List of top Questions asked in CBSE CLASS XII

Sketch the graph of \( y = x|x| \) and hence find the area bounded by this curve, the X-axis, and the ordinates \( x = -2 \) and \( x = 2 \), using integration.
Find the probability that exactly one of them is selected.
If the lines \( \frac{x - 1}{-3} = \frac{y - 2}{2k} = \frac{z - 3}{2} \) and \( \frac{x - 1}{3k} = \frac{y - 1}{1} = \frac{z - 6}{-7} \) are perpendicular to each other, find the value of \( k \) and hence write the vector equation of a line perpendicular to these two lines passing through the point \( (3, -4, 7) \).
Evaluate \( \int_{0}^{\pi/4} \frac{x}{1+\cos 2x + \sin 2x} \, dx \).
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 \( (4, 0, -5) \).
Given that \( x^y + y^x = a^b \), where \( a \) and \( b \) are positive constants, find \( \frac{dy}{dx} \).
If \( y = \cos^3(\sec^2 2t) \), find \( \frac{dy}{dt} \).
Find: \[ I = \int \frac{2x + 3}{x^2(x + 3)} \, dx \]
Find the particular solution of the differential equation \( \left( x (e)^y/^x + y \right) dx = x \, dy \), given that \( y = 1 \) when \( x = 1 \).
The function \( f(x) = kx - \sin x \) is strictly increasing for:
Express \( \tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right) \), where \( -\frac{\pi}{2}<x<\frac{\pi}{2} \), in the simplest form.
The volume of a cube is increasing at the rate of \( 6 \, \text{cm}^3/\text{s} \). How fast is the surface area of the cube increasing, when the length of an edge is \( 8 \, \text{cm} \)?
Let \( \vec{a} \) be any vector such that \( |\vec{a}| = a \). The value of \( |\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2 \) is:
The vectors \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \), \( \vec{b} = \hat{i} - 3\hat{j} - 5\hat{k} \), and \( \vec{c} = -3\hat{i} + 4\hat{j} + 4\hat{k} \) represent the sides of:
If \( e^{x^2y} = c \), then \( \frac{dy}{dx} \) is:
Name the capital city of Vijayanagara empire.
Mention any two centres of the Indian National Movement.
Name any one neighbouring state of Vijayanagara empire.
Name any one ancient city located in the central region of India.
Mention any one Buddhist site of India.

On the same political outline map of India, two places related with the centres of revolt of 1857 one marked as A and B. Identify them and write their names on the lines marked near them. 
(A) Delhi 
(B) Calcutta

On the given political outline map of India, locate and label the following with appropriate signs: 
(i) Amravati – Buddhist site 
(ii) Varanasi – Ancient city 
(iii) (a) Vijayanagara – An Empire of 14th century 
OR 
(iii) (b) Orissa – Territory of 14th century

Why was Gandhiji critical of machines?
Why was wheel considered as an exquisite piece of machinery?
How did Gandhiji glorify charkha?