Case study (4 Marks)

Question 14 Marks

Read the following text carefully and answer the questions that follow:

A manufacturer produces two Models of bikes Model X and Model Y. Model X takes a 6 man hours to make per unit, while Model Y takes 10 man-hours per unit. There is a total of 450 man-hours available per week. Handling and Marketing costs are ₹ 2,000 and ₹ 1,000 per unit for Models X and Y respectively. The total funds available for these purposes are ₹ 80,000 per week. Profits per unit for Models X and Y are ₹ 1,000 and ₹ 500 , respectively. The feasible region of LPP is shown in the following graph.

i. Find the equation of line AB .
ii. Find the equation of line CD.
iii. Find the coordinates of point E.
OR
How many bikes of model $X$ and model $Y$ should the manufacturer produce so as to yield a maximum profit?

Answer

i. From the given graph $\mathrm{OA}=75$ and $\mathrm{OB}=45$
The equation of line AB is $\frac{x}{75}+\frac{y}{45}=1$
i.e., $3 x+5 y=225$

ii. From the given graph $\mathrm{OC}=40$ and $\mathrm{OD}=80$.
The equation of line CD is $\frac{x}{40}+\frac{y}{80}=1$
i.e., $2 \mathrm{x}+\mathrm{y}=80$

iii. On solving the equations of lines AB and CD , we get the coordinates of point E i.e., $(25,30)$.
OR
The objective function for given L.P.P. is Z = 1000x + 500y
From the shaded feasible region, it is clear that coordinates of comer points are (0, 0), (40, 0), (25, 30) and (0, 45)

Corner Points	Value of Z = 1000x + 500y
(0,0)	0
(40, 0)	$40,000 \leftarrow$ Maximum
(25,30)	$25,000+15,000=40,000 \leftarrow$ Maximum
(0,45)	22500

So, the manufacturer should produce 25 bikes of model X and 30 bikes of model Y to get a maximum profit of ₹ $40,000$.

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Question 24 Marks

Read the following text carefully and answer the questions that follow:
A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is ₹ 25 and that from a shade is ₹ 15 .
If x is the number of lamps and y is the number of shades manufactured. Assuming that the manufacturer can sell all the lamps and shades that he produces.
i. In order to maximize the profit, what should be the objective function? (1)
ii. What are the constraints related to the given LPP: (1)
iii. The non-negative constraints associative to the given L.P.P are: (2)
OR
What are the vertices of feasible region of given L.P.P? (2)

Answer

(i). Since profit from the sale of a lamp $=$₹ $25$
And profit from the sale of a shade = ₹ 15
The associative objective function is Max. Z = $25 x+15 y$(ii)

	Lamp (x)	Shade (y)
Cutting/grinding	2	2	12
Sprayer	3	3	20

So, constraints are:
$2 x+y \leq 12$
$3 x+2 y \leq 20$
(iii). The non-negative conditions are: $\mathrm{x} \geq 0, \mathrm{y} \geq 0$
OR
Vertices of feasible region are $\mathrm{O}(0,0), \mathrm{A}(6,0), \mathrm{B}(4,4)$, and $\mathrm{C}(0,10)$.

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Question 34 Marks

Read the text carefully and answer the questions:
The nominal rate of return is the amount of money generated by an investment before factoring in expenses such as taxes, investment fees, and inflation. If an investment generated a $10 \%$ return, the nominal rate would equal $10 \%$. After factoring in inflation during the investment period, the actual return would likely be lower. However, the nominal rate of return has its merits since it allows investors to compare the performance of an investment irrespective of the different tax rates that might be applied for each investment.
(a) A person invests ₹ 10000 in $10 \%$ ₹ 100 shares of a company available at a premium of ₹ 25 . Find his rate of return
(b) A man invests ₹ 22500 in ₹ 50 shares available at $10 \%$ discount. If the dividend paid by the company is $12 \%$, calculate his rate of return.
(c) A person invested ₹200000 in a fund for one year. At the end of the year, the investment was worth ₹216000. Calculate his rate of return.
OR
Balwant invests a sum of money in ₹50 shares paying $10 \%$ dividend quoted at $20 \%$ discount. If his annual dividend is ₹ 600 , calculate his rate of return from the investment.

Answer

The nominal rate of return is the amount of money generated by an investment before factoring in expenses such as taxes, investment fees, and inflation. If an investment generated a $10 \%$ return, the nominal rate would equal $10 \%$. After factoring in inflation during the investment period, the actual return would likely be lower.
However, the nominal rate of return has its merits since it allows investors to compare the performance of an investment irrespective of the different tax rates that might be applied for each investment.
(i) $8 \%$
(ii) $13 \frac{1}{3} \%$
(iii)8\%
OR
$12.5 \%$

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Question 44 Marks

There is a bridge whose length of three sides of a trapezium other than base are equal to 5 cm :

(a) What is the value of DP?
(b) What is the area of the trapezium $\mathrm{A}(\mathrm{x})$ ?
(c) $\quad A^{\prime}(x)=0$ then what is the value of $x$ ?
OR
What is the value of A"(2.5)

Answer

There is a bridge whose length of three sides of a trapezium other than base are equal to 5 cm :

(i) $\sqrt{25-x^{2}}$
(ii) $(\mathrm{x}+5) \sqrt{25-x^{2}}$
(iii)2.5, -5
OR
$
-\frac{15}{\sqrt{18.75}}
$

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Applied Maths Case study (4 Marks)

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