R05222101-mathematics-for-aerospace-engineers

Set No. 1

Code No: R05222101

II B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICS FOR AEROSPACE ENGINEERS (Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Evaluate the following using β − Γ functions. (a)

R1

x7 (1 − x)5 dx

0

(b) (c)

π/2 R

0 ∞ R

sin5 θ cos7/2 θ.dθ

x6 (e−3x ) dx.

[5+5+6]

0

2. (a) Determine the analytic function w = u+iv where u = f(0) =1.

2 cos x cosh y cos 2x +cosh 2y

given that

(b) If cosec ( π4 + i α ) = u + iv prove that (u2 + v 2 )2 = 2(u2 − v 2 ). [8+8] R z dz 3. (a) Evaluate (z2e+Q 2 2 where C is | z | = 4 using Cauchy’s integral formula. ) C

(b) Evaluate

R

C

dz z 3 (z +4)

where C is | z | = 2 using Cauchy’s integral formula. [8+8] 2

(2z+1) 4. (a) Determine the poles and the corresponding residues of the function (4z 3 +z) . R πz 2 +cos πz 2 )dz where C is the circle |z| = 3 using residue theorem. (b) Evaluate (sin(z−1) 2 (z−2) C

[8+8]

5. (a) Show that the image of the hyperbola x2 -y2 =1 under the transformation w=1/z is r2 = cos 2θ. (b) Show that the transformation u = into the straight line 4u+3=0.

2z+3 z−4

changes the circle x2 + y2 –4x = 0 [8+8]

6. If (ds)2 = (dr)2 + r2 (dθ)2 + r2 sin2 θ(dφ)2 , find the values of     (a) 22, 1 and 13, 3     3 1 (b) and 13 22

[8+8]

7. (a) Define a random experiment, sample space, event and mutually exclusive events. Give examples of each. (b) Box A contains 5 red and 3 white marbles and box B contains 2 red and 6 white marbles. 1 of 2

Set No. 1

Code No: R05222101

i. If a marble is drawn from each box, what is the probability that they are both of the same colour? [8+8] 8. (a) The mean life time of light bulbs produced by company is 1500 hours with standard deviation of 150 hours. Determine the probability that lighting will take place for i. at least 5000 hours ii. at most 4200 hours if three bulbs are connected such that when one bulb burns out another bulb will go on. Assume that the life times are normally distributed. (b) Find the probability that random samples of 100 bolts, chosen from a lot of 500 bolts having mean weight of 142.30 gms and standard deviation of 8.50 gms, will have a combined weight of i. between 14061 and 14175 gms ii. more than 14460 gms. ⋆⋆⋆⋆⋆

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[8+8]

Set No. 2

Code No: R05222101

II B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICS FOR AEROSPACE ENGINEERS (Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Prove that

R1

(x2 − 1) P ′n dx =

2n(n+1) . (2n+1)(2n+3))

−1

(b) Prove that J3/2 (x) =

q

2 πx

 sin x x

 − cos x .

[8+8]

2. (a) Find the analytic function whose imaginary part is f(x,y) = x3 y – xy3 + xy +x +y where z = x+iy.  2  ∂ ∂2 (b) Prove that ∂x + |Real f (z)|2 = 2|f ′ (z)|2 where w =f(z) is analytic. 2 ∂y 2 [8+8] R z dz 3. (a) Evaluate log where C: | z − 1 | = 1/2, using Cauchy’s integral Formula. (z−1)3 C

(b) State and prove Cauchy’s Theorem.

4. (a) Evaluate

R2π sin2 θ dθ 0

(b) Evaluate

a+ b cos θ

R∞

[8+8]

using residue theorem.

x2 dx (x2 +1) (x2 +4)

using residue theorem.

[8+8]

−∞

5. (a) Prove that the transformation w=sinz maps the families of lines x=constant and y=constant in to two families of confocal conics. (b) Find the bilinear transformation which maps the points (i, -i, 1) of the z-plane into (0,1, ∞) of the w-plane. [8+8] 6. If (ds)2 = (dr)2 + r2 (dθ)2 + r2 sin2 θ(dφ)2 , find the values of     (a) 22, 1 and 13, 3     3 1 and (b) 13 22

[8+8]

7. (a) If A and B are two events and P(A) = 3/5, P(B)=1/2 prove that i. P(AUB) ≥ 53 1 ≤ P (A ∩ B) ≤ 21 . ii. 10 (b) An integer is chosen at random from the first 200 positive integers. What is the probability that the integer chosen is divisible by 6 or 8.

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Set No. 2

Code No: R05222101

(c) There are 3 boxes. Box-I contains 7 Red, 3 Black, and 4 White balls, box -II contains 9 Red, 2 Black, 4 White, box-III 10 Red, 5 Black, 5 white balls. One box is chosen and one ball is drawn from it. What is the probability that the ball is i. Red ii. Black iii. White.

[5+5+6]

8. (a) If x is a Poisson variate such that 3p (x = 4) = 21 p (x = 2) + p (x = 0) find i. the mean of x ii. p(x≤2). (b) Prove that the mean = mode = median for a normal distribution. ⋆⋆⋆⋆⋆

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[8+8]

Set No. 3

Code No: R05222101

II B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICS FOR AEROSPACE ENGINEERS (Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆

1. (a) Evaluate

π/2 R

sin2 θ cos4 θdθ =

0

(b) Prove that

R∞ √

(c) Show that

R∞

2

x e−x dx = 2

0

0

π 32

R∞

using β − Γ functions. 4

x2 e−x dx using β- Γ functions and evaluate.

