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

Evaluate: \( \int x^9 \cdot \sec^2(x^{10}) \, dx \).

Evaluate: \( \int \frac{1}{25 - 9x^2} \, dx \)

Evaluate: \( \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{1}{1 - \sin x} \, dx \)

Find the area of the region bounded by the parabola \( y^2 = 16x \) and its latus rectum.

Find the general solution of \( \tan^2 \theta = 1 \).

Write the unit vector in the opposite direction to \( \vec{u} = 8\hat{i} + 3\hat{j} - \hat{k} \).

Write the condition for the function \( f(x) \), to be strictly increasing, for all \( x \in \mathbb{R} \).

Write the order of the differential equation \[ \sqrt{1 + \left( \frac{dy}{dx} \right)^2} = \left( \frac{d^2y}{dx^2} \right)^{\frac{3}{2}}. \]

Find the adjoint of the matrix \( \left[ \begin{matrix} 2 & -2 \\ 4 & 3 \end{matrix} \right] \).

Answer the following questions: % Question (i) Write the negation of the statement: \( \exists n \in \mathbb{N} \) such that \( n + 8 > 11 \).

Using truth table, prove that the statement patterns \( p \leftrightarrow q \) and \( (p \land q) \lor (\sim p \land \sim q) \) are logically equivalent.

If \(|\vec{a}| = 5\), \(|\vec{b}| = 13\), and \(|\vec{a} \times \vec{b}| = 25\), then \(|\vec{a} \cdot \vec{b}|\) is equal to ........

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

The value of \( \int x^x(1 + \log x) \, dx \) is equal to ...........

The area bounded by the line \( y = x \), X-axis and the lines \( x = -1 \) and \( x = 4 \) is equal to .......... (in square units).

Let \( f(1) = 3 \), \( f'(1) = -\frac{1}{3} \), \( g(1) = -4 \), and \( g'(1) = -\frac{8}{3} \). The derivative of \( \sqrt{[f(x)]^2 + [g(x)]^2} \) w.r.t. \( x \) at \( x = 1 \) is ........

The vector equation of the line passing through the point having position vector \( 4\hat{i} - \hat{j} + 2\hat{k} \) and parallel to vector \( -2\hat{i} - \hat{j} + \hat{k} \) is given by .........

Select and write the correct answer of the following multiple choice type questions:
(i) If \( A = \{1, 2, 3, 4, 5\} \), then which of the following is not true?

In \(\triangle ABC\), \((a + b) \cdot \cos C + (b + c) \cdot \cos A + (c + a) \cdot \cos B\) is equal to .........