Download AKTU B-Tech 3rd Sem 2018-2019 Mathematics Iii Ras 301 Question Paper

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Paper Id: 199311

Subject Code: RAS301

Roll No:

B TECH

(SEM III) THEORY EXAMINATION 2018-19

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MATHEMATICS-III

Time: 3 Hours

Total Marks: 70

Notes: Assume any Missing Data.

SECTION-A

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1. Attempt ALL parts of the following: 7 X 2 = 14

1. a) The function f(x) = e^x(cos y + i sin y) is holomorphic or not.
2. b) Find the residue of ^z2/_(z-1)(z-2)² at pole z = 2.
3. c) Formula of Measure of Kurtosis ß₂ =
4. d) The first three central moments of a distribution are 0, 15, -31. Find the moment coefficient of skewness.

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5. e) Obtain the function whose first difference is 9x² + 11x + 5.
6. f) Find the normal equation of a curve y = ax + bx²
7. g) Let f(z) = u(r, ?) + iv(r, ?) be an analytic function. If u = -r³ sin 3?, then find v.

SECTION - B

2. Attempt any THREE parts of the following: 3 X 7 = 21

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1. a) From the following table of values of x and y obtain ^dy/_dx for x = 1.2, 2.2, 1.6.

   x:	1.0	1.2	1.4	1.6	1.8	2.0	2.2
   y:	2.7183	3.3201	4.0552	4.9530	6.0496	7.3891	9.0250

2. b) Using Runga-Kutta method of fourth order, find y(0.8) correct to 4 decimal places if ^dy/_dx = y - ^x2, y(0.6) = 1.7379, taking h = 0.1.
3. c) Using complex integration method, evaluate ?₀^2p cos 2? / (5 + 4 cos?) d?.
4. d) The equations of two regression lines, obtained in a correlation analysis of 60 observations are: 5x - 6y = 24, 768x - 100y = 3608. What is the correlation coefficient? Show that the ratio of coefficient of variability of x to that of y is ⁵/₂₄. What is the ratio of variances of x and y?
5. e) The pressure of the gas corresponding to various volumes V is measured, given by the following data:

   V(cm3)	50	60	70	90	100
   P(kg/cm2)	64.7	51.3	40.5	25.9	7.8

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SECTION -C

3. Attempt any TWO parts of the following: 2 X 3.5 = 07

1. a) Find the unique polynomial P(x) of degree 2 such that: P(1) = 1, P(3) = 27, P(4) = 64, use Lagrange method of interpolation.
2. b) Using Simpson's 3/8^th rule on integration, evaluate ?₁^{2 dx}/_1+x
3. c) Expand ¹/_z²-3z+2 in the region 1 < |z| < 2.

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4. Attempt any TWO parts of the following: 2 X 3.5 = 07

1. a) If the probability of hitting a target is 10% and 10 shots are fired independently. What is the probability that the target will be hit at least once?
2. b) A die is thrown 276 times and the results of these throws are given below:

   No. appeared on the die	1	2	3	4	5	6
   Frequency	40	32	29	59	57	59

   Test whether the die is biased or not. [Tabulated value of ?² at 5% level of significance for 5 degree of freedom is 11.09]
3. c) By Residue method, find the inverse Z-transform of ^z/_z²+7z+10

5. Attempt any TWO parts of the following: 2 X 3.5 = 07

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1. a) The following data regarding the heights (y) and weights (x) of 100 college students are given: Sx = 15000, Sx² = 2272500, Sy = 6800, Sy² = 463025, Sxy = 1022250
2. b) Solve x³-5x+3=0 by using Regula-Falsi method correct up to four decimal places.
3. c) From the table, estimate the number of students who obtained marks between 40 and 45.

   Marks:	30-40	40-50	50-60	60-70	70-80
   No.of Students:	31	42	51	35	31

6. Attempt any TWO parts of the following: 2 X 3.5 = 07

1. a) Find the residue of f(z) = ^z3/_{(z-1)⁴(z+2)} at its pole and hence evaluate ?_c f(z) dz if c is the circle |z| = 5/2

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2. b) Determine the largest Eigen value and corresponding eigen vector of the matrix A = -1 & 2 & -1 \\ 0 & -1 & 2 \\ 0 & -1 & 2 till three approximation.
3. c) Verify Cauchy theorem by integrating e^iz along the boundary of the triangle with the vertices at the points 1 + i, -1 + i and -1 - i.

7. Attempt any TWO parts of the following: 2 X 3.5 = 07

1. a) Use Picard's method to obtain y for x = 0.2. Given: ^dy/_dx = x - y with initial condition y = 1 when x = 0 correct up to four decimal places.
2. b) 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/_v(2p) ?₀^t e^-x2/2 dx then f(0.5) = 0.19, f(1.4) = 0.42

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3. c) Prove that hD = log(1+?) = -log(1-?) = sinh^-1(µd)

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