Embibe Experts Solutions for Chapter: Sample Paper, Exercise 1: Exercise

The corner points of the shaded unbounded feasible region of an LPP are $\left(0,4\right),\left(0.6,1.6\right)$  and $\left(3,0\right)$ as shown in the figure. The minimum value of the objective function $Z=4x+6y$ occurs at

$P$ is a point on the line joining the points $A\left(0,5,-2\right)$ and $B\left(3,-1,2\right)$. If the $x-$coordinate of $P$ is $6,$ then its $z-$coordinate is

Solve the following Linear Programming Problem graphically:

Maximize $Z=400x+300y$ subject to $x+y⩽200, x⩽40, x⩾20, y⩾0$

Make a rough sketch of the region $\{\left(x,y\right): 0\le y\le x^2, 0\le y\le x, 0\le x\le 2\}$ and find the area of the region using integration.

Given a non-empty set $X$, define the relation $R$ in $P\left(X\right)$ as follows:

For $A,B\in P\left(X\right),\left(A,B\right)\in R$ iff $A\subset B.$ Prove that $R$ is reflexive, transitive and not symmetric.

An insect is crawling along the line $\overrightarrow{r}=6\hat{i}+2\hat{j}+2\hat{k}+\lambda \left(\hat{i}-2\hat{j}+2\hat{k}\right)$ and another insect is crawling along the line $\overrightarrow{r}=-4\hat{i}-\hat{k}+\mu \left(3\hat{i}-2\hat{j}-2\hat{k}\right)$. At what points on the lines should they reach so that the distance between them is the shortest? Find the shortest possible distance between them.

The equations of motion of a rocket are $:$

$𝑥=2𝑡,𝑦=-4𝑡,𝑧=4𝑡,$ where the time $t$ is given in seconds, and the coordinates of a moving point in km. What is the path of the rocket? At what distances will the rocket be starting from the point $O(0,0,0)$ and from the following lines in $10$ seconds?

$\overrightarrow{r}=20\hat{i}-10\hat{j}+40\hat{k}+\mu \left(10\hat{i}-20\hat{j}+10\hat{k}\right)$

If $A=\left[\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right],$ find $A^{-1}.$ Use $A^{-1}$ to solve the following system of equations:

$2x-3y+5z=11, 3x+2y-4z=-5, x+y-2z=-3$