List of top Questions asked in CBSE CLASS XII

If \( x^y = e^{x-y} \), prove that \(\frac{dy}{dx} = \frac{\log x}{(1 + \log x)^{2}}\). 
 

Evaluate: \[ I = \int_{-\frac{1}{2}}^{\frac{1}{2}} \cos x \cdot \log \left( \frac{1 - x}{1 + x} \right) dx \]

Find the particular solution of the differential equation \( \frac{dy}{dx} = y \cot 2x \), given that \( y\left(\frac{\pi}{4}\right) = 2 \).

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.

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.

Solve the following linear programming problem graphically: Maximise \[ Z = 2x + 3y \] Subject to the constraints: \[ x + y \leq 6 \] \[ x \geq 2 \] \[ y \leq 3 \] \[ x, y \geq 0 \]

Find:  \(\int e^x\) \(\frac{1}{(1+x^2)^{3/2}}\) + \(\frac{x}{\sqrt{1+x^2}}  dx .\) 
 

Use the product of matrices  \(\begin{bmatrix} 1 & 2 & -3 \\ 3 & 2 & -2 \\ 2 & -1 & 1 \end{bmatrix}\) \(\begin{bmatrix} 0 & 1 & 2 \\ -7 & -7 & -7 \\ 7 & 5 & -4 \end{bmatrix}\)  to solve the following system of equations: \[ x + 2y - 3z = 6 \] \[ 3x + 2y - 2z = 3 \] \[ 2x - y + z = 2 \]

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.
 

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.} 
 

Find the probability that exactly two of them are selected.

What rebate in price of the calculator should the store give to maximise the revenue?

Find the probability that exactly one of them is selected.

Find \( P(G \cap \overline{H}) \), where \( G \) is the event of Jaspreet's selection and \( \overline{H} \) denotes the event that Rohit is not selected.

How far is the star \( V \) from star \( A \)?

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 measure of \( \angle VDA \).

What is the projection of vector \( \overrightarrow{DV} \) on vector \( \overrightarrow{DA} \)?

Using integration, find the area bounded by the ellipse \( 9x^2 + 25y^2 = 225 \), the lines \( x = -2, x = 2 \), and the X-axis.

What is the probability that at least one of them is selected?

Find a unit vector in the direction of \( \overrightarrow{DA} \).

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 \( (4, 0, -5) \).

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: \[ I = \int \frac{2x + 3}{x^2(x + 3)} \, dx \]