List of top Mathematics Questions asked in CBSE CLASS XII

Find $ k $ so that the function \[ f(x) = \begin{cases} \frac{x^2 - 2x - 3}{x + 1} & \text{if } x \neq -1 \\ k & \text{if } x = -1 \end{cases} \] is continuous at $ x = -1 $.
A cylindrical water container has developed a leak at the bottom. The water is leaking at the rate of 5 cm$^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.
Find: \[ \int \frac{\sqrt{x}}{1 + \sqrt{x^{3/2}}} \, dx \]
If \[ A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \quad \text{and} \quad A^2 = kA, \quad \text{then find the value of k.} \]
If $ \int e^{-3 \log x} \, dx = f(x) + C $, then $ f(x) $ is:
The solution of the differential equation $ \frac{dy}{dx} = -\frac{x}{y} $ represents family of:
Domain of $ \sin^{-1}(2x^2 - 3) $ is:
The coordinates of the foot of the perpendicular drawn from the point $A(-2, 3, 5)$ on the y-axis are:
If A and B are invertible matrices of order $3 \times 3$ such that $\text{det} = 4$ and $\text{det}([AB]^{-1}) = \frac{1}{20}$, then $\text{det}$ is equal to:
Solve the following Linear Programming Problem graphically:
Minimise \( Z = 3x + 5y \)
subject to the constraints: \[ x + 2y \geq 10, \quad x + y \geq 6, \quad 3x + y \geq 8, \quad x, y \geq 0. \]
Evaluate: \[ I = \int_0^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x} \, dx \]
Find: \[ J = \int \frac{\sqrt{x^2 + 1} \left[ \log(x^2 + 1) - 2 \log x \right]}{x^2} \, dx \]
Find the local maxima and local minima of the function \[ f(x) = \frac{8}{3} x^3 - 12x^2 + 18x + 5. \]
Find: \[ \int \frac{\cos x \, dx}{1 + \cos x + \sin x}. \]
Consider the experiment of tossing a coin. If the coin shows head, toss it again; but if it shows a tail, then throw a die. Find the conditional probability of the event A: `the die shows a number greater than 3' given that B: `there is at least one tail'.

Probability Distribution of Random Variable X

X1232λ3λ4λ
P(X)\(\frac{11}{30}\)\(\frac{1}{15}\)\(\frac{10}{30}\)\(\frac{3\lambda}{10}\)\(\frac{1}{15}\)\(\frac{1}{10}\)

(i) Calculate \( \lambda \), if \( E(X) = 3.2 \)
(ii) Find P(X > 1).

$\int e^x (\cos x - \sin x) \, dx$ is equal to:
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 integrating factor of the differential equation \[ \frac{dy}{dx} + y \tan x - \sec x = 0 \quad \text{is:} \]
If $\left| \begin{array}{ccc} -1 & 2 & 4 \\ 1 & x & 1 \\ 0 & 3 & 3x \end{array} \right| = -57$, the product of the possible values of $x$ 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:
The principal branch of $\cos^{-1} x$ is:
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 \[ \vec{r} = 15\hat{i} + 29\hat{j} + 5\hat{k} + \mu (3\hat{i} + 8\hat{j} - 5\hat{k}). \]