### 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
⋆ ⋆ ⋆ ⋆ ⋆
1. (a) Test the convergence of the series
p2−1
32−1 +
p3−1
42−1 +
p4−1
52−1 + ..... [5]
(b) Examine whether the following series is absolutely convergent or conditionally
convergent 1 − 1
3 ! + 1
5 ! − 1
7 ! + . . . . .. [5]
(c) Verify Rolle’s theorem for f(x) = log h x2+ab
x(a+b)i in [a,b] (x 6= 0). [6]
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. [3]
(b) Solve the differential equation: ( 1+ y2) dx = ( tan −1y – x ) dy. [7]
(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. [6]
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 ] [5]
(b) Find L−1 h s+3
(s2−10s+29)i [6]
1 of 2
Code No: R05010102 Set No. 4
(c) Evaluate
π/4
R0
a Sin
R0
r dr d
pa2 −r2 [5]
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.