Read the following text carefully and answer the questions that f…

Question

Read the following text carefully and answer the questions that follow:
Linear programming is a method for finding the optimal values (maximum or minimum) of quantities subject to the constraints when a relationship is expressed as linear equations or inequations.
i. At which points is the optimal value of the objective function attained? (1)
ii. What does the graph of the inequality $3 x+4 y<12$ look like? (1)
iii. Where does the maximum of the objective function Z = 2x + 5y occur in relation to the feasible region shown in the figure for the given LPP? (2)

OR
What are the conditions on the positive values of p and q that ensure the maximum of the objective function Z = px + qy occurs at both the corner points (15, 15) and (0, 20) of the feasible region determined by the given system of linear constraints? (2? (2)

✓

Answer

When we solve an L.P.P. graphically, the optimal (or optimum) value of the objective function is attained at corner points of the feasible region.
ii. From the graph of 3x + 4y < 12 it is clear that it contains the origin but not the points on the line 3x + 4y = 12

iii. Maximum of objective function occurs at corner points

Corner Points	Value of z = 2x + 5y
(0,0)	0
(7,0)	14
(6,3)	27
(4,5)	$33 \leftarrow$ Maximum
(0,6)	30

OR
Value of $Z=p x+q y$ at $(15,15)=15 p+15 q$ and that at $(0,20)=20 q$. According to given condition, we have $15 p +15 q =20 q \Rightarrow 15 p =5 q \Rightarrow q =3 p$

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If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)] is a differentiable function of x and $\frac{\text{dy}}{\text{dx}}=\frac{\text{dy}}{\text{du}}\times\frac{\text{du}}{\text{dx}}.$ This rule is also known as CHAIN RULE.
Based on the above information, find the derivative of functions w.r.t. x in the following questions.

$\cos\sqrt{\text{x}}$

$\frac{-\sin\sqrt{\text{x}}}{2\sqrt{\text{x}}}$
$\frac{\sin\sqrt{\text{x}}}{2\sqrt{\text{x}}}$
$\sin\sqrt{\text{x}}$
$-\sin\sqrt{\text{x}}$

$7^{\text{x}+\frac{1}{\text{x}}}$

$\Big(\frac{\text{x}^2-1}{\text{x}^2}\Big)\cdot7^{\text{x}+\frac{1}{\text{x}}}\cdot\log7$
$\Big(\frac{\text{x}^2+1}{\text{x}^2}\Big)\cdot7^{\text{x}+\frac{1}{\text{x}}}\cdot\log7$
$\Big(\frac{\text{x}^2-1}{\text{x}^2}\Big)\cdot7^{\text{x}-\frac{1}{\text{x}}}\cdot\log7$
$\Big(\frac{\text{x}^2+1}{\text{x}^2}\Big)\cdot7^{\text{x}-\frac{1}{\text{x}}}\cdot\log7$

$\sqrt\frac{{1-\cos\text{x}}}{1+\cos\text{x}}$

$\frac{1}{2}\sec^2\frac{\text{x}}{2}$
$-\frac{1}{2}\sec^2\frac{\text{x}}{2}$
$\sec^2\frac{\text{x}}{2}$
$-\sec^2\frac{\text{x}}{2}$

$\frac{1}{\text{b}}\tan^{-1}\Big(\frac{\text{x}}{\text{b}}\Big)+\frac{1}{\text{a}}\tan^{-1}\Big(\frac{\text{x}}{\text{a}}\Big)$

$\frac{-1}{\text{x}^2+\text{b}^2}+\frac{1}{\text{x}^2+\text{a}^2}$
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$\frac{1}{\text{x}^2+\text{b}^2}-\frac{1}{\text{x}^2+\text{a}^2}$
None of these.

$\sec^{-1}\text{x}+\text{cosec}^{-1}\frac{\text{x}}{\sqrt{\text{x}^2-1}}$

$\frac{2}{\sqrt{\text{x}^2-1}}$
$\frac{-2}{\sqrt{\text{x}^2-1}}$
$\frac{1}{|\text{x}|\sqrt{\text{x}^2-1}}$
$\frac{2}{|\text{x}|\sqrt{\text{x}^2-1}}$

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In a city there are two factories A and B. Each factory produces sports clothes for boys and girls. There are three types of clothes produced in both the factories, type I, II and III. For boys the number of units of types I, II and III respectively are 80, 70 and 65 in factory A and 85, 65 and 72 are in factory B. For girls the number of units of types I, II and III respectively are 80, 75, 90 in factory A and 50, 55, 80 are in factory B.

