Mathematics I May2006 Rr10102

Set No. 1

Code No: RR10102

I B.Tech Supplementary Examinations, Apr/May 2006 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 Bio-Technology) 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 (n !)2

(2n) !

x2n

(b) Test the following series for absolute /conditional convergence

[5] ∞ P

n=2

(c) Show that sin−1 x = x +

x3 3!

+

12 .32 5!

x5 +

2. (a) If xx yy zz = c, then show that at x = y = z,

12 . 32 . 52 7! ∂2z ∂x∂y

(−1)n n(log n)2

x7 + ....

[6] [5]

= − {x log (ex)}−1

(b) Find the evolute of the hyperbola x2 /a2 -y2 /b2 = 1. Deduce the evolute of a rectangular hyperbola. [8+8] 3. Trace the lemniscate of Bernouli : r2 = a√2 cos2θ. Prove √ that the volume of revoluπ√ a3 tion about the initial line is 6 2 3 log ( 2 + 1) − 2 [16] 4. (a) Form the differential equation by eliminating the arbitrary constant : y2 =4ax. [3] p [7] (b) Solve the differential equation: 1 + y 2dx = ( tan−1 y – x ) dy.

(c) If the air is maintained at 300 C and the temperature of a body cools from 800 C to 600 C in 12 minutes, find the temperature of the body after 24 minutes. [6]

5. (a) Solve the differential equation: (D 2 + 5D + 6)y = ex . (b) Using the method of variation of parameters, solve the differential equation d2 y + 4y = tan2x [8+8] dx2 6. (a) Find the Laplace transformation of e i h − 1 s3 − 3s2 + 6s − 4 (b) Find L (s2 −2s +2)2 (c) Evaluate

R∞ R∞

e−(x

2 +y 2 )

2t

+ 4t3 – 2 sin3t + 3cos3t.

[5] [6] [5]

dxdy

0 0

7. (a) For any vector A, find div curl A. 1 of 2

[6]

Set No. 1

Code No: RR10102 (b) Evaluate

RR

A.n ds where A=z i +x j-3y2 z k and s is the surface of the cylinder

s

x2 + y 2 = 16included in the first octant between z=0 and z=5.

[10]

8. Verify divergence theorem for F = 6zi + (2x + y)j – xk, taken over the region bounded by the surface of the cylinder x2 + y2 = 9 included in z = 0, z = 8, x = 0 and y = 0. [16] ⋆⋆⋆⋆⋆

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Set No. 2

Code No: RR10102

I B.Tech Supplementary Examinations, Apr/May 2006 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 Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Test the convergence of the series

∞ P

n=1

1 2n +3n

[5] x2 2

x3 3

4

+ x4 + .....∞  √ √  (c) Verify Rolls theorem for f(x) = 2x3 + x2 - 4x- 2 in − 2, 2

(b) Find the interval of convergence of the series

+

[5] [6]

2. (a) Find the stationary points of the following function and find the maximum or the minimum u u = x2 + 2xy + 2y2 + 2x + y (b) Considering the evolute of a curve as the envelope of its normals, find the 2 2 evolute of the ellipse xa2 + yb2 = 1 [8+8] 3. (a) Trace the curve xy2 = 4 a2 (2a – x), (a > 0) (b) Find the area of the surface generated of an arch of cycloid x = a(θ - Sinθ), y = a(1- Cosθ) revolving about the tangent at the vertex. [8+8] 4. (a) Form the differential equation by eliminating the arbitrary constant : log y/x = cx. [3] (b) Solve the differential equation: dr + (2r cot θ + sin 2θ) dθ = 0.

[6]

(c) The number N of bacteria in a culture groups at a rate proportional to N. The value of N was initially 100 and increased to 332 in one hour. What was the value of N after 1 1/2 hour. [6] 5. (a) Solve the differential equation:

d3 y dx3

dy + 4 dx = Sin 2x

(b) Solve the differential equation: (D − 2)2 = 8(e2x + sin2x + x2 )   6. (a) Find L e−3t sint 2t  s+2  (b) Find L−1 s2 −4s+13

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[8+8] [5] [6]

Set No. 2

Code No: RR10102

(c) Evaluate the triple integral

R1 R1 1−x R 0 y

x dz dx dy

[5]

0

7. (a) Find curl[rf (r)] where r = xi+yj+zk, r = |r|.

