Deemed University 2011 B.Tech Automobile Engineering University: Lingayas University Term: V Title of the : Applied Mathematics-III - Question Paper

Roll No. ..

 

Lingayas University

B.Tech. 2^nd Year (Term V)

Examination Feb 2011

Applied Mathematics-III (MA - 201)

 

[Time: 3 Hours] [Max. Marks: 100]

 

Before answering the question, candidate should ensure that they have been supplied the correct and complete question paper. No complaint in this regard, will be entertained after examination.

 

Note: Attempt five questions in all. All questions carry equal marks. Question no. 1 is compulsory. Select two questions from Section B and two questions from Section C.

Section A

Q-1. Select the correct answer of the following multiple choice questions. [10x2=20]

(i) The partial differential equation formed from the function is

(a) xp + yp = 0 (b) xp yp = 0

(c) xq + yp = 0 (d) xq yp = 0

(ii) The solution of the differential equation (y z) p + (z x) q = x y is

(a)

(b)

(c)

(d)

(iii) is equal to

(a) (b)

(c) (d)

(iv) The harmonic conjugate of is

(a) (b)

(c) (d)

(v) The invariant point of the transformation is

(a) z = 1 (b) z = - 1

(c) z = i (d) z = 0

(vi) The value of the integral where C is the circle is

(a) 4pie³ (b) 4pie²

(c) 4pie (d) 4pi

(vii) The residue of at its simple pole is

(a) (b)

(c) (d)

(viii) The basic feasible solutions of are

(a) (b)

(c) (d) None of these

(ix) If a sample of size n and mean is drawn from a normal population with standard deviation, then 95% confidence limits for the population mean corresponding to the given sample are

(a) (b)

(c) (d)

(x) The range of t-distribution is

(a) (b)

(c) (d)

Section B

Q-2. (a) Use Charpits method to solve

pxy + pq + qy = yz

(b) A homogeneous rod of conducting material of length has its ends kept of zero temperature and the temperature initially is

= 100 x,

Find the temperature at any time t. [2x10=20]

Q-3. (a) Solve Bessels equation of order n

[10]

(b) State and prove Rodriguezs formula. [2+8]

Q-4. (a) Derive Cauchy-Riemann equations in polar form.

(b) If f(z) is an analytic function of z, show that

[2x10=20]

Section C

Q-5. (a) In a sample of 500 people from a state, 280 take tea and rest take coffee. Can we assume that tea and coffee are equally popular in the state at 5% level of significance?

(b) A group of 10 pigs fed on a diet A and another group of 8 pigs fed on a different diet B, recorded the following increase in weights

Diet A : 5 6 8 1 12 4 3 9 6 10 gms

Diet B : 2 3 6 8 10 1 2 8 gms

Does it show the superiority of diet A order that of B? [2x10=20]

Q-6. (a) Use Cauchy integral formula to evaluate

where c is the circle [10]

(b) Find Laurents series expansion of

for (a) (b) (c)

[3+4+3]

Q-7. (a) Apply calculus of residues to evaluate

(b) Show that the transformation transforms circles with origin at the centre in the z-plane into coaxial concentric confocal ellipses in the w-plane. [2x10=20]

 

Q-8. (a) A firm produces two products A and B on which the profits earned per unit are Rs.3 and Rs.4 respectively. Each product is processed on two machines M₁ and M₂. Product A requires one minute of processing time on M₁ and two minutes on M₂, while B requires one minute on M₁ and one minute on M₂. Machine M₁ is available for not more than 7 hours and 30 minutes, while machine M₂ is available for 10 hours during any working day. Formulate L.P.P. and use graphical method to find the number of units of products A and B to be manufactured to get maximum profit. [10]

(b) A firm manufactures three products A, B and C which are processed on three machines M₁, M₂ and M₃. The relevant data is given below:

 

The profit per unit for products A, B and C is Rs.4, Rs.3 and Rs.6 respectively. Use simplex method to determine the daily number of units to be manufactured for each producer to maximize profit. Assume that all the units produced are consumed in the market.

[10]