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2(a). Evaluate ∫ e−axxm−1 Sinbx dx in terms of Gamma function.

2(b) Show that the area between the parabolas y2=4ax and x2=4ay is 16/3a2

3(a) Evaluate limx→0√xtanx /(ex−1)32

4(a). Test the convergent of the series 1 −1/3+1/32 −1/33 +1/34 … … … . ∞

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4(b). Test the convergent of the series x +22×2/2!+33×3/3!+44×4/4!+55×5/5!………∞

5(a).Obtain the Fourier series for f(x) = e −x in the interval 0

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5(b). Express f(x) = x as a half range cosine series in 0.

6(a). Find the maximum and minimum of the functionf(x) = x5 − 3×4 + 5

6(b). Discuss the continuity of the function
f(x,y) =x3−y3/x2+y2
when x≠0, y≠0
0 when x=0, y=0

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7(b) Expand tan−1x in power of (x-1).

8. Obtain the eigen values and eigen vectors of the matrix
[2 0 1
0 2 0
1 0 2]
and verify that the eigen vectors are orthogonal.

9. Verify cayley-Hamilton theorem for the matrix
A = [
2 1 1
0 1 0
1 1 2]
Also express A8 − 5A7 + 7A6 − 3A5 + A4−5A3 + 8A2 − 2A + I as a quadratic polynomialin A.

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