List of top Mathematics Questions

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

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

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

Find the particular solution of the differential equation \( \left( x (e)^y/^x + y \right) dx = x \, dy \), given that \( y = 1 \) when \( x = 1 \).

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

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

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 \( \frac{dy}{dt} \).

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

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.

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 \( 8 \, \text{cm} \)?

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 \( n \).

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. 

The function \( f(x) = kx - \sin x \) is strictly increasing for:

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:

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:

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