Paper I PYQs-2012

1.(a) Define a function f of two real variables in the plane by: f(x,y)={x3cos1y+y3cos1xx2+y2, for x,y≠00, otherwise 

Check the continuity and differentiability of f at (0,0).

[12M]

1.(b) Let p and q be positive real numbers such that 1p + 1q =1. Show that for real numbers a, b≥0,

[12M]

1.(c) Prove or disprove the following statement: If B={b1,b2,b3,b4,b5} is a basis for R5 and V is a two-dimensional subspace of R5, then V has a basis made of two members of B.

[12M]

1.(d) Let T:R3→R3 be the linear transformation defined by T(α,β,γ)=(α+2β−3γ,2α+5β−4γ,α+4β+γ. Find a basis and the dimension of the image of T and the kernel of T.

[12M]

1.(e) Prove that two of the straight lines represented by the equation x3+bx2y+cxy2+y3=0 will be at right angles, if b+c=−2.

[12M]

2.(a)(i) Let V be the vector space of all 2×2 matrices over the field of real numbers. Let W be the set consisting of all matrices with zero determinant. Is W a subspace of V? Justify your answer?

[8M]

2.(a)(ii) Find the dimension and a basis for the space W of all solutions of the following homogeneous system using matrix notation:

[12M]

2.(b)(i) Consider the linear mapping f:R2→R2 by f(x,y)=(3x+4y,2x−5y). Find the matrix A relative to the basis {(1,0),(0,1)} and the matrix B relative to the basis {(1,2),(2,3)}.

[12M]

2.(b)(ii) If λ is a characteristic root of a non-singular matrix A, then prove that |A|λ is a characteristic root of AdjA.

[8M]

2.(c) Let H=[1i2+i−i21−i2−i1+i2] be a Hermitian matrix. Find a non-singular matrix P such that D=PTH¯P is diagonal.

[20M]

3.(a) Find the points of local extrema and saddle points of the function f for two variable defined by f(x,y)=x3+y3−63(x+y)+12xy.

[20M]

3.(b) Define a sequence sn of real numbers by sn=∑ni=1(log(n+i)−logn)2n+i. Does limn→∞sn exist? If so, compute the value of this limit and justify your answer.

[20M]

3.(c) Find all the real values of p and q so that the integral ∫10xp(log1x)qdx converges.

[20M]

4.(a) Compute the volume of the solid enclosed between the surfaces x2+y2=9 and x2+z2=9.

[20M]

4.(b) A variable plane is parallel to the plane xa+yb+zc=0 and meets the axes in A, B, C respectively. Prove that circle ABC lies on the cone yz(bc+cb)+zx(ca+ac)+xy(ab+ba)=0.

[20M]

4.(c) Show that locus of a point from which three mutually perpendicular tangent lines can be drawn to the paraboloid x2+y2+2z2=0 is x2+y2+4z=1.

[20M]

5.(a) Solve dydx=2xye(x/y)2y2(1+e(x/y)2)+2x2e(x/y)2.

[12M]

5.(b) Find the orthogonal trajectories of the family of curves x2+y2=ax.

[12M]

5.(c) Using Laplace transforms, solve the initial value problem y′′+2y′+y=e−t, y(0)=−1, y′(0)=1.

[12M]

5.(d) A particle moves with an acceleration

towards the origin. It it starts from rest at a distance a from the origin, find its velocity when its distance from the origin is a2.

[12M]

5.(e) If →A=x2yz→i−2xz3→j+xz2→k, →B=2z→i+y→j−x2→k, find the value of ∂2∂x∂y(→A+→B) at (1,0,−2).

[12M]

6.(a) Show that the differential equation (2xylogy)dx+(x2+y2√y2+1)dy=0 is not exact. Find an integrating factor and hence, the solution of the equation.

[20M]

6.(b) Find the general solution of the equation y′′′−y′′=12x2+6x.

[20M]

6.(c) Solve the ordinary differential equation x(x−1)y′′−(2x−1)y′+2y=x2(2x−3).

[20M]

7.(a) A heavy ring of mass m, slides on a smooth vertical rod and is attached to a light string which passes over a small pulley distant a from the rod and has a mass M(>m) fastened to its other end. Show that if the ring be dropped from a point in the rod in the same horizontal plane as the pulley, it will descend a distance 2MmaM2−m2 before coming to rest.

[20M]

7.(b) A heavy hemispherical shell of radius a has a particle attached to a point on the rim, and rests with the curved surface in contact with a rough sphere of radius b at the highest point. Prove that if ba>√5−1, the equilibrium is stable, whatever be the weight of the particle.

[20M]

7.(c) The end links of a uniform chain slide along a fixed rough horizontal rod. Prove that the ratio of the maximum span to the length of the chain is μlog[1+√1+μ2μ], where μ is the coefficient of friction.

[20M]

8.(a) Derive the Frenet-Serret formulae. Define the curvature and torsion for a space curve. Compute them for the space curve x=t, y=t2, z=23t3. Show that the curvature and torsion are equal for this curve.

[20M]

8.(b) Verify Green’s theorem in the plane for ∮C[xy+y2dx+x2dy] where C is the closed curve of the region bounded by y=x and y=x2.

[20M]

8.(c) If →F=y→i+(x−2xz)→j−xy→k, evaluate ∬S(→∇×→F).→nd→s, where S is the surface of the sphere x2+y2+z2=a2 above the xy−plane.

[20M]