# CSVTU Exam Papers – BE I Year – Applied Mathematics-I –Nov-Dec- 2007

**BE (1st Semester)**

**Examination Nov-Dec, 2007**

**Applied Mathematics-II**

**UNIT- I**

1. (a) State De Moire’s theorem.

(b) cos *a* + cos *β* + cos *λ* = sin* a* + sin* β* + sin* λ* = 0 Prove that

(i) cos^{2}* **a* + cos^{2}* **β* + cos^{2}* **λ* = sin^{2}* **a* + sin^{2}* **β* + sin^{2}* **λ*

(ii)cos*2a +* cos2

*β*+ cos 2

*λ*= sin

*2a*+ sin2

*β*+ sin 2

*λ*= 0

(c) If tan (*ө* + iф) = e^{ia}*, *Show that:

*ө – *(n +1) ∏ and log tan(∏ + *a*)

2 2 4 2

(d) Sum the series: ф – 1

2

sin^{2} *ө* 1 sin 2*ө* sin^{2}*ө* + 1 sin^{3}*ө* – 1 sin 4*ө*sin^{4}* ө*+………..* ө*

2 3 4

**UNIT- II**

2. (a) Solve d^{2}y – 3dy – 4y =0

dx^{2 } dx

(b) Solve d^{2}y – 5dy + 6y =sin3x

dx^{2 } dx

(c) Solve by method of variation of parameters d^{2}y + 4y=tan^{2}x

^{ } dx^{2}

(d) Solve the simultaneous equations: dx + 2x +5y=t^{t}

dt

dx + 4x +3y=t

dt

**UNIT- III**

3. (a)Define Bet function .

*a b a *

(b) Evaluate: ʃ ʃ ʃ(x^{2} + y^{2} + z^{2})*dzdydx*

*a b a*

* * (c) Evaluate: _{0}^{1}ʃx^{m}(log)^{n}dx = (-1)^{n} n!

(m+1)^{n+1}

Where n is a positive integer and m>-1.

(d) Find the area included between the parabola y= 4x – x^{2} and the line y = x.

**UNIT- IV**

4. (a) Define Gradient.

(b) Find the constants a and b so that the surface ax^{2} – byz= (a+2)x is orthogonal to the surface

ax^{2}y + z^{3} = 4 at the point (1, -1, 2)

(c) If F = (x^{2}* **+ y ^{2})i + 2xyj + (y^{2}-xy)k,* then find div F and Curl F.

(d) Verify stake’s theorem for :* *

* F* = (x^{2} +* y ^{2})I +* 2

*xyj*Taken round the rectangle bounded by x=

*±a, y = y = b*

* ***UNIT- V**

5. (a)Find the number of roots of the equation* x ^{3} – x^{2}4x* + 4 = 0

(b) Find the condition that the equation* x ^{3} + px^{2}4x + 4 = 0* had roots

*a,*

*β*which satisfy

*a*

*β*+ 1 = 0

(c) If* **a, β, **λ** *are roots of the equation .x^{3} +* qx + r* = 0 find the equation whose roots are

(*a – β*)^{2}, (*β –** λ** *)^{2}, (*λ – **a*)^{2}

(d) Solve the equation* x ^{2}* -15x – 126 = 0 Cardoon’s method.

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