MATHEMATICS-III sem 3rd

Uttarakhand tech. university

b tech

sub:- mathamatics iii

utu previous year question papers

time : 3hr]

Total marks :100

SECTION A

Q1:- Attempt any two of the following :

1. (i) Show that function u = x³ – 3xy² is harmonic and find the corresponding analytic function.
   (ii) If   w = f (z) = u + iv and u - v =  e^x (cosy – siny), find w in terms of z.
2. Evaluate 
                    where C is the circle
   (i) |z| = 2 ,  
3. Apply  calculus of residues to evaluate 
    

SECTION B

Q2:- Attempt any two of the following : 

1. By Newton-Raphson method find the value of (48)^1/3, correct to three decimal places.
2. The table gives the distance in natural miles of the visible horizon for the given heights in feet above the earth's surface.
   
   Find the values of y when x = 218 ft and x = 410 ft.
3. By means of Newton's divided difference formula, find the value of f(8) and f(15) from the following table :
    

SECTION C

Q3:- Attempt any two of the following :

1. Evaluate
                  , by  simpson's 1/3rd rule, using 11 ordinates.
2. Using Runge-kutta method of fourth order, find y for x = 0.1,0.2 and 0.3 given that dy/dx = xy + y^2 , y(0) =1
3. Solve by Crout's method , the following system of equations :
   x + y + z =3 , 2x - y + 3z = 16 , 3x + y -z = -3

SECTION D

Q4:- Attempt any two of the following :

1. Establish the formula δ²_x-y =δ²_x + δ²_y -2rδ_xδ_y where r is the correction coefficient between x and y. Using the above formula calculate the correlation coefficient from the following calculate the correlation coefficient from the following data relating to the marks of 10 candidates in aptitude test (x) and achievement (y) : 
2. The first four moments of a distribution about x = 2 are 1, 2.5, 5.5 and 16. Calculate the first four moments about the mean and about origin.
3. If 10% of the bolts produced by machine are defective, determine the probability that out of 10 bolts chosen at random
   (i) One
   (ii) None
   (iii) at most two bolts, will be defective.

(ONLY FOR CIVIL, ME, IPE, AND AUTOMOBILE ENGG.)

Q5:- Attempt any two of the following :

1. In normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and standard deviation of the distribution.
2. Fit a second degree parabola to the following data by least squares method:
   
3. The following are the mean lengths and ranges of length and ranges of lengths of a finished product from 10 samples each of size 5. The specification limits for length are 200 ± 5 cm. Construct Ᾱ and R-chart and examine whether the process is under control and state your recommendations:
   
   Assume for n=5 , A₂ = 0.58, D₄ = 2.11 and D₃ =0. 
   
   
   
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