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If $y=\log(\sqrt{x}+\frac{1}{\sqrt{x}})^{2}$, then show that $x(x+1)^{2}y_{2}+(x+1)^{2}y_{1}=2$.

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 $y(1)=-2$.

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 $\vec{a}$.

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 $\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}$.

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.

Find: $\int\frac{x+\sin x}{1+\cos x}dx$.

Solve the following linear programming problem graphically: Maximise $Z=x+2y$ Subject to the constraints: $x-y\ge0$, $x-2y\ge-2$, $x\ge0$, $y\ge0$.

The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate its area increasing when the side of the triangle is 15 cm?

Find a vector of magnitude 21 units in the direction opposite to that of $\vec{AB}$ where A and B are the points $A(2,1,3)$ and $B(8,-1,0)$ respectively.

Find the intervals in which function $f(x)=5x^{\frac{3}{2}}-3x^{\frac{5}{2}}$ is (i) increasing (ii) decreasing.

The diagonals of a parallelogram are given by $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$. Find the area of the parallelogram.