Paper I PYQs-2013
Section A
1.(a) Find the dimension and a basis of the solution space W of the system x+2y+2z−s+3t=0, x+2y+3z+s+t=0, 3x+6y+8z+s+5t=0.
[8M]
1.(b) Find the characteristic equation of the matrix A=[211010112] and hence find the matrix represented by A8−5A7+7A6−3A5+A4−5A3+8A2−2A+I.
[8M]
1.(c) Evaluate the integral ∫∞0∫x0xe−x2/ydydx by changing the order of integration.
[8M]
1.(d) Find the surface generated by the straight line which intersects the lines y=z=a and x+3z=a=y+z and is parallel to the plane x+y=0.
[8M]
1.(e) Find C of the mean value theorem, if f(x)=x(x−1)(x−2),a=0,b=12 and C has usual meaning.
[8M]
2.(a) Let V be the vector space of 2×2 matrices over R and let M=[1−1−22]
Let F:V→V be the linear map defined by F(A)=MA. Find a basis and the dimension of
(i) the kernel of W of F
(ii) the image U of F.
[10M]
2.(b) Locate the stationary points of the function x4+y4−2x2+4xy−2y2 and determine their nature.
[10M]
2.(c) Find an orthogonal transformation of co-ordinates which diagonalizes the quadratic form
q(x,y)=2x2−4xy+5y2[10M]
2.(d) Discuss the consistency and the solutions of the equations
x+ay+az=1,ax+y+2az=−4,ax−ay+4z=2for different values of a.
[10M]
3.(a) Prove that if a0,a1,a2,……,an are the real numbers such that
a0n+1+a1n+a2n−1+………+an−12+an=0then there exists at least one real number x between 0 and 1 such that
a0xn+a1xn−1+a2xn−2+…+fn−1x+an=0[10M]
3.(b) Reduce the following equation to its canonical form and determine the nature of the conic 4x2+4xy+y2−12x−6y+5=0.
[10M]
3.(c) Let F be a subfield of complex numbers and T a function from F3→F3 defined by T(x1,x2,x3)=(x1+x2+3x3,2x1−x2,−3x1+x2−x3). What are the conditions on (a,b,c) such that (a,b,c) be in the null space of T? Find the nullity of T.
[10M]
3.(d) Find the equations to the tangent planes to the surface 7x2−3y2−z2+21=0, which pass through the line 7x−6y+9=0, z=3.
[10M]
4.(a) Evaluate
∫π/20xsinxcosxdxsin4x+cos4x[10M]
4.(b) Let H=[1 i 2+i−i21−i2−i1+i2] be a Hermitian matrix. Find a non-singular matrix P such that PtH¯P is diagonal and also find its signature.
[10M]
4.(c) Find the magnitude and the equations of the line of shortest distance between the lines
x−83=y+9−16=z−107x−153=y−298=z−5−5[10M]
4.(d) Find all the asymptotes of the curve
x4−y4+3x2y+3xy2+xy=0[10M]
Section B
5.(a) Solve:
dydx+xsin2y=x3cos2y[8M]
5.(b) A particle is performing a simple harmopiç motion of period T about centre O and it passes through a point P, where OP=(b with velocity v in the direction of OP. Find the time which elapses before it returps to P.
[8M]
5.(c) →F being a vector, prove that
curl curl →F= grad div→F−∇2→F
where ∇2=∂2∂x2+∂2∂y2+∂2∂z2
[8M]
5.(d) A triangular lamina ABC of density ρ floats in a liquid of density σ with its plane vertical, the angle B being in the surface of the liquid, and the angle A not immersed. Find p/σ in terms of the lengths of the sides of the triangle.
[8M]
5.(e) A heavy uniform rod rests with one end against a smooth vertical wall and with a point in its length resting on a smooth peg. Find the position of equilibrium and discuss the nature of equilibrium.
[8M]
6.(a) Solve the differential equation
d2ydx2−4xdydx+(4x2−1)y=−3ex2sin42xby changing the dependent variable.
[13M]
6.(b) Evaluate ∫S→F⋅d→s, where →F=4xi−2y2j+z2→k and s is the surface bounding the region x2+y2=4,z=0 and z=3.
[13M]
6.(c) Two bodies of weights w1 and w2 are placed on an inclined plane and are connected by a light string which coincides with a line of greatest slope of the plane; if the coefficient of friction between the bodies and the plane are respectively μ1 and μ2, find the inclination of the plane to the horizontal when both bodies are on the point of motion, it being assumed that smoother body is below the other.
[14M]
7.(a) Solve
(D3+1)y=ex2sin(√32x) where D=ddx[13M]
7.(b) A body floating in water has volumes v1,v2 and v3 above the surface, when the densities of the surrounding air are respectively ρ1,ρ2,ρ3. Find the value of:
ρ2−ρ3v1+ρ3−ρ1v2+ρ1−ρ2v3[13M]
7.(c) A particle is projected vertically upwards with a velocity u, in a resisting medium which produces a retardation kv2 when the velocity is v. Find the height when the particle comes to rest above the point of projection.
[14M]
8.(a) Apply the method of variation of parameters to solve
d2ydx2−y=2(1+ex)−1[13M]
8.(b) Verify the Divergence theorem for the vector function
→F=(x2−yz)→i+(y2−xz)→j+(z2−xy)ˉktaken over the rectangular parallelopiped
0≤x≤a,0≤y≤b,0≤z≤c[14M]
8.(c) A particle is projected with a velocity v along a smooth horizontal plane in a medium whose resistance per unit mass is double the cube of the velocity. Find the distance it will describe in time t.
[13M]