Code No: R05010102 Set No. 3

I B.Tech Regular Examinations, Apr/May 2007

MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronic Engineering,

Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &

Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Control Engineering, Mechatronics, Computer

Science & Systems Engineering, Electronics & Telematics, Metallurgy &

Material Technology, Electronics & Computer Engineering, Production

Engineering, Aeronautical Engineering, Instrumentation & Control

Engineering and Automobile Engineering)

Time: 3 hours Max Marks: 80

Answer any FIVE Questions

All Questions carry equal marks

⋆ ⋆ ⋆ ⋆ ⋆

1. (a) Test the convergence of the following series P 1

(log log n)n [5]

(b) Find the interval of convergence of the series

x + 1

2 . x3

3 + 1

2 . 3

4 . x5

5 + 1.3.5

2.4.6 . x7

7 + ..... [5]

(c) Show that log (1 + ex) = log 2 + x

2 + x2

8 − x4

192 + ..... and hence deduce that

ex

ex+1 = 1

2 + x

4 − x3

48 + ..... [6]

2. (a) Given that x+y+z=a, find the maximum value of xmynzp.

(b) Find the envelope of the circles through the origin and whose centre lies on

the ellipse x2

a2 + y2

b2 . [8+8]

3. (a) Trace the curve : r = a ( 1 + cos θ ).

(b) Find the length of the arc of the curve x = e sinθ; y = e cosθ from θ = 0 to

θ = π/2. [8+8]

4. (a) Find the differential equation of all parabolas having the axis as the axis and

(a,0) as the focus.

(b) Solve the differential equation x2dy

dx = ey − x.

(c) Find the orthogonal trajection of the family of curves x2/3 + y2/3 = a2/3.

[4+6+6]

5. (a) Solve the differential equation: d2y

dx2 + 4 dy

dx + 5y = −2Coshx given that y(0)

= 0, y′(0) = 1.

(b) Solve the differential equation: (2x − 1)3 d3y

dx3 + (2x − 1) dy

dx − 2y = x. [8+8]

6. (a) Evaluate L{et(cos2t + 1/2 sinh2t)} [5]

(b) Find L−1 1

s2+2s+5 [6]

(c) Evaluate the triple integral

1

R

0

1

R

y

1−x

R

0

x dz dx dy [5]

1 of 2

Code No: R05010102 Set No. 3

7. (a) Evaluate ∇2 log r where r = px2 + y2 + z2

(b) Find constants a, b, c so that the vector A =(x+2y+az)i +(bx-3y-z)j+(4x+cy+2z)k

is irrotational. Also find ϕ such that A = ∇φ . [8+8]

8. Verify Stoke’s theorem for the vector field F=(2x-y)i-yz2j-y2zk over the upper half

surface of x2+y2+z2=1, bounded by the projection of the xy-plane. [16]

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