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Let R be a relation defined on a set N of natural numbers such that $R=\{(x,y): xy \text{ is a square of a natural number, } x, y\in N\}$. Determine if the relation R is an equivalence relation.

Show that the function $f:R\rightarrow R$ defined by $f(x)=4x^{3}-5$, $\forall x\in R$ is one-one and onto.

In a village of 8000 people, 3000 go out of the village to work and 4000 are women. It is noted that 30% of women go out of the village to work. What is the probability that a randomly chosen individual is either a woman or a person working outside the village?

10 identical blocks are marked with '0' on two of them, '1' on three of them, '2' on four of them and '3' on one of them and put in a box. If X denotes the number written on the block, then write the probability distribution of X and calculate its mean.

In a Linear Programming Problem, the objective function $Z=5x+4y$ needs to be maximised under constraints $3x+y\le6$, $x\le1$, $x, y\ge0$. Express the LPP on the graph and shade the feasible region and mark the corner points.

If $-2x^{2}-5xy+y^{3}=76$, then find $\frac{dy}{dx}$.

Differentiate $\left(\frac{5^{x}}{x^{5}}\right)$ with respect to x.

If $A=\begin{bmatrix}2&3\\ -1&2\end{bmatrix}$, then show that $A^{2}-4A+7I=0$.

Let $f:A\rightarrow B$ be defined by $f(x)=\frac{x-2}{x-3}$ where $A=R-\{3\}$ and $B=R-\{1\}$. Discuss the bijectivity of the function.

Let the polished side of the mirror be along the line $\frac{x}{1}=\frac{1-y}{-2}=\frac{2z-4}{6}$. A point $P(1,6,3)$, some distance away from the mirror, has its image formed behind the mirror. Find the coordinates of the image point and the distance between the point P and its image.

Solve the differential equation $(1+x^{2})\frac{dy}{dx}+2xy-4x^{2}=0$ subject to initial condition $y(0)=0$.

Solve the differential equation: $x^{2}y~dx-(x^{3}+y^{3})dy=0$.

Draw a rough sketch for the curve $y=2+|x+1|$. Using integration, find the area of the region bounded by the curve $y=2+|x+1|$, $x=-4$, $x=3$ and $y=0$.

Evaluate: $\int_{0}^{\pi/2}\frac{x}{\sin x+\cos x}dx$.

Find: $\int\frac{x^{2}+1}{(x-1)^{2}(x+3)}dx$.

A person is Head of two independent selection committees I and II. If the probability of making a wrong selection in committee I is 0.03 and that in committee II is 0.01, then find the probability that the person makes the correct decision of selection: (i) in both committees (ii) in only one committee.

Let the position vectors of the points A, B and C be $3\hat{i}-\hat{j}-2\hat{k}$, $\hat{i}+2\hat{j}-\hat{k}$ and $\hat{i}+5\hat{j}+3\hat{k}$ respectively. Find the vector and cartesian equations of the line passing through A and parallel to line BC.

Find the distance of the point $P(2,4,-1)$ from the line $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$.

Consider the Linear Programming Problem, where the objective function $Z=(x+4y)$ needs to be minimized subject to constraints $2x+y\ge1000$, $x+2y\ge800$, $x,y\ge0$. Draw a neat graph of the feasible region and find the minimum value of Z.

Show that the function $f:N\rightarrow N$, where N is a set of natural numbers, given by $f(n) = n-1$, if n is even, $n+1$, if n is odd, is a bijection.