EM III Question Bank IAT II Past Paper PDF

EM III QUESTION BANK IAT II 1. Find Orthogonal Trajectory of the family of curves given by 𝑒−𝑥 𝑐𝑜𝑠 𝑦 + 𝑥𝑦 = 𝐶 2. Show that the function,f(z)=sinh(z) is analytic and find f(z) in terms of z is analytic. 3. Find the Fourier series for f(x)=x in (0,2π). 4. A random variable X has the following probabi...

EM III QUESTION BANK IAT II 1. Find Orthogonal Trajectory of the family of curves given by 𝑒−𝑥 𝑐𝑜𝑠 𝑦 + 𝑥𝑦 = 𝐶 2. Show that the function,f(z)=sinh(z) is analytic and find f(z) in terms of z is analytic. 3. Find the Fourier series for f(x)=x in (0,2π). 4. A random variable X has the following probability density function 𝑓(𝑥) = 1 ; 0 < 𝑥 < 1 find MGF, Mean and Variance. 5. Obtain Half–range cosine series for𝑓(𝑥) = (𝑥 − 1)2 0 < 𝑥 < 1 3 6. If a random variable has the moment generating function 𝑀𝑡 = 3−𝑡 Obtain mean and standard deviation. 7. Discrete random variable has the probability density function given below. Find k, the Mean and Variance. X -2 -1 0 1 2 3 P( X ) 0.1 k 0.2 2k 0.3 k 8. Find the constants a ,b,c,d ,e If F(z) = ( a𝑥4 + 𝑏𝑥2 𝑦2 + 𝑐𝑦4 + 𝑑𝑥2 − 2𝑦2 ) + 𝑖( 4𝑥3 𝑦 − 𝑒𝑥𝑦3 + 4𝑥𝑦 ) is analytic. 9. Show that the function, f(z) = (𝑥3 − 3𝑥𝑦2 + 2𝑥𝑦)+𝑖(3𝑥2 𝑦 − 𝑥2 + 𝑦2 − 𝑦3 ) is analytic and find 𝑓'(𝑧) in terms of z is analytic. 10. Find the Fourier series for 𝑓 𝑥 = 𝑒− 𝑥 in (-π,π) 11. Find half range cosine series for 𝑓 𝑥 = 𝑥 in (0,2). 12. Discrete random variable has the probability density function given below. Find k, P(X > 4), P( X < 5) X 1 2 3 4 5 6 7 P( X=x ) 𝑘 2𝑘 3𝑘 𝑘2 𝑘2 2𝑘2 4𝑘2 +𝑘 13. A random variable X has the following probability density function 𝑓(𝑥) = 𝑘𝑥2 𝑒−𝑥 ; 𝑥 > 0 find k, Mean and Variance. 14. Find the Fourier series for f(x)=x in (0,2π). 15. Obtain Half–range cosine series for𝑓(𝑥) = (𝑥 − 1)2 0 < 𝑥 < 1 16. 17. 18. 19.