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

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

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 \( \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \).

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

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

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

Write the joint equation of co-ordinate axes.

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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If \[ A = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} \] is a non-singular matrix, then find \( A^{-1} \) by elementary row transformations.

In \( \Delta ABC \), if \( c^2 + a^2 - b^2 = ac \), then \( \angle B = \)

If \( p \land q \) is F, \( p \rightarrow q \) is F, then the truth values of \( p \) and \( q \) are

Prove that: \[ \int_0^a f(x) \, dx = \int_0^a f(x) \, dx + \int_0^a f(2a - x) \, dx \]

Verify Lagrange's mean value theorem for the function \( f(x) = \sqrt{x + 4} \) on the interval \([0, 5]\).

Prove that the volume of a tetrahedron with coterminous edges \( \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} \) is \[ \text{Volume} = \frac{1}{6} \left| \overrightarrow{a} \cdot (\overrightarrow{b} \times \overrightarrow{c}) \right| \] Hence, find the volume of the tetrahedron whose coterminous edges are \[ \overrightarrow{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \quad \overrightarrow{b} = -\hat{i} + \hat{j} + 2\hat{k}, \quad \overrightarrow{c} = 2\hat{i} + \hat{j} + 4\hat{k}. \]

Evaluate: \[ \int \frac{2x^2 - 3}{(x^2 - 5)(x^2 + 4)} \, dx \]

If \( y = \cos(m^{-1}x)) \), then show that \[ (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + m^2y = 0 \]

Find the length of the perpendicular drawn from the point \( P(3, 2, 1) \) to the line \[ \overrightarrow{r} = (7\hat{i} + 7\hat{j} + 6\hat{k}) + \lambda(-2\hat{i} + 2\hat{j} + 3\hat{k}) \]

Find the approximate value of \( \sin{30^\circ 30'} \). Given that \( 1^\circ = 0.0175 \) radians and \( \cos{30^\circ} = 0.866 \).