# CSVTU Exams Questions Papers – Ist Year – Applied Mathematics-I – Dec-Jan- 2006-07

BE (1st Semester)

Examination Dec-Jan, 2006-2007

Applied Mathematics-I

UNIT-I

1. (a) If y = (sin-1 x)2, show that:

(1-x2)yn+2-(2n+1)xyn+1-n2yn=0

(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

UNIT-II

2. (a) Find the reduction formula for : I„ = ? sec” xdx

(b) Find the area of the ellipse:  x2 + y2

a2    b2

(c) Find the volume of a sphere of radius.

UNIT-III

3. (a) If u=exyz, find the value of     ?3z

?x?y?z

(b) If u=sin-1 X2+y2  prove that.

x+y

?u + ?u = tan u

?x     ?y

(c) Find the volume of the greatest rectangular parallelepiped that can be inscribed in the ellipsoid

X2 + y2 + z2 = 1

a2    b2    c2

UNIT- IV

4. (a) Change the order of integration in:

I = ? 0?22-x x y dx dy.

(b) Find the volume of the tetrahedron bounded by co-ordinate planes and the plane.

y + = 1.

a    b   c

(c) Show that ?0x/2 ?sin?d?   * ?0x/2 d?      = ?.

?sin?

UNIT- V

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.