Based on the above information, answer the following questions:

If P represents the matrix of number of units of each type produced by factory A for both boys and girls, then P is given by:

$\begin{matrix}&\text{Boys}&\text{Girls}\end{matrix}\\\begin{matrix}\text{I}\\\text{II}\\\text{III}\end{matrix}\begin{bmatrix}85&50\\65&55\\72&80\end{bmatrix}$
$\begin{matrix}&&&\text{I}\ \ \ &\text{II}&\text{III}\end{matrix}\\\begin{matrix}\text{Boys}\\\text{Girls}\end{matrix}\begin{bmatrix}50&55&80\\85&65&72\end{bmatrix}$
$\begin{matrix}&&&\text{I}\ \ \ &\text{II}&\text{III}\end{matrix}\\\begin{matrix}\text{Boys}\\\text{Girls}\end{matrix}\begin{bmatrix}80&75&90\\80&70&65\end{bmatrix}$
$\begin{matrix}&\text{Boys}&\text{Girls}\end{matrix}\\\begin{matrix}\text{I}\\\text{II}\\\text{III}\end{matrix}\begin{bmatrix}80&80\\70&75\\65&90\end{bmatrix}$

If Q represents the matrix of number of units of each type produced by factory B for both boys and girls, then Q is given by:

$\begin{matrix}&\text{Boys}&\text{Girls}\end{matrix}\\\begin{matrix}\text{I}\\\text{II}\\\text{III}\end{matrix}\begin{bmatrix}85&50\\65&55\\72&80\end{bmatrix}$
$\begin{matrix}&&&\text{I}\ \ \ &\text{II}&\text{III}\end{matrix}\\\begin{matrix}\text{Boys}\\\text{Girls}\end{matrix}\begin{bmatrix}80&75&90\\80&70&65\end{bmatrix}$
$\begin{matrix}&&&\text{I}\ \ \ &\text{II}&\text{III}\end{matrix}\\\begin{matrix}\text{Boys}\\\text{Girls}\end{matrix}\begin{bmatrix}80&75&90\\80&70&65\end{bmatrix}$
$\begin{matrix}&\text{Boys}&\text{Girls}\end{matrix}\\\begin{matrix}\text{I}\\\text{II}\\\text{III}\end{matrix}\begin{bmatrix}80&80\\70&75\\65&90\end{bmatrix}$

The total production of sports clothes of each type for boys is given by the matrix.

$\begin{matrix}\ \ \text{I}&\ \ \ \ \text{II}&\ \ \ \text{III}\end{matrix}\\\begin{matrix}[165&130&137]\end{matrix}\\$
$\begin{matrix}\ \ \text{I}&\ \ \ \ \text{II}&\ \ \ \text{III}\end{matrix}\\\begin{matrix}[130&165&137]\end{matrix}\\$
$\begin{matrix}\ \ \text{I}&\ \ \ \ \text{II}&\ \ \ \text{III}\end{matrix}\\\begin{matrix}[165&135&137]\end{matrix}\\$
$\begin{matrix}\ \ \text{I}&\ \ \ \ \text{II}&\ \ \ \text{III}\end{matrix}\\\begin{matrix}[137&135&165]\end{matrix}\\$

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$\begin{matrix}\ \ \text{I}&\ \ \ \ \text{II}&\ \ \ \text{III}\end{matrix}\\\begin{matrix}[170&130&130]\end{matrix}\\$
$\begin{matrix}\ \ \text{I}&\ \ \ \ \text{II}&\ \ \ \text{III}\end{matrix}\\\begin{matrix}[130&170&130]\end{matrix}\\$
None of these

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$\begin{bmatrix}165 & 135 & 137\\130 & 130 & 170 \end{bmatrix}$
$\begin{bmatrix}130 & 130 & 170\\165 & 135 & 138 \end{bmatrix}$
$\begin{bmatrix}165 & 132 \\135 & 130 \\137 & 170 \end{bmatrix}$
$\begin{bmatrix}130 & 168 \\130 & 135 \\170 & 137 \end{bmatrix}$

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A tin can manufacturer designs a cylindrical tin can for a company making sanitizer and disinfector. The tin can is made to hold $3$ litres of sanitizer or disinfector.

