Code No: R05010102 Set No. 1

I B.Tech Supplimentary Examinations, Aug/Sep 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 Pn2

2n + 1

n2 [5]

(b) Find the interval of convergence of the series whose nth term is P (−1)n(x+2)

(2n +5)

[5]

(c) If a < b prove that b−a

(1+b2) < tan−1b − tan−1a < b−a

(1+a2) using Lagrange’s Mean

value theorem. Deduce the following [6]

i.

4 + 3

25 < tan−1 4

3 <

4 + 1

6

ii. 5+4

20 < tan−1 2 < +2

4

2. (a) If x = r sinθ cosφ, y = r sinθ sinφ and z = r cosθ prove that

@(x,y,z)

@(r,,) = r2 sin θ. [6]

(b) Find the radius of curvature at any point on the curve y = c coshx

c . [10]

3. Trace the curve y = a cosh (x/a) and find the volume got by rotating this curve

about the x-axis between the ordinates x = ± a. [16]

4. (a) Form the dierential equation by eliminating the arbitrary constant

secy + secx = c + x2/2.

(b) Solve the dierential equation:

(2y sin x + cos y ) dx = (x siny + 2 cosx + tany ) dy.

(c) Find the orthogonal trajectories of the family: rn sin nθ = bn. [3+7+6]

5. (a) Solve the dierential equation: y′′ - 4y′ + 3y = 4e3x,

y(0) = - 1, y′(0) = 3.

(b) Solve the dierential equation: (1 + x)2 d2y

dx2 + (1 + x) dy

dx + y = 4 cos log(1 + x)

[8+8]

6. (a) Find the Laplace Transformation of the following function: t e−t sin2t.

Code No: R05010102 Set No. 1

(b) Using Laplace transform, solve y′′+2y′+5y = e−t sint, given that

y(0) = 0, y′(0) = 1.

(c) Evaluate

5

R0

x2

R0

x(x2 + y2) dxdy [5+6+5]

7. (a) Prove that div(A×B)=B.curlA - A.curlB.

(b) Find the directional derivative of the scalar point function φ (x,y,z) = 4xy2 +

2x2yz at the point

A(1, 2, 3) in the direction of the line AB where B = (5,0,4). [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]

⋆ ⋆ ⋆ ⋆ ⋆

set-2

Code No: R05010102 Set No. 2

I B.Tech Supplimentary Examinations, Aug/Sep 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 series

∞

P

n=1

x2n

(n+1)√n. [5]

(b) Find the interval of convergence of the series x2

2 + x3

3 + x4

4 + .....∞. [5]

(c) Write Taylor’s series for f(x) = (1 − x)5/2 with Lagrange’s form of remainder

upto 3 terms in the interval [0,1]. [6]

2. (a) Locate the stationary points and examine their nature of the following func-

tions:

u = x4 + y4 - 2x2 + 4xy - 2y2, (x > 0, y > 0).

(b) From any point of the ellipse x2

a2 + y2

b2 = 1, perpendiculars are drawn to the

coordinates axes. Prove that the envelope of the straight line joining the feet

of these perpendiculars is the curve.

Code No: R05010102 Set No. 2

7. (a) If φ1 = x2 y and φ2 = xz – y2 find ∇× ( ∇φ1 × ∇φ2 )

(b) If F = (3x2 + 6y) i − 14yz j + 20 x z2 k evaluate the line integral R

C

F . d r

from (0,0,0), (1,1,1) along x = t, y = t, z = t3. [8+8]

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]

set-3

Code No: R05010102 Set No. 3

I B.Tech Supplimentary Examinations, Aug/Sep 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 for convergence of the series

1

P

1

√n4 + 1 − √n4 − 1 [5]

(b) Find the interval of convergence of the following series

1

1−x + 1

2(1−x)2 + 1

3(1−x)3 + ........ [5]

(c) Prove that

3 − 1

5p3

> cos−1 3

5 >

3 − 1

8 using Lagrange’s mean value theorem.

[6]

2. (a) Find the volume of the largest rectangular parallelopiped that can be inscribed

in the ellipsoid of revolution 4x2 + 4y2 + 9z2 = 36. [8+8]

(b) Find the envelope of the family of curves ax

cos − by

sin = a2−b2, α is a parameter.

3. (a) Trace the curve r=a sin2θ.

(b) Find the whole length of the curve 8a2y2=x2(a2–x2). [8+8]

4. (a) Form the dierential equation by eliminating the arbitrary constants,

y=a secx+ b tan x.

(b) Solve the dierential equation (y4+2y)dx+(xy3+2y4-4x)dy=0.

(c) If 30% of a radio active substance disappears in 10 days, how long will it take

for 90% to disappear. [3+7+6]

5. (a) Solve the dierential equation: (D2 + 4D + 4)y = 18 coshx.

(b) Solve the dierential equation: (D2 + 4)y = cosx. [8+8]

6. (a) Show that L{tn f(t)}= (−1)n dn

dsn f(s) where n = 1,2,3, . . . .

(b) Evaluate: L−1 h 1

s2(s+2)i

(c) Evaluate ∫ ∫r sinθ dr dθ over the cardioid r = a(1 - cosθ) above the initial line.

[5+6+5]

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 Green’s theorem for HC

[(3x − 8y2)dx + (4y − 6xy)dy]

where C is the region bounded by x=0, y=0 and x + y = 1. [16]

⋆ ⋆ ⋆ ⋆ ⋆

set-4

Code No: R05010102 Set No. 4

I B.Tech Supplimentary Examinations, Aug/Sep 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

1 + 3

1x + 3.6

7.10x2 + 3.6.9

7.10.13 x3 + .... x > 0 [5]

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

P (−1)n.n

3n2 −2 [5]

(c) Expand ex secx as a power series in x up to the term containing x3 [6]

2. (a) Find the points on the surface z2=xy+1 that are nearest to the origin.

(b) Prove that if the centre of curvature of the ellipse x2

a2 + y2

b2 = 1 at one end of

the minor axis lies at the other end, then the eccentricity of the ellipse is 1

√2

.

[8+8]

3. (a) Trace the lemniscate of Bernoulli : r2 = a2 cos2θ.

(b) The segment of the parabola y2 = 4ax which is cut o by the latus rectum

revolves about the directrix. Find the volume of rotation of the annular region.

[8+8]

4. (a) Form the dierential equation by eliminating the arbitrary constant : y2=4ax.

(b) Solve the dierential equation: dy

dx

−

tan y

1+x = (1 + x) ex sec y.

(c) In a chemical reaction a given substance is being converted into another at a

rate proportional to the amount of substance unconverted. If (1/5)th of the

original amount has been transformed in 4 minutes how much time will be

required to transform one half. [3+7+6]

5. (a) Solve the dierential equation y′′ − y′ − 2y = 3e2x, y(0) = 0, y′(0) = 2

(b) Solve the dierential equation: (D2+1)y = cosec x by variation of parameters

method. [8+8]

6. (a) Find the Laplace Transformations of the following functions

e−3t(2cos5t – 3sin5t)

Code No: R05010102 Set No. 4

(b) Find L−1 log

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