[6]

(b) Find the work done in moving a particle in the force field F=3x2 i + (2xz − y)j + zk along the straight line from (0, 0, 0) to (2, 1, 3). [10]

8. Verify divergence theorem for F = 4xi - 2y2 j + z2 k taken over the Surface bounded by the region x2 +y2 =4, z = 0 and z = 3. [16] ⋆⋆⋆⋆⋆

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Set No. 3

Code No: RR10102

I B.Tech Supplementary Examinations, Apr/May 2006 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 Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) P Test the convergence of the series n (n + 1)n nxn+1 , (x > 0) [5] ∞ P 2 √ nx converges absolutely for all values of x. (b) Prove that the series (−1)n Cos n n 1

[6] (c) Verify Roll’s theorem for f (x) = 2x3 + x2 − 4x − 2

√ √ in (− 3, + 3)

2. (a) If xx yy zz = c, then show that at x = y = z,

[5] ∂2z ∂x∂y

= − {x log (ex)}−1

(b) Find the evolute of the hyperbola x2 /a2 -y2 /b2 = 1. Deduce the evolute of a rectangular hyperbola. [8+8] 3. (a) Trace the curve: y(x 2 + 4a2 ) = 8a2 . (b) Obtain the surface area of the solid of revolution of the curve r = a (1 + cosθ) about the initial line. [8+8] 4. (a) Form the differential equation by eliminating the arbitrary constant tan x tan y = c. (b) Solve the differential equation:

dy dx

+

y x log x

=

sin 2x log x

(c) Obtain the orthogonal trajectories of the family : rn = an cos nθ.

[3] [7] [6]

5. (a) Solve the differential equation: (D 3 + 1)y = Sin3x - Cos2 x. (b) Solve the differential equation: (D 2 − 4)y = (1 + ex )2 . 6. (a) Find the Laplace transformation of e h i − 1 s3 − 3s2 + 6s − 4 (b) Find L (s2 −2s +2)2 1 of 2

2t

+ 4t3 – 2 sin3t + 3cos3t.

[8+8] [5] [6]

Set No. 3

Code No: RR10102 (c) Evaluate

R∞ R∞

e−(x

2 +y 2 )

[5]

dxdy

0 0

7. (a) Find the angle between the surfaces x2 +y2 + z2 =9 and z=x2 + y 2 − 3 at the point (2, -1, 2) (b) Evaluate ∇.(r/r3 ) where r =xi+yj+zk and r =|r|

[8+8]

8. Verify divergence theorem for F = 4xi - 2y2 j + z2 k taken over the Surface bounded by the region x2 +y2 =4, z = 0 and z = 3. [16] ⋆⋆⋆⋆⋆

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Set No. 4

Code No: RR10102

I B.Tech Supplementary Examinations, Apr/May 2006 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 Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Test for convergence of the series

1 √ P ∞

n4 + 1 −

√

(b) State and prove Cauchy Mean value theorem. b−a (1+b2 )

< tan−1 b − tan−1 a (c) If a < b prove that value theorem. Deduce the following i. ii.

Π 3 + 25 4 5Π+4 < 20

< tan−1 34 < Π4 + tan−1 2 < Π+2 4

 n4 − 1

b−a < (1+a 2)

[5] [5]

using Lagrange’s Mean [6]

1 6

2. (a) If µ = log (x3 +y3 +z3 -3xyz) prove that  ∂ −1 µx + µy + µz = 3(x+y+z) and + ∂x

∂ ∂y

+

∂ ∂z

2

µ=

−9 (x+y+z)2

(b) Find the radius of curvature at any point of the parabola y2 = 4ax. Prove that the square of the radius of curvature at any point of the curve varies as the cube of the focal distance of the point. [8+8]

3. (a) Trace the curve : x4 + y4 = 2a2 xy. (b) Find the surface area got by rotating one loop of the curve r2 = a2 cos2θ about the initial line. [8+8] 4. (a) Form the differential equation by eliminating the arbitrary constant ‘c’: √ 2 y = 1 + x + c 1 + x2 . (b) Solve the differential equation: dy + (y – 1) cos x = e−sinx cos2 x. dx

[3] [7]

0

(c) An object whose temperature is 75 C cools in an atmosphere of constant temperature 250 C at the rate kθ , θ being the excess temperature of the body over the atmosphere. If after 10 minutes the temperature of the objects falls to 650 . Find its temperature after 20 minutes. Find the time required to cool down to 550 C. [6] 1 of 2

Set No. 4

Code No: RR10102

5. (a) Solve the differential equation: y′′ + 4y′ + 20y = 23 sint - 15cost, y(0) = 0, y′ (0) = -1 (b) Solve the differential equation: (2x + 5)2

6. (a) Prove that L



 (b) Find L − 1 log

Rt

f (u) du =

0

s+1 s−1

f (s) s



d2 y dx2

dy + 6 (2x + 5) dx + 8y = 4(2x + 5) [8+8]

, where L{f(t)}= f (s).




[5] [6]

(c) Evaluate ∫ ∫ r sinθdr dθ over the cardioid r = a(1 - cosθ) above the initial line. [5]

7. (a) If φ1 = x2 z, φ2 = xy – 3z2 , then find ∇( ∇φ1 . ∇φ2 ) R (b) Evaluate F . d r where F = z i + x j + y k and C is x = a Cost, y = a Sint, z=

t 2π

C

from z = 0 to z = 1.

[8+8]

H 8. State Green’s theorem and verify Green’s theorem for [(xy + y2 )dx + x2 dy], where C

c is bounded by y = x and y = x2 .

[16]

⋆⋆⋆⋆⋆

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