Paper I PYQs-2012

1.(a) Let V=R3 and α1=(1,1,2),α2=(0,1,3) α3=(2,4,5) and α4=(−1,0,−1)⋅ be the elements of V. Find a basis for the intersection of the subspace spanned by {α1,α2} and {α3,α4}

[8M]

1.(b) Show that the set of all functions which satisfy the differential equation, d2fdx2+3dfdx=0 is a vector space.

[8M]

1.(c) If the three thermodynamic variables P,V,T are connected by a relation f(P,V,T)=0 show that, (∂P∂T)V⋅(∂T∂V)P(∂ˆV∂P)T≅−1.

[8M]

1.(d) If u=Ae−gxsin(nt−gx), where A,g,n are positive constants, satisfies the heat conduction equation, ∂u∂t=μ∂2u∂x2 then show that g=√(n2μ).

[8M]

1.(e) Find the equations to the lines in which the plane 2x+y−z=0 cuts the cone

[8M]

2.(a) Let f:R→R3 be a linear transformation defined by f(a,b,c)=(a,a+b,0).

Find the matrices A and B respectively of the linear transformation f with respect to the standard basis (e1,e2,e3) and the basis (e′1,e′2,e′3) where e′1=(1,1,0),e′2=(0,1,1) e′3=(1,1,1).

Also, show that there exists an invertible matrix P such that

[10M]

2.(b) Verify Cayley-Hamilton theorem for the matrix A=(1423) and find its inverse. Also express A5−4A4−7A3+11A2−A−10I as a linear polynomial in A.

[10M]

2.(c) Find the equations of the tangent plane to the ellipsoid 2x2+6y21+3z2=27 which passes through the line x−y−z=0=x−y+2z−9.

[10M]

2.(d) Show that there are three real values of λ for which the equations: (a−λ)x+by+cz=0, bx+(c−λ)y+az=0, cx+ay+(b−λ)z=0 are simultaneously true and that the product of these values of λ is D=|abcbcacab|.

[10M]

3.(a) Find the matrix representation of linear transförmation T on V3(IR) defined as T(a,b,c)=(2b+c,a−4b,3a) corresponding to the basis B={(1,1,1),(1,1,0),(1,0,0)}.

[10M]

3.(b) Find the dimensions of the rectangular box, open at the top, of maximum capacity whose surface is 432 sq. cm.

[10M]

3.(c) If 2C is the shortest distance between the lines

and ym+zn=1,x=0

then show that

[10M]

3.(d) Show that the function defined as

has removable discontinuity at the origin.

[10M]

4.(a) Find by triple Integration the volume cut off from the cylinder x2+y2=ax by the planes z=mx and z=nx.

[10M]

4.(b) Show that afl the spheres, that can be drawn through the origin and each set of points where planes parallel to the plane xa+yb+zc=0 cut the co-ordinate axes, form a system of spheres which are cut orthogonally by the sphere

if af+bg+ch=0

[10M]

4.(c) A plane makes equal intercepts on the positive parts of the axes and touches the ellipsoid x2+4y2+9z2=36. Find its equation.

[10M]

4.(d) Evaluate the following in terms of Gamma function:

[10M]

5.(a) Solve dydx−tany1+x=(1+x)exsecy

[8M]

5.(b) Solve and find the singular solution of x3p2+x2py+a3=0.

[8M]

5.(c) A particle is projected vertically upwards from the earth’s surface with a velocity just sufficient to carry it to infinity. Prove that the time it takes to reach a height h is 13√(2ag)[(1+ha)3/2−1].

[8M]

5.(d) A triangle ABC is immersed in a liquid with the vertex C in the surface and the sides AC, BC equally inclined to the surface. Show that the vertical C divides the triangle into two others, the fluid pressures on which are as b3+3ab2:a3+3a2b where a and b are the sides BC & AC respectively.

[8M]

5.(e) If u=x+y+z, v=x2+y2+z2, w=yz+zx+xy, prove that grad u, grad v and grad w are coplanar.

[8M]

6.(a) Solve:

[10M]

6.(b) Find the value of ∬s(→∇×→F)⋅→ds taken over the upper portion of the surface x2+y2−2ax+az=0 and the bounding curve lies in the plane z=0, when

[10M]

6.(c) A particle is projected with a velocity u and strikes at right angle on a plane through the plane of projection inclined at an angle β to the horizon. Show that the time of flight is 2u ¯g√(1+3sin2β) range on the plane is 2u2g⋅sinβ1+3sin2β and the vertical height of the point struck is 2u2sin2βg(1+3sin2β) above the point of projection.

[10M]

6.(d) Solve d4ydx4+2d2ydx2+y=x2c.

[10M]

7.(a) A particle is moving with central acceleration μ[r5−c4r] being projected from an apse at. a distance c with velocity √(2μ3)c3, show that its path is a curve, x4+y4=c4.

[13M]

7.(b) A thin equilateral rectangular plate of uniform thickness and density rests with one end of its base on a rough horizontal plane and the other against a small vertical wall. Show that the least angle, its base can make with the horizontal plane is given by cotθ=2μ+1√3 μ, being the coefficient of friction.

[14M]

7.(c) A semicircular area of radius a is immersed vertically with its diameter horizontal at a depth b. If the circumference be below the centre, prove that the depth of centre of pressure is 143π(a2+4b2)+32ab4a+3πb

[13M]

8.(a) Solve x=ydydx−(dydx)2.

[10M]

8.(b) Find the value of the line integral aver a circular path given by x2+y2=a2,z=0 where the vector field, →F=(siny)→i+x(1+cosy)→j.

[10M]

8.(c) A heavy elastic string, whose natural length is 2πa, is placed round a smooth cone whose axis is vertical and whose semi vertical angle is α. If W be the weight and λ the modulus of elasticity of the string, prove that it will be in equilibrium when in the form of a circle whose radius is

[10M]

8.(d) Solve x2d2ydx2+3xdydx+y=(1−x)−2.

[10M]