Paper I PYQs-2009

1.(a) Find a Hermitian and skew-Hermitian matrix each whose sum is the matrix \(\begin{bmatrix}{2 i} & {3} & {-1} \\ {1} & {2+3 i} & {2} \\ {-i+1} & {4} & {5 i}\end{bmatrix}\).

[12M]

1.(b) Prove that the set \(V\) of the vectors \((x_{1}, x_{2}, x_{3}, x_{4})\) in which \(\mathbb{R}^{4}\) satisfy the equation \(x_{1}+x_{2}+x_{3}+x_{4}=0\) and \(2 x_{1}+3 x_{2}-x_{3}+x_{4}=0\), is a subspace of \(\mathbb{R}^{4}\). What is dimension of this subspace? Find one of its bases.

[12M]

1.(c) Suppose that \(f^{\prime}\) is continuous on \([1,2]\) and that \(f\) has three zeroes in the interval \((1,2)\), show that \(f^{\prime \prime}\) has at least one zero in the interval \((1,2)\).

[12M]

1.(d) If \(f\) is the derivative of some function defined on \([a, b]\), prove that there exists a number \(\eta \in [a, b]\) such that \(\int_{a}^{b} f(t) d t=f(\eta)(b-a)\).

[12M]

1.(e) A line is drawn through a variable point on the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1\), \(z$=0\) to meet two fixed lines \(y=mx\), \(z=c\) and \(y=-mx\), \($z=-c\). Find the locus of the line.

[12M]

1.(f) Find the equation of the sphere having its centre on the plane \(4x- 5y- z = 3\), and passing through the circle

[12M]

2.(a) Let \(\beta=\{(1,1,0),(1,0,1),(0,1,1)\}\) and \(\beta^{\prime}=\{(2,1,1),(1,2,1),(-1,1,1)\}\) be the two ordered bases of \(R^{3}\). Then, find a matrix representing the linear transformation \(T: R^{3} \rightarrow R^{3}\) which transforms \(\beta\) into \(\beta^{\prime}\). Use this matrix representation to find \(T(x)\), where \(x=(2,3,1)\).

[20M]

2.(b) If \(x=3 \pm 0.01\) and \(y=4 \pm 0.01\) with approximately what accuracy can you calculate the polar coordinates \(r\) and \(\theta\) of the point \(P(x, y)\). Express your estimates as percentage changes of the value that \(r\) and \(\theta\) have at the point (3,4).

[20M]

2.(c) Find a \(2 \times 2\) real matrix \(A\) which is both orthogonal and skew-symmetric. Can there exist a \(3 \times 3\) real matrix which is both orthogonal and skew-symmetric? Justify your answer.

[20M]

3.(a) Let \(L: \mathbb{R}^{4} \rightarrow \mathbb{R}^{3}\) be a linear transformation defined by \(L\left(x_{1}, x_{2}, x_{3}, x_{4}\right) =\left(x_{3}+x_{4}-x_{1}-x_{2}, x_{3}-x_{2}, x_{4}-x_{1}\right)\). Then find the rank and nullity of \(L\). Also, determine null space and range space of \(L\).

[20M]

3.(b) Let \(f: R^2 \rightarrow R\) be defined as:

Is \(f\) continuous at (0,0)? Compute partial derivatives of \(f\) at any point \((x,y)\), if it exists.

[20M]

3.(c) A space probe in the shape of the ellipsoid \(4 x^{2}+y^{2}+4 z^{2}=16\) enters the earth’s atmosphere and its surface begins to heat. After one hour, the temperature at the point \((x, y, z)\) on the probe surface is given by \(T(x, y, z)=8 x^{2}+4 y z-16 z+1600\). Find the hottest point on the probe surface.

[20M]

4.(a) Prove that the set \(V\) of all \(3 \times 3\) real symmetric matrices forms a linear subspace of the space of all \(3 \times 3\) real matrices. What is the dimension of this subspace? Find at least one of the bases for \(V\).

[20M]

4.(b) Evaluate \(I=\iint_{s} (x d y d z+d z d x+x z^{2}) dx dy\), where \(S\) is the outer side of the part of the sphere \(x^{2}+y^{2}+z^{2}=1\) in the first octant.

[20M]

4.(c) Prove that the normals from the point \((\alpha, \beta, \gamma)\) to the paraboloid \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=2z\) lie on the cone:

[20M]

5.(a) A body is describing an ellipse of eccentricity \(e\) under the action of a central force directed towards a focus and when at the nearer apse, the centre of force is transferred to the other focus. Find the eccentricity of the new orbit in terms of the eccentricity of the original orbit.

