Paper I PYQs-2017

1.(a) Let A=[2213]. Find a non-singular matrix P such that P−1AP is diagonal matrix.

[10M]

1.(b) Show that similar matrices have the same characteristic polynomial.

[10M]

1.(c) Integrate the function f(x,y)=xy(x2+y2) over the domain R:{−3≤x2−y2≤3,1≤xy≤4}

[10M]

1.(d) Find the equation of the tangent at the point (1,1,1) to the conicoid 3x2−y2=2z.

[10M]

1.(e) Reduce the following equation to the standard form and hence determine the conicoid:

[15M]

2.(a) Find the volume of the solid above the xy−plane and directly below the portion of the elliptic paraboloid x2+y24=z which is cut off by the plane z=9.

[15M]

2.(b) A plane passes through a fixed point (a,b,c) and cuts the axes at the points A, B, C respectively. Find the locus of the center of the sphere which passes through the origin O and A, B, C.

[15M]

2.(c) Show that the plane 2x−2y+z+12=0 touches the sphere x2+y2+z2−2x−4y+2z−3=0. Find the point of contact.

[10M]

2.(d) Suppose U and W are distinct four dimensional subspaces of a vector space V, where dimV=6. Find the possible dimensions of subspace U∩W.

[10M]

3.(a) Consider the matrix mapping A:R4→R3, where A=[1231135−23813−3]. Find a basis and dimension of the image of A and those of the kernel A.

[15M]

3.(b) Prove that distance non-zero eigenvectors of a matrix are linearly independent.

[10M]

3.(c) If f(x,y)={xy(x2−y2)x2+y2,(x,y)≠(0,0)0,(x,y)=(0,0)
Calculate ∂2f∂x∂y and ∂2f∂y∂x at (0,0).

[15M]

3.(d) Find the locus of the points of intersection of three mutually perpendicular tangent planes to ax2+by2+cz2=1.

[10M]

4.(a) Reduce the following equation to the standard form and hence determine the nature of the conicoid: x2+y2+z2−yz−zx−xy−3x−6y−9z+21=0.

[15M]

4.(b) Consider the following system of equation in x, y, z:
x+2y+2z=1
x+ay+3z=3
x+11y+az=b

i) For which values of a does the system have a unique solution?
ii) For which of values (a,b) does the system have more than one solution?

[15M]

4.(c) Examine if the improper integral ∫302xdx(1−x2)2/3 exists.

[10M]

4.(d) Prove that π3≤∬dxdy√x2+(y−2)2≤π, where D is the unit disc.

[10M]

5.(a) Find the differential equation representing the entire circle in the xy−plane.

[10M]

5.(b) Suppose that the streamlines of the fluid are given by a family of curves xy=c. Find the equipotential lines, that is, the orthogonal trajectories of the family of curves representing the streamlines.

[10M]

5.(c) A fixed wire is in the shape of the cardiod r=a(1+cosθ), the initial line being the downward vertical. A small ring of mass m can slide on the wire and is attached to point r=0 of the cardiod by an elastic string of natural length a and modulus of elasticity 4 mg. The string is released from rest when the string is horizontal. Show by laws of conservation of energy that aθ2(1+cosθ)−gcosθ(1−cosθ)=0, g being the acceleration due to gravity.

[10M]

5.(d) For what values of the constant a, b and c the vector ¯V=(x+y+az)ˆi+(bx+2y−z)j+(−x+cy+2z)k is irrotational. Find the divergence in cylindrical coordinates of the vector with these values.

[10M]

5.(e) The position vector of a moving point at time t is ¯r=sintˆi+cos2tj+(t2+2t)k. Find the components of acceleration ¯a in the direction parallel to the velocity vector ¯v and perpendicular to the plane of ¯r and ¯v at time t=0.

[10M]

6.(a)(i) Solve the following simultaneous liner differential equations:
(D+1)y=z+ex and (D+1)z=y+ex
where y and z are functions of independent variable x and D≡ddx.

[8M]

6.(a)(ii) If the growth rate of the population of bacteria at time t is proportional to the amount present at the time t and population doubles in one week, then how much bacteria can be expected after 4 weeks?

[8M]

6.(b)(i) Consider the differential equation xyp2−(x2+y2−1)p+xy=0 where p=dydx substituting u=x2 and v=y2 reduce the equation to Clairaut’s form in terms of u, v and p′=dvdu hence or otherwise solve the equation.

[10M]

6.(b)(ii) Solve the following initial value problem using Laplace transform: d2ydx2+9y=r(x), y(0)=0, y′(0)=4 where r(x)={8sinx if 0<x<π0 if x≥π.

[17M]

6.(c) A uniform solid hemisphere rests on a rough plane inclined to the horizon at an angle ϕ with its curved surface touching the plane. Find the greatest admissible value of the inclination ϕ for equilibrium. If ϕ be less than this value, is the equilibrium stable?

[17M]

7.(a) Find the curvature vector and its magnitude at any point ¯r=(θ) of the curve ¯r=(acosθ,asinθ,aθ). Show that the locus of the feet of the perpendicular from the origin to the tangent is a curve that completely lies on the hyperboloid x2+y2−z2=a2.

[16M]

7.(b)(i) Solve the differential equation: xd2ydx2−dydx−4x3y=8x3sin(x2)

[8M]

7.(b)(ii) Solve the following initial value differential equations: 20y′′+4y′+y=0, y(0)=3.2, y′(0)=0.

[7M]

7.(c) A particle is free to move on a smooth vertical circular wire of radius a. At time t=0 it is projected along the circle from its lowest point A with velocity just sufficient to carry it to the highest point B. Find the time T at which the reaction between the particle and the wire is zero.

[17M]

8.(a) A spherical shot of W gm weight and radius r cm, lies at the bottom of cylindrical bucket of radius Rcm. The bucket is filled with water up to a depth of hcm(h>2r). Show that the minimum amount of work done in lifting the shot just clear of the water must be [W(h−4r33R2)]+W′(r−h+2r33R2)cmgm. W′gm is the weight of water displaced by the shot.

[17M]

8.(b) Solve the following initial value problem using Laplace transform:

where

[17M]

8.(c)(i) Evaluate the integral ∬SFnds, where ¯F=3xy2ˆi+(yx2−y3)j+3zx2K and S is a surface of the cylinder y2+z2≤4,−3≤x≤3 using divergence theorem.

[9M]

8.(c)(ii) Using Green theorem, evaluate the ∫CF(→r).d→r counterclockwise where F(r)=(x2+y2)ˆi+(x2−y2)j and d→r=xˆi+dyj and the curve C is the boundary off the region R={(x,y)|1≤y≤2−x2}.

[8M]