List of top Questions asked in Maharashtra Class XII

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.

Solve the differential equation \( \cos x \cos y \, dy - \sin x \sin y \, dx = 0 \).

Find the mean of a number randomly selected from 1 to 15.

Find the area of the region bounded by the curve \( y = x^2 \) and the line \( y = 4 \).

Find the general solution of \( \sin \theta + \sin 3\theta + \sin 5\theta = 0 \).

If \( -1 \leq x \leq 1 \), then prove that \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \).

Solve the following inequations graphically and write the corner points of the feasible region: \( 2x + 3y \leq 6 \), \( x + y \geq 2 \), \( x \geq 0 \), \( y \geq 0 \).

Show that the function \( f(x) = x^3 + 10x + 7 \), \( x \in \mathbb{R} \), is strictly increasing.

Evaluate: \( \int_0^{\pi/2} (1 - \cos 4x) \, dx \).

Find the area of the region bounded by the curve \( y^2 = 4x \), the X-axis, and the lines \( x = 1 \), \( x = 4 \) for \( y \geq 0 \).

Find the cartesian coordinates of the point whose polar coordinates are \( \left( 2, \frac{\pi}{4} \right) \).

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.

Write the degree of the differential equation \( \frac{d}{dx} e^y + \frac{dy}{dx} = x \).

Write inverse and contrapositive of the following statement: If \( x<y \), then \( x^2<y^2 \).

If \( A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix} \) is a non-singular matrix, then find \( A^{-1} \) by elementary row transformations. Hence write the inverse of \( \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \).

Find the values of c which satisfy \( c \mathbf{u} = 3 \) where \( \mathbf{u} = \hat{i} + 2\hat{j} + 3\hat{k} \).

Write \( \int \cot x \, dx \).

Let the probability mass function (p.m.f.) of a random variable X be \( P(X = x) = \binom{4{x} \left( \frac{5}{9} \right)^x \left( \frac{4}{9} \right)^{4-x} \), for \( x = 0, 1, 2, 3, 4 \), then E(X) is equal to _______.
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Write the joint equation of co-ordinate axes.