Based on the above in formation, answer the following questions.

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$2\pi\text{r}^2$
$2\pi\text{r}^2+6000$
$2\pi\text{r}^2+\frac{5000}{\text{r}}$
$2\pi\text{r}^2+\frac{6000}{\text{r}}$

The radius that will minimize the cost of the material to manufacture the tin can is.

$\sqrt[3]{\frac{600}{\pi}}\text{cm}$
$\sqrt{\frac{500}{\pi}}\text{cm}$
$\sqrt[3]{\frac{1500}{\pi}}\text{cm}$
$\sqrt{\frac{1500}{\pi}}\text{cm}$

The height that will minimize the cost of the material to manufacture the tin can is.

$\sqrt[3]{\frac{600}{\pi}}\text{cm}$
$2\sqrt[3]{\frac{1500}{\pi}}\text{cm}$
$\sqrt{\frac{1500}{\pi}}$
$2\sqrt{\frac{1500}{\pi}}$

If the cost of material used to manufacture the tin can is $₹\frac{100}{\text{m}^2}$ and $\sqrt[3]{\frac{1500}{\pi}}\approx7.8,$ then minimum cost is approximately.

$₹\ 11.538$
$₹\ 12$
$₹\ 13$
$₹\ 14$

To minimize the cost of the material used to manufacture the tin can, we need to minimize the.

Volume.
Curved surface area.
Total surface area.
Surface area of the base.

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The Government declare that farmers can get ₹ 300 per quintal for their onions on 1st July and after that, the price will be dropped by ₹ 3 per quintal per extra day. Govind's father has 80 quintals of onions in the field on 1st July and he estimates that the crop is increasing at the rate of 1 quintal per day.

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(ii) Find the number of days after 1st July, when Govind's father attains maximum revenue.

(iii) On which day should Govind's father harvest the onions to maximize his revenue?

Find the maximum revenue collected by Govind's father.

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Ramesh, the owner of a sweet selling shop, purchased some rectangular card board sheets of dimension 25 cm by 40 cm to make container packets without top. Let x cm be the length of the side of the square to be cut out from each corner to give that sheet the shape of the container by folding up the flaps.
Based on the above information answer the following questions.
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OR
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Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold handmade fans, mats, and plates from recycled material at a cost of ₹ 25 , ₹ 100 and ₹ 50 each. The number of articles sold by school A, B, C are given below.

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(ii) Find the funds collected by school A, B and C by selling the given articles.

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Find the total funds collected for the required purpose after $20 \%$ hike in price.

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A trust fund has ₹ 35000 that must be invested in two different types of bonds, say X and Y. The first bond pays 10% interest p.a. which will be given to an old age home and second one pays 8% interest p.a. which will be given to WWA (Women Welfare Association).
Let A be a 1 × 2 matrix and B be a 2 × 1 matrix, representing the investment and interest rate on each bond respectively.

Based on the above information, answer the following questions.

If ₹ 15000 is invested in bond X, then

$\begin{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \text{investment}\end{matrix}\begin{matrix}&&&\text{X}&&\text{Y}\end{matrix}\\\begin{matrix}\text{A}\ =\text{X}\\\ \ \ \ \ \ \ \ \ \ \text{Y}\end{matrix}\begin{bmatrix}\ \ 15000\ \ \\\ \ 20000\ \end{bmatrix};\text{B}=\begin{bmatrix}0.1&0.08\end{bmatrix}\text{Interest rate.}$
$\begin{matrix}&&&&&&&&\text{X}&\ \ \ \ \ \ \ \text{Y}\end{matrix}\begin{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{investment}\end{matrix}\\\text{A = Investment}\begin{bmatrix}15000&20000\end{bmatrix};\ \begin{matrix}\text{B}\ =\text{X}\\\ \ \ \ \ \ \ \ \ \ \text{Y}\end{matrix}\begin{bmatrix}\ \ 0.1\ \ \\\ \ 0.08\ \end{bmatrix}$
$\begin{matrix}&&&&&&&&\text{X}&\ \ \ \ \ \ \ \text{Y}\end{matrix}\begin{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{investment}\end{matrix}\\\text{A = Investment}\begin{bmatrix}20000&15000\end{bmatrix};\ \begin{matrix}\text{B}\ =\text{X}\\\ \ \ \ \ \ \ \ \ \ \text{Y}\end{matrix}\begin{bmatrix}\ \ 0.08\ \ \\\ \ 0.1\ \end{bmatrix}$
$\text{None of these}$

