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Vectors \(\vec{a}=3\hat{i}-2\hat{j}+2\hat{k}\) and \(\vec{b}=\hat{i}+2\hat{k}\) represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.

Find the vector of magnitude \(14\) in the direction of \(\overrightarrow{QP}\), where \(P=(1,3,2)\) and \(Q=(-1,0,8)\) respectively.

A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume in the bottle decreases at the steady rate of 1 mm3/min. Find the rate at which level of perfume is dropping at an instant when level of perfume in the bottle is 10 mm, if the semi-vertical angle of conical bottle is $\frac{\pi}{6}$

If $ \sqrt{3}\,(x^2+y^2)=4xy$, then find \(\dfrac{dy}{dx}\) at $\left(\frac{1}{2},\,\frac{\sqrt{3}}{2}\right).$

Check whether function \(f(x)\) defined as

\[
f(x)=
\begin{cases}
\dfrac{|x-3|}{2(x-3)}, & x<3, \\[6pt]
\dfrac{x-6}{6}, & x\ge 3
\end{cases}
\]

is continuous at \(x=3\) or not?

The general solution for the differential equation $\frac{dy}{dx} = e^{3x-y}$ is

Evaluate the following integral $\int x^2 dx$

Which of the following cannot be the order of a row-matrix ?

If $2 \cos^{-1} x = y$, then

A matrix B = $[b_{ij}]_{m \times m}$ is said to be a diagonal matrix, if :

For any square matrix A with real entries, if $A + A'$ is a symmetric matrix then :

Define reproduction.

Find the point Q on the line $\frac{2x+4}{6}=\frac{y+1}{2}=\frac{-2z+6}{-4}$ at a distance of $3\sqrt{2}$ from the point $P(1,2,3)$.

Solve the differential equation $\frac{dy}{dx}=\cos x-2y$.

Evaluate: $\int_{0}^{\pi}\frac{x\tan x}{\sec x+\tan x}dx$

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

Find the shortest distance between the lines: $\vec{r}=(2\hat{i}-\hat{j}+3\hat{k})+\lambda(\hat{i}-2\hat{j}+3\hat{k})$ and $\vec{r}=(\hat{i}+4\hat{k})+\mu(3\hat{i}-6\hat{j}+9\hat{k})$.

The scalar product of the vector $\vec{a}=\hat{i}-\hat{j}+2\hat{k}$ with a unit vector along sum of vectors $\vec{b}=2\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{c}=\lambda\hat{i}-2\hat{j}-3\hat{k}$ is equal to 1. Find the value of $\lambda$.

In the Linear Programming Problem for objective function $Z=18x+10y$ subject to constraints $4x+y\ge20$, $2x+3y\ge30$, $x,y\ge0$ find the minimum value of Z.

Amongst all pairs of positive integers with product as 289, find which of the two numbers add up to the least.