Using L’ Hospital’s rule, evaluate l i m x → 1 1 + x l o − g x x x − x .03
[ marks]Define Beta function and evaluate 10 x ( 1 − x ) d x04 .
[5 marks]Find the Fourier series of f ( x ) x x 0 x x 0 = + − −
[7 marks]Show that the sequence u n , where u n = s i n n n converges to zero.
[3 marks]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
[7 marks]obtained by revolving it about a tangent at one of its ends.
[ marks]Find the length of the loop of the curve
[ marks]9 a y 2 = ( x − 2 a ) ( x − 5 a ) 2 07 .
[ marks]Evaluate 0 ( 1 + v 2 ) ( d1 v + t a n − 1 v )03 . Test the convergence of the series
[ marks] n = 1 n 22 ( n n + +1 1 ) 204 .
[ marks]If t n e r4 t = − then find n so that1 r 2 r r r t . = 07
[2 marks]Check the convergence of 50 1 x d x03 .
[4 marks]Evaluate 3−4x2 dx.0 Find the Fourier series of f(x)= (−x) in the interval (0,2). Hence, deduce2
[ marks] 1 1 1 that =1− + − +....
[ marks]Prove that t a n − 1 x = x − x3 + x5 − x7 + . . . .03
[ marks]Find the Fourier sine series of f ( x ) = e x in 0 x .
[4 marks]Find the extreme values of the function f ( x , y ) = x 3 + y 3 − 3 x − 1 2 y + 2 0 .
[7 marks]Find the directional derivative of f(x,y,z)=x2yz+4xz2 at (1, -2, -1) in the
[ marks]direction of 2 ˆi − ˆj − 2 ˆk03 . Solve the following system by Gauss – Jordan method:
[ marks]−3 x x y + + +6 y y z − + =3 z z = = −5 Change the order of integration and evaluate
[ marks]10 1 − 0 y 1 − x c2 o s1 − 1 − x x 2 − y d x d y07 .
[2 marks]Evaluate 30 10 ( x 2 + 3 y 2 ) d y d x .03
[ marks]Apply Cayley – Hamilton theorem to A = − 1 and deduce that A8 =625I. 2 2x−x2 x
[7 marks]Evaluate dydxby changing to polar coordinates. x2 + y200
[ marks]Evaluate 20 21 y z0 x y z d x d y d z03 .
[ marks]Using Gauss Jordan method, find inverse of A = 24 .
[4 marks]Find the eigen values and eigen vectors of the matrix A = 41 − 1 −6 − .
[7 marks]