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The relation between the height of the plant (y cm) with respect to exposure to sunlight is governed by the equation $y=4x-\frac{1}{2}x^{2}$, where x is the number of days exposed to sunlight. (i) Find the rate of growth of the plant with respect to sunlight. (ii) In how many days will the plant attain its maximum height? What is the maximum height?

If A is a $3\times3$ invertible matrix, show that for any scalar $k\ne0$, $(kA)^{-1}=\frac{1}{k}A^{-1}$. Hence calculate $(3A)^{-1}$, where $A=\begin{bmatrix}2&-1&1\\ -1&2&-1\\ 1&-1&2\end{bmatrix}$.

Solve the following Linear Programming Problem using graphical method: Maximise $Z=100x+50y$ subject to the constraints $3x+y\le600$, $x+y\le300$, $y\le x+200$, $x\ge0$, $y\ge0$.

Find the distance of the point $(-1, -5, -10)$ from the point of intersection of the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$.

For the curve $y=5x-2x^{3}$ if x increases at the rate of $2\text{ units/s}$, then how fast is the slope of the curve changing when $x=2$?

If $A=\begin{bmatrix}1&2&0\\ -2&-1&-2\\ 0&-1&1\end{bmatrix}$, then find $A^{-1}$. Hence, solve the system of linear equations: $x-2y=10$, $2x-y-z=8$, $-2y+z=7$.

Given $A=\begin{bmatrix}-4&4&4\\ -7&1&3\\ 5&-3&-1\end{bmatrix}$ and $B=\begin{bmatrix}1&-1&1\\ 1&-2&-2\\ 2&1&3\end{bmatrix}$, find AB. Hence, solve the system of linear equations: $x-y+z=4$, $x-2y-2z=9$, $2x+y+3z=1$.

Find the image A' of the point A(2, 1, 2) in the line $l:\vec{r}=4\hat{i}+2\hat{j}+2\hat{k}+\lambda(\hat{i}-\hat{j}-\hat{k})$. Also, find the equation of line joining AA'. Find the foot of perpendicular from point A on the line l.

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

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

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'.

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.

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.

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.

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
$L_{2}:\vec{r}=\hat{i}+(2\mu+1)\hat{j}-(\mu+2)\hat{k}$