Sameer, I think there is a problem in your statement as far as the computation of t_1 goes. t_1(x) is clearly a function of the sample and only the sample, whereas you include mu as a parameter. I believe the problem definition states that t1 is sufficient for theta1 given theta2, ie P_theta1(t_1(x), theta2) = P_theta2(x, theta2). So theta2 may not be used in computing t_1(x), just for establishing its sufficiency.