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During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by $\vec{B}=2\hat{i}+8\hat{j}$, $\vec{W}=6\hat{i}+12\hat{j}$ and $\vec{F}=12\hat{i}+18\hat{j}$ respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.

Verify that lines given by $\vec{r}=(1-\lambda)\hat{i}+(\lambda-2)\hat{j}+(3-2\lambda)\hat{k}$ and $\vec{r}=(\mu+1)\hat{i}+(2\mu-1)\hat{j}-(2\mu+1)\hat{k}$ are skew lines. Hence, find shortest distance between the lines.

Evaluate: $\int_{0}^{\frac{\pi}{4}}\frac{dx}{\cos^{3}x\sqrt{2\sin 2x}}$.

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.

Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors $\vec{a}=3\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=2\hat{i}-2\hat{j}+4\hat{k}$. Determine the angle formed between the kite strings. Assume there is no slack in the strings.

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.

Evaluate: $\tan^{-1}\left[2\sin\left(2\cos^{-1}\frac{\sqrt{3}}{2}\right)\right]$.

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

Differentiate $2^{\cos^{2}x}$ w.r.t $\cos^{2}x$.

Solve the following L.P.P. graphically: Maximise $Z=60x+40y$ Subject to $x+2y\le12$, $2x+y\le12$, $4x+5y\ge20$, $x,y\ge0$

Find the shortest distance between the lines $L_{1}$ & $L_{2}$ given below :
$L_{1}$: The line passing through (2, -1, 1) and parallel to $\frac{x}{1}=\frac{y}{1}=\frac{z}{3}$ and
$L_{2}:\vec{r}=\hat{i}+(2\mu+1)\hat{j}-(\mu+2)\hat{k}$

Find the co-ordinates of the foot of the perpendicular drawn from the point (2, 3, -8) to the line $\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}$ Also, find the perpendicular distance of the given point from the line.

Find the area of the region bounded by the curve $4x^{2}+y^{2}=36$ using integration.

Find the product of the matrices $[\begin{bmatrix}1&2&-3\\ 2&3&2\\ 3&-3&-4\end{bmatrix}][\begin{bmatrix}-6&17&13\\ 14&5&-8\\ -15&9&-1\end{bmatrix}]$ and hence solve the system of linear equations: $x+2y-3z=-4$, $2x+3y+2z=2$, $3x-3y-4z=11$

If $A=[\begin{bmatrix}1&2&-3\\ 2&0&-3\\ 1&2&0\end{bmatrix}],$ then find $A^{-1}$ and hence solve the following system of equations: $x+2y-3z=1$, $2x-3z=2$, $x+2y=3$

The random variable X has the following probability distribution where a and b are some constants: $P(X)$ for X=1 is 0.2, X=2 is a, X=3 is a, X=4 is 0.2, X=5 is b. If the mean $E(X)=3$, then find values of a and b and hence determine $P(X\ge3)$