Mathematics Model paper Jntu

Code No: R05010102 Set No. 4
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
All Questions carry equal marks
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1. (a) Test the convergence of the series
p2−1
32−1 +
p3−1
42−1 +
p4−1
52−1 + ..... 
(b) Examine whether the following series is absolutely convergent or conditionally
convergent 1 − 1
3 ! + 1
5 ! − 1
7 ! + . . . . .. 
(c) Verify Rolle’s theorem for f(x) = log h x2+ab
x(a+b)i in [a,b] (x 6= 0). 
2. (a) Show that the functions u = x+y+z , v = x2+y2+z2-2xy-2zx-2yz and
w = x3+y3+z3-3xyz are functionally related. Find the relation between them.
(b) Find the centre of curvature at the point 􀀀a
4 , a
4 of the curve √x +√y = √a.
Find also the equation of the circle of curvature at that point. [8+8]
3. (a) In the evolute of the parabola y2= 4ax, show that the length of the curve from
its cusp x = 2a to the point where it meets the parabola y2 = 4ax is 2a(3√3
- 1)
(b) Find the length of the arc of the curve y = log ex
−1
ex+1 from x = 1 to x = 2
[8+8]
4. (a) Form the differential equation by eliminating the arbitrary constant : log y/x
= cx. 
(b) Solve the differential equation: ( 1+ y2) dx = ( tan −1y – x ) dy. 
(c) The temperature of the body drops from 1000 C to 750C in ten minutes when
the surrounding air is at 200C temperature. What will be its temperature
after half an hour. When will the temperature be 250C. 
5. (a) Solve the differential equation: d3y
dx3 + 4dy
dx = Sin 2x.
(b) Solve the differential equation: x2 d2y
dx2 − 2x dy
dx − 4y = x4. [8+8]
6. (a) Find L [ t2 Sin2t ] 
(b) Find L−1 h s+3
(s2−10s+29)i 
1 of 2
Code No: R05010102 Set No. 4
(c) Evaluate
π/4
R0
a Sin
R0
r dr d
pa2 −r2 
7. (a) Prove that ∇ × A¯×¯r
rn = (2−n)A¯
rn +
n(r¯.A¯)r¯
rn+2
(b) If ¯ F = (x2 − 27) i−6yzj +8xz2k evaluate RC
¯ F.d¯r from the point (0,0,0) to the
point (1,1,1) along the straight line from (0,0,0) to (1,0,1), (1,0,0) to (1,1,0)
and (1,1,0) to (1,1,1) [8+8]
8. Verify Stokes theorem f=x2i-yzj+k integrated around the square x=0, y=0, z=0,
x=1, y=1 and z=1.