Paper II PYQs-2017

1.(a) Let x1=2 and xn+1=√xn+20,n=1,2,3,…, show that the sequence x1,x2,x3.. is convergent.

[10M]

1.(b) Let G be a group of order n. Show that G is isomorphic to a subgroup of the permutation group Sn.

[10M]

1.(c) Find the supremum and infimum of xsinx on the interval (0,π2]

[10M]

1.(d) Determine all entire functions f(z) such that 0 is a removable singularity of f(1z).

[10M]

1.(e) Using graphical method, find the maximum value of

subject to

4x+3y≤12 4x+y≤8 4x−y≤8 x,y≥0

[10M]

2.(a) Let f(t)=∫t0[x]dx where [x] denotes the largest integer less than or equal to x.
i) Determine all the real numbers t at which f is differentiable.
ii) Determine all the real numbers t at which f is continuous but not differentiable.

[15M]

2.(b) Using contour integral method, prove that ∫∞0xsinmxa2+x2dx=π2e−ma.

[15M]

2.(c) Let F be a field and F[x] denote the ring of polynomial over F in a single variable X. For f(X),g(X)∈F[X] with g(X)≠0, show that there exist q(X),r(X)∈F[X] such that degree r(X)< degree g(X) and f(X)=q(X)⋅g(X)+r(X).

[20M]

3.(a) Show that the groups Z5×Z7 and Z35 are isomorphic.

[15M]

3.(b) Let f=u+iv be analytic function on the unit disc D={z∈C:|z|<1}. Show that ∂2u∂x2+∂2u∂y2=0=∂2v∂x2+∂2v∂y2 at all points of D.

[15M]

3.(c) Solve the following linear programming problem by simplex method. Maximize z=3x1+5x2+4x3, subject to:
2x2+3x2≤8
2x1+5x2≤10
3x1+2x2+4x3≤15x3
x1,x2,x3≥0

[20M]

4.(a) For a function f:C→C and n≥1, let f(n) denotes the nth derivative of f and f(0)=f. Let f be an entire function such that for some n≥1, f(n)(1k)=0 for all k=1, 2, 3, show that f is a polynomial.

[15M]

4.(b) Find the initial basic feasible solution of the following transportation problem using Vogel’s approximation methods and find the cost.
Destinations D1D2D3D4D5O14703614O212−3389O33−140517838138

[15M]

4.(c) Let ∑∞n=1xn be a conditionally convergent series of real numbers. Show that there is a rearrangement ∑∞n=1xπ(n) of the series ∑∞n=1xn that converges to 100.

[20M]

5.(a) Solve (D2−2DD′−D2)z=ex+2y+x3+sin2x where D≡∂∂x, D′≡∂∂y, D2≡∂2∂x2, D′2≡∂2∂y2.

[10M]

5.(b) Explain the main steps of the Gauss-Jordan method and apply this method to find the inverse of the matrix [266286268].

[10M]

5.(c) Write the Boolean expression z(y+z)(x+y+z) in the simplest form using Boolean postulate rules. Mention the rules used during simplification. Verify your result by constructing the truth table for the given expression and for its simplest form.

[10M]

5.(d) Let Γ be a closed curve in xy−plane and let S denote the region bounded by the curve Γ. Let ∂2w∂x2+∂2w∂y2=f(x,y)∀(x,y)∈S. If f is prescribed at each point (x,y) of S and w is prescribed on the boundary Γ of S then prove that any solution w=w(x,y), satisfying these conditions, is unique.

[10M]

5.(e) Show that the moment of inertia of an elliptic area of mass M and semi-axis a and b about a semi-diameter of length r is 14Ma2b2r2. Further, prove that the moment of inertia about a tangent is 5M4p2, where p is the perpendicular distance from the centre of the ellipse to the tangent.

[10M]

6.(a) Find a complete integral of the partial differential equation 2(pq+yp+qx)+x2+y2=0.

[15M]

6.(b) For given equidistant values u−1, u0, u1 and u2, a value is interpolated by Lagrange’s formula. Show that it may be written in the form ux=yu0+y(y2−1)3!Δ2u−1+x(x2−1)3!Δ2u0, where x+y=1.

[15M]

6.(c) Two uniform rods AB and AC, each of mass m and length 2a, are smoothly hinged together at A and move on a horizontal plane. At time t, the mass centre of the road is at the point (ξ,η) referred to fixed perpendicular axes Ox, Oy in the plane, and the rods make angles θ±ϕ with Ox. Prove that the kinetic energy of the system is m[ξ2+η2+(13+sin2ϕ)a2θ2+(13+cos2ϕ)a2ϕ2]. Also derive Lagrange’s equation of motion for the system if an external force with components [X,Y] along the axes acts at A.

[20M]

7.(a) Reduce the equation y2∂2z∂x2−2xy∂2z∂x∂y+x2∂2z∂y2=y2∂x∂z∂x+x2∂y∂z∂y to canonical form and hence solve it.

[15M]

7.(b) Derive the formula ∫baydx=3h8[(y0+yn)+3(y1+y2+y4+y5+…+yn−1)+2(y3+y6+yn−3)]. Is there any restriction on n? State that condition. What is the error bounded in the case of Simpson’s 38th rule?

[20M]

7.(c) A stream is rushing from a boiler through a conical pipe, the diameters of the ends of which are D and d. If V and v be the corresponding velocities of the stream and if the motion is assumed to steady and diverging from the vertex of the cone, then prove that vV=D2d2e(u2−v2)/2K, where K is the pressure divided by the density and is constant.

[15M]

8.(a) Given the one-dimensional wave equation ∂2y∂t2=c2∂2y∂x2; t>0, where c2=Tm, T the constant tension in the string and m is the mass per unit length of the string.

(i) Find the appropriate solution of the wave equation.
(ii) Find also the solution under the conditions y(0,t)=0, y(l,t)=0 for all t and [∂y∂t]t=0=0, y(x,0)=asinπxt, 0<x<l, a>0.

[20M]

8.(b) Write an algorithm in the form of a flow chart for Newton-Raphson method. Describe the cases of failure of this method.

[15M]

8.(c) If the velocity of an incompressible fluid at the point (x,y,z) is given by (3xzr5,3yzr5,3z2−r2r5), r2=x2+y2+z2 then prove that the liquid motion is possible and that the velocity potential is zr3. Further, determine the the streamlines.

[15M]