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In the Linear Programming Problem (LPP), find the point/points giving maximum value for $Z=5x+10y$ subject to constraints $x+2y\le120$, $x+y\ge60$, $x-2y\ge0$, $x, y\ge0$.

Find the particular solution of the differential equation $\left[x\sin^{2}\left(\frac{y}{x}\right)-y\right]dx+x~dy=0$ given that $y=\frac{\pi}{4}$ when $x=1$.

Find: $\int\frac{2x}{(x^{2}+3)(x^{2}-5)}dx$.

A man needs to hang two lanterns on a straight wire whose end points have coordinates $A(4,1,-2)$ and $B(6,2,-3)$. Find the coordinates of the points where he hangs the lanterns such that these points trisect the wire AB.

Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three vectors such that $\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}$ and $\vec{a}\times\vec{b}=\vec{a}\times\vec{c}$, $\vec{a}\ne\vec{0}$. Show that $\vec{b}=\vec{c}$.

Find a vector of magnitude 5 which is perpendicular to both the vectors $3\hat{i}-2\hat{j}+\hat{k}$ and $4\hat{i}+3\hat{j}-2\hat{k}$.

If $y=5\cos x-3\sin x$, prove that $\frac{d^{2}y}{dx^{2}}+y=0$.

Differentiate $\frac{\sin x}{\sqrt{\cos x}}$ with respect to x.

Surface area of a balloon (spherical), when air is blown into it, increases at a rate of $5\text{ mm}^{2}/\text{s}$. When the radius of the balloon is 8 mm, find the rate at which the volume of the balloon is increasing.

Find the domain of the function $f(x)=\cos^{-1}(x^{2}-4)$.

Find the equation of a line in vector and cartesian form which passes through the point $(1,2,-4)$ and is perpendicular to the lines $\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}$ and $\vec{r}=15\hat{i}+29\hat{j}+5\hat{k}+\mu(3\hat{i}+8\hat{j}-5\hat{k})$.

Show that the area of a parallelogram whose diagonals are represented by $\vec{a}$ and $\vec{b}$ is given by $\frac{1}{2}|\vec{a}\times\vec{b}|$. Also find the area of a parallelogram whose diagonals are $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+3\hat{j}-\hat{k}$.

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

Find: $\int\frac{\cos x}{(4+\sin^{2}x)(5-4\cos^{2}x)}dx$.

A coin is tossed twice. Let X be a random variable defined as number of heads minus number of tails. Obtain the probability distribution of X and also find its mean.

Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.

Evaluate: $\int_{\pi/2}^{\pi}e^{x}\left(\frac{1-\sin x}{1-\cos x}\right)dx$.

Check the differentiability of function $f(x)=x|x|$ at $x=0$.

Find k so that $f(x)=\begin{cases}\frac{x^{2}-2x-3}{x+1},&x\ne-1\\ k,&x=-1\end{cases}$ is continuous at $x=-1$.

Let $A=\{1,2,3\}$ and $B=\{4,5,6\}$. A relation R from A to B is defined as $R=\{(x,y):x+y=6, x\in A, y\in B\}$. (i) Write all elements of R. (ii) Is R a function? Justify. (iii) Determine domain and range of R.