MHT CET 2026 May 13 Shift 2 PCM Question Paper- Download PDF with Solutions

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

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

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

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

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

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

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

• (A) 4
• (B) 3
• (C) 1
• (D) 2

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

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

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 Br\o{nsted--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) \(3I_0/8\)

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) \(\dfrac{2K}{K+1}\,C\)
• (B) \(\dfrac{K+1}{2K}\,C\)
• (C) \(\dfrac{K}{K+1}\,C\)
• (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 \((2a,\,0,\,0)\) is:

• (A) \(\dfrac{q\lambda}{2\pi\varepsilon_0}\ln 2\)
• (B) \(\dfrac{q\lambda}{4\pi\varepsilon_0}\ln 2\)
• (C) \(\dfrac{q\lambda}{2\pi\varepsilon_0}\ln\!\left(\tfrac{1}{2}\right)\)
• (D) zero