It might be appreciated that you help me to figure out answers and steps too..

I don't know where to start with.. so please give me detailed steps.

Question 1 (22 marks) 6 -2 - Given that a 3x3 real matrix A= -2 5 (a) Show that the eigenvalues of matrix A are 3, 6 and 8, and hence, find their corresponding eigenvectors; (14 marks) (b) check whether the eigenvectors found in part (a) form an orthogonal set; (2 marks) (c) find, if any, a diagonal matrix D and an invertible matrix P such that P- AP= D. Also, find P-1. (6 marks) Question 2 (28 marks) (a) Find the stationary points of the function f (x, y) = x' +3xy? - y3 -3x and determine their nature. (18 marks) (b) If f = g(x, y), x=red and y =re-, calculate of and of 20' and show that 2x- of _of of ax Or 30 and 2yay of of of =rar 30 of Hence, let or of find all the elements a, (i=1, 2, j =1, 2) a22 in terms of r and 0 (10 marks)Question 3 (20 marks) Let =x'y+2xz' +zy? be a scalar field, (a) find Vo and div (Vo); (b) find the directional derivatives of $ at the point (1, 2, 0) in the direction of the vector (6 marks) a = 37 +4] -k ; (c) calculate the magnitude of the maximum rate of change of @ at the point (1, 2, 0). (10 marks) (4 marks) Question 4 (20 marks) (a) Verify that the vector field F = yi + (x+zcosy)] +sin yk is irrotational, and find a scalar function f (x, y, z) for which F = Vf (15 marks) (b) From the result in (a), evaluate J F . dr along any path from (0, 0, 0) to (3, 2, 0) , where F=xi + yj +zk is the position vector of a moving particle. (5 marks) Question 5 (20 marks) Compute the following integrals using suitable method. (a) ff xy dx dy, where R is the region bounded by y=1, y=1+x, and y=3-x. (10 marks) (b) SIT- V9-x? -y? edx dy dz, where V is the region bounded above by a sphere x2+y? +z? =9 and bounded below by a plane z=1. (10 marks)