Paper I PYQs-2017

1.(a) Let \(\mathrm{A}\) be a square matrix of order 3 such that each of its diagonal elements is ‘\(a\)’ and each of its off-diagonal elements is 1. If \(\mathrm{B}=\mathrm{bA}\) is orthogonal, determine the values of \(a\) and \(b\).

[8M]

1.(b) Let \(\mathrm{V}\) be the vector space of all \(2 \times 2\) matrices over the field \(\mathrm{R}\). Show that \(W\) is not a subspace of \(\mathrm{V},\) where
(i) \(W\) contains all \(2 \times 2\) matrices with zero determinant.
(ii) \(W\) consists of all \(2 \times 2\) matrices \(A\) such that \(\mathrm{A}^{2}=\mathrm{~A}\).

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1.(c) Using the Mean Value Theorem, show that
(i) \(f(x)\) is constant in \([a, b],\) if \(f^{\prime}(x)=0\) in \([a, b]\)
(ii) \(f(x)\) is a decreasing function in \((a, b),\) if \(f^{\prime}(x)\) exists and is \(<0\) everywhere in \((\mathrm{a}, \mathrm{b})\)

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1.(d) Jacobian \(\mathrm{J}=\dfrac{\partial(\mathrm{u}, \mathrm{v})}{\partial(\mathrm{x}, \mathrm{y})},\) and hence show that \(\mathrm{u}, \mathrm{v} .\) are independent unless

\[\dfrac{a}{A}=\dfrac{b}{B}=\dfrac{h}{H}\]

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1.(e) Find the equations of the planes parallel to the plane \(3 x-2 y+6 z+8=0\) and at a distance 2 from it.

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2.(a) State the Cayley-Hamilton theorem. Verify this theorem for the matrix
\(\mathbf{A}=\left[\begin{array}{ccc}1 & 0 & 2 \\ 0 & -1 & 1 \\ 0 & 1 & 0\end{array}\right] .\) Hence find \(A^{-1}\)

[10M]

2.(b) Show that

\[\int_{0}^{\pi / 2} \sin ^{p} \theta \cos ^{q} \theta \mathrm{d} \theta=\dfrac{1}{2} \dfrac{\Gamma\left(\dfrac{\mathrm{p}+1}{2}\right) \Gamma\left(\dfrac{\mathrm{q}+1}{2}\right)}{\Gamma\left(\dfrac{\mathrm{p}+\mathrm{q}+2}{2}\right)}, \mathrm{p}, \mathrm{q}>-1\]

Hence evaluate the following integrals:

(i) \(\int_{0}^{\pi / 2} \sin ^{4} x \cos ^{5} x d x\)
(ii) \(\int_{0}^{1} x^{3}\left(1-x^{2}\right)^{5 / 2} d x\)
(iii) \(\int_{0}^{1} x^{4}(1-x)^{3} d x\)

[10M]

2.(c) Find the maxima and minima for the function

\[f(x, y)=x^{3}+y^{3}-3 x-12 y+20\]

Also find the saddle points (if any) for the function.

[10M]

2.(d) Show that the angles between the planes given by the equation \(2 x^{2}-y^{2}+3 z^{2}-x y+7 z x+2 y z=0\) is \(\tan ^{-1} \dfrac{\sqrt{50}}{4}\).

[10M]

3.(a) Reduce the following matrix to a row-reduced echelon form and hence find its rank:

\[\mathbf{A}=\left[\begin{array}{rrrr} -1 & 2 & -1 & 0 \\ 2 & 4 & 4 & 2 \\ 0 & 0 & 1 & 5 \\ 1 & 6 & 3 & 2 \end{array}\right]\]

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3.(b) Given that the set \(\{u, v, w\}\) is linearly independent, examine the sets

(i) \(\quad\{\mathbf{u}+\mathbf{v}, \mathbf{v}+\mathbf{w}, \mathbf{w}+\mathbf{u}\}\)
(ii) \(\quad\{\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v}, \mathbf{u}-2 \mathbf{v}+2 \mathbf{w}\}\) for linear independence.

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3.(c) Evaluate the integral \(\int_{0}^{\infty} \int_{0}^{\infty} \mathrm{e}^{-\left(x^{2}+y^{2}\right)} d x d y,\) by changing to polar cöordinates. Hence show-that \(\int_{0}^{\infty} \mathrm{e}^{-\mathrm{x}^{2}} \mathrm{dx}=\dfrac{\sqrt{\pi}}{2}\).

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3.(d) Find the angle between the lines whose direction cosines are given by the relations \(l+\mathrm{~m}+\mathrm{n}=0\) and \(2 \mathrm{lm}+2 \ln -\mathrm{mn}=0\).

[10M]

4.(a) Find the eigenvalues and the corresponding eigenvectors for the matrix \(A=\left[\begin{array}{ll}0 & -2 \\ 1 & 3\end{array}\right]\). Examine whether the matrix \(A\) is diagonalizable. Obtain a matrix \(\mathrm{D}\) (if it is diagonalizable) such that \(\mathrm{D}=\mathrm{P}^{-1} \mathrm{~A}P\).

