MHT CET 2026 May 20 Shift 2 PCM Question Paper- Download PDF with Solutions

Given \( f(x) = x - 1 \), \( h(1) = 4 \), and \( h'(1) = -2 \). If \( g(x) = (f[4(h(x)) + 3])^2 \), find the value of \( g'(1) \).

• (A) \( -288 \)
• (B) \( 288 \)
• (C) \( -144 \)
• (D) \( 144 \)

The area in sq. units bounded by the curve \( y = 2\sqrt{1 - x^2} \) and the x-axis is :

• (A) \( \pi \)
• (B) \( 2\pi \)
• (C) \( \pi/2 \)
• (D) \( 4\pi \)

The objective function is \( Z = 3x + 5y \). The difference between the maximum and minimum values of \( Z \) is \( 3a \), subject to constraints: \( x + y \ge 1 \), \( 5x + 10y \le 50 \), \( x, y \ge 0 \). Find the value of \( a \).

• (A) \( 3 \)
• (B) \( 6 \)
• (C) \( 9 \)
• (D) \( 27 \)

The surrounding temperature is \( 20^°C \). A body cools from \( 100^°C \) to \( 60^°C \) in 20 minutes. Find the time taken for the body to cool down to \( 30^°C \).

• (A) \( 40 \) minutes
• (B) \( 50 \) minutes
• (C) \( 60 \) minutes
• (D) \( 80 \) minutes

For the parabola \( y^2 = 16x \), find the distance between the focus and the directrix.

• (A) \( 2 \)
• (B) \( 4 \)
• (C) \( 8 \)
• (D) \( 16 \)

The lines \( \vec{r} \times \vec{a} = \vec{b} \times \vec{a} \) and \( \vec{r} \times \vec{b} = \vec{a} \times \vec{b} \) intersect at a point, where \( \vec{a} = \hat{i} + \hat{j} \) and \( \vec{b} = \hat{i} - \hat{k} \). Find the point of intersection.

• (A) \( (2,1,-1) \)
• (B) \( (1,1,-1) \)
• (C) \( (2,0,-1) \)
• (D) \( (1,0,-1) \)

The statement \( \neg(p \leftrightarrow q) \) is logically equivalent to:

• (A) \( \neg p \leftrightarrow \neg q \)
• (B) \( \neg p \to q \)
• (C) \( \neg(p \to \neg q) \)
• (D) \( p \leftrightarrow \neg q \)

If \( A=\begin{bmatrix}\sec\theta & -\tan\theta
-\tan\theta & \sec\theta\end{bmatrix} \) and \( A+\operatorname{adj}A=4I \), then find the value of \( \theta \).

• (A) \( \frac{\pi}{6} \)
• (B) \( \frac{\pi}{4} \)
• (C) \( 0 \)
• (D) \( \frac{\pi}{3} \)

Let \( f(x)=\sqrt{x^2+1} \), \( g(x)=\frac{x+1}{x^2+1} \), \( h(x)=2x-3 \). Then, \( f'(h'(g'(x)))=? \)

• (A) \( \frac{2}{\sqrt5} \)
• (B) \( \frac{-2}{\sqrt5} \)
• (C) \( 0 \)
• (D) \( \frac{1}{\sqrt5} \)

Evaluate: \[ \int \frac{x+1}{x(1+xe^x)^2}\,dx \]

• (A) \[ \ln\left|\frac{xe^x}{1+xe^x}\right|+\frac{1}{1+xe^x}+C \]
• (B) \[ \ln|xe^x|-\ln|1+xe^x|-\frac{1}{1+xe^x}+C \]
• (C) \[ \ln\left|\frac{1+xe^x}{xe^x}\right|+\frac{1}{1+xe^x}+C \]
• (D) \[ \frac{1}{1+xe^x}+C \]