List of top Questions asked in CBSE CLASS XII

$\int e^x (\cos x - \sin x) \, dx$ is equal to:
The integrating factor of the differential equation \[ \frac{dy}{dx} + y \tan x - \sec x = 0 \quad \text{is:} \]
Using matrices and determinants, find the value(s) of $k$ for which the pair of equations \[ 5x - ky = 2; \quad 7x - 5y = 3 \] has a unique solution.
The principal branch of $\cos^{-1} x$ is:
If $f(x) = \left\{ \begin{array}{ll} \frac{1 - \sin^3 x}{3 \cos^2 x} & \text{for} \, x \neq \frac{\pi}{2}, \\ k & \text{for} \, x = \frac{\pi}{2}, \end{array} \right. $ is continuous at $x = \frac{\pi}{2}$, then the value of $k$ is:
If the direction cosines of a line are $\lambda, \lambda, \lambda$, then $\lambda$ is equal to:
Show that the area of a parallelogram whose diagonals are represented by \( \vec{a} \) and \( \vec{b} \) is given by \[ \text{Area} = \frac{1}{2} | \vec{a} \times \vec{b} |. \] Also, find the area of a parallelogram whose diagonals are \( 2\hat{i} - \hat{j} + \hat{k} \) and \( \hat{i} + 3\hat{j} - \hat{k} \).
Find the equation of a line in vector and Cartesian form which passes through the point \( (1, 2, -4) \) and is perpendicular to the lines \[ \frac{x - 8}{3} = \frac{y + 19}{-16} = \frac{z - 10}{7}. \] and \[ \vec{r} = 15\hat{i} + 29\hat{j} + 5\hat{k} + \mu (3\hat{i} + 8\hat{j} - 5\hat{k}). \]
Find: \[ \int \frac{\cos x}{(4 + \sin^2 x)(5 - 4 \cos^2 x)} \, dx. \]
The relation between the height of the plant (\(y\) cm) with respect to exposure to sunlight is governed by the equation \[ y = 4x - \frac{1}{2} x^2, \] where \(x\) is the number of days exposed to sunlight.
(i) Find the rate of growth of the plant with respect to sunlight.
(ii) In how many days will the plant attain its maximum height? What is the maximum height?
Find $k$ so that \[ f(x) = \begin{cases} \frac{x^2 - 2x - 3}{x + 1}, & \text{if } x \neq -1 \\ k, & \text{if } x = -1 \end{cases} \] is continuous at $x = -1$.
Check the differentiability of the function $f(x) = |x|$ at $x = 0$.
Evaluate: \[ \int_{\frac{\pi}{2}}^{\pi} \frac{e^{x} \left(1 - \sin x \right)}{1 - \cos x} \, dx. \]
Find the domain of $\sin^{-1} \sqrt{x - 1}$.
For the curve \( y = 5x - 2x^3 \), if \( x increases at the rate of 2 units/s, then how fast is the slope of the curve changing when x = 2?
If $f : \mathbb{R}^+ \to \mathbb{R}$ is defined as $f(x) = \log_a x$ where $a>0$ and $a \neq 1$, prove that $f$ is a bijection. (R$^+$ is the set of all positive real numbers.)
Let $A = \{1, 2, 3\}$ and $B = \{4, 5, 6\}$. A relation $R$ from $A$ to $B$ is defined as $R = \{(x, y) : x + y = 6, x \in A, y \in B \}$. (i) Write all elements of $R$.
(ii) Is $R$ a function? Justify.
(iii) Determine domain and range of $R$.
Let A and B be two square matrices of order 3 such that $\text{det} = 3$ and $\text{det} = -4$. Find the value of $\text{det}(-6AB)$.
If $f(x) = x + \frac{1}{x}, \, x \geq 1$, show that $f$ is an increasing function.
Simplify $\sin^{-1} \left( \frac{x}{\sqrt{1 + x^2}} \right)$.
The area of the region enclosed by the curve $y = \sqrt{x}$ and the lines $x = 0$ and $x = 4$ and the x-axis is :
Assertion : If A and B are two events such that $P(A \cap B) = 0$, then A and B are independent events.
Reason (R): Two events are independent if the occurrence of one does not affect the occurrence of the other.
Assertion : In a Linear Programming Problem, if the feasible region is empty, then the Linear Programming Problem has no solution.
Reason (R): A feasible region is defined as the region that satisfies all the constraints.
The value of \[ \int_0^1 \frac{dx}{e^x + e^{-x}} \] is :
The order and degree of the differential equation \[ \left( \frac{d^2y}{dx^2} \right)^2 + \left( \frac{dy}{dx} \right)^2 = x \sin \left( \frac{dy}{dx} \right) \] are :