Paper I PYQs-2018

1.(a) Show that the maximum rectangle inscribed in a circle is a square.

[8M]

1.(b) Given that Adj A=[220251011] and det A=2. Find the matrix A.

[8M]

1.(c) If f:[a,b]→R be continuous in [a,b] and derivable in (a,b), where 0<a<b, show that for c∈(a,b)

[8M]

1.(d) Find the equations of the tangent planes to the ellipsoid

which pass through the line

[8M]

1.(e) Prove that the eigenvalues of a Hermitian matrix are all real.

[8M]

2.(a) Find the equation of the cylinder whose generators are parallel to the line x1=y−2=z3 and whose guiding curve is x2+y2=4, z=2.

[10M]

2.(b) Show that the matrices A=[11−1121−113] and B=[103022320] are congruent.

[10M]

2.(c) If ϕ and ψ be two functions derivable in [a,b] and ϕ(x)ψ′(x)−ψ(x)ϕ′(x)>0 for any x in this interval, then show that between two consecutive roots of ϕ(x)=0 in [a,b], there lies exactly one root of ψ(x)=0

[10M]

2.(d) Show that the vectors α1=(1,0,−1),α2=(1,2,1),α3=(0,−3,2) form a basis for R3. Express each of the standard basis vectors as a linear combination of α1,α2,α3

[10M]

3.(a) Find the equation of the tangent plane that can be drawn to the sphere

through the straight line

[10M]

3.(b) If f=f(u,v), where u=excosy and v=exsiny, show that

[10M]

3.(c) Let T:V2(R)→V2(R) be a linear transformation defined by T(a,b)=(a,a+b). Find the matrix of T, taking {e1,e2} as a basis for the domain and {(1,1),(1,-1)} as a basis for the range.

[10M]

3.(d) Evaluate ∬R(x2+xy)dx dy over the region R bounded by xy=1, y=0 y=x and x=2.

[10M]

4.(a) Find the equations of the straight lines in which the plane 2x+y−z=0 cuts the cone 4x2−y2+3z2=0. Find the angle between the two straight lines.

[10M]

4.(b) Show that the functions u=x+y+z,v=xy+yz+zx and w=x3+y3+z3−3xyz are dependent and find the relation between them.

[10M]

4.(c) Find the locus of the point of intersection of the perpendicular generators of the hyperbolic paraboloid x2a2−y2b2=2z.

[10M]

4.(d) If (n+1) vectors α1,α2,…,αn,α form a linearly dependent set, then show that the vector α is a linear combination of α1,α2,…,αn; provided α1,α2,…,αn form a linearly independent set.

[10M]

5.(a) Find the complementary function and particular integral for the equation

and hence the general solution of the equation.

[8M]

5.(b) Solve d2ydx2−2dydx+y=xexlogx(x>0) by the method of variation of parameters.

[8M]

5.(c) If the velocities in a simple harmonic motion at distances a, b and c from a fixed point on the straight line which is not the centre of force, are u, v and w respectively, show that the periodic time T is given by

[8M]

5.(d) From a semi-circle whose diameter is in the surface of a liquid, a circle is cut out, whose diameter is the vertical radius of the semi-circle. Find the depth of the centre of pressure of the remainder part.

[8M]

5.(e) If →r=xˆi+yˆj+zˆk and f(r) is differentiable, show that

Hence or otherwise show that div(→rr3)=0

[8M]

6.(a) Solve the differential equation (y2+2x2y)dx+(2x3−xy)dy=0.

[10M]

6.(b) Let T1 and T2 be the periods of vertical oscillations of two different weights suspended by an elastic string, and C1 and C2 are the statical extensions due to these weights and g is the acceleration due to gravity. Show that g=4π2(C1−C2)T21−T22.

[15M]

6.(c) Show that →F=(2xy+z3)ˆi+x2ˆj+3xz2ˆk is a conservative force.

Hence, find the scalar potential. Also find the work done in moving a particle of unit mass in the force field from (1,−2,1) to (3,1,4).

[15M]

7.(a) 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

[10M]

7.(b) Solve:

[10M]

7.(c) A frame ABC consists of three light rods, of which AB,AC are each of length a, BC of length 32a, freely jointed together. It rests with BC horizontal, A below BC and the rods AB,AC over two smooth pegs E and F, in the same horizontal line, at a distance 2b apart. A weight W is suspended from A. Find the thrust in the rod BC.

[10M]

7.(d) Let α be a unit-speed curve in R3 with constant curvature and zero torsion. Show that α is (part of a circle.

[10M]

8.(a) A solid hemisphere floating in a liquid is completely immersed with a point of the rim joined to a fixed point by means of a string. Find the inclination of the base to the vertical and tension of the string.

[15M]

8.(b) A snowball of radius r(t) melts at a uniform rate. If half of the mass of the snowball melts in one hour, how much time will it take for the entire mass of the snowball to melt, correct to two decimal places? Conditions remain unchanged for the entire process.

[15M]

8.(c) For a curve lying on a sphere of radius a and such that the torsion is never 0, show that

[10M]