Paper I PYQs-2014

1.(a) Find one vector in R3 which generates the intersection of V and W, where V is the xy−plane and W is the space generated by the vectors (1,2,3) and (1,−1,1).

[10M]

1.(b) Using elementary row or column operations, find the rank of the matrix [01−3−10011310211−20]

[10M]

1.(c) Prove that between two real roots of excosx+1=0, a real root of exsinx+1=0 lies.

[10M]

1.(d) Evaluate ∫10loge(1+x)1+x2dx

[10M]

1.(e) Examine whether the plane x+y+z=0 cuts the cone yz+zx+xy=0 in perpendicular lines.

[10M]

2.(a) Let V and W be the following subspaces of R4:V={(a,b,c,d):b−2c+d=0} and W={(a,b,c,d):a=d,b=2c}. Find a basis and the dimension of:
(i) V
(ii) V∩W.

[15M]

2.(b)(i) Investigate the values of λ and μ so that the equations x+y+z=6, x+2y+3z=10, x+2y+λz=μ have:
(i) no solution
(ii) a unique solution,
(iii) an infinite number of solutions.

[10M]

2.(b)(ii) Verify Cayley-Hamilton theorem for the matrix A=[1423] and hence find its inverse. Also, find the matrix represented by A5−4A4−7A3+11A2−A−10I.

[10M]

2.(c) By using the transformation x+y=u, y=uv, evaluate the integral ∬{xy(1−x−y)}1/2dxdy taken over the area enclosed by the straight lines x=0, y=0 and x+y=1.

[15M]

3.(a) Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.

[15M]

3.(b) Find the maximum or minimum values of x2+y2+z2 subject to the condition ax2+by2+cz2=1 and lx+my+nz=0 interpret result geometrically.

[20M]

3.(c)(i) Let A=[−22−321−6−1−20]. Find the Eigen values of A and the corresponding Eigen vectors.

[8M]

3.(c)(ii) Prove that eigen values of a unitary matrix have absolute value 1.

[7M]

4.(a)(i) Find the co-ordinates of the points on the sphere x2+y2+z2−4x+2y=4 the tangent planes at which are parallel to the plane 2x−y+2z=1.

[10M]

4.(a)(ii) Prove that equation ax2+by2+cz2+2ux+2vy+d=0 represents a cone if u2a+v2b+w2c=d.

[10M]

4.(b) Show that the lines drawn from the origin parallel to the normals to the central conicoid ax2+by2+cz2=1, at its points of intersection with the plane lx+my+nz=p generate the cone p2(x2a+y2b+z2c)=(lxa+myb+nzc)2.

[15M]

4.(c) Find the equations of the two generating lines through any point (acosθ,bsinθ,0) of the principal elliptic section x2a2+y2b2=1, z=0 of the hyperboloid by the plane z=0.

[15M]

5.(a) Justify that a differential equation of the form: [y+xf(x2+y2)]dx+[yf(x2+y2)−x]dy=0 where f(x2+y2) is an arbitrary function of (x2+y2), is not an exact differential equation and 1x2+y2 is an integrating factor for it. Hence solve this differential equation for f(x2+y2)=(x2+y2)2.

[10M]

5.(b) Find the curve for which the part of the tangent cut-off by the axes is bisected at the point of tangency.

[10M]

5.(c) A particle is performing a simple harmonic motion (SHM) of period T about a centre O with amplitude a and it passes through a point P, where OP=b in the direction OP. Prove that the time which elapses before it returns to P is Tπcos′(ba).

[10M]

5.(d) Two equal uniform rods AB and AC, each of length l, are freely jointed at A and rest on a smooth fixed vertical circle of radius r. If 2θ is the angle between the rods, then find the relation between l, r and θ, by using the principle of virtual work.

[10M]

5.(e) Find the curvature vector at any point of the curve ¯r(t)=tcostˆi+tsintˆj, 0≤t≤2π. Give its magnitude also.

[10M]

6.(a) Solve by the method of variation of parameters: dydx−5y=sinx.

[10M]

6.(b) Solve the differential equation: x3d3ydx3+3x2d2ydx2+xdydx+8y=65cos(logex).

[20M]

6.(c) Evaluate by Stokes’ theorem:

where is the curve given by x2+y2+z2−2ax−2ay=0, x+2y=2a, starting from (2a,0,0) and then going below the z-plane.

[20M]

7.(a) Solve the following differential equation: xd2ydx2−2(x+1)dydx+(x+2)y=(x−2)e2x, when ex is a solution to its corresponding homogeneous differential equation.

[15M]

7.(b) A particle of mass m, hanging vertically from a fixed point by a light inextensible cord of length l, is struck by a horizontal blow which imparts to it a velocity 2√gl. Find the velocity and height of the particle from the level of its inition when the cord becomes slack.

[15M]

7.(c) A regular pentagon ABCDE, formed of equal heavy uniform bars jointed together, is suspended from the joint A, and is maintained in form by a light rod joining the middle points of BC and DE. Find the stress in this rod.

[20M]

8.(a) Find the sufficient condition for the differential equation M(x,y)dx+N(x,y)dy=0 to have an integrating factor as a function of (x+y). What will be the integrating factor in that case? Hence find the integrating factor for the differential equation of (x2+xy)dx+(y2+xy)dy=0 and solve it.

[15M]

8.(b) A particle is acted on by a force parallel to the axis of y whose acceleration (always towards the axis of x) is μy−2 and when y=a, it is projected parallel to the axis of X with velocity √2μa. Find the parametric equation of the path of the particle. Here μ is a constant.

[15M]