List of top Mathematics Questions asked in CBSE CLASS XII

Let \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) be three vectors such that $\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c} \text{ and } \mathbf{a} \times \mathbf{b} \neq 0 \text{ Show that } \mathbf{b} = \mathbf{c}$.

Find a vector of magnitude 5 which is perpendicular to both the vectors \( 3\hat{i} - 2\hat{j} + \hat{k} \text{ and } 4\hat{i} + 3\hat{j} - 2\hat{k} \).
If $y = 5 \cos x - 3 \sin x, \text{ prove that } \frac{d^2y}{dx^2} + y = 0$.
Differentiate \(\frac{\sin x}{\sqrt{\cos x}}\) \(\text{ with respect to } x\).

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

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

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

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.
Check the differentiability of the function f(x) = |x| at x = 0.
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 $.
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:
Domain of $ \sin^{-1}(2x^2 - 3) $ is:
Evaluate: \[ I = \int_0^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x} \, dx \]
Solve the following Linear Programming Problem graphically:
Minimise \( Z = 3x + 5y \)
subject to the constraints: \[ x + 2y \geq 10, \quad x + y \geq 6, \quad 3x + y \geq 8, \quad x, y \geq 0. \]
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).

Find the local maxima and local minima of the function \[ f(x) = \frac{8}{3} x^3 - 12x^2 + 18x + 5. \]
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.
$\int e^x (\cos x - \sin x) \, dx$ is equal to:
If the direction cosines of a line are $\lambda, \lambda, \lambda$, then $\lambda$ is equal to:
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:
The principal branch of $\cos^{-1} x$ is:
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 \[ \vec{r} = 15\hat{i} + 29\hat{j} + 5\hat{k} + \mu (3\hat{i} + 8\hat{j} - 5\hat{k}). \]