Paper I PYQs-2010

1.(a) If \(\lambda_{1}\), \(\lambda_{2}\), \(\ldots \ldots_{3}\) are the Eigen values of the matrix \(\mathrm{A}= \begin{bmatrix}{26} & {-2} & {2} \\ {2} & {21} & {4} \\ {44} & {2} & {28}\end{bmatrix}\), show that \(\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}} \leq \sqrt{1949}\)

[12M]

1.(b) What is the null space of the differentiation transformation \(\dfrac{d}{d x}: p_{n} \rightarrow p_{n}\) where \(p_{n}\) is the space of all polynomials of degree \(\leq{n}\) over the real numbers? What is the null space of the second derivative as a transformation of? What is the null space of the \(k^{th}\) derivative \(p_{n}\)?

[12M]

1.(c) A twice differentiable function \(f(x)\) is such that \(f(a)=0=f(b)\) and \(f(c)>0\) for \(a<c<b\). Prove that there be is at least one point \(\xi\), \(a<\xi< b\) for which \(f^{\prime \prime}(\xi)< 0\).

[12M]

1.(d) Does the integral \(\int_{-1}^{1} \sqrt{\dfrac{1+x}{1-x}}dx\) exist? If so, find its value.

[12M]

1.(e) Show that the plane \(x+y-2 z=3\) cuts the sphere \(x^{2}+y^{2}+z^{2}-x+y=2\) in a circle of radius 1 and find the equation of the sphere which has this circle as a great circle.

[12M]

1.(f) Show that the function

\(f(x) =\)[x^2] + \vert x-1 \vert\(\)

is Riemann integrable m the interval [0, 2], where \([\alpha]\) denotes the greatest integer less than or equal to \(\alpha\). Can you give an example of a function that is not Riemann integrable on [0, 2]? Compute \(\int_0^2 f(x) dx\), where \(f(x)\) is as above.

[12M]

2.(a) Let \(M= \begin{bmatrix}{4} & {2} & {1} \\ {0} & {1} & {3}\end{bmatrix}\). Find the unique linear transformation \(T : \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}\) so that \(M\) is the matrix of \(T\) with respect to the basis \(\beta=\left\{v_{1}=(1,0,0) v_{2}=(1,1,0) v_{3}=(1,1,1)\right\}\) of \(\mathbb{R}^{3}\) and \(\beta^{\prime}=\left\{w_{1}=(1,0), w_{2}=(1,1)\right\}\) of \(\mathbb{R}^{2}\). Also find \(T(x, y, z)\).

[20M]

2.(b) Show that a box (rectangular parallelepiped) of maximum volume \(V\) with prescribed surface area is a cube.

[20M]

2.(c) Show that the plane \(3 x+4 y+7 z+\dfrac{5}{2}=0\) touches the paraboloid \(3 x^{2}+40 z\) and find the point of contact.

[20M]

3.(a) Let \(A\) and \(B\) be \(n \times n\) matrices over reals. Show that \(I-BA\) is invertible if \(I-A B\) is invertible. Deduce that \(A B\) and \(BA\) have the same Eigen values.

[20M]

3.(b) Let \(D\) be the region determine by the inequalities \(x>0, y>0, z< 8\) and \(z>x^{2}+y^{2}\). Compute \(\iiint_{D} 2 x d x d y d z\).

[20M]

3.(c) Show that every sphere through the circle \(x^{2}+y^{2}-2a x+r^{2}=0, z=0\) cuts orthogonally every sphere through the circle \(x^{2}+z^{2}=r^{2}, y=0\).

[20M]

4.(a)(i) In the space \(R^{n}\) determine whether or not the set \(\left\{e_{1}-e_{2}, e_{2}-e_{3}, \ldots \ldots, e_{n-1}-e_{n}, e_{n}-e_{1}\right\}\) is linearly independent.

[10M]

4.(a)(ii) Let \(T\) be a linear transformation from a vector space \(V\) over reals into \(V\) such that \(T-T^{2}=I\). Show that \(T\) is invertible.

[10M]

4.(b) If \(f(x, y)\) is a homogeneous function of degree \(n\) in \(x\) and \(y\), and has continuous first and second order partial derivatives, then show that:-
i) \(x \dfrac{\partial f}{\partial x}+y \dfrac{\partial f}{\partial y}=n f\)
ii) \(x^{2} \dfrac{\partial^{2} f}{\partial x^{2}}+2 x y \dfrac{\partial^{2} f}{\partial x \partial y}+y^{2} \dfrac{\partial^{2} f}{\partial y^{2}}=n(n-1) f\).

[10M]

4.(c) Find the vertices of the skew quadrilateral formed by the four generators of the hyperboloid \(\dfrac{x^{2}}{4}+y^{2}-z^{2}=49\) passing through \((10,5,1)\) and $(14,2,-2)$$.

