List of top Questions asked in CBSE CLASS XII

Assertion (A): A line in space cannot be drawn perpendicular to $ x $, $ y $, and $ z $ axes simultaneously.

Reason (R): For any line making angles $ \alpha, \beta, \gamma $ with the positive directions of $ x $, $ y $, and $ z $ axes respectively, \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1. \]

In the given figure, ABCD is a parallelogram. If $ \vec{AB} = 2\hat{i} - 4\hat{j} + 5\hat{k} $ and $ \vec{DB} = 3\hat{i} - 6\hat{j} + 2\hat{k} $, then find $ \vec{AD} $ and hence find the area of parallelogram ABCD.

A function $ f $ is defined from $ R \to R $ as $ f(x) = ax + b $, such that $ f(1) = 1 $ and $ f(2) = 3 $. Find the function $ f(x) $. Hence, check whether the function $ f(x) $ is one-one and onto.

If $y = (\tan x)^x$, then find $\frac{dy}{dx}$.

Find the particular solution of the differential equation: \[ x^2 \frac{dy}{dx} - xy = x^2 \cos^2\left(\frac{y}{2x}\right), \] given that when $x = 1$, $y = \frac{\pi}{2}$.

E and F are two independent events such that $P(\overline{E}) = 0.6$ and $P(E \cup F) = 0.6$. Find $P(F)$ and $P(\overline{E} \cup \overline{F})$.

Which of the following statements is true for the function \[ f(x) = \begin{cases} x^2 + 3, & x \neq 0, \\ 1, & x = 0? \end{cases} \]

Evaluate: \[ \int_{1}^{3} \left(|x - 1| + |x - 2| + |x - 3|\right) \, dx. \]

If $A = \begin{bmatrix} 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 \end{bmatrix}$, find $A^{-1}$ and use it to solve the following system of equations: \[ x - 2y = 10, \quad 2x - y - z = 8, \quad -2y + z = 7. \]

If $ A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix} $ and $ A^{-1} = \begin{bmatrix} 1 & -1 & 1 \\ -8 & 7 & -5 \\ b & y & 3 \end{bmatrix} $, find the value of $ (a + x) - (b + y) $.

Evaluate: \[ \int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9 + 16 \sin 2x} \, dx. \]

Evaluate: \[ \int_{0}^{\frac{\pi}{2}} \sin 2x \tan^{-1}(\sin x) \, dx. \]

The image of point $P(x, y, z)$ with respect to the line: \[ \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}, \] is $P'(1, 0, 7)$. Find the coordinates of point $P$.

According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are $55\%$, $37\%$ and $17\%$ due to severe, moderate and light turbulence respectively.
On the basis of the above information, answer the following questions:
(i) Find the probability that an airplane reached its destination
(ii)If the airplane reached its destination late, find the probability that it was due to moderate turbulence.

If \[ \begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}, \] then the value of \[ \left( \frac{24}{x} + \frac{24}{y} \right) \] is:

Why is chlorobenzene less reactive towards nucleophilic substitution reaction?

Compound A with molecular formula (C₂H₆O) on oxidation by PCC gives compound B, which on treatment with dilute alkali forms compound C (a β-hydroxy aldehyde). B on oxidation by potassium permanganate forms C.
Identify A, B, C and D and write all the chemical equations involved.

Assertion (A): Quadrilateral formed by vertices A(0, 0, 0), B(3, 4, 5), C(8, 8, 8) and D(5, 4, 3) is a rhombus.
Reason (R): ABCD is a rhombus if $AB = BC = CD = DA$, and $AC \ne BD$.

If A is a square matrix of order 3 and $|A| = 6$, then the value of $|\text{adj} A|$ is:

If ABCD is a parallelogram and AC and BD are its diagonals, then $\overrightarrow{AC} + \overrightarrow{BD}$ is:

Function f is defined as: \[ f(x) = \begin{cases} 2x + 2, & x<2 \\ k, & x = 2 \\ 3x, & x>2. \end{cases} \text{Find the value of } k \text{ for which } f \text{ is continuous at } x = 2. \]