Paper I PYQs-2018
Section A
1.(a) Let A be a 3×2 matrix and B a 2×3 matrix. Show that C=A⋅B is a singular matrix.
[10M]
1.(b) Express basis vectors e1=(1,0) and e2=(0,1) as linear combination of α1=(2,−1) and α2=(1,3).
[10M]
1.(c) Determine if limz→1(1−z)tanπz2 exists or not. If the limit exists, then find its value.
[10M]
1.(d) Find the limit limn→∞1n2∑n−1r=0√n2−r2.
[10M]
1.(e) Find the projection of the straight line x−12=y−13=z+1−1 on the plane x+y+2z=6.
[10M]
2.(a) Show that if A and B are similar n×n matrices, then they have the same eigenvalues.
[12M]
2.(b) Find the shortest distance from the point (1,0) to the parabola y2=4x.
[13M]
2.(c) The ellipse x2a2+y2b2=1 revolves about the x−axis. Find the volume of the solid of revolution.
[13M]
2.(d) Find the shortest distance between the lines
a1x+b1y+c1z+d1=0 a2x+b2y+c2z+d2=0and the z−axis.
[12M]
3.(a) For the system of linear equations
x+3y−2z=−1
5y+3z=−8
x−2y−5z−7
determine which of the following statements are true and which are false:
(i) The system has no solution.
(ii) The system has a unique solution.
(iii) The system has infinitely many solutions.
[13M]
3.(b) Let
f(x,y)=xy2, if y>0 and
f(x,y)=−xy2, if y≤0
Determine which of ∂f∂x(0,1) and ∂f∂y(0,1) exists and which does not exist.
[12M]
3.(c) Find the equations to the generating lines of the paraboloid (x+y+z)(2x+y−z)=6z which pass through the point (1,1,1).
[13M]
3.(d) Find the equation of the sphere in xyz−plane passing through the points (0,0,0), (0,1,−1), (−1,2,0), (1,2,3).
[12M]
4.(a) Find the maximum and the minimum value of x4−5x2+4 on the interval [2,3].
[13M]
4.(b) Evaluate the integral ∫a0∫xx/axdydxx2+y2.
[12M]
4.(c) Find the equation of the cone with (0,0,1) as the vertex and 2x2−y2=4, z=0 as the guiding curve.
[13M]
4.(d) Find the equation of the plane parallel to 3x−y+3z=8 and passing through the point (1,1,1).
[12M]
Section B
5.(a) Solve:
y″[10M]
5.(b) Find the angle between the tangent and a general point of the curve whose equations are x=3t, y=3t^2, z=3t^3 and the line y=z-x=0.
[10M]
5.(c) Solve:
y'''-6y''+12y'-8y=12e^{2x}+27e^{-x}[10M]
5.(d)(i) Find the Laplace transform of f(t)=\dfrac{1}{\sqrt{t}}.
[5M]
5.(d)(ii) Find the inverse Laplace transform of \dfrac{5s^2+3s-16}{(s-1)(s-2)(s-3)}.
[5M]
5.(e) A particle projected from a given point on the ground just clears a wall of height h at a distance from the point of projection. If the particle moves in a vertical plane and if the horizontal range is R, find the elevation of the projection.
[10M]
6.(a) Solve:
\left( \dfrac{dy}{dx} \right)^2y+2\dfrac{dy}{dx}-y=0[13M]
6.(b) A particle moving with simple harmonic motion in a straight line has velocities v_1 and v_2 at the distances x_1 and x_2 respectively from the center of its path. Find the period of its motion.
[12M]
6.(c) Solve:
y^{\prime \prime}+16y=32sec 2x[13M]
6.(d) If S is the surface of the sphere x^2+y^2+z^2=a^2, then evaluate
\iint_S [(x+y) dydz+(y+z) dzdx+(x+y) dxdy]using Gauss’s divergence theorem.
[12M]
7.(a) Solve:
(1+x)^2y''+(1+x)y'+y=4cos(log(1+x))[13M]
7.(b) Find the curvature and torsion of the curve
\vec{r}=a(u-sin u)\hat{i}-a(1-cos u)\hat{j}+bu\hat{k}[12M]
7.(c) Solve the initial value problem
y''-5y'+4y=e^{2t}
y(0)=\dfrac{19}{20}, y'(0)=\dfrac{8}{3}
[13M]
7.(d) Find \alpha and \beta such that x^{\alpha}y^{\beta} is an integrating factor of (4y^2+3xy)dx-(3xy+2x^2)dy=0 and solve the equation.
[12M]
8.(a) Let \vec{v}=v_1\hat{i}+v_2\hat{j}+v_3\hat{k}. Show that curl(curl \vec{v})=grad(div \vec{v})-\nabla^2 \vec{v}
[12M]
8.(b) Evaluate the line integral \int_C -y^3dx +x^3dy +z^3dz using Stokes’ theorem. Here C is the intersection of the cylinder x^2+y^2=1 and the plane x+y+z=1. The orientation on C corresponds to counterclockwise motion in the xy-plane.
[13M]
8.(c) Let \vec{F}=xy^2\hat{i}+(y+x)\hat{j}. Integrate (\nabla\times\vec{F})\cdot\vec{k} over the region in the first quadrant bounded by the curve y=x^2 and y=x using Green’s theorem.
[12M]
8.(d) Find f(y) such that (2xe^y+3y^2)dy+(3x^2+f(y))dx=0 is exact and hence solve.
[13M]