[20M]

5.(a) Consider the differential equation \(y^{\prime}=\alpha y, x>0\) where \(\alpha\) is a constant. Show that:

i) If \(\phi(x)\) is any solution and \(\psi(x)=\phi(x) e^{-\alpha x}\), then \(\psi(x)\) is a constant.

ii) If \(\alpha<0\), then every solution tends to zero as \(x \rightarrow \infty\).

[12M]

5.(b) Show that the differential equation \(\left(3 y^{2}-x\right)+2 y\left(y^{2}-3\right) y^{\prime}=0\) admits an integrating factor which is a function of \(\left(x+y^{2}\right)\). Hence solve the equation.

[12M]

5.(c) Find \(\kappa / \tau\) for the curve

[12M]

5.(d) If \(v_{1}, v_{2}, V_{3}\) are the velocities at three points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) of the path of a projectile, where the inclinations to the horizon are \(\alpha, \alpha-\beta, \alpha-2 \beta\), and if \(\mathrm{t}_{1}, \mathrm{t}_{2}\) are the times of describing the arcs \(\mathrm{AB}, \mathrm{BC}\), respectively, prove that \(v_{3} t_{1}=v_{1} t_{2}\) and \(\dfrac{1}{v_{1}}+\dfrac{1}{v_{3}}=\dfrac{2 \cos \beta}{v_{2}}\).

[12M]

5.(e) Find the directional derivative of \(f(x, y)=x^{2} y^{3}+x y\) at the point \((2,1)\) in the direction of a unit vector which makes an angle of \(\dfrac{\pi}{3}\) with the \(x-axis\).

[10M]

5.(f) Show that the vector field defined by the vector function \(\vec{v}=x y z(y \vec{z} \vec{i}+x y \vec{j}+x y \vec{k})\) is conservative.

[12M]

6.(a) Verify that \(\dfrac{1}{2}(M x+N y) d\left[\log _{e}(x y)\right]+\dfrac{1}{2}(M x-N y) d\left[\log _{e}(x / y)\right]=M d x+N d y\). Hence show that:

i) If the differential equation \(M d x+N d y=0\) is homogeneous, then \((M x+N y)\) is an integrating factor unless \(M x+N y \equiv 0\).

ii) If the differential equation \(M d x+N d y=0\) is not exact but is of the form \(f_{1}(x y) y d x+f_{2}(x y) x d y=0\) then \((M x-N y)^{-1}\) is an integrating factor unless \(M x+N y \equiv 0\).

[20M]

6.(b) A particle slides down the arc of a smooth cycloid whose axis is vertical and vertex lowest. Prove that the time occupied in falling down the first half of the vertical height is equal to the the time of falling down the second half.

[20M]

6.(c) Prove that \(\operatorname{div}(f \vec{V})=f(d i v \overline{V})+(\operatorname{grad} . f) \vec{V}\), where \(\mathrm{f}\) is a scalar function.

[20M]

7.(a) Show that the set of solutions of the homogeneous linear differential equation

on an interval \(I=[a,b]\) forms a vector subspace \(W\) of the real vector space of continuous functions on \(I\). What is the dimension of $$$W$?

[20M]

7.(b) A particle moves with a central acceleration \(\mu\left(r^{5}-9 r\right)\), being projected from and apse at a distance \(\sqrt{3}\) with velocity 3\(\sqrt{(2 \mu)}\). Show that its path is the curve \(x^{4}+y^{4}=9\).

[20M]

7.(c) Use the divergence theorem to evaluate \(\iint_{s} \vec{V}ndA\), where \(\vec{V}=x^{2} z \vec{i} y \vec{j}-x z^{2} \vec{k}\) and \(S\) is he boundary of the region bounded by the paraboloid \(z=x^{2}+y^{2}\) and the plane \(z=4y\).

[20M]

8.(a) Use the method of undetermined coefficients to find the particular solutions of \(y^{\prime \prime}+y=\sin x+\left(1+x^{2}\right) e^{x}\) and hence find its general solution.

[20M]

8.(b) A solid hemisphere is supported by a string fixed to a point on its rim and to a point on a smooth vertical wall with which the curved surface of the hemisphere is in contact. If \(\theta\) and \(\phi\) are the inclinations of the string and the plane base of the hemisphere to the vertical, prove by using the principle of virtual work that tan \(\phi=\dfrac{3}{8}+\tan \theta\).

[20M]

8.(c) Verify Green’s theorem for \(e^{-x} \sin y d x+e^{-x} \cos y\) by the path of integration being the boundary of the square whose vertices are \((0,0)\), \(\left(\dfrac{\pi}{2},0\right)\), \(\left(\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)\) and \(\left(0, \dfrac{\pi}{2}\right)\).

[20M]