List of top Mathematics Questions asked in Maharashtra Class XII

Prove by vector method, the angle subtended on a semicircle is a right angle.
Prove that: \[ \int_0^{2a} f(x) dx = \int_0^a f(x) dx + \int_0^a f(2a - x) dx. \] Hence show that: \[ \int_0^\pi \sin x \,dx = 2 \int_0^{\pi/2} \sin x \,dx. \]

The perpendicular distance of the plane \( r \cdot (3\hat{i} + 4\hat{j} + 12\hat{k}) = 78 \) from the origin is __________.

Express the following switching circuit in symbolic form of logic. Construct the switching table.

Find \( \frac{dy}{dx} \), if \( y = (\log x)^x \).

The principal solutions of the equation \( \cos\theta = \frac{1}{2} \) are _________.

Evaluate: \[ I = \int \frac{5e^x}{(e^x + 1)(e^{2x} + 9)} \, dx. \]

If \( \alpha, \beta, \gamma \) are direction angles of a line and \( \alpha = 60^\circ, \beta = 45^\circ \), then \( \gamma \) is _________.

Construct the truth table for the statement pattern: \[ [(p \rightarrow q) \land q] \rightarrow p \]
Find the approximate value of \( \tan^{-1} (1.002) \). \[ {Given: } \pi = 3.1416. \]

The dual of statement \( t \lor (p \lor q) \) is _________.

A particle is moving along the X-axis. Its acceleration at time \( t \) is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
If two coins are tossed simultaneously, write the probability distribution of the number of heads.
If \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are the position vectors of the points \( A, B, C \) respectively and \[ 5\mathbf{a} - 3\mathbf{b} - 2\mathbf{c} = \mathbf{0}, \] then find the ratio in which the point \( C \) divides the line segment \( BA \) externally.
Solve: \[ 1 + \frac{dy}{dx} = {cosec}(x + y); \quad {put } x + y = u. \]
If the vectors \(2\hat{i} - 3\hat{j} + 4\hat{k}\) and \( p\hat{i} + 6\hat{j} - 8\hat{k} \) are collinear, then find the value of \( p \).
Evaluate: \[ \int \frac{1}{x^2 + 25} dx. \]

The slope of the tangent to the curve \( x = \sin\theta \) and \( y = \cos 2\theta \) at \( \theta = \frac{\pi}{6} \) is ___________.

If the mean and variance of a binomial distribution are \( 18 \) and \( 12 \) respectively, then the value of \( n \) is __________.

In \( \triangle ABC \), if \( a = 18 \), \( b = 24 \), and \( c = 30 \), then find the value of \[ \sin \left(\frac{A}{2} \right). \]
A box with a square base is to have an open top. The surface area of the box is 147 cm\(^2\). What should its dimensions be in order that the volume is largest?
If \( x = f(t) \) and \( y = g(t) \) are differentiable functions of \( t \), so that \( y \) is a function of \( x \) and \( \frac{dx}{dt} \neq 0 \), then prove that: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}. \] Hence find \( \frac{dy}{dx} \), if \( x = at^2 \), \( y = 2at \).

Solve the following L.P.P. by graphical method: 
Maximize: 
\[ z = 10x + 25y. \] Subject to: \[ 0 \leq x \leq 3, \quad 0 \leq y \leq 3, \quad x + y \leq 5. \]

Find the volume of the parallelepiped whose vertices are
\( A(3,2,-1), B(-2,2,-3), C(3,5,-2) \) and \( D(-2,5,4) \).
Prove that the acute angle \( \theta \) between the lines represented by the equation \[ ax^2 + 2hxy + by^2 = 0 \] is \[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right|. \] Hence find the condition that the lines are coincident.