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ISC Class 12 Mathematics Solved Question Paper 2014

Long Answer Type

31.
Solve the following differential equation.
$(3 xy + y^{2}) dx + (x^{2} + xy) dy = 0$

$(3 xy + y^{2}) dx + (x^{2} + xy) dy = 0$

$\Rightarrow \frac{d y}{d x} = - \frac{(3 xy + y^{2})}{(x^{2} + xy)}$

Put y = vx

$\Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}$

$∴ v + x \frac{d v}{d x} = - \frac{3 x . vx + v^{2} x^{2}}{x^{2} + x . vx}$

= $- \frac{3 v + v^{2}}{1 + v}$

$\Rightarrow x \frac{d v}{d x} = - \frac{3 v + v^{2}}{1 + v} - v$

= $\frac{- 3 v - v^{2} - v - v^{2}}{1 + v}$

$\Rightarrow x \frac{d v}{d x} = \frac{- 4 v - 2 v^{2}}{1 + v}$

$\Rightarrow x \frac{d v}{d x} = \frac{- 2 v (v + 2)}{1 + v}$

$\Rightarrow \frac{(1 + v)}{v (v + 2)} dv = - 2 \frac{dx}{x}$

Integrating both sides, we get

$\int \frac{(1 + v)}{v (v + 2)} dv = - 2 \int \frac{dx}{x}$

$\Rightarrow \int (\frac{1}{2 v} + \frac{1}{2 (v + 2)}) dv = - 2 \int \frac{dx}{x}$

$\Rightarrow \int (\frac{1}{v} + \frac{1}{(v + 2)}) dv = - 4 \int \frac{dx}{x}$

$\Rightarrow \log |v| + \log |v + 2| = - 4 \log |x| + \log |c|$

$\Rightarrow \log |v (v + 2) x^{4}| = \log (c)$

$\Rightarrow v (v + 2) x^{4} = c$

$\Rightarrow \frac{y}{x} (\frac{y}{x} + 2) x^{4} = c$

$\Rightarrow x^{2} y (y + 2 x) = c$

32.

In a triangle ABC, using vectors , prove that c²= a² +b² - 2ab.cos(C)

33.

A person borrows 68,962 on the condition that he will repay the money with compound interest at 5 % per annum, installments, the first one being payable at the end of the first year. Find the value of each installment.

34.

A company manufacturers two types of toys A and B. a toy of type a requires 5 minutes for cutting and 10 minutes for assembling. A toy of type B requires 8 minutes for cutting and 8 minutes for assembling. There are 3 hours available for cutting and 4 hours available for assembling the toys in a day. The profit is 50 each on a toy of type A and 60 each on a toy of type B.. How many toys of each type should the company manufacture in a day to maximize the profit? Use linear programming to find the solution.

35.

The piece of six different commodities for years 2009 and year 2011 are as follows:

Commodities	A	B	C	D	E	F
Price in 2009()	35	80	25	30	80	x
Price in 2011()	50	y	45	70	120	105

The index number for the year 2011 taking 2009 a sthe base year for the aove data was calculated to be 125.. Find the values of x and y if the total price in 2009 is 360.

36.

The number of road accidents in the city due to rash driving, over a period of 3 years, is given in the following table:

Year	Jan - Mar	April - June	July - Sept.	Oct. - DEc.
2010	70	60	45	72
2011	79	56	46	84
2012	90	64	45	82

Calculate four quarterly moving averages and illustrate them and original figures on one graph using the same axes for both.

37.

A firm has the cost function C = $\frac{x^{3}}{3} - 7 x^{2} + 111 x + 50$ and demand function x = 100 - p

(i) Write the total revenue function in terms of x.

(ii) Formulate the total profit function P in terms of x.

(iii) Find the profit maximising level of output x.

38.

A bill of Rs. 5050 is drawn on 13^th April 2013. It was discounted on 4^th July 2013 at 5 % per annum. If the banker's gain on the transaction is Rs. 0.50, find the nominal date of the maturity of the bill.