If ₹ 15000 is invested in bond X, then total amount of interest received on both bonds is:

₹ 2000
₹ 2100
₹ 3100
₹ 4000

If the trust fund obtains an annual total interest of ₹ 3200, then the investment in two bonds is:

₹ 15000 in X, ₹ 20000 in Y
₹ 17000 in X, ₹ 18000 in Y
₹ 20000 in X, ₹ 15000 in Y
₹ 18000 in X, ₹ 17000 in Y

The total amount of interest received on both bonds is given by:

AB
A' B
B' A
None of these

If the amount of interest given to old age home is ₹ 500, then the amount of investment in bond Y is:

₹ 20000
₹ 30000
₹ 15000
₹ 25000

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Consider $2$ families $A$ and $B$. Suppose there are $4$ men$,4$ women and $4$ children in family $A$ and $2$ men$, 2$ women and $2$ children in family $B$. The recommend daily amount of calories is $2400$ for a man, $1900$ for a woman$, 1800$ for a children and $45$ grams of proteins for a man$, 55$ grams for a woman and $33$ grams for children.

Based on the above information, answer the following questions.

The requirement of calories and proteins for each person in matrix form can be represented as:

$\begin{matrix}& \ \ \ \ \ \ \ \ \ \ \ \ \ \text{Calorise}&\text{Proteins}\end{matrix}\\\begin{matrix}\text{Man}\\\text{Woman}\\\text{Children}\end{matrix}\begin{bmatrix} \ \ 2400 \ \ \ & 45\\1900 & 55\\1800& \ 33&\end{bmatrix}$
$\begin{matrix}& \ \ \ \ \ \ \ \ \ \ \ \ \ \text{Calorise}&\text{Proteins}\end{matrix}\\\begin{matrix}\text{Man}\\\text{Woman}\\\text{Children}\end{matrix}\begin{bmatrix} \ \ 1900 \ \ \ & 55\\2400 & 45\\1800& \ 33&\end{bmatrix}$
$\begin{matrix}& \ \ \ \ \ \ \ \ \ \ \ \ \ \text{Calorise}&\text{Proteins}\end{matrix}\\\begin{matrix}\text{Man}\\\text{Woman}\\\text{Children}\end{matrix}\begin{bmatrix} \ \ 1800 \ \ \ & 33\\1900 & 55\\2400& \ 45&\end{bmatrix}$
$\begin{matrix}& \ \ \ \ \ \ \ \ \ \ \ \ \ \text{Calorise}&\text{Proteins}\end{matrix}\\\begin{matrix}\text{Man}\\\text{Woman}\\\text{Children}\end{matrix}\begin{bmatrix} \ \ 2400 \ \ \ & 33\\1900 & 55\\1800& \ 45&\end{bmatrix}$

Requirement of calories of family $A$ is:

$24000$
$24400$
$15000$
$15800$

Requirement of proteins for family $B$ is:

$560$ grams
$332$ grams
$266$ grams
$300$ grams

If $A$ and Bare two matrices such that $AB = B$ and $BA = A,$ then $A^2 + B^2$ equals.

$2AB$
$2BA$
$A + B$
$AB$

If $\text{A}=(\text{a}_\text{ij})_{\text{m}\times\text{n}},\ \ \text{B}=(\text{b}_\text{ij})_{\text{n}\times\text{p}}$ and $\text{C}=(\text{c}_\text{ij})_{\text{p}\times\text{q}}$ then the product $(BC) A$ is possible only when.