Mathematics – I — Winter 2018

State Cayley– Hamilton theorem. Find eigen values of Aand A  1 , where A   03

State L’ Hospital’s Rule. Use it to evaluate l i m x  0  1 x  s i1 n 2 x 

Investigate convergence of the following integrals:

  5  1  5 x x 2  d x (ii)  0 x 1 0  1   x  x2  d x07

Test the convergence of series  n  1 4 n  6 n 5 n

 n1 State the p-series test. Discuss the convergence of the series n3 3n2 n2

State D’Alembert’s ratio test and Cauchy’s root test. Discuss the convergence of the following series:

 n  1 4 n  n n n   1 1  ! (ii) n    l o n g n  n07

Test the convergence of the series 1 1 2   4  x 5  7 x  82  9  1 0  x1 1  1    07 ; x0 1 5 3 2  

Reduce matrix A 2 0 4 1 to row echelon form and find its rank.   4 8 9 1  

Derive half range sine series of f xx, 0 x   

Find the eigen values and corresponding eigen vectors for the matrix A  1 2 2   where A 0 2 1   1 2 2  

Expand exsin(x) in power of x up to the terms containing x6.

Solve system of linear equation by Gauss Elimination method, if solution exists.2 x  y  2 z  9 ; 2 x  4 y  3 z  1 ; 3 x  6 y  5 z 

Find Fourier series of f  x    x2 , ,    x x  0

Discuss the continuity of the function f defined as f  x , y    x x3  0 y y3 ;   x x , , y y      0 , ,0  03

Define gradient of a function. Use it to find directional derivative of f  x , y , z   x 3  x y 2  z at P  1 , 1 , 0  in the direction of a  2 ˆi  3 ˆj  6 ˆk04 .

Find the shortest and largest distance from the point  1 , 2 ,  1  to the sphere x 2  y 2  z 2  2 407 .

Find the extreme values of x 3  3 x y 2  3 x 2  3 y 2 

Evaluate  0 4  x 2  x 2  y 2  d y d x 04 by changing into polar coordinates.

If ux2yy2zz2x then find out   u x    u y    u z (ii) If x 3  y 3  6 x y then find d d y x and d d2 x y2

Evaluate  Ry s i n  x y  d A , where Ris the region bounded by x  1 , x  2 , y   and y  .2

By changing the order of integration, evaluate 30 3y x x d2 x  d y y 204

Find the volume below the surface z  x2  y2, above the plane z  0 , and inside the cylinder x 2  y 2  2 y07 .

Evaluate integral R r d a r2 d r     over the region Rwhich is one loop of r 2 a 2 c o s 2  03

Evaluate the integral  e1  lo1 g y  e1 x  x2  y 2  d z d x d y04 .

Find the volume of the solid obtained by rotating the region Renclosed by the curves yx and y  x 207 about the line y 2.