# Engineering Mathematics-III-Engineering-Pune University-November2019

**S.E. (Comp. & IT) (Second Semester) EXAMINATION, 2019**

**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) Use of electronic pocket calculator is allowed.

(iv) Assume suitable data, if necessary.

__________________________________________________________________________________________________________________________________________

1. (a) Solve any two differential equations : [8]

(i)

(ii)

(iii)

, by using the method of variation

of parameters.

(b) Solve the integral equation :** [4]**

**Or**

2. (a) A capacitor of 10^{–3} farads and inductor of (0.4) henries are connected in series with an applied emf 20 volts in an electrical circuit. Find the current and charge at any time t. **[4]**

(b) Solve any one of the following : **[4]**

(i) Obtain z[ke^{-k} ], k ≥0

(ii) Obtain

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

3. (a) The first three moments of a distribution about the value 2 are 1, 16 and –40. Find the first three central moments, standard deviation and β_{1}. **[4]**

(b) Fit a straight line of the form X = aY + b to the following data by the least square method : **[4]**

X | Y |

2 | 2 |

5 | 3 |

8 | 4 |

11 | 5 |

17 | 7 |

20 | 8 |

(c) On an average, there are 2 printing mistakes on a page of

a book. Using Poisson distribution, find the probability that

a randomly selected page from the book has at least one printing

mistake. **[4]**

**Or**

4. (a) 200 students appeared for an examination. Average marks were 50% with standard deviation 5%. How many students are expected to score at least 60% marks assuming that marks are normally distributed. [Given : Z = 2, A = 0.4772]. **[4]**

(b) On an average, a box containing 10 articles is likely to have 2 defectives. If we consider a consignment of 100 boxes, how many of them are expected to have at the most one defective? **[4]**

(c) Find the regression equation of Y on X for bivariate data with the following details.

**[4]**

5. (a) Find the directional dervative of Φ(x,y,z)=xy^{3}+yz^{3 }at the point (2, –1, 1) in the direction of vector is irrotational. Hence find the scalar potential Φ such that . **[4]**

(c) Evaluate where k and C is the boundary of the rectangle 0 ≤ x ≤π and 0 ≤ y ≤ 1 and z = 3. **[5]**

**Or**

6. (a) Show that (any one) : **[4]**

(i)

(ii)

(b) Find the directional derivative of at (1, –1, 1)

towards the point (2, 1, –1). **[4]**

(c) If : ** [5]**

Evaluate :

where C is the curve x = t, y = t^{2}, z = t^{3} joining the points (0, 0, 0) and (1, 1, 1).

7. (a) Determine the analytic function f(z) = u + iv if u =4xy – 3x + 2. **[4]**

(b) Find the bilinear tranformation which maps the point z = i, –1, 1 into the point ω = 0, 1 ,∞. **[4]**

(c) Evaluate :

where C is the circle |z-1|=3/2 ** [5]**

**Or**

8. (a) Determine the analytic function f(z) = u + iv if u = x^{2}– y^{2} – 2xy – 2x – y – 1. **[4]**

(b) Under the transformation ω=1/z , find the image of |z – 3i|= 3. **[4]**

(c) Evaluate : **[5]**

where C is the circle |z|=3/2