Paper I PYQs-2018
Section A
1.(a) Let \(A\) be a \(3\times 2\) matrix and \(B\) a \(2\times 3\) matrix. Show that \(C=A\cdot B\) is a singular matrix.
[10M]
1.(b) Express basis vectors \(e_1=(1,0)\) and \(e_2=(0,1)\) as linear combination of \(\alpha_1=(2,-1)\) and \(\alpha_2=(1,3)\).
[10M]
1.(c) Determine if \(\lim_{z \to 1} (1-z) \tan\dfrac{\pi z}{2}\) exists or not. If the limit exists, then find its value.
[10M]
1.(d) Find the limit \(\lim_{n \to \infty} \dfrac{1}{n^2}\sum_{r=0}^{n-1} \sqrt{n^2-r^2}\).
[10M]
1.(e) Find the projection of the straight line \(\dfrac{x-1}{2}=\dfrac{y-1}{3}=\dfrac{z+1}{-1}\) on the plane \(x+y+2z=6\).
[10M]
2.(a) Show that if \(A\) and \(B\) are similar \(n\times n\) matrices, then they have the same eigenvalues.
[12M]
2.(b) Find the shortest distance from the point \((1,0)\) to the parabola \(y^2=4x\).
[13M]
2.(c) The ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) revolves about the \(x-axis\). Find the volume of the solid of revolution.
[13M]
2.(d) Find the shortest distance between the lines
\[a_1x+b_1y+c_1z+d_1=0\] \[a_2x+b_2y+c_2z+d_2=0\]and 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
\(f(x,y) = xy^2\), if \(y>0\) and
\(f(x,y) = -xy^2\), if \(y \leq 0\)
Determine which of \(\dfrac{\partial f}{\partial x}(0,1)\) and \(\dfrac{\partial f}{\partial y}(0,1)\) 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 \(x^4-5x^2+4\) on the interval \([2,3]\).
[13M]
4.(b) Evaluate the integral \(\int^a_0 \int^x_{x/a} \dfrac{xdydx}{x^2+y^2}\).
[12M]
4.(c) Find the equation of the cone with \((0,0,1)\) as the vertex and \(2x^2-y^2=4\), \(z=0\) 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:
\[y''-y=x^2e^{2x}\][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]