List of top Mathematics Questions asked in CBSE CLASS XII

The perimeter of a rectangular metallic sheet is \( 300 \, {cm} \). It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that the volume of the cylinder so formed is maximum.
The chances of \( P \), \( Q \), and \( R \) getting selected as CEO of a company are in the ratio \( 4 : 1 : 2 \), respectively. The probabilities for the company to increase its profits from the previous year under the new CEO, \( P, Q, \) or \( R \), are \( 0.3, 0.8, \) and \( 0.5 \), respectively. If the company increased the profits from the previous year, find the probability that it is due to the appointment of \( R \) as CEO.
It is given that function \( f(x) = x^4 - 62x^2 + ax + 9 \) attains a local maximum value at \( x = 1 \). Find the value of \( a \), hence obtain all other points where the given function \( f(x) \) attains local maximum or local minimum values.
Evaluate: \[ \int_{0}^{\pi/4} \frac{1}{\sin x + \cos x} \, dx \]
Find: \[ \int \frac{x^2 + 1}{(x^2 + 2)(x^2 + 4)} \, dx \]
Find: \[ \int \frac{2 + \sin 2x}{1 + \cos 2x} e^x \, dx \]
Solve the following linear programming problem graphically: \[ {Maximise } z = 4x + 3y, \quad {subject to the constraints:} \] \[ x + y \leq 800, \quad 2x + y \leq 1000, \quad x \leq 400, \quad x, y \geq 0. \]
Let \( \vec{a} \) and \( \vec{b} \) be two non-zero vectors. Prove that \( |\vec{a} \times \vec{b}| \leq |\vec{a}| |\vec{b}| \). State the condition under which equality holds, i.e., \( |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \).
If \( x \cos(p + y) + \cos p \sin(p + y) = 0 \), prove that \( \cos p \frac{dy}{dx} = -\cos^2(p + y) \), where \( p \) is a constant.
Find the position vector of point \( C \) which divides the line segment joining points \( A \) and \( B \) having position vectors \( \hat{i} + 2\hat{j} - \hat{k} \) and \( -\hat{i} + \hat{j} + \hat{k} \), respectively, in the ratio 4:1 externally. Further, find \( | \overrightarrow{AB} | : | \overrightarrow{BC} | \).
Evaluate: \[ \int_{0}^{\pi/2} \sin 2x \cos 3x \, dx \]
Evaluate: \[ \sec^2 \left( \tan^{-1} \frac{1}{2} \right) + \csc^2 \left( \cot^{-1} \frac{1}{3} \right) \]
If a line makes an angle of \( \frac{\pi}{4} \) with the positive directions of both \( x \)-axis and \( z \)-axis, then the angle which it makes with the positive direction of \( y \)-axis is:
If \( |a| = 2 \) and \( -3 \leq k \leq 2 \), then \( |a| |k| \in: \)
The derivative of \( 2^x \) w.r.t. \( 3^x \) is:
Let \( E \) and \( F \) be two events such that \( P(E) = 0.1, P(F) = 0.3, P(E \cup F) = 0.4 \). Then \( P(F \,|\, E) \) is:
If \( A \) and \( B \) are two skew-symmetric matrices, then \( AB + BA \) is:
If \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} - 2\hat{j} + \hat{k} \), then \( \vec{a} \) and \( \vec{b} \) are:
If \( \alpha, \beta \), and \( \gamma \) are the angles which a line makes with the positive directions of \( x, y, z \) axes respectively, then which of the following is not true?
The restrictions imposed on decision variables involved in an objective function of a linear programming problem are called:
If the sides of a square are decreasing at the rate of \( 1.5 \, \mathrm{cm/s} \), the rate of decrease of its perimeter is:
\( x \log x \frac{dy}{dx} + y = 2 \log x \) is an example of a:
\( \int_{-a}^a f(x) \, dx = 0 \), if:
A function \( f(x) = |1 - x + |x|| \) is:
Let \( \mathbb{R}_+ \) denote the set of all non-negative real numbers. Then the function \( f : \mathbb{R}_+ \to \mathbb{R}_+ \) defined as \( f(x) = x^2 + 1 \) is: