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PreDegree Maths Final Exam Summer 2008 Time: 2 hours. Answer any 5 of the 6 questions. Calculators and log tables are allowed. The table of Laplace transforms and formula sheet is attached. Q1. a.) Find an example of a 2 2 matrix A such that A 2 A . b.) Use Cramer’s Rule to find z in terms of a, b and c, where a, b and c are constants, given that: ax ay 2 z 5 bx by cz 2 cx 2cy 4cz c Q2. Recall that L( f (t )) e st f (t )dt . 0 a.) Prove that L(sin( wt )) w , where w is a constant, without using the s w2 2 formula sheet. b.) Prove the Formula 9 without using Formula 14 (i.e. prove that L(te at ) 1 without using the First Shifting ( s a) 2 Theorem). Q3. a.) Use Laplace transforms to solve the differential equation d2y dy 6 5 y sin( 2 x) , given that y(0) 1 and y' (0) 0 . 2 dx dx b.) Without using Laplace transforms, solve the differential equation d2y dy 6 5 y sin( 2 x) , given that y(0) 1 and y' (0) 0 . 2 dx dx Q4. Q5. 2 w w y and evaluate it at the 3 cos(xyz) . Find xy z x point ( x, y, z, w) (1,0,0,3) . a.) Let w f ( x, y, z ) b.) Find a.) Solve the separable differential equation b.) Show that the derivative of cos 1 x is 1 x 2 1 dx 2 x 2x 3 d5y 0. dx5 1 1 x2 (without using the formula in the log tables). Q6. a.) Draw on an Argand diagram the set of complex numbers z , where | 2 i z | 3 . b.) Let z 1 i . Write z in polar form, find z 25 and write your answer in standard (Cartesian) form. c.) Let z1 3i and let z 2 2 ei . Find z1 . z2