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Find the equation of a line (in vector and cartesian form) that passes through the point of intersection of lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ and is parallel to the vector $3\hat{i}+2\hat{j}-8\hat{k}$.

Opposite sides of a square are along the lines : $\vec{r}=\hat{i}+2\hat{j}-4\hat{k}+\lambda(2\hat{i}+3\hat{j}+6\hat{k})$ and $\vec{r}=3\hat{i}+3\hat{j}-5\hat{k}+\mu(2\hat{i}+3\hat{j}+6\hat{k})$. Find the area of the square if direction ratios of other pair of opposite sides of the square are given by <-3, 6, p>. Also, find the value of p.

Represent the equations of lines $l_{1}$ and $l_{2}$ in vector form and check whether they are intersecting or not. $l_{1}:\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$, $l_{2}:\frac{x+1}{-1}=\frac{2-y}{-2}=\frac{z-5}{5}$

A line passing through the points $A(1,2,3)$ and $B(5,8,11)$ intersects the line $\vec{r}=4\hat{i}+\hat{j}+\lambda(5\hat{i}+2\hat{j}+\hat{k})$. Find the co-ordinates of the point of intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines.

Find the general solution of the differential equation $(x^{3}-3xy^{2})dx=(y^{3}-3x^{2}y)dy$.

Solve the differential equation $y~e^{y}dx = (y^{3} + 2x~e^{y}) dy$, when $y(0)=1$.

Sketch the curve $\{(x, y): 100x^{2}+25y^{2}=2500\}$ and find the area of the region enclosed by it, using integration.

Using integration, find the area of the region enclosed by the curve $y=|x-6|$, the x-axis, and between $x=4$ and $x=8$.

Evaluate: $\int_{0}^{1}\frac{x~\tan^{-1}x}{(1+x^{2})^{3/2}}dx$

Find: $\int\frac{x}{(x-1)(x^{2}+4)}dx$

A rectangle of perimeter 36 cm is revolved around one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.

Find the sub intervals in which $f(x)=\cot^{-1}(\sin x+\cos x)$, $x \in (0,\pi)$ is increasing and decreasing.

Find the differential of $x^{\cot x}+\frac{2x^{2}-3}{2x^{2}-x+2}$ with respect to x.

If $y\sqrt{x^{2}+1}=\log\sqrt{x^{2}+1}-x$, show that $(x^{2}+1)\frac{dy}{dx}+xy+1=0$.

A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost ₹ 60; 2 apples, 4 oranges and 6 bananas cost ₹ 90; whereas 6 apples, 2 oranges and 3 bananas cost ₹ 70. Using matrix method, find the cost of one fruit of each kind.

Given that $P=\begin{bmatrix}2&-1\\3&4\end{bmatrix}$, $Q=\begin{bmatrix}5&2\\7&4\end{bmatrix}$ and $R=\begin{bmatrix}2&5\\3&8\end{bmatrix}$, find a matrix S such that PQ - RS is a null matrix.

On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.

Obtain the value of $\Delta=\begin{vmatrix}1+x&1&1\\1&1+y&1\\1&1&1+z\end{vmatrix}$ in terms of x, y, and z. Further, if $\Delta=0$ and x, y, z are non-zero real numbers, prove that $x^{-1}+y^{-1}+z^{-1}=-1$

If $P=\begin{bmatrix}1&-1&0\\2&3&4\\0&1&2\end{bmatrix}$ and $Q=\begin{bmatrix}2&2&-4\\-4&2&-4\\2&-1&5\end{bmatrix}$, find (QP) and hence solve the following system of equations using matrices: $x-y=3$, $2x+3y+4z=17$, $y+2z=7$

Find the domain of $g(x)=\cos^{-1}(x^{2}-1)$. Hence, find the value of x for which $g(x)=\frac{\pi}{3}$. Also, write the range of $\cos^{-1}x$ other than its principal branch.