# 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) cos2 a + cos2 β + cos2 λ = sin2 a + sin2 β + sin2 λ

(ii)cos2a + cos2 β + cos 2 λ = sin2a + sin2 β + sin 2 λ = 0

(c) If tan (ө + iф) = eia, Show that:

ө – (n +1)  and                 log tan(  +  a)

2    2                                        4       2

(d) Sum the series: ф – 1

2

sin2 ө 1 sin 2ө sin2ө + 1 sin3ө1 sin 4өsin4 ө+……….. ө

2                         3              4

UNIT- II

2. (a) Solve d2y  – 3dy – 4y =0

dx2       dx

(b) Solve d2y  – 5dy + 6y =sin3x

dx2       dx

(c) Solve by method of variation of parameters d2y + 4y=tan2x

dx2

(d) Solve the simultaneous equations: dx + 2x +5y=tt

dt

dx + 4x +3y=t

dt

UNIT- III

3. (a)Define Bet function .

a   b    a

(b) Evaluate:   ʃ    ʃ     ʃ(x2 + y2 + z2)dzdydx

a   b     a

(c) Evaluate:   01ʃxm(log)ndx =     (-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 – x2 and the line y = x.

UNIT- IV

(b) Find the constants a and b so that the surface ax2 – byz= (a+2)x is orthogonal to the surface

ax2y + z3 = 4 at the point (1, -1, 2)

(c) If F = (x2 + y2)i + 2xyj + (y2-xy)k, then find div F and Curl F.

(d) Verify stake’s theorem for :

F = (x2 + y2)I + 2xyj  Taken round the rectangle bounded by  x= ±a, y = y = b

UNIT- V

5. (a)Find the number of roots of the equation x3 – x24x + 4 = 0

(b) Find the condition that the equation x3 + px24x + 4 = 0 had roots a, β which satisfy aβ + 1 = 0

(c) If a, β, λ are roots of the equation .x3 + qx + r = 0 find the equation whose roots are

(a – β)2, (β – λ )2, (λ – a)2

(d) Solve the equation x2  -15x – 126 = 0 Cardoon’s method.