List of top Mathematics Questions asked in CBSE CLASS XII

Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

Find:
Let $A=[a_{ij}]$ be a $2\times2$ matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix $A'$.

Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at $x=0$, then find the value of $k$.

Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of $p$ and $q$.

Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box i is selected and a cap picked out is i/6, where i = 1, 2, 3.
Based on the above information, answer the following questions :

Find the vector and cartesian equations of the line passing through the point of intersection of the lines $ \vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j}) $ and $ \vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + 3\hat{k}) $ and parallel to the line $ \frac{x - 1}{-2} = \frac{7 - y}{-3} = z $.

Using integration, find the area of the region bounded by the curve $ y = x|x| $, the x-axis, and the vertical lines $ x = -2 $ and $ x = 2 $.

Solve the following Linear Programming Problem graphically :
Minimize $Z = 20x + 10y$
subject to the constraints:
$x + 2y \le 40$
$3x + y \ge 30$
$4x + 3y \ge 60$
$x, y \ge 0$

If $ xy = e^{x - y} $, then find $ \frac{dy}{dx} $.

Find the general solution of the differential equation $ (x^3 - 3xy^2) dx = (y^3 - 3x^2y) dy $.

Find a point on the line $ \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} $ at a distance of $ \sqrt{2} $ units from the point $ (1, 2, 3) $.

Differentiate $ \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) $ with respect to $ \cos^{-1}(x^2) $.

Solve the differential equation $ y e^y dx = (y^3 + 2x e^y) dy $, given that $ y(0) = 1 $.

Let $n$ be a fixed positive integer. A relation $R$ is defined in the set of integers $\mathbb{Z}$ such that $R = \{(x, y) : (x - y) \text{ is divisible by } n, x, y \in \mathbb{Z}\}$. Determine if $R$ is an equivalence relation.

Let $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$. A function $f : A \to B$ is defined by $f(x) = \frac{x - 2}{x - 3}$. Find whether $f$ is one-one and onto.

Show that the function $f(x) = \begin{cases} \frac{\cos x}{\frac{\pi}{2} - x}, & x \neq \frac{\pi}{2} \\ 1, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$.

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$, then compute $A^2 - 4A - 5I$.