Paper I PYQs-2019
Section A
1.(a) Let f:[0,π2→R] be a continuous function such that
f(x)=cos2x4x2−π2,0≤x≤π2Find the value of f(π2).
[10M]
1.(b) Let f:D(⊆R2)→R be a function and (a,b)∈D. If f(x,y) is continuous at (a,b), then show that the function f(x,b) and f(a,y) are continuous at x=a and at y=b respectively.
[10M]
1.(c) Let T:R2→R2 be a linear map such that that T(2,1)=(5,7) and T(1,2)+(3,3). If A is the matrix corresponding to T with repect to the standard bases e1, e2, then find Rank(A).
[10M]
1.(d) If A=[1211−4130−3] and B=[2111−1021−1], then show that AB=6I3. Use this result to solve the following system of equations:
2x+y+z=5
x−y=0
2x+y−z=1
[10M]
1.(e) Show that the lines x+1−3=y−32=z+21 and x1=y−7−3=z+72 intersect. Find the coordinates of the point of intersection and the equation of the plane containing them.
[10M]
2.(a) Is f(x)=|cosx|+|sinx| is differentiable at x=π2? If yes, then find its derivative at x=π2. If no, then give a proof of it.
[15M]
2.(b) Let A and B be two orthogonal matrices of same order and det A + det B=0.Show that A+B is a singular matrix.
[15M]
2.(c)(i) The plane x+2y+3z=12 cuts the axes of coordinates in A, B, C. Find the equations of the circle circumscribing the triangle ABC.
[10M]
2.(c)(ii) Prove that the plane z=0 cuts the enveloping cone of the sphere x2+y2+z2=11 which has the vertex at (2,4,1) in a rectangulat hyperbola.
[10M]
3.(a) Find the maximum and the minimum value of the function f(x)=2x3−9x2+12x+6 on the interval (2,3).
[15M]
3.(b) Prove that, in general, three normals can be drawn from a given point to the paraboloid x2+y2=2az, but if the point lies on the surface
27a(x2+y2)+8(a−z)3=0then two of the three normal coincide.
[10M]
3.(c) Let A=[572111−81235034−31].
(i) Find the rank of matrix A.
[15M]
(ii) Find the dimension of the subspace
V={(x1,x2,x3,x4)∈R4|A(x1x2x3x4)=0}[5M]
4.(a) State the Cayley-Hamilton theorem. Use this theorem to find A100, where
A=[100101010][15M]
4.(b) Find the length of the normal chord through a point P of the ellipsoid
x2a2+y2b2+z2c2=1and prove that if it is equal to 4PG3, where G3 is the point where the normal chord through P meets the xy−plane, then P lies on the cone
x6a6(2c2−a2)+y6b6(2c2−b2)+z4c4=0[15M]
4.(c)(i) If
u=sin−1√x1/3+y1/3x1/2+y1/2,then show that sin2u is homogeneous function of x and y of degree −16. Hence show that
x2∂2u∂x2+2xy∂2u∂x∂y+y2∂2u∂y2=tanu12(1312+tan2u12)
[12M]
4.(c)(ii) Using the Jacobian method, show that if f′(x)=11+x2 and f(0)=0, then
f(x)+f(y)=f(x+y1−xy)[8M]
Section B
5.(a) Solve the differential equation
(2ysinx+3y4sinxcosx)dx−(4y3cos2x+cosx)dy=0[10M]
5.(b) Determine the complete solution of the differential equation
d2ydx2−4dydx+4y=3x2e2xsinx[10M]
5.(c) One end of a heavy uniform rod AB can slide along a rough horizontal rod AC, to which it is attracted by a ring. B and C are joined by a string. When the rod is on the point of sliding, then AC2−AB2=BC2. If θ is the angle between AB and the horizontal line, then pove that the coefficient of friction is cotθ2+cot2θ.
[10M]
5.(d) The force of attraction of a particle by the earth in inversely proportional to the square of its distance from the earth’s centre. A particle, whose weight on the surface of the earth is W, falls to the surface of the earth from a height 3h above it. Show that the magnitude of work done by the earth’s attraction force is 34hW, where h is the radius of the earth.
[10M]
5.(e) Find the directional derivative of the function xy2+yz2+zx2 along the tangent to the curve x=t, y=t2, z=t2 at the point (1,1,1).
[10M]
6.(a) A body consists of a cone and underlying hemisphere. The base of the cone and the top of the hemisphere have same radius a. The whole body rests on a rough horizontal table with hemisphere in contact with the table. Show that the greatest height of the cone, so that the equilibrium may be stable, is √3a.
[15M]
6.(b) Find the circulation of →F round the curve C, where →F=(2x+y2)ˆi+(3y−4x)ˆj and C is the curve y=x2 from (0,0) to (1,1) and the curve y2=x from (1,1) to (0,0).
[15M]
6.(c)(i) Solve the differential equation
d2ydx2+(3sinx−cotx)dydx+2ysin2x=e−cosxsin2x[10M]
6.(c)(ii) Find the Laplace transforms of t−1/2 and t−1/2. Prove that the Laplace transform of tn+1/2, where n∈N, is
Γ(n+1+12)sn+1+12[10M]
7.(a) Find the linearly independent solutions of the corresponding homogeneous differential equation of the equation x2y″−2xy′+2y=x3sinx and then find the general solution of the given equation by the method of variation of parameters.
[15M]
7.(b) Find the radius of curvature and radius of torsion of the helix x=acosu, y=asinu, z=autanα.
[15M]
7.(c) A particle moving along the y−axis has an acceleration Fy towards the origin, where F is a psoitive and even function of y. The periodic time, when the particle vibrates between y=−a and y=a, is T. Show that
2π√F1<T<2π√F2where F1 and F2 are the greatest and the least values of F within the range [−a,a]. Further, show that when a simple pendulum of length l oscillates through 30∘ on either side of the vertical line, T lies between 2π√l/g and 2π√l/g√π/3.
[20M]
8.(a) Obtain the singular solution of the differential equation
(dydx)2(yx)2cot2α−2(dydx)(yx)+(dydx)2cosec2α=1Also find the complete primitive of the given differential equation. Give the geometrical interpretations of the complete primitive and singular solution.
[15M]
8.(b) Prove that the path of a planet, which is moving so that its acceleration is always directed to a fixed point (star) and is equal to μ(distance)2 is a conic section. Find the conditions under which the path becomes
(i) ellipse,
(ii) parabola and
(iii) hyperbola
[15M]
8.(c)(i) State Gauss divergence theorem. Verify this theorem for →F=4xˆi−2y2ˆj+z2ˆk, taken over the region bounded by x2+y2=4, z=0 and z=3.
[15M]
8.(c)(ii) Evalute by Stokes’ theorem ∮Cexdx+2ydy−dz, where C is the curve x2+y2=4, z=2.
[5M]