0 xm−1 dx (x+a)m+n

= a−n β(m, n).

[5+6+5]

2. (a) Define analyticity of a complex function at a point P and in a domain D. Prove that the real and imaginary parts of an analytic function satisfy Cauchy Riemann Equations. 3

3

(1−i) at z 6= 0 and f(0) = 0 (b) Show that the function defined by f (z) = x (1+i)−y x2 +y 2 is continuous and satisfies C-R equations at the origin but f ′ (0) does not exist. [8+8] R where C :| z + 1 + i | = 2. 3. (a) Evaluate using Cauchy’s integral formula z(z+1)dz 2 +2z+4

(b) Evaluate

R

C

−

z dz f rom z = 0 to 4 + 2i along the curve C given by

C

i. z=t2 +it ii. Along the line z=0 to z=2 and then from z=2 to 4+2i.

[8+8]

4. (a) Find the poles and residues at each pole of the function cosec2 z. R zeiz dz (b) Evaluate (z 2 +9)2 where c is the circle |z | = 4 by residue theorem. C

(c) Evaluate

R

C

z 3 dz (z−1)2 (z−3)

where c is | z | = 2 by residue theorem.

[5+5+6]

5. (a) Show that the image of the hyperbola x2 -y2 =1 under the transformation w=1/z is r2 = cos 2θ. (b) Show that the transformation u = into the straight line 4u+3=0.

2z+3 z−4

changes the circle x2 + y2 –4x = 0 [8+8]

6. (a) Prove that metric tensor is a covariant symmetric tensor of order two and conjugate tensor is a contra variant symmetric tensor of order two. (b) Find g and gij corresponding to metric ds2 =

dr2 2 1− r 2 a

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+ r2 (dθ2 + sin2 θdφ2 ).[8+8]

Set No. 3

Code No: R05222101

7. (a) There are 12 cards numbered 1 to 12 in a box. If two cards are selected what is the probability that the sum in odd i. With replacement ii. Without replacement (b) Suppose colored balls are distributed in three indistinguishable boxes as follows: Box-I Red 2 White 3 Blue 5

Box-II 4 1 3

Box-III 3 4 3

A box is selected at random from which a ball is selected at a random. What is the probability that the ball is colored (a) red (b) blue

[6+10]

8. (a) If x is a Poisson variate such that 3p (x = 4) = 12 p (x = 2) + p (x = 0) find i. the mean of x ii. p(x≤2). (b) Prove that the mean = mode = median for a normal distribution. ⋆⋆⋆⋆⋆

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[8+8]

Set No. 4

Code No: R05222101

II B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICS FOR AEROSPACE ENGINEERS (Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆

1. (a) Evaluate

π/2 R

sin2 θ cos4 θdθ =

0

(b) Prove that

R∞ √

(c) Show that

R∞

2

x e−x dx = 2

0

0

π 32

R∞

using β − Γ functions. 4

x2 e−x dx using β- Γ functions and evaluate.

0 xm−1 dx (x+a)m+n

= a−n β(m, n).

[5+6+5]

2. (a) Show that w = z n (n , a positive integer) is analytic and find it’s derivative. (b) If w = f(z) is an analytic function, then prove that the family of curves defined by u(x,y) = constant cuts orthogonally the family of curves v(x,y) = constant. (c) If α + iβ = tanh (x + i π/4) prove that α2 + β 2 =1. [6+5+5] R dz 3. (a) Evaluate (z(z+2) 2 +2z + 5) where C is the circle |z + 1 − i| = 2 using Cauchy’s inteC

gral formula. z=1+i R (x − y + x2 )dz along closed path bounded by y = x2 and y = (b) Evaluate z=0

x.

[8+8]

z 4. (a) Find the poles and residue at each pole zsin 0 ≤ z ≤ π. cos z R (2z+1)dz (b) Evaluate (z+2)z2 where C is | z | = 1 by residue theorem. C

(c) Evaluate

R

C

e−z dz z3

where c is |z | = 1 by residue theorem.

[5+6+5]

5. (a) Discuss the transformation w=cos z. (b) Find the bilinear transformation which maps the points (l, i, -l) into the points (o,1,∞). [8+8] 6. (a) Determine the fundamental and reciprocal tensor in (ds)2 = 3(x1 )2 + 3(x3 )2 + 4x1 x2 + 8x1 x3 + 8x2 x3 . 2

2

2

(b) If ds2 = 5 (dx1 ) + (dx2 ) + 4 (dx3 ) − 6 (dx1 ) (dx2 ) + 4 (dx2 ) (dx3 ) find the values of gij and g ij . [8+8] 7. (a) State and prove Baye’s theorem.

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Set No. 4

Code No: R05222101

(b) In a certain college 25% of boys and 10% of girls are studying mathematics. The girls constitute 60% of the students. If a student is selected at random and is found to be studying mathematics, find the probability that the student is a i. girl ii. boy.

[8+8]

8. (a) Find the variance of the binomial distribution. (b) Determine the probability distribution of the number of bad eggs in a basket containing 6 eggs given that 10% of eggs are bad in a large consignment.[8+8] ⋆⋆⋆⋆⋆

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