If A and B are non-singular matrices, prove that AB is non-singular.

Evaluate \(\int _{ 0 }^{ \pi }{ { sin }^{ 2 } } x\ dx\)

Find the consumer's surplus for the demand function p = 25 - x -x² when P_o = 19

Solve: \(\frac { dy }{ dx } \)+ ay = e^x (where a ≠ -1)

When h = 1, find Δ (x³).

A continuous random variable. X has the p.d.f. defined by \(f(x)=\left\{\begin{array}{l} C e^{-a x}, \quad 0<x<\infty \\ 0, \quad \text { elsewhere } \end{array}\right.\) Find the value of C if a> 0

If you buy a lottery ticket in 50 lotteries, in each which your chance of winning a prize is \(\frac { 1 }{ 100 } \). What is the approximate probability that you will win a prize at least once (e-0.5 = 0.6066). 

A sample of 400 students is found to have mean height of 171.38 cms, Can it reasonable be regarded as a sample from a large population with mean height of 171.17 cms and standard deviation of 3.3 cms (Test at 5% level)

The following data shows the value of sample mean (\(\bar{X}\)) and the range R for 10 samples of size 5 each. Calculate the control limits for : mean chart and range chart.

Sample No.	1	2	3	4	5	6	7	8	9	10
Mean \(\bar{X}\)	11.2	11.8	10.8	11.6	11.0	9.6	10.4	9.6	10.6	10.0
Range	7	4	8	5	7	4	8	4	7	9

(Given for n = 5, A₂ = .577, D₃ = 0, D₄ = 2.115)

The following is the pay-off matrix (in rupees) for three strategies and three states of nature. Select a strategy using maximin principle.