MHT CET 2025 25 April Shift 2 Question Paper (Available): Download Question Paper (PCM) with Answers PDF

Principal Solution

• (1) \( (5 + 3 \sin \theta) \)
• (2) \( (2 \cos \theta + 1) \)

Curves Represented

x = 3(cos t + sin t)
y = (cos t − sin t)

Principal Solution

• (1) \( \frac{2\pi}{3} \), \( \frac{4\pi}{3} \)
• (2) \( \frac{2\pi}{3} \), \( \frac{1}{3} \), \( \frac{2\pi}{3} \)

Find the solution

• (1) \( y^3 = 3y^3 \sin m \)
• (2) \( y^3 = 3x^2 \sin m \)

Series Expansion

• (1) \( 9 \)
• (2) \( 8 \)
• (3) \( 10 \)

Binomial Expansion Series

If \( y = \frac{b}{a} \), then \( \frac{dy}{dx} \) is:

• (1) \( -\frac{b^4}{a} \)
• (2) \( \frac{b^5}{a} \)
• (3) \( -\frac{b^5}{a^2 y^3} \)
• (4) \( \frac{b^5}{a^2} \)

The eccentricity of the curve represented by
\( x = 3 (\cos t + \sin t) \), \( y = 4 (\cos t - \sin t) \) is:

• (1) \( \frac{\sqrt{7}}{4} \)
• (2) \( \frac{1}{16} \)
• (3) \( \frac{\sqrt{7}}{3} \)
• (4) \( \frac{\sqrt{8}}{4} \)

The integral \[ \int e^x \left( \frac{x + 5}{(x + 6)^2} \right) dx \]
is:

• (1) \( \frac{e^x}{x + 6} \)
• (2) \( -\frac{e^x}{x + 6} \)
• (3) \( \frac{e^x}{(x + 6)} \)
• (4) \( -\frac{e^x}{(x + 6)^2} \)

The integral \[ \int_0^1 \frac{1}{2 + \sqrt{2e}} \, dae \]
is:

• (1) \( \frac{1}{2} \ln (2 + \sqrt{2e}) \)
• (2) \( \frac{1}{\sqrt{2}} \)
• (3) \( \frac{1}{\sqrt{2}} \ln (2 + \sqrt{2e}) \)
• (4) \( \frac{1}{2 \sqrt{2}} \)

The integral \[ \int e^x \left( \frac{x + 5}{(x + 6)^2} \right) dx \]
is:

• (1) \( \frac{e^x}{x + 6} \)
• (2) \( -\frac{e^x}{x + 6} \)
• (3) \( \frac{e^x}{(x + 6)^2} \)
• (4) \( -\frac{e^x}{(x + 6)} \)

If \( \mathbf{a} \) and \( \mathbf{b} \) are non-coplanar unit vectors such that \[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \frac{\mathbf{b}}{2} \]
then the angle between \( \mathbf{a} \) and \( \mathbf{b} \) is:

• (1) \( \frac{\pi}{4} \)
• (2) \( \frac{\pi}{3} \)
• (3) \( \frac{\pi}{2} \)
• (4) \( \frac{\pi}{6} \)

If a random variable \( X \) has the following probability distribution values

Then \( P(X \geq 6) \) has the value

• (1) \( \frac{16}{100} \)
• (2) \( \frac{81}{100} \)
• (3) \( \frac{1}{100} \)
• (4) \( \frac{91}{100} \)

In \( \triangle ABC \), with usual notations, \[ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{C}{2} \right) = \sin \left( \frac{B}{2} \right) \quad and \quad 2s is the perimeter of the triangle. Find the value of s. \]
Then the value of \( s \) is:

• (1) \( 2b \)
• (2) \( 6b \)
• (3) \( 3b \)
• (4) \( 4b \)

Evaluate the limit \[ \lim_{n \to \infty} \frac{6^n - 9x - 7^n + 1}{\sqrt{2} - \sqrt{11} + \cos n} \]