List of top Mathematics Questions asked in CBSE CLASS XII

Surface area of a balloon (spherical), when air is blown into it, increases at a rate of 5 mm²/s. When the radius of the balloon is 8 mm, find the rate at which the volume of the balloon is increasing.
Differentiate \(\frac{\sin x}{\sqrt{\cos x}}\) \(\text{ with respect to } x\).

The respective values of \( |\vec{a}| \) and} \( |\vec{b}| \), if given \[ (\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 512 \quad \text{and} \quad |\vec{a}| = 3 |\vec{b}|, \] are:

Assertion (A): The shaded portion of the graph represents the feasible region for the given Linear Programming Problem (LPP).
Reason (R): The region representing \( Z = 50x + 70y \) such that \( Z < 380 \) does not have any point common with the feasible region.

For a Linear Programming Problem (LPP), the given objective function \( Z = 3x + 2y \) is subject to constraints: \[ x + 2y \leq 10, \] \[ 3x + y \leq 15, \] \[ x, y \geq 0. \]

The correct feasible region is:

The sum of the order and degree of the differential equation \[ \left( 1 + \left( \frac{dy}{dx} \right)^2 \right) \frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^3 \] is:
Let \( \vec{a} \) be a position vector whose tip is the point (2, -3). If \( \overrightarrow{AB} = \vec{a} \), where coordinates of A are (–4, 5), then the coordinates of B are:
The integral \[ \int_0^{\frac{\pi}{2}} \cos x \cdot e^{\sin x} \, dx \] is equal to:
The integrating factor of the differential equation \[ e^{-\frac{2x}{\sqrt{x}}} \, \frac{dy}{dx} = 1 \] is:

The area of the region enclosed between the curve \( y = |x| \), x-axis, \( x = -2 \)} and \( x = 2 \) is:

The integral \[ \int \sqrt{1 + \sin x} \, dx \] is equal to:
The slope of the curve \( y = -x^3 + 3x^2 + 8x - 20 \) is maximum at:
If \[ \begin{bmatrix} 4 + x & x - 1 \\ -2 & 3 \end{bmatrix} \] is a singular matrix, then the value of \( x \) is:
If \( f(x) = \lfloor x \rfloor \) is the greatest integer function, then the correct statement is:
If \[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & -2 \end{bmatrix}, \] then \( |A| \) is:
The principal value of \( \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right) \) is:
If \[ f(x) = \begin{cases} \frac{\sin^2(ax)}{x^2}, & x \neq 0 \\ 1, & x = 0 \end{cases} \] is continuous at \( x = 0 \), then the value of \( a \) is:
If \[ \left| \frac{2x}{5} \right| = \left| \frac{6 - 5}{4} \right|, \quad \text{then the value of } x \text{ is:} \]

If \( P(A \cup B) = 0.9 \) and} \( P(A \cap B) = 0.4 \), then \( P(A) + P(B) \) is:

If \[ A = \begin{bmatrix} 1 & 2 & 3 \\ -4 & 3 & 7 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 3 \\ -1 & 2 \\ 0 & 5 \end{bmatrix}, \] then the correct statement is:
Find: \[ I = \int (\sqrt{\tan x} + \sqrt{\cot x}) dx. \]
For the given graph of a Linear Programming
For the given graph of a Linear Programming Problem, write all the constraints satisfying the given feasible region.

Evaluate:
$\displaystyle \int_{0}^{3} x \cos(\pi x) \, dx$

Find:$\displaystyle \int \dfrac{dx}{\sin x + \sin 2x}$

A cylindrical water container has developed a leak at the bottom. The water is leaking at the rate of 5 cm$^3$/s from the leak. If the radius of the container is 15 cm, find the rate at which the height of water is decreasing inside the container, when the height of water is 2 meters.