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4.(b) A function \(f(x, y)\) is defined as follows:

\[f(x, y)=\left\{\begin{array}{cl} \dfrac{x^{2} y^{2}}{x^{2}+y^{2}}, & \text { if }(x, y) \neq(0,0) \\ 0, & \text { if }(x, y)=(0,0) \end{array}\right.\]

Show that \(\mathrm{f}_{\mathrm{xy}}(0,0)=\mathrm{f}_{\mathrm{yx}}(0,0)\)

[10M]

4.(c) Find the equation of the right circular cone with vertex at the origin and whose axis makes equal angles with the coordinate axes and the generator is the line passing through the origin with direction ratios \((1,-2,2)\).

[10M]

4.(d) Find the shortest distance and the equation of the line of the shortest distance between the lines \(\dfrac{x-3}{3}=\dfrac{y-8}{-1}=\dfrac{z-3}{1}\) and \(\dfrac{x+3}{-3}=\dfrac{y+7}{2}=\dfrac{z-6}{4}\).

[10M]

5.(a) Solve

\(\left(2 D^{3}-7 D^{2}+7 D-2\right) y=e^{-8 x}\) where \(D=\dfrac{d}{d x}\)

[8M]

5.(b) Solve the differential equation
\(x^{2} \dfrac{d^{2} y}{d x^{2}}-2 x \dfrac{d y}{d x}-4 y=x^{4}\)

[8M]

5.(c) A particle is undergoing simple harmonic motion of period T about a centre \(O\) and it passes through the position \(P(O P=b)\) with velocity \(v\) in the direction \(OP\). Prove that the time that elapses before it returns to \(\mathrm{P}\) is \(\dfrac{\mathrm{T}}{\pi} \tan ^{-1}\left(\dfrac{\mathrm{vT}}{2 \pi \mathrm{b}}\right)\).

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5.(d) A heavy uniform cube balances on the highest point of a sphere whose radius is \(r\). If the sphere is rough enough to prevent sliding and if the side of the cube be \(\dfrac{\pi r}{2},\) then prove that the total angle through which the cube can swing without falling is \(90^{\circ}\).

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5.(e) Prove that \({\nabla}^{2} r^{n}=n(n+1) r^{n-2}\)

and that \(\mathrm{r}^{\mathrm{n}} \overrightarrow{\mathrm{r}}\) -is irrotational, where \(\mathrm{r}=\vert \overrightarrow{\mathrm{r}} \vert=\sqrt{\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}}\).

[8M]

6.(a) Solve the differential equation

\[-\left(\dfrac{d y}{d x}\right)^{2}+2 \cdot \dfrac{d y}{d x} \cdot y \cot x=y^{2}\]

[15M]

6.(b) A string of length \(a\), forms the shorter diagonal of a rhombus formed of four uniform rods, each of length \(\mathrm{b}\) and weight \(\mathrm{W},\) which are hinged together. If one of the rods is supported in a horizontal position, then prove that the tension of the string is \(\dfrac{2 \mathrm{W}\left(2 \mathrm{b}^{2}-\mathrm{a}^{2}\right)}{\mathrm{b} \sqrt{4 \mathrm{b}^{2}-\mathrm{a}^{2}}}\).

[10M]

6.(c) Using Stokes’ theorem, evaluate

\[\int_{\mathrm{C}}[(x+y) d x+(2 x-z) d y+(y+z) d z]\]

where \(\mathrm{C}\) is the boundary of the triangle with vertices at \((2,0,0)\), \((0,3,0)\) and \((0,0,6)\).

[15M]

7.(a) Solve the differential equation

\[\mathrm{e}^{3 \mathrm{x}}\left(\dfrac{\mathrm{dy}}{\mathrm{dx}}-1\right)+\left(\dfrac{\mathrm{dy}}{\mathrm{dx}}\right)^{3} \mathrm{e}^{2 \mathrm{y}}=0\]

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7.(b) A planet is describing an ellipse about the Sun as a focus. Show that its velocity away from the Sun is the greatest when the radius vector to the planet is at a right angle to the major axis of path and that-the velocity then is \(\dfrac{2 \pi \text { ae }}{\mathrm{T} \sqrt{1-\mathrm{e}^{2}}},\) where \(2 \mathrm{a}\) is the major axis, e is the eccentricity and \(\mathrm{T}\) is the periodic time.

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7.(c) A semi-ellipse bounded by its minor axis is just immersed in a liquid, the density of which varies as the depth. If the minor axis lies on the surfare, then find the eccentricity in order that the focus may be the centre of pressure.

[10M]

7.(d) Evaluate \(\iint_{\mathrm{S}}(\nabla \times \overrightarrow{\mathrm{f}}) \cdot \hat{\mathrm{n}} \mathrm{d} \mathrm{S}\) \(y=3=\dfrac{-3}{2}(3 \cdot 0)\) where \(S\) is the surface of the cone, \(z=2-\sqrt{x^{2}+y^{2}}\) above xy-plane and \(\vec{f}=(x-z) \hat{i}+\left(x^{3}+y z\right) \hat{j}-3 x y^{2} \hat{k}\)

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8.(a) Solve \(\dfrac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+4 \mathrm{y}=\tan 2 \mathrm{x}\) by using the method of variation of parameter.

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