aeronautical-mathematics-aeronautical engineering-old jntu 2-1 papers- M2

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