MHT CET 2026 May 21 Shift 1 PCM Question Paper- Download PDF with Solutions

If \( \sin x \cos x = \frac{1}{4} \), then the general solution is:

• (A) \( \dfrac{n\pi}{2} + (-1)^n \dfrac{\pi}{12} \)
• (B) \( \dfrac{n\pi}{2} + \dfrac{\pi}{4} \)
• (C) \( n\pi \pm \dfrac{\pi}{6} \)
• (D) \( \dfrac{n\pi}{2} + (-1)^n \dfrac{\pi}{6} \)

If \( n \in \mathbb{Z} \), then the expression \[ \frac{2^n}{(1-i)^{2n}} + \frac{(1+i)^{2n}}{2^n} \]
is equal to:

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

The value of \[ \int \frac{x^2-1}{(x^4+3x^2+1)\tan^{-1}\left(x+\frac1x\right)}\,dx \]
is:

• (A) \( \log \left|\tan^{-1}\left(x+\frac1x\right)\right| + C \)
• (B) \( \tan^{-1}\left(x+\frac1x\right) + C \)
• (C) \( -\log \left|\tan^{-1}\left(x+\frac1x\right)\right| + C \)
• (D) \( \dfrac12 \log \left|x+\dfrac1x\right| + C \)

If \[ f(x)=\int \frac{x^2}{(1-x^2)(1+\sqrt{1-x^2})}\,dx \]
and \( f(0)=2 \), then \( f\left(\frac12\right) \) is:

• (A) \(2+\dfrac12\log 3-\dfrac{\sqrt3}{2}\)
• (B) \(2+\log 3-\sqrt3\)
• (C) \(2+\sqrt3-\log 3\)
• (D) \(2+\dfrac{\sqrt3}{2}-\dfrac12\log 3\)

If \[ A= \begin{bmatrix} \sin\alpha & -\cos\alpha
\cos\alpha & \sin\alpha \end{bmatrix} \]
and \( \alpha\in\left(\frac{\pi}{2},\frac{3\pi}{2}\right) \). If \( A+A^T=I \), then \( \alpha= \)

• (A) \( \dfrac{2\pi}{3} \)
• (B) \( \dfrac{5\pi}{6} \)
• (C) \( \dfrac{\pi}{3} \)
• (D) \( \dfrac{4\pi}{3} \)

The range of the function \[ y=\log(\sin x) \]
where \( \sin x>0 \) is:

• (A) \([0,\infty)\)
• (B) \((-\infty,0]\)
• (C) \((-\infty,\infty)\)
• (D) \([-1,1]\)

Let \(f(x)\) be defined by: \[ f(x)= \begin{cases} \displaystyle \int_x^6 (|t-2|+3)\,dt, & x>4
2x+8, & x\le4 \end{cases} \]

Then at \(x=4\), \(f(x)\) is:

• (A) Continuous but not differentiable
• (B) Differentiable
• (C) Discontinuous
• (D) None of these

If \[ y=(x-1)(x-2)(x-3)\cdots(x-100) \]
and the value of \( \dfrac{dy}{dx} \) at \(x=0\) is equal to \[ \lambda\left(\frac{100!}{^{100}C_5}\right) \]
then \( \lambda \) is:

• (A) \( \dfrac1{120} \)
• (B) \(120\)
• (C) \(1\)
• (D) \( \dfrac1{24} \)

Let \[ \vec a=\hat i+\hat j+\hat k \] \[ \vec b=\hat i-\hat j+2\hat k \]

If a vector \( \vec c \) is coplanar with \( \vec a \) and \( \vec b \) such that \[ \vec c\cdot \vec a=1 \]
and \[ \vec c\cdot \vec b=2 \]
then \( \vec c \) is:

• (A) \( \dfrac13(-\hat i+5\hat j+3\hat k) \)
• (B) \( \dfrac13(5\hat i-\hat j+3\hat k) \)
• (C) \( \hat i-\hat j+\hat k \)
• (D) \( \dfrac12(\hat j+\hat k) \)

If \(n\) is an odd natural number and \[ I_n=\int_0^1 e^x(x-1)^n\,dx \]
then \( I_n+nI_{n-1} \) is equal to:

• (A) \(1\)
• (B) \(-1\)
• (C) \(e\)
• (D) \(0\)