# CSVTU Exam Papers – BE I 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-x^{2})y_{n+2}-(2n+1)xy_{n+1}-n^{2}y_{n}=0

(b) Using Taylor’s theorem expresses the polynomial

2x* ^{3} + 7x^{2} 4x—6* in powers of (x-1).

(c) Test the convergence or divergence of the series:

∞

∑ [√^{ }n^{2}=1 – n]n=1

**UNIT-II**

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

* *(b) Find the area of the ellipse: x^{2} + y^{2}

a^{2} b^{2}

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

**UNIT-III**

3. (a) If u=e^{xyz}, find the value of ∂^{3}z

∂x∂y∂z

(b) If u=sin^{-1} X^{2}+y^{2} prove that.

x+y

X ∂*u* + ∂*u* = tan *u*

∂x ∂y

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

X^{2} + y^{2} + z^{2} = 1

a^{2} b^{2} c^{2}

**UNIT- IV**

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

*I** = *ʃ* *_{0}^{1 }ʃ_{2}^{2-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 ʃ_{0}^{x/2 }√sinᶿdᶿ * ʃ_{0}^{x/2 }dᶿ = ∏.

√sinᶿ

**UNIT- V**

5. (a) Show that: V^{2} (r^{n}) =n (n+1) r^{n-2}.

(b) Verify stokes theorem for F = (x^{2}+y^{2})*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 *f*_{S} *FNdS*=(a+b+c)^{V}.

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