07a4bs04-mathematicsforaerospaceengineers

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Code No: 07A4BS04 R07 Set No. 2

II B.Tech II Semester Regular Examinations,May 2010MATHEMATICS FOR AEROSPACE ENGINEERS

Aeronautical EngineeringTime: 3 hours Max Marks: 80

Answer any FIVE QuestionsAll Questions carry equal marks

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1. (a) A random variable X has the following distribution

X=x 0 1 2 3 4 5 6 7 8P(X=x) a 3a 5a 7a 9a 11a 13a 15a 17a

Determine a and find P ( x<3), P( X<3) < P(0< x < 5 ). and the smallestvalue of x for which P(x ≤ x) > 0.5

(b) If X is a normal variate show that E (X) = µ, and Var (X) =σ2

2. (a) If w=u+i ϑ = z3, prove that the curves u=c1, and v=c2 where C1 and C2 areconstants, cut eachother orthogonally.

(b) If u=x2 − y2, ν = − yx2+y2

, then show that both u and v are harmonic but

u+iv is not analytic. [8+8]

3. (a) Evaluate the equation∫c

(z2−z−1)z(z−1)2

dz with c :∣∣z − 1

2

∣∣ = 1 using Cauchy’s inte-

gral formula.

(b) Using Cauchy’s integral formula, evaluate∫c

e2z

(z2+π2)3dz where c is |z| = 4

(c) Evaluate∫ (1,1)

(0,0)(3x2 + 4XY + ix2)dz along y = x2 [5+6+5]

4. (a) Two dice, one green and the other red, are thrown. Let A be the event thatthe sum of the points on the faces is odd and B the event of at least one ace(number 1 on the face of die). Find the probabilities of the events

i. A U (Ac ∩ B)

ii. (A|B)

iii. (Ac|Bc)

iv. A U(Ac U B)

(b) A vendor has 25 gas filled balloons tied to srings. 10 balloons are yellow, 8are red and 7 are green. Find the probability that two balloons selected atrandom by a boy are both yellow. [8+8]

5. (a) Find the expansion of by Taylor’s series about z=1.

(b) Expand f(z) = z(z−1)(2−z) in a Laurent’s series for 1 < |z| < 2

(c) Expand f(z) = z e2z in a Taylor’s series about z=-1. [5+6+5]

6. (a) Show that the transformation w = z + 1/z maps the circle r = c into an ellipse.Discuss the case when c= 1. Draw rough sketches in each case.

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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD II B.TECH II SEM–REGULAR/SUPPLEMENTARY EXAMINATIONS MAY – 2010

[8+8]

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(b) Show that w = (i /4 ) [ (z +2) / (z +1 ) ] transforms the real axis in z -planeto a circle in w -plane. Find the pre-image of the center of such circle. [8+8]

7. (a) Evaluate∫∞

0xm−1

(a+bx)m+ndx using β and Γ s.

(b) Prove that∫ 1

0(1− xn)

1ndx =

[Γ( 1n)]

2

2nΓ( 2n)

(c) Show that

β (m,n) = 2∫ π

2

0sin2m−1 (θ) cos2n−1 (θ) dθ

Deduce that∫ π

2

0sinn (θ) dθ =

∫ π2

0cosn (θ) dθ =

Γ(n+12 )Γ( 1

2)2 Γn+2

2

[5+5+6]

8. Define covariant tensor of order one. Give an example. A covariant tensor has com-ponents 2x - z, x2y , yz in rectangular coordinates. Find its covariant componentsspherical coordinates. [16]

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1. Evaluate the following, using β and Γ Functions

(a)∫∞

0y−3/2 (1− e−y) dy

(b)∫ 1

0xm (log x)n dx where m > -1 and n is a positive integer

(c)∫∞

0e−x

2x7/2 dx. [6+5+5]

2. (a) the components of a tensor are zero in one coordinate system, then prove thatthe components are zero in all coordinate systems.

(b) With the usual notation, prove that {iij}=∂/∂xi(log√

g)

[8+8]

3. (a) Find the image and sketch the mapping of the region 2 ≤ x ≤ 3 and 3 ≤ y ≤ 4.under the transformation w = ez.

(b) Show that a bilinear transformation preserves the cross ratio of four points.[8+8]

4. (a) Find the analytic function whose imaginary part is 2 sinx sin ycos 2x+cosh 2y

(b) If tan [(x+ iy)] = a+ ib, than show that 2a1−a2−b2 = tan [log(x2 + y2)] [8+8]

5. (a) Evaluate∫c

ez

z(1−z)3dz if

i. z=1 lies inside c and z=0 lies outside and

ii. Z=0 and z=1 both lie inside c.

(b) Using Cauchy’s integral formula, evaluate∫c

z3−2z+1z2(z−i)2 dz where c is the circle

|z| = 2 [8+8]

6. (a) Find the poles and residues at each pole of f (z) = 1−ezz4

(b) Evaluate∫c

(z−3)z2+2z+5

dz where C is the circle

i. |z| = 1

ii. |z + 1− i| = 2, by using residue theorem. [6+10]

7. (a) If X is a random variable with distribution function given by,F (x) = 1− eλx for 0 ≤ x ≤ ∞

= 0 otherwiseFind p.d.f of X. Determine the mean and variance of the distribution.

(b) Show that Poisson distribution is a limiting case of binomial distribution.[8+8]

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8. (a) A person takes 4 tests in succession. The probability of his passing the firsttest is p, while that of his passing each succeeding test is p or p/2 accordingas he passes or fails in the preceding test. He qualifies provided he passes atleast three tests. Find the probability of his qualifying.

