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

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at least 5 successes.

If \( y = \sqrt{\tan{x} + \sqrt{\tan{x} + \sqrt{\tan{x} + \dots}}} \), then show that \[ \frac{dy}{dx} = \sec^2{x} \cdot \frac{2y - 1}{y^2 - 1} \] Find \( \frac{dy}{dx} \) at \( x = 0 \).

Find the particular solution of the differential equation: \[ \frac{dy}{dx} = e^{2y} \cos{x}, \quad \text{when} \quad x = \frac{\pi}{6}, \, y = 0 \]

Evaluate: \[ \int x \tan^{-1}{x} \, dx \]

For the following probability density function of a random variable \( X \), find: \[ f(x) = \frac{x + 2}{18}, \quad \text{for} \, -2<x<4 \]

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 mean of numbers randomly selected from 1 to 15.

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

If \( \theta \) is the acute angle between the lines represented by \( ax^2 + 2hxy + by^2 = 0 \), then prove that \[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| \]

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

Find the shortest distance between lines: \[ \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}, \quad \frac{x-2}{4} = \frac{y-4}{5} = \frac{z-5}{6} \]

Solve the differential equation: \[ \cos{x} \cos{y} \, dy - \sin{x} \sin{y} \, dx = 0 \]

Lines \[ \overrightarrow{r} = (i + j - k) + \lambda(2i - 2j + k), \quad \overrightarrow{r} = (4i - 3j + 2k) + \mu(i - 2j + 2k) \] are coplanar. Find the equation of the plane determined by them.

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

If \( \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} \) are the position vectors of the points A, B, C respectively and \( 5\overrightarrow{a} + 3\overrightarrow{b} - 8\overrightarrow{c} = 0 \), then find the ratio in which the point C divides the line segment AB.

Find the Cartesian co-ordinates of the point whose polar co-ordinates are \( \left( \sqrt{2}, \frac{\pi}{4} \right) \).

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

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.

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

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

Evaluate: \[ \int_0^{\frac{\pi}{2}} \sqrt{1 - \cos{4x}} \, dx \]

Solve the following inequalities graphically and write the corner points of the feasible region: \[ 2x + 3y \leq 6, \quad x + y \geq 2, \quad x \geq 0, \quad y \geq 0 \]

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

i. Write the joint equation of coordinate axes.