Paper II PYQs-2018

1.(a) Let R be an integral domain with unit element. Show that any unit in R(x) is a unit in R.

[10M]

1.(b) Prove the inequality: π29<∫π2π6xsinxdx<2π29

[10M]

1.(c) Prove that the function: u(z,y)=(x−1)3−3xy2+3y2 is harmonic and find its harmonic conjugate and the corresponding analytic function f(z) in terms of z.

[10M]

1.(d) Find the range of p(>0) for which the series:

1(1+a)p−1(2+a)p+1(3+a)p−....,a>0, is

(i) absolutely convergent and (ii) conditionally convergent.

[10M]

1.(e) An agricultural firm has 180 tons of nitrogen fertilizer, 250 tons of phosphate and 220 tons of potash.It is able to sell a mixture of these substances in their respective ratio 3:3:4 at a profit of Rs. 1500 per ton and a mixture in the ratio 2:4:2 at a profit of Rs. 1200 per ton. Pose a linear programming problem to show how many tons of these two mixtures should be prepared to obtain the maximum profit.

[10M]

2.(a) Show that the quotiet group of (R,+) modulo Z is isomorphic to the multiplicative group of complex numbers on the unit circle in the complex numbers on the unit circle in the complex plane. Here R is the set of real numbers and Z is the set of integers.

[15M]

2.(b) Solve the following linear programming problem by Big M-method and show that the problem has finite optimal solutions. Also, find the value of the objective function: Minimize z=3x1+5x2 subject to x1+2x2≥8
3x1+2x2≥12,
5x1+6x2≥60,
x1,x2≥0

[20M]

2.(c) Show that if a function f defined on the open interval (a,b) of R is convex, then f is continuous. Show by example, if the condition of open interval is dropped, then the convex function need not be continuous.

[15M]

3.(a) Find all the proper subgroups of the multiplicative group of the field (Z13,+13,×13), where +13 and ×13 represent addition modulo 13 and multiplication modulo 13 respectively.

[20M]

3.(b) Show by applying the residue theorem that ∫∞0dx(x2+a2)2=π4a3, a>0.

[15M]

3.(c) How many basic solutions are there in the following linearly independent set of equations? Find all of them.

2x1−x2+3x3+x4=6 4x1−2x2−x3+2x4=10

[15M]`

4.(a) Suppose R be the set of all real numbers and f:R→R is a function such that the following equations hold for all x, y∈R:

f(x+y)=f(x)+f(y)
f(xy)=f(x)f(y)

Show that that ∀c∈R, either f(x)=0 or f(x)=x.

[20M]

4.(b) Find the Laurent’s series which represent the function 1(1+z2)(z+2) when

(i) |z|<1 (ii) 1<|z|<2 (iii) |z|>2

[15M]

4.(c)

In a factory there are five operator O1, O2, O3, O4, O5 and five machine M1, M2, M3, M4, M5. The operating costs are given when the Oi operator operates the Mj machine (i,j=1,2,..,5). But there is a restriction that O3 cannot be allowed to operate the third machine M3 and O2 cannot be allowed to operate the fifth machine M5. The cost matrix is given above. Find the optional assignment and the optimal assignment cost also.

[15M]

5.(a) Find the partial differential equation of the family of all tangent planes to the ellipsoid: x2+4y2+4z2=4, which are not prependicular the the xy plane.

[10M]

5.(b) Using Newton’s forward difference formula find the lowest degree polynomial ux when it is given that u1=1, u2=9, u3=25, u4=55,u5=105.

[10M]

5.(c) For an incompressible fluid flow, two components of velocity (u,v,w) are given by u=x2+2y2+3z2, v=x2y−y2z+zx. Determine the third component w so that they satisfy the equation of continuity. Also, find the z−component of acceleration.

[10M]

5.(d)

Starting from rest in the beginning, the speed(in Km.h) of a train at different times(in minutes) is given by the above table.

Using Simpson’s 13rd rule, find the approximate distance travelled (in Km) in 20 minutes fro the beginning.

[10M]

5.(e) Write down the basic algorithm for solving the equation: xex−1=0 by bisection method, correct to 4 decimal places.

[10M]

6.(a) Find the general solution of the partial differential equation

(y3x−2x4)p+(2y4−x3y)q=9z(x3−y3),

where p=∂z∂x, q=∂z∂y, and its integral surface that passes through the curve: x=t, y=t2, z=1.

[15M]

6.(b) Find the equivalent of numbers given in a specified number system to the system mentioned against them.

(i) (111011.101)2 to decimal system
(ii) (1000111110000.00101100)2 to hexadecimal system
(iii) (C4F2)16 to decimal system
(iv) (418)10 to binary system

[15M]

6.(c) Suppose the Lagrangian of a mechanical system is given by L=12m(a˙x2+2b˙x˙y+c˙y2)−12k(ax2+2bxy+cy2),

where a, b, c(>0), k(>0) are constant and b2≠ac. Write down the Lagrangian Equations of motion and identify the system.

[20M]

7.(a) Solve the partial differential equation:

where D≡∂∂x, D′≡∂∂y.

[15M]

7.(b) Find the values of the constant a,b,c such that the quadrature formula:

∫hof(x)dx=h[af(o)+bf(h3)+cf(h)] is exact for polynomial of as high degree as possible, and hence find the order of the truncation error.

[15M]

7.(c) The Hamiltonian of a mechanical system is given by,

H=p1q1−aq21+bq22=p2q2, where a, b are constants. Solve the Hamiltonian equations and show that p2−bq2q1=constant.

[20M]

8.(a) Simplify the boolean expression:

(a+b)⋅(ˉb+c)+b⋅(ˉa+ˉc) by using the laws of boolean algebra. From its truth table write it in minterm normal form.

[15M]

8.(b) For a two-dimensional potential flow, the velocity potential is given by ϕ=x2y=xy2+13(x3−y3). Determine the velocity components along the direction x and y. Also, detrmine the stream function ψ and check whether ϕ represents a possible case of flow or not.

[15M]

8.(c) A thin annulus occupies the region 0<a≤r≤b, 0≤θ≤2π. The faces are insulated. Along the inner edge the temperature is maintained at 0∘, while along the outer egde the temperature is held at T=Kcosθ2, where K is a constant. Determine the temperature distribution in the annulus.

[20M]