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The order and degree of the differential equation $\frac{d}{dx}(e^y) = 0$ respectively are

The area bounded by the curve $y = x|x|$, x-axis and the ordinates $x = -1$ and $x = 1$ is given by

The value of $\int_{-1}^{1} \frac{x^3}{x^2 + 2|x| + 1} dx$ is

If $\int \frac{3ax}{b^2 + c^2x^2} dx = A \log |b^2 + c^2x^2| + K$, then the value of $A$ is

The least value of $f(x) = x^3 - 12x$, $x \in [0, 3]$ is

If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew symmetric ?

One of the values of $x$ for which $\begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = 1$ is

If $\Delta_{1}=\begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix}$ and $\Delta_{2}=\begin{vmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 6 \end{vmatrix}$, then

In an experiment of throwing an unbiased die, the probability of getting a prime number given that number appearing on the die being odd is $\frac{2}{3}$.

Lines given by $x = py + q, z = ry + s$ and $x = p'y + q', z = r'y + s'$ are perpendicular to each other when $pp' + rr' = 1$.

If $x = \cos t, y = \cos mt$, prove that $\left(1-x^2\right) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + m^2y = 0$.

Solve the following Linear Programming Problem graphically: Maximise $Z=600x+400y$ subject to the constraints $x+2y\le12$, $2x+y\le12$, $4x+5y\ge20$, $x, y\ge0$

Find the value of p if the shortest distance between the lines $\vec{r}=(\hat{i}+2\hat{j}+\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})$ and $\vec{r}=(p\hat{i}-\hat{j}-\hat{k})+\mu(2\hat{i}+\hat{j}+2\hat{k})$ is $\frac{3}{\sqrt{2}}$ units.

Find the foot of the perpendicular from the point (0, 2, 3) on the line $\frac{-x-3}{-5}=\frac{1-y}{-2}=\frac{3z+12}{9}$ and hence find the length of the perpendicular.

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 $y(0)=1$.