# Engineering Mathematics-III-Engineering-Pune University-November2017

**S.E. (COMP/IT) (II Semester) EXAMINATION, 2017**

**ENGINEERING MATHEMATICS–III**

**(2015 PATTERN)**

**Time: Two Hours Maximum Marks: 50**

N.B.:— ( i) Neat diagrams must be drawn wherever necessary.

(ii) Figures to the right indicate full marks.

(iii) Your answers will be valued as a whole.

(iv) Use of an electronic pocket calculator is allowed.

(v) Assume suitable data, if necessary.

_______________________________________________________________________________________________________________________________________________

Q1(a).Solve any two: **[8]
**(i)

(ii)(D

^{2}+4D+4)Y=x

^{-3}e

^{-2x }(iii)

(b)Find the Fourier transform of:

f(x)=1, |x|≤1

=0, |x|>1

and evaluate

**Or**

Q2a)An inductor of 0.5 henries is connected in series with a register of 6 ohms, a capacitor of 0.02 farads, a generator having alternative voltage given by 24sin 10t, t>0, and switch k. Set up a differential equation for this circuit and find charge at time t. **[4]
**b)Solve any one of the following:

**[4]**

(i)Find z{f(k)},where f(k)=3

^{k},k<0

=2

^{k}k≥0

(ii)Find:

by using inversion integral method.

(c)Solve the following difference equation:**[4]**

y(k+2)-5y(k+1)+6y(k)=36

y(0)=y(1)=0

Q3a)Calculate thr=e first four central moments from the following data and hence β1 and β2:

x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

f | 5 | 15 | 17 | 25 | 19 | 14 | 5 |

(b)Fit a straight line to the following data by least square method:

x | 0 | 5 | 10 | 15 | 20 | 25 |

y | 12 | 15 | 17 | 22 | 24 | 30 |

(c)The number of breakdowns of computer in a week is a Poisson variable with λ=np=0.3.What is the probability that the computer will operate:

(i)with no breakdown and

(ii)at the most one breakdown in a week.

**Or**

Q4a)The average test mark in a particular class is 79 and the standard deviation is 5.If the marks are normally distributed, how many students in a class of 200, did not receive marks between 75 and 82.Given z=0.8,Area=0.2881 andz=0.6,Area=0.2257 **[4]
**b)An insurance agent accepts policies of 5men of identical age and in good health. The probability that a man of this age will be alive 30 years hence is 2/3. Find the probability that in 30 years:

**[4]**

i)all five men and

ii)at least one man will be alive.

c)The two variables x and y have regression lines:

**[4]**

3x+2y-26=0 and 6x+y-31=0

Find:

i)the mean values of x and y and

ii)correlation coefficient between x and y.

Q5a)Find the directional derivative of a scalar point function at (2,-1,1) in the direction of a vector 4i+2j+4k. **[4]
**b)Show that the vector field :

**[4]**

is irrotational and hence find a scalar potential function Φ such that

**Or**

Q6a)Find the directional derivative of a scalar point function at (1,0,2) in the direction of 4i-j+2k. **[4]
**b)Show that (any one):

I) ,where is a constant vector.

ii)

c)Evaluate the integral ,along the curve x=2t,y=t,z=3t from t=0 to t=1 ,where

**[5]**

Q7a)If:

find v such that f(z)=u+iv is analytic.Determine f(z) in terms of z. **[4]
**b)Evaluate where c is the contour |z+1|=1/2.

c)Find the Bilinear transformation which maps the point -i,0,2+i of the z-plane onto the points 0,-2i,4 of the w-plane.

**[4]**

**Or**

Q8a)If:

find v such that f(z)=u+iv is analytic.Determine f(z) in terms of z. **[4]
**b)Evaluate ,where c is the circle |z|=4

**[5]**

c)Find the image of the circle (x-3)

^{2}+y

^{2}=2 under the transformation w=1/z

**[4]**