BE (1st Semester)
Examination Dec-Jan, 2006-2007
1. (a) If y = (sin-1 x)2, show that:
(b) Using Taylor’s theorem expresses the polynomial
2x3 + 7x2 4x—6 in powers of (x-1).
(c) Test the convergence or divergence of the series:
∑ [√ n2=1 – n]n=1
2. (a) Find the reduction formula for : I„ = ʃ sec” xdx
(b) Find the area of the ellipse: x2 + y2
(c) Find the volume of a sphere of radius.
3. (a) If u=exyz, find the value of ∂3z
(b) If u=sin-1 X2+y2 prove that.
X ∂u + ∂u = tan u
(c) Find the volume of the greatest rectangular parallelepiped that can be inscribed in the ellipsoid
X2 + y2 + z2 = 1
a2 b2 c2
4. (a) Change the order of integration in:
I = ʃ 01 ʃ22-x x y dx dy.
(b) Find the volume of the tetrahedron bounded by co-ordinate planes and the plane.
x + y + z = 1.
a b c
(c) Show that ʃ0x/2 √sinᶿdᶿ * ʃ0x/2 dᶿ = ∏.
5. (a) Show that: V2 (rn) =n (n+1) rn-2.
(b) Verify stokes theorem for F = (x2+y2)j-2xyj taken around the rectangle bounded by the lines.
X=±a, y=0, y=h.
(c) If is any closed surface enclosing a volume V and
F = axi+byj+czk. Prove that fS FNdS=(a+b+c)V.