MHT CET 2025 April 27 Shift 1 Question Paper (Available): Download Question Paper (PCM) with Answers PDF

Evaluate the integral:
\[ \int \frac{\sqrt{\tan x}}{\sin x \cos x} \, dx \]

• (A) \( \frac{2}{\cos^2 x} \)
• (B) \( \frac{2}{\sin^2 x} \)
• (C) \( \frac{2}{\cos x} \)
• (D) \( \frac{2}{\sin x} \)

Population of Town A and B was 20,000 in 1985.
In 1989, the population of Town A was 25,000, and Town B had 28,000.
What will be the difference in population between the two towns in 1993?

• (A) 5950
• (B) 6950
• (C) 4500
• (D) 0

A die was thrown \( n \) times until the lowest number on the die appeared.
If the mean is \( \frac{n}{g} \),
then what is the value of \( n \)?

• (A) \( 2 \)
• (B) \( 3 \)
• (C) \( 4 \)
• (D) \( 5 \)

There are 6 boys and 4 girls. Arrange their seating arrangement on a round table such that 2 boys and 1 girl can't sit together.

• (A) \( 6! \times 4! \)
• (B) \( 6! \times 3! \times 4! \)
• (C) \( 5! \times 4! \)
• (D) \( 5! \times 3! \times 4! \)

Choose a randomly selected leap year, in which 52 Saturdays and 53 Sundays are to be there.

Find the mean and standard deviation.

• (A) Mean = 2.7, Standard Deviation = 1.5
• (B) Mean = 2.5, Standard Deviation = 1.2
• (C) Mean = 2.4, Standard Deviation = 1.4
• (D) Mean = 3.0, Standard Deviation = 1.6

If \( \tan^{-1} (\sqrt{\cos \alpha}) - \cot^{-1} (\cos \alpha) = x \),
then what is \( \sin \alpha \)?

• (A) \( \tan \left( \frac{x}{2} \right) \)
• (B) \( \cot \left( \frac{x}{2} \right) \)
• (C) \( \cot^2 \left( \frac{x}{2} \right) \)
• (D) \( \tan^2 \left( \frac{x}{2} \right) \)

If \( \tan(\pi \cos x) = \cot(\pi \sin x) \),
then what is \( \sin \left( \frac{\pi}{2} + x \right) \)?

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

Evaluate the integral: \[ \int \frac{1}{\sin^2 2x \cdot \cos^2 2x} \, dx \]

• (A) \( \frac{1}{2} \tan 2x \)
• (B) \( \frac{1}{2} \cot 2x \)
• (C) \( \frac{1}{4} \cot 2x \)
• (D) \( \frac{1}{4} \tan 2x \)

Given the equation: \[ 81 \sin^2 x + 81 \cos^2 x = 30 \]
Find the value of \( x \).

• (A) \( x = \frac{\pi}{4} \)
• (B) \( x = \frac{\pi}{6} \)
• (C) \( x = \frac{\pi}{3} \)
• (D) \( x = \frac{\pi}{2} \)

The angle between the lines whose direction cosines satisfy the equations: \[ l + m + n = 0 \quad and \quad m^2 + n^2 - l^2 = 0 \]
Find the angle between the two lines.

• (A) 30°
• (B) 45°
• (C) 60°
• (D) 90°

Let \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) be vectors of magnitude 2, 3, and 4 respectively. If:
- \( \mathbf{a} \) is perpendicular to \( (\mathbf{b} + \mathbf{c}) \),
- \( \mathbf{b} \) is perpendicular to \( (\mathbf{c} + \mathbf{a}) \),
- \( \mathbf{c} \) is perpendicular to \( (\mathbf{a} + \mathbf{b}) \),

then the magnitude of \( \mathbf{a} + \mathbf{b} + \mathbf{c} \) is equal to:

• (A) 29
• (B) \( \sqrt{29} \)
• (C) 26
• (D) \( \sqrt{26} \)

A boy tries to message his friend.
Each time, the chance the message is delivered is \( \frac{1}{6} \), and the chance it fails is \( \frac{5}{6} \).
He sends 6 messages.
Find the probability that exactly 5 messages are delivered.

• (A) \( \frac{1}{6} \)
• (B) \( \frac{5}{6} \)
• (C) \( \binom{6}{5} \left( \frac{1}{6} \right)^5 \left( \frac{5}{6} \right) \)
• (D) \( \frac{5}{36} \)

Given that \( \cot \left( \frac{A+B}{2} \right) \cdot \tan \left( \frac{A-B}{2} \right) = \),
and the equation \( \frac{x}{2} + \frac{y}{3} + \frac{2}{6} - 1 = 0 \),
find the area of \( \Delta ABC = 2 \).

• (A) \( 2 \)
• (B) \( 3 \)
• (C) \( 4 \)
• (D) \( 5 \)

Evaluate the following integrals:
\[ \int \frac{(x^4 + 1)}{x(2x + 1)^2} \, dx \]

and
\[ \int \frac{1}{x^4 + 5x^2 + 6} \, dx \]

• (A) \( \frac{1}{(2x + 1)} \)
• (B) \( \frac{1}{(x^4 + 5x^2 + 6)} \)
• (C) \( \frac{1}{2} \left( \ln \left| \frac{x^2 + 3}{x + 2} \right| \right) \)
• (D) \( \frac{1}{3} \left( \ln |x^2 + 5x + 6| \right) \)

Given that: \[ \cot \left( \frac{A + B}{2} \right) \cdot \tan \left( \frac{A - B}{2} \right) \]

and the equation involving coordinates: \[ \frac{x}{2} + \frac{y}{3} + \frac{2}{6} - 1 = 0 \]

Find the area of \( \Delta ABC = 2 \).

• (A) 2
• (B) 3
• (C) 4
• (D) 5