# 4th Sem Mathematics-4 July 2011

UNIT-I

1 (a) Find f(4) from the following table :

 X 0 1 2 5 fix) 2 5 7 8

(b) From the following table, find the number of students who obtain less than 45 marks ?

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

OR

1 (a) Use Stirling’s formula to find y28 given y20=49225, y25=48316, y30=47236, y3 =45926, y4O=44306.

(b) Show that :

(i) (1 +A)(l-V) = I

(ii) p.2 = 1 + 52 / 4

UNIT-II

(a) Calculate the value of the integral J J

dx+ xUsing Simpson’s 1 /3rd rule by dividing the interval (2,10) into eight equal parts upto 4 decimal places.

(b) Given “ = v – x with v(0) = 2 . find >'(0.1) correct to 4 decimal places using Runge-Kutta 4th order method.

OR

(a) Let ^=TTT^> with boundary conditions v = 1 when

# = 0. Find approximately y for x – 0.1 by Euler’s modified method (3 steps).

(b) Find f(1.5) and f'(1.5) from the following table :

 X 1.5 2 2.5 3 3.5 4 fix) 3.375 7 13.625 24 38.875 59

UNIT – III

3 (a) If a and p are the roots of Jn(x) = 0 then prove that ^xJn(QLx)Jn(\$x)dx =

(b) Prove thatxnJ (x) n\ /xnJ Ax)n~lv ’10

(ii)ddxx nJ (x) n 7-x-”Jn+

OR

Prove that :

(i) (2rl^l)xPn(x)^(n^l)Pn^(x) + nPn_l(x)

(ii) nPn{x) * Xx)~P’nl{x)

Prove that Pn(x) is the coefficient of in the expansion of (\-2xh + h2)~^t hi ascending powers of h.

UNIT – IV

Show that the angle 0, between the two lines of regression is given by

Also interpret the cases when r = 0,± 1.

Two random variables have the least square regression lines with equations 3x + 2y-26 = 0 and 6x+>>-31 = 0. Find coefficient of correlation between x and y.

Obtain the rank correlation coefficient for the following data :

 X 85 74 85 50 65 78 74 60 74 90 y 78 91 78 58 60 72 80 55 68 70

Write statement of Bay’s theorem.

Define normal probability distribution. If the mean of a normal distribution is ft and its variance , find its moment generating function.

(c)  In a bolt manufacturer factory, machine A, B and C manufacture 25%, 35% and 40% of the total product respectively. Of their output 5%, 4% and 2% are defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by machine B ?

UNIT-V

5 (a) Find the curve through two points (x[ty) and (x2,y2) which when rotated about the x-axis, given minimum surface area.

(b) Find the externals of the functional

Vi , I[y(x),z(x)]= j [y’2 + z’2+2yzjdx where

V(0) = 0, yf|J = l;z(0) = 0 and z(f) =1

OR

(a) Find the path on which a particle, in the absence of friction, will slide down from one fixed point to another fixed point in the shortest time.

(b) Find the external of the functional