AMIETE – ET/CS/IT (NEW SCHEME)   –   Code: AE56/AC56/AT56

Subject: ENGINEERING MATHEMATICS - II

Time: 3 Hours                                                                                                     Max. Marks: 100

NOTE: There are 9 Questions in all.

· Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1       Choose the correct or the best alternative in the following:                                  (210)

             a.  Image of   under the mapping w = 1/z is 

(A) 2v+1=0                                          (B) 2v-1=0                                                                       

                  (C) 2u+1=0                                          (D) 2u-1=0                                                          

             b. The value of  is equal to 

(A) -1                                                  (B) 1

(C) 2                                                    (D) 0

             c.  is equal to

(A) 0                                                    (B) -1

(C) (D)

             d.  If  at (1,-2,-1) is  equal to

(A) -(12i+9j+16k)                               (B) (12i+5j+8k)

(C) -(12i-5j+8k)                                  (D) -(12i+5j-8k)

             e.  If  is such that then is called

(A) irrotational                                      (B) solenoidal

(C) rotational                                        (D) none of these

             f.   The solution of the partial differential equation is

(A) (B)

(C) (D)

             g. If and 0 elsewhere, is a  p.d.f. then the value of  k is equal to

(A)  4                                                   (B) 2

(C)  3                                                   (D)  1

             h.  If X is a binomial variate with p = 1/5, for the experiment of 50 trials, then the standard deviation is equal to

(A) 6                                                    (B) -8

(C) 8                                                    (D)

             i.   The value of

(A) (B)

(C) (D) None of these

             j.   The value of by Simpson’s 1/3^rd rule (taking n = ¼)   is equal to

(A) -0.7845                                         (B) 0.7854                                                                        

                  (C) 0.8745                                           (D) 0

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

  Q.2     a.   Show that the function is harmonic. Find its conjugate harmonic function u(x,y) and the corresponding analytic function f(z).      (8)

b.   Show that the bilinear transformation  transform the real axis in the z plane onto a circle in the w plane. Find the center and radius of the circle in the w plane. Find the point  in the z plane which is  mapped onto the centre of the circle in the w plane.                                            (8)

Q.3     a.   Evaluate the integral (8)

b.   Obtain the first three terms of the Laurent series expansion of the function about the point z = 0 valid in the region (8)

Q.4     a.   Show that vector field is conservative. Find its scalar potential and the work done in moving a particle from (-1,2,1) to (2,3,4)                                                      (8)

             b.   Find the values of constants  and  so that the surfaces  intersect orthogonally at the point (1,-1,2).         (8)

Q.5    a.   The cylinderintersect the sphere . Find the surface area of the portion of the sphere cut by the cylinder above the yz plane and within the cylinder.                                  (8)

             b.   Use the Divergence theorem to evaluate  taken over the sphere  and l, m, n are the direction cosines of the external normal to the sphere.                                                                                                                           (8)

Q.6     a.   Solve by method of separation of variables (8)

             b.   Solve (8)

Q.7      a.         The following are data from the steam table:

Temp C(t)	140	150	160	170	180
Pressure kgf/cm^2 (P)	3.685	4.854	6.302	8.076	10.225

                   Using Newton’s formula, find the pressure of steam for temperature 142and 175.                     (8)

            b.  A curve is drawn to pass through the following points:

x	1	1.5	2	2.5	3	3.5	4
y	2	2.4	2.7	2.8	3	2.6	2.1

                   Estimate the area bounded by the curve, x-axis and lines x = 1, x = 4. Also find the volume of solid generated by revolving this area using Simpson’s 3/8 rule.                                                               (8)

  Q.8     a.   The probability that a man aged 60 will live to be 70 is 0.65. What is the probability that out of 10 men, now 60, at least 7 will live to be 70?            (8)

             b.   X is a continuous random variable with probability density function given by

                   Find k and mean value of X.                                                                              (8)

Q.9     a.   X is a continuous random variable with probability density function given by

Find the standard deviation and also the mean deviation about the mean.             (8)

             b.   A car hire firm has two cars which it hires out day by day. The number of demand for a car on each day is distributed as a Poisson Distribution with mean 1.5.  Calculate the proportion of days on which car is not used and the proportion of days on which some demand is refused. (given that )                                                                  (8)