List of top Questions asked in CBSE CLASS XII

Evaluate \[ I = \int_0^\pi \frac{x \, \tan x}{\sec x + \tan x} \, dx \]
(a) 10 identical blocks are marked with '0' on two of them, '1' on three of them, and '2' on four of them. If $X$ denotes the number written on the block, then write the probability distribution of $X$ and calculate its mean.
(b) In a village of 8000 people, 3000 go out of the village to work and 4000 are women. It is noted that 30% of women go out of the village to work. What is the probability that a randomly chosen individual is either a woman or a person working outside the village?
Show that the function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 4x^3 - 5 \), \( \forall x \in \mathbb{R} \), is one-one and onto.
Let \( 2x + 5y - 1 = 0 \) and \( 3x + 2y - 7 = 0 \) represent the equations of two lines on which the ants are moving on the ground. Using matrix method, find a point common to the paths of the ants.
A shopkeeper sells 50 Chemistry, 60 Physics and 35 Maths books on day I and sells 40 Chemistry, 45 Physics and 50 Maths books on day II. If the selling price for each subject book is \( \text{Rs } 150 \) (Chemistry), \( \text{Rs } 175 \) (Physics) and \( \text{Rs } 180 \) (Maths), then find his total sale in two days, using matrix method. If cost price of all the books together is \( \text{Rs } 35,000 \), what profit did he earn after the sale of two days?
Assertion (A): $f(x) = \begin{cases} x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$ is continuous at $x = 0$.
Reason (R): When $x \to 0$, $\sin \frac{1}{x}$ is a finite value between -1 and 1.
Assertion (A): The set of values of $\sec^{-1} \left( \frac{\sqrt{3}}{2} \right)$ is a null set.
Reason (R): $\sec^{-1}x$ is defined for $x \in \mathbb{R} - (-1, 1)$.
In a Linear Programming Problem, the objective function $Z = 5x + 4y$ needs to be maximised under constraints $3x + y \leq 6$, $x \leq 1$, $x \geq 0$, $y \geq 0$. Express the LPP on the graph and shade the feasible region, and mark the corner points.
$\int \frac{e^x}{\sqrt{4 - 2x}} dx$ is equal to:
If $|\mathbf{a} + \mathbf{b}| = |\mathbf{a} - \mathbf{b}|$ for any two vectors, then vectors $\mathbf{a}$ and $\mathbf{b}$ are:
If $\begin{bmatrix} 2x-1 & 3x \\ 0 & y^2 - 1 \end{bmatrix}$ is a matrix, then the value of $(x - y)$ is:
If $f(x) = \begin{cases} ax + b, & x \leq 3 \\ 7, & 5<x \end{cases}$ is continuous in $\mathbb{R}$, then the values of $a$ and $b$ are:
If $f(x) = -2x^2$, then the correct statement is:
If $f: \mathbb{R} \to \mathbb{R}$ is defined as $f(x) = 2x - \sin x$, then $f$ is:
$\int \frac{e^{\log x}}{e^{6 \log x} - e^{5 \log x}} dx$ is equal to:
If $A$ and $B$ are two square matrices each of order 3 with $|A| = 3$ and $|B| = 5$, then $|2AB|$ is:
Let $A$ be a square matrix of order 3. If $|A| = 5$, then $|adj A|$ is:
Domain of $f(x) = \cos^{-1} x + \sin x$ is:
Write the scientific name of the source organism of the thermostable DNA polymerase used in PCR.


A ladder of fixed length \( h \) is to be placed along the wall such that it is free to move along the height of the wall.
Based upon the above information, answer the following questions:

(ii)} Find the derivative of the area \( A \) with respect to the height on the wall \( x \), and find its critical point.


A ladder of fixed length \( h \) is to be placed along the wall such that it is free to move along the height of the wall.
Based upon the above information, answer the following questions:

(iii) (a) Show that the area \( A \) of the right triangle is maximum at the critical point.


A ladder of fixed length \( h \) is to be placed along the wall such that it is free to move along the height of the wall.
Based upon the above information, answer the following questions:

(iii) (b) If the foot of the ladder, whose length is 5 m, is being pulled towards the wall such that the rate of decrease of distance \( y \) is \( 2 \, \text{m/s} \), then at what rate is the height on the wall \( x \) increasing when the foot of the ladder is 3 m away from the wall?

Three students, Neha, Rani, and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads, and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads, and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads, and 3 erasers.
Based upon the above information, answer the following questions:

(ii) Find \( |A| \) and confirm if it is possible to find \( A^{-1} \).

Three students, Neha, Rani, and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads, and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads, and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads, and 3 erasers.
Based upon the above information, answer the following questions:

(iii) Find \( A^{-1} \), if possible, and write the formula to find \( \mathbf{X} \).