Using Laplace transform, solve the initial value problem ty'' + 2ty' + 2y = 2; y(0) = 1 and y'(0) is arbitrary. Does this problem have a unique solution?

2Medium15 marks

Find the equation of the cylinder whose generators are parallel to the line \frac{x}{1} = \frac{y}{-2} = \frac{z}{3} and whose guiding curve is x^2 + y^2 = 4, z = 2.

3Medium10 marks

Let S3 and Z3 be permutation group on 3 symbols and group of residue classes module 3 respectively. Show that there is no homomorphism of S3 in Z3 except the trivial homomorphism.

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