List of top Mathematics Questions asked in CBSE CLASS XII

The number of arbitrary constants in the particular solution of the differential equation \[ \log \left( \frac{dy}{dx} \right) = 3x + 4y; \quad y(0) = 0 \] is/are:

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: 
 

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: 
 

The integrating factor of the differential equation \( (x + 2y^2) \frac{dy}{dx} = y \, (y>0) \) is:} 
 

If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( |\vec{a}| = 1 \), \( |\vec{b}| = 2 \), and \( \vec{a} \cdot \vec{b} = \sqrt{3} \), then the angle between \( 2\vec{a} \) and \( -\vec{b} \) is:

Assertion (A): The corner points of the bounded feasible region of a L.P.P. are shown below. The maximum value of \( Z = x + 2y \) occurs at infinite points. 
Reason (R): The optimal solution of a LPP having bounded feasible region must occur at corner points. 

Assertion (A): The relation \( R = \{(x, y) : (x + y) \text{ is a prime number and } x, y \in \mathbb{N}\} \) is not a reflexive relation.
Reason (R): The number \( 2n \) is composite for all natural numbers \( n \).

Find the principal value of \( \tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(-\frac{1}{\sqrt{2}}\right) \).

Show that \( f(x) = \frac{4 \sin x}{2 + \cos x} - x \) is an increasing function of \( x \) in \( \left[ 0, \frac{\pi}{2} \right] \).

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?

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

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