List of top Mathematics Questions asked in Maharashtra Board Class XII Exam

Prove that the volume of a tetrahedron with coterminous edges \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) is \( \frac{1}{6} |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| \). Hence, find the volume of tetrahedron whose coterminous edges are \( \mathbf{a} = \hat{i} + 2\hat{j} + 3\hat{k} \), \( \mathbf{b} = -\hat{i} + \hat{j} + 2\hat{k} \), and \( \mathbf{c} = 2\hat{i} + \hat{j} + 4\hat{k} \).
For the following probability density function of a random variable X, find (a) P(X<1) and (b) P(|X|<1).
\[ f(x) = \begin{cases} \frac{x + 2}{18}, & -2<x<4
0, & \text{otherwise} \end{cases} \]

Simplify the given circuit by writing its logical expression. Also write your conclusion.

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at least 5 successes.
If \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \), verify that \( A (\text{adj } A) = (\text{adj } A) A = |A| I \).
If \( y = \tan x + \tan x + \tan x + \dots \infty \), then show that \( \frac{dy}{dx} = \frac{2 \sec^2 x}{2 - y} \). Find \( \frac{dy}{dx} \) at \( x = 0 \).
Find the approximate value of \( \sin (30^\circ 30') \). Given that \( 1^\circ = 0.0175^c \) and \( \cos 30^\circ = 0.866 \).
Evaluate \( \int \tan^{-1} x \, dx \).
Find the particular solution of the differential equation \( \frac{dy}{dx} = e^{2y} \cos x \), when \( x = \frac{\pi}{6} \), \( y = 0 \).
If \( \theta \) is the acute angle between the lines represented by \( ax^2 + 2hxy + by^2 = 0 \), then prove that \( \tan \theta = \frac{2 \sqrt{h^2 - ab}}{a + b} \).
Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are \( -2, 1, -1 \) and \( -3, -4, 1 \).
Find the shortest distance between lines \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \) and \( \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5} \).
Lines \( \mathbf{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda (2\hat{i} - 2\hat{j} + \hat{k}) \) and \( \mathbf{r} = (4\hat{i} - 3\hat{j} + 2\hat{k}) + \mu (\hat{i} - 2\hat{j} + 2\hat{k}) \) are coplanar. Find the equation of the plane determined by them.
Find the area of the region bounded by the curve \( y = x^2 \) and the line \( y = 4 \).
Solve the differential equation \( \cos x \cos y \, dy - \sin x \sin y \, dx = 0 \).
Find the general solution of \( \sin \theta + \sin 3\theta + \sin 5\theta = 0 \).
Find the mean of a number randomly selected from 1 to 15.
If \( -1 \leq x \leq 1 \), then prove that \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \).
Evaluate: \( \int_0^{\pi/2} (1 - \cos 4x) \, dx \).
Solve the following inequations graphically and write the corner points of the feasible region: \( 2x + 3y \leq 6 \), \( x + y \geq 2 \), \( x \geq 0 \), \( y \geq 0 \).
Find the area of the region bounded by the curve \( y^2 = 4x \), the X-axis, and the lines \( x = 1 \), \( x = 4 \) for \( y \geq 0 \).
Show that the function \( f(x) = x^3 + 10x + 7 \), \( x \in \mathbb{R} \), is strictly increasing.
If \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of lines and \( h^2 = ab \neq 0 \), then find the ratio of their slopes.
If \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are the position vectors of the points A, B, C respectively and \( 5\mathbf{a} + 3\mathbf{b} - 8\mathbf{c} = 0 \), then find the ratio in which the point C divides the line segment AB.
Find the cartesian coordinates of the point whose polar coordinates are \( \left( 2, \frac{\pi}{4} \right) \).