# RTU Previous Year Question Papers BE CE

# 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 y_{28} given y_{20}=49225, y_{25}=48316, y_{30}=47236, y_{3} =45926, y_{4O}=44306.

(b) Show that :

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

(ii) p.^{2} = 1 + 5^{2} / 4

**UNIT-II**

(a) Calculate the value of the integral J J

dx+ xUsing Simpson’s 1 /3^{rd} 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 4^{th} 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 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 |

fix) | 3.375 | 7.00 | 13.625 | 24.00 | 38.875 | 59.00 |

**UNIT – III**

3 (a) If a and p are the roots of J_{n}(x) = 0 then prove that ^xJ_{n}(QLx)J_{n}($x)dx =

(b) Prove thatx^{n}J (x) _{n}\ /x^{n}J Ax)n~l^{v} ’10

(ii)ddxx ^{n}J (x) n ^{7}-x-”J_{n+}

**OR**

Prove that :

(i) (2r_{l}^l)xP_{n}(x)^(n^l)P_{n}^(x) + nP_{n}__{l}(x)

(ii) nP_{n}{x) * Xx)~P’_{nl}{x)

Prove that P_{n}(^{x}) is the coefficient of in the expansion of (\-2xh + h^{2})~^^{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_{[t}y) and (x_{2},y_{2}) 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