7/29/2019 Engg Mathematics - 4 June 2012 NEW

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USN I I I I I I I I I I I 10MAT41Fourth Semester B.E. Degree Examination, June 2012

Engineering Mathematics - IVTime: 3 hrs. Max. Marks: I00

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Note: Answer FIVE full questions, selectingat least TWO questions from each part.

PART-Aa. Using the Taylor's method, find the third order approximate solution at x = 0.4 of the

problem dy =x 2y +I , with y(O) =0 . Consider terms upto fourth degree. (06 Marks)dxb. Solve the differential equation dy = -xy " under the initial condition y(O) =2, by using thedxmodified Euler's method, at the points x = 0.1 and x =0.2. Take the step size h = 0. 1 and

carry out two modifications at each step. (07 Marks)c. G iven dy = xy + y" ; y(O) = I, y(O. I) = 1.1169, y(0.2) = 1.2773 , y(0.3) = 1.5049, find y(0.4)dx

co tTect to three decimal places, using the Milne's predictor-con-ector method. Apply theconector formula twice. (07 Marks)a. Employing the Picard's method, obtain the second order approximate so lution of the

following problem at x =0.2.dy dz- = x + yz - = y + zx dx ' dx ' y(O) = I, z(O) =-I . (06 Marks)b. Using the Runge-Kutta method, solve the following differentiJl equoti::m at x =0.1 under the

given condition:d 2y "'( dy)= X y+-dx 2 dx ' y(O) = I , y'(O) =0.5.

Take step length h =0.1. (07 Marks)c. Using the Milne's method , obtain an approximate solution at the point x = 0.4 of the

d 2 dproblem ~ + 3x 2 .- 6y =0, y(O) = I , y'(O) = 0.1. Given y(O .I) = 1.03995,dx - dxy'(O.I) =0.6955, y(0.2) = 1.138036, y'(0.2) = 1.258, y(0 .3) = 1.29865, y'(0.3) = 1.873.(07 Marks)a. Derive Cauchy-Riemann equations in polar form. (06 Marks)b. I f f(z) is a regular function of z. prove that (! ', + : , ) If (zJI' =4lf'( zJI' . (07 Marks)

1 1 Xc. If w =$ + iy represents the complex potential for an electric field and y = x- - y- + " ,x - + y-determine the function$. Also find the complex potential as a function of z. (07 Marks)

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7/29/2019 Engg Mathematics - 4 June 2012 NEW

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a. Discuss the transformation of w = z + k 2 .z

10MAT41

(06 Ma1ks)b. Find the bilinea r transformation that transforms the points z 1 = i, z2 = I , z3 = -I on to theC.

a.b.c.

po int s w 1 = I, w 2 = , w3 = xJ respectively. (07 Marks)0 , )

E I Jsm nz - +co snz - dva ua te , z where cis the circle lzl =3, using Cauchy's integral fo rmula.c (z - lt ( z -2 )

PART-BObtain the so lution of x 2y" + xy ' + (x 2 - n 2 )y = 0 in terms of J11(x) and ] _1(x).Express f(x) =x4 + 3x3 - x2 +5x - 2 in terms of Legendre polynomials.

+I 2Prove that JP111 (x) P11 (x)dx =- - , n1 =n._1 2n + I

(07 Ma rk s)

(06 Marks)(07 Marks)(07 Ma rks)

a. From five posit ive and seven nega t ive numbers, five numbers are cho sen at rando m andmultiplied. What is the probability that the product is a ( i) nega tive number and ( ii) pos it ivenumber? (06 Marks)

b. If A and Bare two events with P(A) = ~ P(B) = ~ P(A n B) = find P(A/B) , P(B/A) .- - - - -P(A / B) , P(B / A) and P(A / B) . (07 Ma rks)

c. In a certain co llege , 4% of boy students and I% of girl students are ta ller than 1.8 m.

a.

Furthermore, 60% of the students are girls. If a student is selected at random and is foundta ller than 1.8 m, what is the probability that the student is a g irl? (07 Marks)

d h h d . f . {Kx 2, 0 S: X S: 3A ran om variab le x as t e ens1ty unct1on P(x) = . Evaluate K. and0, elsewherefind: i) P( x S: I), (ii) P( I S: x S: 2), (iii) P(x S: 2) , iv) P(x > I), (v) P(x > 2). (06 Marks)

b. Obtain the mean and standard dev iat ion of binomial distributio n. (07 Ma ks)C. In an exam ination 7% of students score less than 35 % marks and 89 % of students score less

than 60% marks. Find the mean and standard dev iat ion if the marks a re nor mally di stributed.It is given that P(O < z < 1.2263) = 0.39 and P(O < z < 1.4757) =0.43 . (07 Ma rks)

8 a. A random sample of 400 items chose n from an infinite population is found to have a meanof 82 and a standard deviat ion of 18. Find the 95 % confidence limits for the mea n of thepopulation from which the sa mple is drawn. (06 Marks)b. In the past, a machine has produced washers hav ing a th ickness of 0.50 mm . To determinewhether the machine is in proper working order, a sample of I0 washers is chosen fo r whic hthe mean thickness is found as 0.53 mm wi th standard dev iat ion 0.03 mm. Tes t thehypothesis that the machine is in proper working order , using a level of significance of(i) 0.05 and (ii) 0.0 1. (07 Marks)

c. Genetic theory s tates that children having o ne parent of blood type M and the other of bloodtype N wi ll a lways be one of the three types M, MN , N and tha t the proport ions of thesetypes will on an average be 1 : 2 : I. A report states that out of 300 ch ildren ha ving one Mparent and one N parent, 30 % were found to be of type M , 45 % of type MN and there mainder of type N. Test the theo ry by X2 (Chi square) test. (07 Ma rks)

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