Engineering Mathematics-III-Engineering-Pune University-November2019
S.E. (Comp. & IT) (Second Semester) EXAMINATION, 2019
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 : 
, by using the method of variation
(b) Solve the integral equation : 
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. 
(b) Solve any one of the following : 
(i) Obtain z[ke-k ], k ≥0
(c) Solve the difference equation : 
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 β1. 
(b) Fit a straight line of the form X = aY + b to the following data by the least square method : 
(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
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]. 
(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? 
(c) Find the regression equation of Y on X for bivariate data with the following details.
5. (a) Find the directional dervative of Φ(x,y,z)=xy3+yz3 at the point (2, –1, 1) in the direction of vector is irrotational. Hence find the scalar potential Φ such that . 
(c) Evaluate where k and C is the boundary of the rectangle 0 ≤ x ≤π and 0 ≤ y ≤ 1 and z = 3. 
6. (a) Show that (any one) : 
(b) Find the directional derivative of at (1, –1, 1)
towards the point (2, 1, –1). 
(c) If : 
where C is the curve x = t, y = t2, z = t3 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. 
(b) Find the bilinear tranformation which maps the point z = i, –1, 1 into the point ω = 0, 1 ,∞. 
(c) Evaluate :
where C is the circle |z-1|=3/2 
8. (a) Determine the analytic function f(z) = u + iv if u = x2– y2 – 2xy – 2x – y – 1. 
(b) Under the transformation ω=1/z , find the image of |z – 3i|= 3. 
(c) Evaluate : 
where C is the circle |z|=3/2