List of top Questions asked in CBSE CLASS XII

Find the distance between the line \( \frac{x}{2} = \frac{2y - 6}{4} = \frac{1 - z}{-1} \) and another line parallel to it passing through the point \( (4, 0, -5) \).
Check whether the relation \( S \) in the set of real numbers \( \mathbb{R} \), defined by \( S = \{(a, b) : a - b + \sqrt{2} \text{ is an irrational number} \} \), is reflexive, symmetric, or transitive.
Let \( A = \mathbb{R} - \{5\} \) and \( B = \mathbb{R} - \{1\} \). Consider the function \( f : A \to B \), defined by \( f(x) = \frac{x - 3}{x - 5} \). Show that \( f \) is one-one and onto.
Find: \( \int e^x \left[ \frac{1}{(1 + x^2)^{3/2}} + \frac{x}{\sqrt{1 + x^2}} \right] dx \).
Evaluate \( \int_{0}^{\pi/4} \frac{x}{1+\cos 2x + \sin 2x} \, dx \).

The corner points of the feasible region determined by the system of linear constraints are as shown in the following figure: 
(i) If \( Z = 3x - 4y \) be the objective function, then find the maximum value of \( Z \). 
(ii) If \( Z = px + qy \) where \( p, q>0 \) be the objective function, find the condition on \( p \) and \( q \) so that maximum value of \( Z \) occurs at \( B(4, 10) \) and \( C(6, 8) \). 


 

A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.
A card from a well-shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.
Find: \[ \int \sec^3 \theta \, d\theta \]
Find the particular solution of the differential equation \( \left( x e^{\frac{y}{x}} + y \right) dx = x \, dy \), given that \( y = 1 \) when \( x = 1 \).
Find the particular solution of the differential equation \( \frac{dy}{dx} = y \cot 2x \), given that \( y\left(\frac{\pi}{4}\right) = 2 \).
Find \( \frac{dy}{dx} \), if \( y = (\cos x)^x + \cos^{-1}(\sqrt{x}) \) is given:
Find: \[ \int \frac{2x}{(x^2 + 1)(x^2 - 4)} \, dx. \]
Find the principal value of \( \tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(-\frac{1}{\sqrt{2}}\right) \):
Express \( \tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right) \), where \( -\frac{\pi}{2}<x<\frac{\pi}{2} \), in the simplest form.
Show that the function \( f(x) = \sin x + \cos x \) is strictly decreasing in the interval \( \left( \frac{\pi}{4}, \frac{5\pi}{4} \right) \).
The volume of a cube is increasing at the rate of \( 6 \, {cm}^3/{s} \). How fast is the surface area of the cube increasing, when the length of an edge is \( 8 \, {cm} \)?
If \( x^y = e^{x - y} \), prove that \( \frac{dy}{dx} = \frac{\log x}{(1 + \log x)^2} \):
If \( y = \cos^3(\sec^2 2t) \), find \( \frac{dy}{dt} \).

ASSERTION-REASON BASED QUESTIONS:-
Questions number 19 and 20 are Assertion and Reason based questions. 
Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). 
Select the correct answer from the codes (A), (B), (C) and (D) as given below.

The integrating factor of the differential equation \( (x + 2y^2) \frac{dy}{dx} = y \, (y > 0) \) is:
If \( A = \begin{bmatrix} 2 & 1 \\ -4 & -2 \end{bmatrix} \), then the value of \( I - A + A^2 - A^3 + \ldots \) is:
Given that \( A^{-1} = \frac{1}{7} \begin{bmatrix} 2 & 1 \\ -3 & 2 \end{bmatrix} \), matrix \( A \) is:
If \[ \begin{bmatrix} a & c & 0 \\ b & d & 0 \\ 0 & 0 & 5 \end{bmatrix} \] is a scalar matrix, then the value of \( a + 2b + 3c + 4d \) is:

The Cartesian equation of a line passing through the point with position vector \( \vec{a} = \hat{i} - \hat{j} \) and parallel to the line 
\( \vec{r} = \hat{i} + \hat{k} + \mu(2\hat{i} - \hat{j}) \), is: