Question

In a parliament election, a political party hired a public relations firm to promote its candidates in three ways — telephone, house calls and letters. The cost per contact (in paise) is given in matrix A as

Find the particular solution of the differential equation ${\frac{d y}{d x}}={\frac{x\left(2\log x+1\right)}{\sin y+y\cos y}}$ given that $y={\frac{\pi}{2}}$ where x = 1.

If Z is the set of all integers and R is the relation on Z defined as R = {(a, b) : a, b ∈ Z and a – b is divisible by 5}. Prove that R is an equivalence relation.

Find the particular solution of the differential equation $\log\left(\frac{d y}{d x}\right)=3x+4y,$ given that y = 0, when x = 0.

Three schools A, B and C organised a fete (mela) collecting funds for flood victims in which they sold hand-helds fans, mats and toys made from recycled material, the sale price of each being ₹ 25, ₹ 100 and ₹ 50 respectively. The following table shows the number of articles of each type sold :

Evaluate : $\large\textstyle\int\!{\frac{d x}{x(x^{3}+1)}}\,.$

A and B throw a pair of dice alternatively, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first.