List of top Mathematics Questions asked in Maharashtra Class XII

Solve the following system of equations by the method of reduction: \[ x + y + z = 6, \quad y + 3z = 11, \quad x + z = 2y. \]
A die is thrown 6 times. If "getting an odd number" is a success, find the probability of 5 successes.
Find \( k \), if the probability density function is given by: \[ f(x) = kx^2(1 - x), \quad {for } 0<x<1, \] \[ = 0, \quad {otherwise.} \]
Solve the differential equation: \[ x \frac{dy}{dx} - y + x \sin \left(\frac{y}{x}\right) = 0. \]
Prove that: \[ \int \frac{1}{a^2 - x^2} dx = \frac{1}{2a} \log \left( \frac{a + x}{a - x} \right) + C. \]
If \( y = \sin^{-1} x \), then show that: \[ (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} = 0. \]
Find the angle between the line \[ \mathbf{r} = ( \hat{i} + 2\hat{j} + \hat{k} ) + \lambda( \hat{i} + \hat{j} + \hat{k} ) \] and the plane \[ \mathbf{r} \cdot (2\hat{i} + \hat{j} + \hat{k}) = 8. \]
Find the shortest distance between the lines \[ \mathbf{r} = (4\hat{i} - \hat{j}) + \lambda( \hat{i} + 2\hat{j} - 3\hat{k} ) \] and \[ \mathbf{r} = ( \hat{i} - \hat{j} - 2\hat{k} ) + \mu ( \hat{i} + \hat{j} - 5\hat{k} ). \]
In \( \triangle ABC \), prove that: \[ \frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{a^2 + b^2 + c^2}{2abc}. \]
Prove that: \[ \tan^{-1} \left(\frac{1}{2}\right) + \tan^{-1} \left(\frac{1}{3}\right) = \frac{\pi}{4}. \]
Find the area of the region bounded by the curve \( y = x^2 \), and the lines \( x = 1 \), \( x = 2 \), and \( y = 0 \).
Evaluate: \[ \int_{0}^{\frac{\pi}{2}} \cos^2 x \,dx. \]
Evaluate: \[ \int \log x \,dx. \]
Find the vector equation of the line passing through the point having position vector \[ 4\hat{i} - \hat{j} + 2\hat{k} \] and parallel to the vector \[ -2\hat{i} - \hat{j} + \hat{k}. \]
Find \( k \), if the sum of the slopes of the lines represented by \[ x^2 + kxy - 3y^2 = 0 \] is twice their product.

Check whether the matrix 


 is invertible or not.

Write the compound statement ‘Nagpur is in Maharashtra and Chennai is in Tamilnadu’ symbolically.

The integrating factor of the linear differential equation \[ x \frac{dy}{dx} + 2y = x^2 \log x \] is __________.

If \( \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} x^3 \sin^4 x \, dx = k \), then \( k \) is ____________.

If \[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \] verify that \[ A(\text{adj}A) = (\text{adj}A)A = |A|I \]
Prove that: \( \int_0^{2a} f(x) \, dx = \int_0^a f(x) \, dx + \int_0^a f(2a - x) \, dx \).
Evaluate: \( \int \frac{2x^2 - 3}{(x^2 - 5)(x + 4) \, dx \).
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Find the length of the perpendicular drawn from the point \( P(3, 2, 1) \) to the line \( \mathbf{r} = (7\hat{i} + 7\hat{j} + 6\hat{k}) + \lambda (-2\hat{i} + 2\hat{j} + 3\hat{k}) \).
If \( y = \cos (m \cos^{-1} x) \), then show that \( (1 - x^2) \frac{d^2 y{dx^2} - x \frac{dy}{dx} + m^2 y = 0 \).
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Verify Lagrange’s mean value theorem for the function \( f(x) = x + 4 \) on the interval [0, 5].