(b) A consulting firm rents cars from three agencies in the following manner. 20%of cars from agency D, 20% of cars from agency E, 60% of cars from agency F.If 10% of the cars from D, 12% of the cars from E and 4% of the cars from Fhave bad tyres. If a car received by the firm is found to have bad tyres, whatis the probability that the car was supplied by the agency F? [8+8]

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1. (a) For the conformal transformation w = z2 ,show that the circle |z− 1| = 1transforms the cardioid r = 2( 1 + cos θ ) where w = r eiθ in the w- plane .

(b) Find the condition for the transformation w = ( az + b) / ( cz + d ) to makethe circle |w| = 1 correspond to straight line in the z-plane. [8+8]

2. (a) Determine the analytic function f(z)=u+iv where u= 2 cosx cosh ycos 2x+cosh 2y

, given that

f(0)=1

(b) If u is a harmonic function, show that w=u2 is not a harmonic function unlessu is a constant. [8+8]

3. (a) For the function f(z) = 2z3+1z(z+1)

, find Taylor’s series valid in a neighbourhood ofz=1.

(b) Expand f(x) = 1z2−3z+2

in the region.

i. 0 < |z − 1| < 1

ii. 1 < |z| < 2 [8+8]

4. Prove that∫ a

0xJn (αx) Jn (βx) dx =

{0 if α 6= β

a2

2J2n+1(a α) if α = β

Where α and β are the roots of the equation Jn (ax) = 0 [16]

5. (a) Prove that∫c

dzz−a = 2πi where c is given by the equation |z − a| = r

(b) Evaluate∫c

zezdz(z+2)3

where c is |z| = 3 using Cauchy’s integral formula

(c) Evaluate∫c

(ez

z3+ z4

(z+i)2

)dz where c is |z| = 2 using Cauchy’s integral theorem.

[5+5+6]

6. (a) Explain summation convention in tensor analysis Write out in full, the follow-ing

i. aij xi xj ( i, j = 1,2,3 )

ii. gij dxi dxj ( i, j = 1,2,3 )

(b) Define Christoffel symbol of first and second kind. If (ds)2 = (dr)2 +r2 (dθ)2

+ r2sin2 θ ( dϕ)2, then find the value of [22,1]and [1, 22 ] [8+8]

7. (a) Two dice are thrown together. Find the probability that

i. the sum of numbers on their faces is 9

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ii. the numbers on their faces are both odd

iii. the numbers on their faces are same.

(b) A distributor receives 20%, 15%, 35% and 30% of eggs from four poultriesA,B,C,D which contains rotten eggs of 1%, 2%, 2% and 1% in the suppliesfrom A,B,C,D respectively. A randomly chosen egg was found to be rotten.What is the probability that such egg came from the poultry C? [8+8]

8. (a) A continuous random variable X has the p.d.f given by f(x) = K e −b(x−a) fora ≤ x ≤ ∞where a, b, K are constants. Find K, mean and standard deviation in termsof a and b.

(b) If X is a Poisson random variable such that P(X = 1) = 3 /10, P(X = 2) = 1/5. Find P(X = 0) and P(X = 3). [8+8]

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1. (a) A random variable X has the following distribution.

X=x: -2 -1 0 1 2 3p(X=x): 0.1 k 0.2 2k 0.3 k

Find the value of k, and calculate mean and variance.

(b) The mean and standard deviation of intelligence quotient (I Q) of group of500 children is 90 and 20 respectively. If the IQ is normally distributed, findthe number children with IQ

i. greater than 100

ii. less than 60 and

iii. between 80 and 110. [8+8]

2. (a) 500 people were asked about their morning vitamin intake. It was found that150 Take vitamin B, 200 take vitamin C, 165 take vitamin E, 57 take bothB and C, 125 take both B and E, 82 take all three vitamins. What is theprobability that a person takes none of the vitamins?

(b) A bag contains 5 red, 3 blue and 4 black balls. If three balls are drawn atrandom, what is the probability that

i. the three balls are of different colours

ii. two balls are of the same colour

iii. all balls are of same colour. [8+8]

3. (a) Expand 1(z2+1)(z2+2)(z2+3)

in Positive and negative power of z if 1 < |z| <√

2

(b) Expand f(z) = sin z in Taylor’s series about z = π4

and find the region ofconvergence. [8+8]

4. (a) write down the the law of transformation for the tensors

i. Akji

ii. Cmn

(b) Define Christoffel symbol of second kind. If (ds)2 = (dr)2 +r2(dθ)2 + r2sin2 θ( dϕ)2, then find the value of [1 ,22] and [3,13 ] [8+8]

5. (a) Find a and b if f(z) = (x2 − 2xy + ay2) + i (bx2 − y2 + 2xy) is analytic. Hencefind f(z) in terms of z.

(b) Find the general and principal values of

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i. log (1 + i) + log (1− i)ii. (1 + i)i [8+8]

6. (a) Evaluate∫c

e−2zz2

(z−1)3(z+2)dz where c is |z + 2| = 1 using cauchy’s integral formula.

(b) Evaluate∫c

(z3+z2+2z−1)(z−1)3

dz where c is |z| = 3 using Cauchy’s integral formula.

(c) Evaluate∫c

(y2 + 2xy)dx + (x2 − 2xy) dy where c is the boundary of region

bounded by y=x2 and x=y2. [5+5+6]

7. (a) Prove that ex2 (t− 1

t )=∞∑

n=−∞tnJn(x)

(b) Prove that nPn (x) = xP ′n (x)− P ′n−1 (x) [8+8]

8. (a) Show that the transformation w =z + a /z maps the circle in |z| = (a + b)/2z -plane into an ellipse of the semi-axes a, b in the w - plane.

(b) Find the bilinear transformation which maps the points 1, -1,∞ of the z-planeinto 1+i, 1- i , i of the w-plane. Find the invariant points of the transformation.

[8+8]

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