Nr10102-mathematics-i

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NR

Code No: NR10102

I B.Tech (NR) Supplementary Examinations, June 2009 MATHEMATICS-I (Common to Civil Engineering, Electrical & Electronic Engineering, Mechanical Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Electronics & Instrumentation Engineering, Bio-Medical Engineering, Information Technology, Electronics & Control Engineering, Computer Science & Systems Engineering and Electronics & Computer Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Test the convergence of the series ³ 2 ´−1 ³ 3 ´−2 ³ 4 ´−3 2 2 3 3 4 4 − + − + − ..... 12 1 23 2 34 3

[5]

(b) Find the interval of convergence of the following series 1 1 1 + 2(1−x) + ..... 2 + 1−x 3(1−x)3

[5]

(c) If 0 < a < b < 1, using Lagranges mean value theorem, prove that √b−a < sin−1 b − sin−1 a < √b−a 1−a2 1−b2

[6]

2. (a) Expand f(x,y) = ey log(1+x) in powers of x and y. ¡ ¢ (b) Show that the evolute of x = a Cosθ + log tan 2θ , y = a Sinθ is the catenary y = a Cos h xa [8+8] 3. (a) Trace the Folium of Decartes : x3 + y3 = 3axy. (b) Determine the volume of the solid generated by revolving the limacon r = a + b cosθ (a>b) about the initial line. 4. (a) Obtain the differential equation of the coaxial circles of the system x2 +y2 +2ax + c2 = 0 where c is a constant and a is a variable. (b) Solve the differential equation: (x2 – 2xy + 3y2 ) dx + (y2 + 6xy – x2 ) dy = 0. (c) Find the orthogonal trajectory of the family of the cardioids r = a ( 1 + cos θ)

[8+8] [3] [7] [6]

5. (a) Solve the differential equation: (D4 − 5D2 + 4)y = 10 cosx. (b) Solve the differential equation: (x2 D2 − x3D + 1)y =

log x sin (log x)+1 x

6. (a) Find the of the rectified semi-wave function defined by ½ Laplace Transformation π Sin ωt 0 < t < ω f (t) = π 0 < t < 2π ω ω h −1

(b) Find L

S 2 +2S−4 (S 2 +9)(S−5)

[8+8]

[5]

i

[6]

(c) Change the order of integration and evaluate

R1 2−x R

xy dx dy

[5]

0 x2

7. (a) Find curl[rf (r)] where r = xi+yj+zk, r = |r|. [6] 2 (b) Find the work done in moving a particle in the force field F=3x i + (2xz − y)j + zk along the straight line from (0, 0, 0) to (2, 1, 3). [10] 8. Verify divergence theorem for F =2x zi + yzj + z2 k over upper half of the sphere x2 +y2 +z2 =a2 . ?????

[16]