Determine the maximum and minimum values of the following functio… - Vidyadip

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Determine the maximum and minimum values of the following functions:
$f(x)=x \cdot \log x$

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Find the inverse of matrix $A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1\end{array}\right]$ by using elementary row transformations

Evaluate $\int_0^1 e ^{x^2} \cdot x ^3 d x$

x	y	$x-\bar{x}$	$y-\bar{y}$	$(x-\bar{x})(y-\bar{y})$	$(x-\bar{x})^2$	$(y-\bar{y})^2$
1	5	-2	-4	8	4	16
2	7	-1	-2	$\square$	1	4
3	9	0	0	0	0	0
4	11	1	2	2	4	4
5	13	2	4	8	1	16
Total = 15	Total = 45	Total = 0	Total = 0	Total =$\square$	Total = 10	Total = 40

Mean of $x =\bar{x}=$$\square$
Mean of $y =\bar{y}=$$\square$
$ b_{x y}=\frac{\square}{\square}$
$b_{y x}=\frac{\square}{\square} $
Regression equation of $x$ on $y$ is $(x-\bar{x})= b _{x y}(y-\bar{y})$
$\therefore$ Regression equation $x$ on $y$ is$\square$
Regression equation of $y$ on $x$ is $(y-\bar{y})= b _{y x}(x-\bar{x})$
$\therefore$ Regression equation of $y$ on $x$ is$\square$

Find the area of the circle $x^2 + y^2 = 16$

Following table shows the amount of sugar production (in lac tons) for the years 1971 to 1982

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Fit a trend line by the method of least squares

Solve the following differential equation $y x \frac{ d y}{ d x}= x ^2+2 y ^2$

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance of X.

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Consider the problem of assigning five operators to five machines. The assignment costs are given in the following table.

Operator	Machine
Operator	1	2	3	4	5
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Operator A cannot be assigned to machine 3 and operator C cannot be assigned to machine 4. Find the optimal assignment schedule.

Find the general solution of the equation $\frac{ d y}{ d x}-y=2 x$.
Solution: The equation $\frac{ d y}{ d x}-y=2 x$
is of the form $\frac{ d y}{ d x}+ P y= Q$
where $P =\square$ and $Q =\square$
$\therefore \text { I.F. }= e ^{\int- d x}= e ^{- x }$
$\therefore$ the solution of the linear differential equation is
$ ye ^{- x }=\int 2 x \cdot e ^{-x} d x+ c$
$\therefore ye ^{- x }=2 \int x \cdot e ^{-x} d x+ c$
$=2\left\{x \int e ^{-x} d x-\int \square d x \cdot \frac{ d }{ d x} \square d x\right\}+ c$
$=2\left\{x \frac{ e ^{-x}}{-1}-\int \frac{ e ^{-x}}{-1} \cdot 1 d x\right\}+ c$
$\therefore ye ^{- x }=-2 x \cdot e ^{-x}+2 \int e ^{-x} d x+ c$
$\therefore e ^{- x } y =-2 x \cdot e ^{-x}+2 \square+ c $
$\therefore y+\square+\square= ce ^{ x }$ is the required general solution of the given differential equation