JNTU Mathematics Previous paper
Code No: R05010102 Set No. 2
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 n2
2n + 1
n2 [5]
(b) Find the interval of convergence of the series whose n th term is P (−1)n(n+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 u=x2-y2, v=2xy where x=r cosθ, y=rsinθ. Show that @(u,v)
@(r, ) = 4r3.
(b) For the cardioid r=a(1+cosθ) Prove that 2
r is constant where rho is the radius
of curvature. [8+8]
3. (a) Find the volume of the solid generated by revolution of y2 = x3
(2a−x) about its
asymptote.
(b) Find the area of the loop of the curve r=a(1+cos θ). [8+8]
4. (a) Form the differential equation by eliminating the arbitrary constant y = a+x
x2+1 .
[3]
(b) Solve the differential equation: (1-x 2) dy
dx - xy = y3sin −1x. [7]
(c) Prove that the family of confocal conics x2
a2+ + y2
b2+ = 1 are self orthogonal
(λ the parameter) [6]
5. (a) Solve the differential equation: (D3 − 7D2 + 14D − 8)y = excos2x.
(b) Solve the differential equation: (x2D2 − x3D + 1)y = log x sin (log x)+1
x . [8+8]
6. (a) Solve the differential equation d2x
dx2 + 9x = Sin t using Laplace transforms
given that x(0) = 1, x(π/2) =1
1 of 2
Code No: R05010102 Set No. 2
(b) Change the order of integration hence evaluate
1
R0
2−x
Rx2
xdy dx [8+8]
7. (a) Prove that ∇x(∇xA) = -∇2A+∇(∇.A).
(b) If φ = 2xy2z +x2y, evaluate RC
φ dr where C consists of the straight lines from
(0, 0, 0) to (1, 0, 0) then to (1, 1, 0) and then to (1, 1, 1). [8+8]
8. Verify Green’s theorem for HC
(y − Sin x ) dx + Cos x dy where C is the triangle
formed by the points (0,0) (π/2, 0) and (π/2, 1). [16]
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