Download AKTU B-Tech 3rd Sem 2015-2016 AS 303 Mathematics Iii Question Paper

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Printed Pages: 6                                                                              4 AS-303

(Following Paper ID and Roll No. to be filled in your Answer Book)

Paper ID :199312                                                                              Roll No.

B.Tech.

(SEM. III) THEORY EXAMINATION, 2015-16

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MATHEMATICS-III

[Time:3 hours]                                                                              [Maximum Marks:100]

Note: Attempt all questions from each Section as indicated.

The symbols have their usual meaning.

Section-A

1. Attempt all parts of this section. Each part carries 2 marks. (2×10=20)

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1. Show that w=iz is the rotation of the z-plane through an angle p/2 in the counterclockwise direction.
2. Determine and classify all the singularity of
   1/(z(z-2)⁵) + 1/(z-2)
3. Define Fourier Transform of a function f(x).
4. Find the Z- Transform of {(-1)ⁿ}.

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5. Define Probability density function.
6. What is Karl Pearson's coefficient of skewness.
7. Show that ?² - A = -?A.
8. Define Bisection method.
9. What is cubic spline?

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10. Find missing value in following table:
    

    X	45	50	55	60
    Y	3.1	16		-2.4

Section-B

Attempt any five questions from this section. (5×10=50)

1. (a) Show that the function defined by f(z) = vxy is not regular at origin, although Cauchy-Riemann equations are satisfied.
   (b) Determine the analytic function f(z)=u+iv, in terms of z, whose u - v = e^x (cos y - sin y).

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2. (a) Find inverse Z-Transform of (z-5)^-3, when |z|>5
   (b) Solve the following difference equation using Z-transform u_n+2+2u_n+1+u_n= n, u₀ = u₁ = 0.
3. (a) In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and standard deviation of the distribution. It is given that if f(t) = (1/v2p) ?^t_-8 e^-x²/2dx, when f(0.5)=0.19, and f (1.4)=0.42.
   (b) In a bombing action, there is a 50% chance that any bomb will strike the target. Two direct hits are needed to destroy the target completely. How many bombs are required to be dropped to give a 99% chance of completely destroying the target.
4. (a) Find to four places of decimal, the smallest root the equation e^-x = sin x.
   

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   (b) From the following table find the value of e^0.24.
   

   X	0.1	0.2	0.3	0.4	0.5
   Y	1.10517	1.2214	1.34986	1.49182	1.64872

5. (a) The distance covered an athlete for the 50 meter race is given as:
   

   Time (sec)	0	1	2	3	4	5	6
   Distance (meter)	0	2.5	8.5	15.5	24.5	36.5	50

   Determine speed of the athlete at t=5 sec correct to two decimal.
   (b) Evaluate ?¹₀ dx/(1+x²) using Simpson's 3/8th rule, by taking h=1/6.

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6. (a) Evaluate ?_c (sin(pz)+cos(pz))/((z-1)(z-2)) dz, where C is the circle |z| =4, using Cauchy integral formula.
   (b) Find the Fourier Sine transform of: f(x) = e^-ax, for x > 0 and a > 0. hence show that ?₀⁸ (a sin(ax))/(x²+a²) dx = p/2
7. (a) using Milne's method, solve dy/dx =1+y² with initial conditions. y(0)=0, y(0.2)=0.2027, y(0.4)= 0.4228,y(0.6)=0.6841, find y(0.8).
   (b) Find the value of y (0.6) by Ranga Kutta fourth order method taking h=0.2 for the initial value problem:
8. (a) Six coins are tossed 6400 times. Using the Poisson distribution, determine the probability of getting six heads x times.
   

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   (b) Using Newton's divided difference formula find a polynomial which takes the values 3, 12, 15, -21 when x has the values 3, 2, 1 and -1 respectively.

Section-C

Attempt any two parts of this Section.

1. (a) Apply calculus of residues to evaluate.
   (b) Solve the equation ?u/?x = ?u/?t , X>0,t>0 Subject to the conditions: (i) u=0 when x = 0, (ii) u(x,0)=1 e^-x x<1 (iii) u (x,t) is bounded.
   (c) The first four moments about working mean 28.5 of a distribution are 0.294, 7.144, 42.409, and 454.98. Calculate the moments about mean. Also calculate ß₁ and ß₂ and comment upon the skewness and kurtosis of the distribution.
   

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   (d) Use Gauss-Seidal method to solve the following equations,
   2x+10y+z=51
   10x+y+2z=44
   x+2y+10z=61

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