List of top Questions asked in CBSE CLASS XII

Find $ \frac{dy}{dx} $ if $ x^3 + y^3 + x^2 = a^b $, where $ a $ and $ b $ are constants.

Find the foot of the perpendicular drawn from the point $ (1, 1, 4) $ on the line $ \frac{x+2}{5} = \frac{y+1}{2} = \frac{z-4}{-3} $.

Sketch a graph of $ y = x^2 $. Using integration, find the area of the region bounded by $ y = 9 $, $ x = 0 $, and $ y = x^2 $.

A furniture workshop produces three types of furniture: chairs, tables, and beds each day. On a particular day, the total number of furniture pieces produced is 45. It was also found that the production of beds exceeds that of chairs by 8, while the total production of beds and chairs together is twice the production of tables. Determine the units produced of each type of furniture, using the matrix method.

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

Evaluate: \[ \int_1^5 \left( |x-2| + |x-4| \right) \, dx \]

Find the particular solution of the differential equation: \[ x \sin^2 \left( \frac{y}{x} \right) \, dx + x \, dy = 0 \quad \text{given that} \quad y = \frac{\pi}{4}, \text{ when } x = 1. \]

In the Linear Programming Problem (LPP), find the point/points giving the maximum value for $ Z = 5x + 10y$ subject to the constraints:
\[x + 2y \leq 120 \\ x + y \geq 60 \\ x - 2y \geq 0 \\ x \geq 0, y \geq 0\]

Find a vector of magnitude 5 which is perpendicular to both the vectors $ 3\hat{i} - 2\hat{j} + \hat{k} \text{ and } 4\hat{i} + 3\hat{j} - 2\hat{k} $.

Find the value of $ a $ for which } $ f(x) = \sqrt{3} \sin x - \cos x - 2ax + 6 $ $\text{ is decreasing in } \mathbb{R}$.

Assertion (A): Let $ A = \{ x \in \mathbb{R} : -1 \leq x \leq 1 \} $. If $ f : A \to A $ be defined as $ f(x) = x^2 $, then $ f $ is not an onto function.
Reason (R): If $ y = -1 \in A $, then $ x = \pm \sqrt{-1} \notin A $.

Find the domain of the function $ f(x) = \cos^{-1}(x^2 - 4) $.

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 $\frac{\sin x}{\sqrt{\cos x}}$ $\text{ with respect to } x$.

If $y = 5 \cos x - 3 \sin x, \text{ prove that } \frac{d^2y}{dx^2} + y = 0$.

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.

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:

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 integral \[ \int_0^{\frac{\pi}{2}} \cos x \cdot e^{\sin x} \, dx \] is equal to:

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

The integrating factor of the differential equation \[ e^{-\frac{2x}{\sqrt{x}}} \, \frac{dy}{dx} = 1 \] is:

The slope of the curve $ y = -x^3 + 3x^2 + 8x - 20 $ is maximum at:

The integral \[ \int \sqrt{1 + \sin x} \, dx \] is equal to:

If $ f(x) = \lfloor x \rfloor $ is the greatest integer function, then the correct statement is: