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Solve the following linear programming problem graphically: Minimise $Z=x-5y$ subject to the constraints: $x-y\ge0$, $-x+2y\ge2$, $x\ge3$, $y\le4$, $y\ge0$.

Let R be a relation defined over N, where N is set of natural numbers, defined as "mRn if and only if m is a multiple of n, m, $n\in N$." Find whether R is reflexive, symmetric and transitive or not.

Solve the following differential equation: $(1+x^{2})\frac{dy}{dx}+2xy=4x^{2}$.

Solve the differential equation $2(y+3)-xy\frac{dy}{dx}=0;$ given $y(1)=-2$.

Check the differentiability of f(x) at $x=-2$ if $f(x)=\begin{cases}2x-3,-3\le x\le-2\\ x+1,-2<x\le0\end{cases}$.

If $x=e^{\frac{x}{y}}$, then prove that $\frac{dy}{dx}=\frac{x-y}{x\log x}$.

If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors, then find x, such that $\vec{\alpha}=(x-2)\vec{a}+\vec{b}$ and $\vec{\beta}=(3+2x)\vec{a}-2\vec{b}$ are collinear.

Find the values of 'a' for which $f(x)=\sin x-ax+b$ is increasing on R.

Evaluate: $\int_{0}^{\frac{\pi}{4}}\sqrt{1+\sin 2x}dx$.

If $\vec{\alpha}$ and $\vec{\beta}$ are position vectors of two points P and Q respectively, then find the position vector of a point R in QP produced such that $QR=\frac{3}{2}QP$.

A vector $\vec{a}$ makes equal angles with all the three axes. If the magnitude of the vector is $5\sqrt{3}$ units, then find $\vec{a}$.

A school wants to allocate students into three clubs Sports, Music and Drama, under following conditions: The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club. The number of students in Music club should be 20 more than half the number of students in Sports club. The total number of students to be allocated in all three clubs are 180. Find the number of students allocated to different clubs, using matrix method.

Find a point P on the line $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$ such that its distance from point $Q(2,4,-1)$ is 7 units. Also, find the equation of line joining P and Q.

Find the image A' of the point $A(1,6,3)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$. Also, find the equation of the line joining A and A'.

Find the absolute maximum and absolute minimum of function $f(x)=2x^{3}-15x^{2}+36x+1$ on $[1, 5]$.

If $x=a\left(\cos\theta+\log\tan\frac{\theta}{2}\right)$ and $y=\sin\theta$, then find $\frac{d^{2}y}{dx^{2}}$ at $\theta=\frac{\pi}{4}$.

If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$, then prove that $\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}$.

Sketch the graph of $y=|x+3|$ and find the area of the region enclosed by the curve, x-axis, between $x=-6$ and $x=0$, using integration.

For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data it was revealed that two third of the total applicants were females and other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.

The probability distribution for the number of students being absent in a class on a Saturday is as follows: X: 0, 2, 4, 5; $P(X)$: p, 2p, 3p, p. Where X is the number of students absent. (i) Calculate p. (ii) Calculate the mean of the number of absent students on Saturday.