(iii) Two variables x and y have zero means, the same variance 2σ and zero correlation, show that u = xcosα + ysinα, v = xsinα –

ycosα have the same variance 2σ and zero correlation. 4. Attempt any Two parts from the following [2 x 5 = 10] (i) The income of a group of 10, 000 persons was found to be normally distributed with mean Rs 750 p.m/ and standard deviation of Rs 50. Show that of this group about 95% had income exceeding Rs 668 and 5% had income exceeding Rs 832. Also find the lowest income among the richest 100.

(ii) By using 2χ test , find whether there is any association between income level and type of schooling. Income Public School Govt. School Low 200 400 High 1000 400

(Given that degree of freedom 84.3205.0 =χ )

(iii)The number of defective in 20 samples each of 2000 items are given below 425 430 216 341 225 322 280 306 337 305 356 402 216 264 126 409 193 280 326 389 Calculate the value of the central line UCL and LCL.

No. of Pages – 4 EAS - 401

B.Tech.

(SEM. IV) EVEN SEMESTER THEORY EXAMINATION 2013 -14

MATHEMATICS – III Time: 3 Hours Total marks: 100 Note: (1) Attempt ALL questions (2) Provide Graph paper, Chi – square table and Normal table. 1. Attempt any Four parts of the following [4 x 5 = 20] (i) If ψφ iw += represents the complex potential for an electric field and 22

22

yxxyx+

+−=ψ , determine the function φ

(ii) Evaluate the line integral ∫C

dzz 2 where C is the boundary of a triangle with vertices 0, 1 + i, -1 + i clockwise.

(iii) Evaluate ( ) ( )dz

zzzz

C∫

++−

412

22

2

where C is the circle 11 =−z

(iv) Obtain the Taylor or Laurent series which represent the

function ( )( )211)( 2 ++

=zz

zf when 21 << z

(v) Evaluate ∫−+

Cdz

zzz

12

2

2

where C is the circle 11 =−z

(vi) Show by the method of residue, that ∫

+=

+

π πθ

θ0 222 1sin ad

aa

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2. Attempt any Four parts of the following: [4 x 5 = 20]

(i) Find a positive real root of 01cos3 =−− xx using the method of false position correct to four decimal places.

(ii) Prove that (a) ( ) ( ) 2/12/1 11 −− ∇−∇=∆+∆ (b) ( )( ) ∆+=∆++ − 21 2/12/12/1 EE (iii) Determine the missing term from the following table x 0 5 10 15 20 25 y 6 10 - 17 - 31

(iv) Use Newton’s Forward Interpolation formula to otain the interpolating polynomial f(x) satisfying the following table X 1 2 3 4 f(x) 26 18 4 1 If another point x = 5, f(x) = 26 is added to the above data, will the interpolating polynomial be same as before or different. Explain why? (v)Use Newton’s Divided Difference method to compute f(5.5) from the table x 0 1 4 5 6 f(x) 1 14 15 6 3 3. Attempt any Two parts from the following [2 x 5 = 10] (i) Solve by Guass – Seidel Method 6x – y – z = 19, 3x + 4y + z = 26, x + 2y + 6z = 22

(ii) (a) The distance covered by an athelete for the 50m race is given by the table Time(s) 0 1 2 3 4 5 6 Dist (m) 0 2.5 8.5 15.5 24.5 36.5 50 Determine the speed of the athelete at t = 5 sec correct to two decimal places. (b) The velocity v of a particle at a distance s from a point on its path is given by the table below s (in m) 0 10 20 30 40 50 60 v(in m/sec) 47 58 64 65 64 52 38 Estimate the time taken to travel 60 metre by usng Simpson’s 1/3rd Rule.

(iii) If 2yxdxdy

+= , we use Runge Kutta Method of fourth order to find an approximate value of y for x = 0.2 given that y = 1 when x = 0 (Take h = 0.1) 4. Attempt any Two parts from the following [2 x 5 = 10] (i) The first two moments about the working mean 28.5 of a distribution are 0.294, 7.144, 42.409, 454.98. Calculate the moments about the mean. Also evaluate 21 , ββ and comment upon the skewness and kurtosis of the distribution.

(ii) Use the method of least square to fit the curve xcx

cy 1

0 +=

to the following data y 0.1 0.2 0.4 0.5 1 2 x 21 11 7 6 5 6

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