s201 Mathematics i Set1
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JNTUWORLD Code No: S201 OR I B.Tech.(CCC) Supplimentary Examinations, June 2009 MATHEMATICS-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 Questions All Questions carry equal marks 1. (a) Trace the curve y(x 2 +a 2 )=a 3 . (b) Determine the surface area of the solid generated when laminiscate of Bernoulli r 2 = a 2 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 - π 2 ). [10+10] 3. (a) If u = tan -1 ( x 3 +y 3 x-y ), Prove that x ∂u ∂x + y ∂u ∂y = sin 2u (b) If u = sin (x 2 + y 2 ), where a 2 + x 2 + b 2 y 2 = c 2 , find du dx . (c) If the horse power to propel a steamer varies as the cube of the velocity and square of the length, prove that a 3% increase in velocity and 4% increase in length 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 -6 3 -3 1 2 1 1 1 2 (b) Prove that the matrix A = -2 3 1 3 2 3 2 3 2 3 1 3 1 3 -2 3 2 3 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 function f (x) =1+ 2x π , -π ≤ x ≤ 0 =1 - 2x π , 0 ≤ x ≤ π Deduce that 1 1 2 + 1 3 2 + 1 5 2 + · · ·· = π 2 8 . [10+10] 6. (a) i. Solve (1 + y 2 )+ x - e tan -1y dy dx =0 ii. x 2 ydx - (x 3 + y 3 ) 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] 1 of 2 www.jntuworld.com www.jntuworld.com
Transcript of s201 Mathematics i Set1
JNTUWORLD
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
? ? ? ? ?
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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JNTUWORLD
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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