Solve the following differential equation.

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

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

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.

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. 

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

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.

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

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

A firm has the cost function C = $\frac{{\mathrm{x}}^3}{3} - 7{\mathrm{x}}^2 + 111\mathrm{x} + 50$ 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.