Paper I PYQs-2014

1.(a) Find one vector in \(R^{3}\) which generates the intersection of \(V\) and \(W\), where \(V\) is the \(x y-plane\) 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 \(\left[ \begin{array}{cccc}{0} & {1} & {-3} & {-1} \\ {0} & {0} & {1} & {1} \\ {3} & {1} & {0} & {2} \\ {1} & {1} & {-2} & {0}\end{array}\right]\)

[10M]

1.(c) Prove that between two real roots of \(e^{x} \cos x+1=0\), a real root of \(e^{x} \sin x+1=0\) lies.

[10M]

1.(d) Evaluate \(\int_{0}^{1} \dfrac{\log _{e}(1+x)}{1+x^{2}} dx\)

[10M]

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

[10M]

2.(a) Let \(V\) and \(W\) be the following subspaces of \(R^{4}: V=\{(a, b, c, d) : b-2 c+d=0\}\) and \(W=\{(a, b, c, d): a=d, b=2 c\}\). Find a basis and the dimension of:
(i) \(V\)
(ii) \(V \cap W\).

[15M]

2.(b)(i) Investigate the values of \(\lambda\) and \(\mu\) so that the equations \(x+y+z=6\), \(x+2 y+3 z=10\), \(x+2 y+\lambda z=\mu\) 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 \(A=\begin{bmatrix}{1} & {4} \\ {2} & {3}\end{bmatrix}\) and hence find its inverse. Also, find the matrix represented by \(A^{5}-4 A^{4} - 7A^3+11 A^{2}-A-10 I\).

[10M]

2.(c) By using the transformation \(x+y=u\), \(y=u v\), evaluate the integral \(\iint\{x y(1-x-y)\}^{1/2} d x d y\) 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 \(x^{2}+y^{2}+z^{2}\) subject to the condition \(a x^{2}+b y^{2}+c z^{2}=1\) and \(l x+m y+n z=0\) interpret result geometrically.

[20M]

3.(c)(i) Let \(A= \begin{bmatrix}{-2} & {2} & {-3} \\ {2} & {1} & {-6} \\ {-1} & {-2} & {0}\end{bmatrix}\). 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 \(x^{2}+y^{2}+z^{2}-4 x+2 y=4\) the tangent planes at which are parallel to the plane \(2 x-y+2 z=1\).

[10M]

4.(a)(ii) Prove that equation \(a x^{2}+b y^{2}+c z^{2}+2 u x+2 v y+d=0\) represents a cone if \(\dfrac{u^{2}}{a}+ \dfrac{v^{2}}{b}+ \dfrac{w^{2}}{c}=d\).

[10M]

4.(b) Show that the lines drawn from the origin parallel to the normals to the central conicoid \(a x^{2}+b y^{2}+c z^{2}=1\), at its points of intersection with the plane \(l x+m y+n z=p\) generate the cone \(\quad p^{2}\left(\dfrac{x^{2}}{a}+\dfrac{y^{2}}{b}+\dfrac{z^{2}}{c}\right)=\left(\dfrac{l x}{a}+\dfrac{m y}{b}+\dfrac{n z}{c}\right)^{2}\).

[15M]

4.(c) Find the equations of the two generating lines through any point \((a \cos \theta, b \sin \theta, 0)\) of the principal elliptic section \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\), \(z=0\) of the hyperboloid by the plane \(z=0\).

[15M]

5.(a) Justify that a differential equation of the form: \(\left[y+x f\left(x^{2}+y^{2}\right)\right] d x+\left[y f\left(x^{2}+y^{2}\right)-x\right] d y=0\) where \(f\left(x^{2}+y^{2}\right)\) is an arbitrary function of \(\left(x^{2}+y^{2}\right)\), is not an exact differential equation and \(\dfrac{1}{x^{2}+y^{2}}\) is an integrating factor for it. Hence solve this differential equation for \(f\left(x^{2}+y^{2}\right)=\left(x^{2}+y^{2}\right)^{2}\).

[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 \(\dfrac{T}{\pi} \cos ^{\prime}\left(\dfrac{b}{a}\right)\).

[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 2\(\theta\) is the angle between the rods, then find the relation between \(l\), \(r\) and \(\theta\), by using the principle of virtual work.

[10M]

5.(e) Find the curvature vector at any point of the curve \(\overline{r}(t)=t \cos t \hat{i}+t \sin t \hat{j}\), \(0 \leq t \leq 2 \pi\). Give its magnitude also.

[10M]

6.(a) Solve by the method of variation of parameters: \(\dfrac{d y}{d x}-5 y=\sin x\).

[10M]

6.(b) Solve the differential equation: \(x^{3} \dfrac{d^{3} y}{d x^{3}}+3 x^{2} \dfrac{d^{2} y}{d x^{2}}+x \dfrac{d y}{d x}+8 y=65 \cos \left(\log _{e} x\right)\).

[20M]

6.(c) Evaluate by Stokes’ theorem:

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

[20M]

7.(a) Solve the following differential equation: \(x \dfrac{d^{2} y}{d x^{2}}-2(x+1) \dfrac{d y}{d x}+(x+2) y=(x-2) e^{2 x}\), when \(e^{x}\) is a solution to its corresponding homogeneous differential equation.

[15M]

7.(b) A particle of mass \(\mathrm{m}\), 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\(\sqrt{g l}\). 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 \(\mathrm{ABCDE}\), formed of equal heavy uniform bars jointed together, is suspended from the joint \(\mathrm{A}\), and is maintained in form by a light rod joining the middle points of \(\mathrm{BC}\) and \(DE\). Find the stress in this rod.

[20M]

8.(a) Find the sufficient condition for the differential equation \(M(x, y) d x+N(x, y) d y=0\) 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 \(\left(x^{2}+x y\right) d x+\left(y^{2}+x y\right) d y=0\) 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 \(\mu y-2\) and when \(y=a\), it is projected parallel to the axis of \(X\) with velocity \(\sqrt{\dfrac{2 \mu}{a}}\). Find the parametric equation of the path of the particle. Here \(\mu\) is a constant.

[15M]