[12M]

5.(b) Find the Wronskian of the set of functions: \(\left\{3 x^{3},\left \vert 3 x^{3}\right \vert \right\}\) on the interval \([-1,1]\) and determine whether the set is linearly dependent on \([-1,1]\).

[12M]

5.(c) A uniform rod \(AB\) is movable about a hinge at \(A\) and rests with one end in contact with a smooth vertical wall. If the rod is inclined at an angle of \(30^{\circ}\) with the horizontal, find the reaction at the hinge in magnitude and direction.

[12M]

5.(d) A shot fired with a velocity \(V\) at an elevation \(\alpha\) strikes a point \(P\) in a horizontal plane through the point of projection. If the point \(P\) is receding from the gun with \(V\), show that the elevation must be changed \(\theta\), where \(\sin 2 \theta=\sin 2 \alpha+\dfrac{2 v}{V} \sin \theta\).

[12M]

5.(e) Show that \(\operatorname{div}\left(\operatorname{grad} r^{n}\right)=n(n+1) r^{n-2}\), where \(r=\sqrt{x^{2}+y^{2}+z^{2}}\).

[12M]

5.(f) Find the directional derivative of
i) \(4 x z^{3}-3 x^{2} y^{2} z^{2}\) at \((2,-1,2)\) along \(z-axis\)
ii) \(-x^{2} y z+4 x z^{2}\) at \((1,-2,1)\) in the direction of \(2 \hat{i}-\hat{j}-2 \hat{\kappa}\).

[12M]

6.(a) Find the differential equation of the family of circles in the \(xy-plane\) passing through \((-1,1)\) and \((1,1)\).

[20M]

6.(b) Find the inverse Laplace transform of \(F(s)=1 n\left(\dfrac{s+1}{s+s}\right)\).

[20M]

6.(c) Solve:

[20M]

7.(a) One end of a light elastic string of natural length \(I\) and modulus of elasticity 2 \(\mathrm{mg}\) is attached to a fixed point \(\mathrm{O}\) and the other end to a particle of mass \(\mathrm{m}\). The particle initially held at rest at \(\mathrm{O}\) is let fall. Find the greatest extension of the string during the motion and show that the particle will reach \(O\) again after a time \(\left(\pi+2-\tan ^{1} 2\right) \sqrt{\dfrac{21}{g}}\).

[20M]

7.(b) A particle is projected with velocity \(V\) from the cusp of a smooth inverted cycloid down the arc. Show that the time of reaching the vertex is 2\(\sqrt{\dfrac{a}{g}} \cot ^{\prime}\left(\dfrac{V}{2 \sqrt{a g}}\right)\), where \(\alpha\) is the radius of the generating circle.

[10M]

7.(c) On a rigid body, the forces \(10(\hat{i} +2\hat{j}+2\hat{k})N\), \(5(-2\hat{i} -\hat{j}+2\hat{k})N\) and \(6(2\hat{i} +2\hat{j}-\hat{k})N\) are acting at points with position vectors \(\hat{i}-\hat{j}\), \(2\hat{i}+5\hat{k}\) and \(4\hat{i}-\hat{k}\) respectively. Reduce this system to a single force \(\vec{R}\) acting at the point \((4\hat{i}+2\hat{j})\) together with a couple \(\vec{G}\) whose axis passes through this point. Does the point \((4\hat{i}+2\hat{j})\) lie on the central axis?

[15M]

7.(d) Find the length of an endless chain which will hang over a circular pulley of radius \('a'\) so as to be in contact with the three-fourth of the circumference of the pulley.

[15M]

8.(a) Find the work done in moving the particle once round the ellipse \(\dfrac{x^{2}}{25}+\dfrac{y^{2}}{16}=1\), \(z=0\) under the field of force of given by \(\overline{F}=(2 x-y+z) \hat{i}+\left(x+y-z^{2}\right) \hat{j}+(3 x-2 y+4 z) \hat{k}\).

[20M]

8.(b) Using divergence theorem, evaluate \(\iint_{s} \overline{A} \cdot d \overline{S}\) where \(\overline{A}=x^{3} \hat{i}+y^{3} \hat{j}+z^{3} \hat{k}\) and \(S\) is the surface of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\).

[20M]

8.(c) Find the value of \(\iint_{s}(\vec{\nabla} \times \vec{f}) . d s\) taken over the upper portion of the surface \(x^{2}+y^{2}-2 a x+a z=0\) and the bounding curve lies in the plane \(z=0\), when \(\vec{F}=\left(y^{2}+z^{2}-x^{2}\right) \hat{i}+\left(z^{2}+x^{2}-y^{2}\right) \hat{j}+\left(x^{2}+y^{2}-z^{2}\right) \hat{k}\).

[20M]