) 1
• (B) \( \frac{13}{15} \)
• (C) \( \frac{1}{4} \)
• (D) \( \frac{8}{15} \)

In a \( \Delta ABC, a=1, b=\sqrt{3} \) and \( \angle C = \frac{\pi}{6} \). Then the measure of the third side \( c = \)

• (A) 4
• (B) 3
• (C) 1
• (D) 2

The integral \( \int \frac{1}{\sqrt[4]{(x-1)^{3}(x+2)^{5}}} d x \) is equal to: (where \( C \) is a constant of integration)

• (A) \( \frac{3}{4}\left(\frac{x+2}{x-1}\right)^{1/4}+C \)
• (B) \( \frac{3}{4}\left(\frac{x+2}{x-1}\right)^{5/4}+C \)
• (C) \( \frac{4}{3}\left(\frac{x-1}{x+2}\right)^{1/4}+C \)
• (D) \( \frac{4}{3}\left(\frac{x-1}{x+2}\right)^{5/4}+C \)

Calculate the pH of 0.02 M monobasic acid having 2% dissociation.

• (A) 3.4
• (B) 4.5
• (C) 5.1
• (D) 5.8

What is the pH of \( 2 \times 10^{-3} \) M solution of monoacidic weak base if it ionises to the extent of 5%?

• (A) 14
• (B) 6
• (C) 4
• (D) 2

Identify base\(_2\) for the following equation according to Brønsted–Lowry theory:
\[ HCl_{(aq)} + H_2O_{(l)} \rightleftharpoons H_3O^+_{(aq)} + Cl^-_{(aq)} \]

• (A) \( H_3O^+_{(aq)} \)
• (B) \( H_2O_{(l)} \)
• (C) \( Cl^-_{(aq)} \)
• (D) \( HCl_{(aq)} \)

Calculate the pH of 0.01 M sulphuric acid.

• (A) 1.699
• (B) 2.00
• (C) 0.699
• (D) 3.398

What is the pH of a weak dibasic acid that is 2% dissociated in its M/100 solution at 298 K?

• (A) 1.6990
• (B) 2.3979
• (C) 3.3970
• (D) 4.6990

An open pipe resonates with a tuning fork of frequency 500 Hz. It is observed that two successive nodes are separated by 34 cm. The velocity of sound in air is:

• (A) 330 m/s
• (B) 340 m/s
• (C) 350 m/s
• (D) 360 m/s

A train moving at 20 m/s approaches a stationary observer. The frequency of the whistle emitted by the train is 640 Hz. If the velocity of sound is 340 m/s, the apparent frequency heard by the observer is:

• (A) 680 Hz
• (B) 600 Hz
• (C) 720 Hz
• (D) 640 Hz

Unpolarized light of intensity \( I_0 \) is incident on two polaroids placed coaxially. The transmission axis of the second polaroid is at an angle of 60\(^\circ\) to the first. The intensity of emerging light is:

• (A) \( I_0/2 \)
• (B) \( I_0/4 \)
• (C) \( I_0/8 \)
• (D) \( 3 I_0/8 \)

A parallel plate capacitor has capacitance \( C \). If a dielectric slab of dielectric constant \( K \) and thickness equal to half the plate separation is introduced parallel to the plates, the new capacitance is:

• (A) \( \frac{2K}{K+1} C \)
• (B) \( \frac{K+1}{2K} C \)
• (C) \( \frac{K}{K+1} C \)
• (D) \( KC \)

An infinite line charge of uniform linear charge density \( \lambda \) is placed along the z-axis. The work done in moving a charge \( q \) from \( (a, 0, 0) \) to \( (2 a, 0, 0) \) is:

• (A) \( \frac{q \lambda}{2 \pi \epsilon_0} \ln 2 \)
• (B) \( \frac{q \lambda}{4 \pi \epsilon_0} \ln 2 \)
• (C) \( \frac{q \lambda}{2 \pi \epsilon_0} \ln (1/2) \)
• (D) zero