Please answer to all questions with proper explanation step by step. Thankyou :)

1. A quadratic surface has equation a 2 ' '- : 33'1 3'2: 3'12; + 4.11.1\"; + 41:13:; + 12:1'23'3 1 . (a) [2 marks] YWrite this equation in the form DETAX 1. (b) [4 marks] Find eigenvalues of the matrix fl and the correspol'uling unit eigenvee- tors. (c) [3 marks] Write down the spectral matrix. the new (canonical) variables yl. y; and ya; in terms of the original variables .1? 1. 1:2 and 3. and the canonical form of the equation to the surface in terms of the new variables. ((1) [1 mark] Identify the surface. 2. [12 marks] Ttil-"rite the system of differential equations (11: 'J- a ._..l'- + U a (11; _ _ " _ in: 2': (it J in a matrix brin. compute exp[At). where A is the matrix of equation coefcients. and use this matrix exponential to nd the solution of the system subject to the initial conditions .1'([l) 2. 13(0) l