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Code No: S201 OR

I B.Tech.(CCC) Supplimentary Examinations, June 2009MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronic Engineering,Mechanical Engineering, Electronics & Communication Engineering and

Computer Science & Engineering)Time: 3 hours Max Marks: 100

Answer any FIVE QuestionsAll Questions carry equal marks

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1. (a) Trace the curve y(x2+a2)=a3.

(b) Determine the surface area of the solid generated when laminiscate of Bernoullir2 = a2 cos 2θ revolves about the initial line. [10+10]

2. (a) Find the C.G of the arc of the astroid lying in the first quadrant.

(b) Expand sinx in powers of (x− π2). [10+10]

3. (a) If u = tan−1 (x3+y3

x−y), Prove that x∂u

∂x+ y ∂u

∂y= sin 2u

(b) If u = sin (x2 + y2), where a2 + x2 + b2y2 = c2, find dudx

.

(c) If the horse power to propel a steamer varies as the cube of the velocity andsquare of the length, prove that a 3% increase in velocity and 4% increase inlength will require an increase of about 17% in horse power. [7+7+6]

4. (a) Determine the rank of the matrix A =

2 1 −3 −63 −3 1 21 1 1 2

(b) Prove that the matrix A =

−23

13

23

23

23

13

13

−23

23

is orthogonal.

(c) Show that every square matrix can be uniquely expressed as a sum of a sym-metric and a skew symmetric. [7+7+6]

5. (a) Obtain Fourier series expansion of the function f(x) = x sin x in the interval−π < x < π.

(b) Find the Fourier series for the functionf(x) = 1 + 2x

π, −π ≤ x ≤ 0

= 1− 2xπ

, 0 ≤ x ≤ π

Deduce that 112 + 1

32 + 152 + · · ·· = π2

8. [10+10]

6. (a) i. Solve (1 + y2) +(x− etan−1y

)dydx

= 0

ii. x2ydx− (x3 + y3) dy = 0.

(b) Find the orthogonal trajectories of the family of cardioids r=a(1-cos θ) when‘a’ is the parameter. [7+6+7]

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Code No: S201 OR

7. (a) solve (D3 − 3D2 + 3D − 1) y = (x + 1)ex + cos x

(b) The differential equation of a simple pendulum is d2xdt2

+w20 x = δ0 sin nt, where

w0 and F0 are constants. If initially x=0, dxdt

= 0, determine the motion whenwo=n. [10+10]

8. (a) Find the value of Γ(12).

(b) Find L(e−2t cos2 t).

(c) Find L−1[

5s−13s2−6s+15

]. [6+6+8]

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