Mathematics – I — Summer 2024

Using L’ Hospital’s rule, evaluate l i m x → 1 1 + x l o − g x x x − x .03

Define Beta function and evaluate 10 x ( 1 − x ) d x04 .

Find the Fourier series of f ( x ) x x 0 x x 0     =  + − −    

Show that the sequence  u n  , where u n = s i n n n converges to zero.

Express f(x)=2x3+3x2−8x+7in terms of ( x − 2 ) . Find the area of the surface of revolution of a quadrant of a circular arc as

obtained by revolving it about a tangent at one of its ends.

Find the length of the loop of the curve

9 a y 2 = ( x − 2 a ) ( x − 5 a ) 2 07 .

Evaluate 0 ( 1 + v 2 ) ( d1 v + t a n − 1 v )03 . Test the convergence of the series

 n = 1 n 22 ( n n + +1 1 ) 204 .

If t n e r4 t  = − then find n so that1 r 2 r r r t .         =  07

Check the convergence of 50 1 x d x03 . 

Evaluate 3−4x2 dx.0 Find the Fourier series of f(x)= (−x) in the interval (0,2). Hence, deduce2

 1 1 1 that =1− + − +....

Prove that t a n − 1 x = x − x3 + x5 − x7 + . . . .03

Find the Fourier sine series of f ( x ) = e x in 0 x    .

Find the extreme values of the function f ( x , y ) = x 3 + y 3 − 3 x − 1 2 y + 2 0 .

Find the directional derivative of f(x,y,z)=x2yz+4xz2 at (1, -2, -1) in the

direction of 2 ˆi − ˆj − 2 ˆk03 . Solve the following system by Gauss – Jordan method:

−3 x x y + + +6 y y z − + =3 z z = = −5 Change the order of integration and evaluate

10 1 − 0 y 1 − x c2 o s1 − 1 − x x 2 − y d x d y07 .

Evaluate 30 10 ( x 2 + 3 y 2 ) d y d x .03

Apply Cayley – Hamilton theorem to A =  − 1  and deduce that A8 =625I. 2 2x−x2 x

Evaluate   dydxby changing to polar coordinates. x2 + y200

Evaluate 20 21 y z0 x y z d x d y d z03 .

Using Gauss Jordan method, find inverse of A =  24  .

Find the eigen values and eigen vectors of the matrix A =  41 − 1 −6 −  .