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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.

Differentiate $y=\sqrt{\log\left\{\sin\left(\frac{x^{3}}{3}-1\right)\right\}}$ with respect to x.

Let $2x+5y-1=0$ and $3x+2y-7=0$ represent the equations of two lines on which the ants are moving on the ground. Using matrix method, find a point common to the paths of the ants.

Let R be a relation defined on a set N of natural numbers such that $R=\{(x,y): xy \text{ is a square of a natural number, } x, y\in N\}$. Determine if the relation R is an equivalence relation.

Show that the function $f:R\rightarrow R$ defined by $f(x)=4x^{3}-5$, $\forall x\in R$ is one-one and onto.

10 identical blocks are marked with '0' on two of them, '1' on three of them, '2' on four of them and '3' on one of them and put in a box. If X denotes the number written on the block, then write the probability distribution of X and calculate its mean.

If $-2x^{2}-5xy+y^{3}=76$, then find $\frac{dy}{dx}$.

Differentiate $\left(\frac{5^{x}}{x^{5}}\right)$ with respect to x.

If $A=\begin{bmatrix}2&3\\ -1&2\end{bmatrix}$, then show that $A^{2}-4A+7I=0$.

Let $f:A\rightarrow B$ be defined by $f(x)=\frac{x-2}{x-3}$ where $A=R-\{3\}$ and $B=R-\{1\}$. Discuss the bijectivity of the function.

Solve the differential equation $(1+x^{2})\frac{dy}{dx}+2xy-4x^{2}=0$ subject to initial condition $y(0)=0$.

Solve the differential equation: $x^{2}y~dx-(x^{3}+y^{3})dy=0$.