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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 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: $x+2y-3z=1$, $2x-3z=2$, $x+2y=3$

Find a vector of magnitude 4 units perpendicular to each of the vectors $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\hat{k}$ and hence verify your answer.

The area of the circle is increasing at a uniform rate of $2~cm^{2}/sec$. How fast is the circumference of the circle increasing when the radius $r=5$ cm?

If the lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$ and $\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-7}$ are perpendicular to each other, find the value of k and hence write the vector equation of a line perpendicular to these two lines and passing through the point (3, -4, 7).

Find the distance between the line $\frac{x}{2}=\frac{2y-6}{4}=\frac{1-z}{-1}$ and another line parallel to it passing through the point (4, 0, -5).

If $A=[\begin{bmatrix}2&1&-3\\ 3&2&1\\ 1&2&-1\end{bmatrix}],$ find $A^{-1}$ and hence solve the following system of equations: $2x+y-3z=13$, $3x+2y+z=4$, $x+2y-z=8$

Sketch the graph of $y=x|x|$ and hence find the area bounded by this curve, X-axis and the ordinates $x=-2$ and $x=2,$ using integration.

A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.

A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.

Find the particular solution of the differential equation $(xe^{\frac{y}{x}}+y)dx=x~dy$, given that $y=1$ when $x=1$

Evaluate $\int_{0}^{\frac{\pi}{4}}\frac{x~dx}{1+cos~2x+sin~2x}$

The volume of a cube is increasing at the rate of $6~cm^{3}/s.$ How fast is the surface area of cube increasing, when the length of an edge is 8 cm?

The image of point $P(x,y,z)$ with respect to line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ is $P^{\prime}(1,0,7)$ Find the coordinates of point P.

If $A=[\begin{bmatrix}1&-2&0\\ 2&-1&-1\\ 0&-2&1\end{bmatrix}],$ find $A^{-1}$ and use it to solve the following system of equations: $x-2y=10$, $2x-y-z=8$, $-2y+z=7$

E and F are two independent events such that $P(\overline{E})=0\cdot6$ and $P(E\cup F)=0\cdot6$ Find $P(F)$ and $P(\overline{E}\cup\overline{F})$

Solve the following linear programming problem graphically: Maximise $z=500x+300y,$ subject to constraints $x+2y\le12$, $2x+y\le12$, $4x+5y\ge20$, $x\ge0$, $y\ge0$

Find the particular solution of the differential equation given by $x^{2}\frac{dy}{dx}-xy=x^{2}cos^{2}(\frac{y}{2x})$ given that when $x=1$, $y=\frac{\pi}{2}$

In the given figure, ABCD is a parallelogram. If $\vec{AB}=2\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{DB}=3\hat{i}-6\hat{j}+2\hat{k}$ , then find $\vec{AD}$ and hence find the area of parallelogram ABCD.