List of top Questions asked in CBSE CLASS XII

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.

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: \[ \int \frac{\cos x}{(4 + \sin^2 x)(5 - 4 \cos^2 x)} \, dx. \]

Solve the following Linear Programming Problem using graphical method : Maximize \( Z = 100x + 50y \) subject to the constraints \[ 3x + y \leq 600, \quad x + y \leq 300, \quad y \leq x + 200, \quad x \geq 0, \quad y \geq 0. \]

A coin is tossed twice. Let $X$ be a random variable defined as the number of heads minus the number of tails. Obtain the probability distribution of $X$ and also find its mean.

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 the distance of the point $(-1, -5, -10)$ from the point of intersection of the lines \[ \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}, \quad \frac{x - 4}{5} = \frac{y - 1}{2} = z. \]

Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.

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 $k$ 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 $x = -1$.

Calculate the area of the region bounded by the curve \[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \] and the x-axis using integration.

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?

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 $R$.

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.)