MHT CET 2025 20 April Shift 1 Question Paper (Available): Download Question Paper (PCM) with Answers PDF

Water is being poured at the rate of 36 m\(^3\)/min into a cylindrical vessel whose circular base is of radius 3 meters. Then the water level in the cylinder increases at the rate of:

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

If \( \mathbf{a} = 2\hat{i} + 2\hat{j} + 3\hat{k}, \mathbf{b} = -\hat{i} + 2\hat{j} + \hat{k} \) and \( \mathbf{c} = 3\hat{i} + \hat{j} \) are the vectors such that \( \mathbf{a} + \lambda \mathbf{b} \) is perpendicular to \( \mathbf{c} \), then the value of \( \lambda \) is:

• (1) 6
• (2) -6
• (3) 8
• (4) -8

If \( \cos^{-1}\left( \frac{12}{13} \right) + \sin^{-1}\left( \frac{3}{5} \right) = \sin^{-1} P \), then the value of \( P \) is:

• (1) \( \frac{63}{65} \)
• (2) \( \frac{56}{65} \)
• (3) \( \frac{48}{65} \)
• (4) \( \frac{36}{65} \)

The area enclosed between the parabola \( y^2 = 4x \) and the line \( y = 2x - 4 \) is:

• (1) \( \frac{17}{3} \) sq. units
• (2) 15 sq. units
• (3) \( \frac{19}{3} \) sq. units
• (4) 9 sq. units

If \( y = \tan^{-1} \left( \frac{2 + 3x}{3 - 2x} \right) + \tan^{-1} \left( \frac{4x}{1 + 5x^2} \right) \), then
\[ \frac{dy}{dx} = \]

• (1) \( \frac{1}{1 + 25x^2} \)
• (2) \( \frac{5}{1 + 25x^2} \)
• (3) \( \frac{1}{1 + 5x^2} \)
• (4) \( \frac{5}{1 + 5x^2} \)

The particular solution of the differential equation, \[ x y \frac{dy}{dx} = x^2 + 2y^2 \quad when \quad y(1) = 0 \quad is: \]

• (1) \( \frac{x^2 + y^2}{x^3} = 1 \)
• (2) \( x^2 + y^2 = x \)
• (3) \( x^2 + y^2 = x^4 \)
• (4) \( x^2 + 2y^2 = x^4 \)

If the angle between the line \( 2(x + 1) = y = z \) and the plane \( 2x - y + \sqrt{2} z + 4 = 0 \) is \( \frac{\pi}{6} \), then the value of \( \lambda \) is:

• (1) \( \frac{135}{7} \)
• (2) \( \frac{45}{11} \)
• (3) \( \frac{45}{7} \)
• (4) \( \frac{135}{11} \)

If the volume of the tetrahedron, whose vertices are \( A(1, 2, 3), B(-3, -1, 1), C(2, 1, 3) \) and \( D(-1, 2, x) \), is \( \frac{11}{6} \) cubic units, then the value of \( x \) is:

• (1) 3
• (2) -2
• (3) 4
• (4) -1

A 5-ohm resistor is connected to a 10 V battery. Calculate the current flowing through the resistor.

• (A) 1.0 A
• (B) 2.0 A
• (C) 0.5 A
• (D) 0.2 A

A charge of 2 \(\mu C\) is placed in an electric field of intensity \( 4 \times 10^3 \, N/C \). What is the force experienced by the charge?

• (A) \( 8 \times 10^{-3} \, N \)
• (B) \( 8 \times 10^{-6} \, N \)
• (C) \( 4 \times 10^{-3} \, N \)
• (D) \( 4 \times 10^{-6} \, N \)