Paper I PYQs-2012
Section A
1.(a) Let \(V=\mathbb{R}^{3}\) and \(\alpha_{1}=(1,1,2), \alpha_{2}=(0,1,3)\) \(\alpha_{3}=(2,4,5)\) and \(\alpha_{4}=(-1,0,-1) \cdot\) be the elements of \(V\). Find a basis for the intersection of the subspace spanned by \(\left\{\alpha_{1}, \alpha_{2}\right\}\) and \(\left\{\alpha_{3}, \alpha_{4}\right\}\)
[8M]
1.(b) Show that the set of all functions which satisfy the differential equation, \(\dfrac{d^{2} f}{d x^{2}}+3 \dfrac{d f}{d x}=0\) is a vector space.
[8M]
1.(c) If the three thermodynamic variables \(P, V, T\) are connected by a relation \(f(P, V, T)=0\) show that, \(\left(\dfrac{\partial P}{\partial T}\right)_{V} \cdot \left(\dfrac{\partial T}{\partial V}\right)_{P}\left(\dfrac{\partial \hat{V}}{\partial P}\right)_{T} \cong-1\).
[8M]
1.(d) If \(u=A e^{-g x} \sin (n t-g x),\) where \(A, g, n\) are positive constants, satisfies the heat conduction equation, \(\dfrac{\partial u}{\partial t}=\mu \dfrac{\partial^{2} u}{\partial x^{2}}\) then show that \(g=\sqrt{\left(\dfrac{n}{2 \mu}\right)}\).
[8M]
1.(e) Find the equations to the lines in which the plane \(2 x+y-z=0\) cuts the cone
\[4 x^{2}-y^{2}+3 z^{2}=0\][8M]
2.(a) Let \(f: \mathbb{R} \rightarrow \mathbb{R}^{3}\) be a linear transformation defined by \(f(a, b, c)=(a, a+b, 0)\).
Find the matrices \(A\) and \(B\) respectively of the linear transformation \(f\) with respect to the standard basis \(\left(e_{1}, e_{2}, e_{3}\right)\) and the basis \(\left(e_{1}^{\prime}, e_{2}^{\prime}, e_{3}^{\prime}\right)\) where \(e_{1}^{\prime}=(1,1,0), e_{2}^{\prime}=(0,1,1)\) \(e_{3}^{\prime}=(1,1,1)\).
Also, show that there exists an invertible matrix \(P\) such that
\[B=P^{-1} A P\][10M]
2.(b) Verify Cayley-Hamilton theorem for the matrix \(A=\left(\begin{array}{ll}1 & 4 \\ 2 & 3\end{array}\right)\) and find its inverse. Also express \(A^{5}-4 A^{4}-7 A^{3}+11 A^{2}-A-10 I\) as a linear polynomial in \(A\).
[10M]
2.(c) Find the equations of the tangent plane to the ellipsoid \(2 x^{2}+6 y_{1}^{2}+3 z^{2}=27\) which passes through the line \(x-y-z=0=x-y+2 z-9\).
[10M]
2.(d) Show that there are three real values of \(\lambda\) for which the equations: \((a-\lambda) x+b y+c z=0\), \(b x+(c-\lambda) y+a z=0\), \(c x+a y+(b-\lambda) z=0\) are simultaneously true and that the product of these values of \(\lambda\) is \(D=\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}\).
[10M]
3.(a) Find the matrix representation of linear transförmation \(T\) on \(V_{3}(I R)\) defined as \(T(a, b, c)=(2 b+c, a-4 b, 3 a)\) corresponding to the basis \(B=\{(1,1,1),(1,1,0),(1,0,0)\}\).
[10M]
3.(b) Find the dimensions of the rectangular box, open at the top, of maximum capacity whose surface is 432 sq. \(\mathrm{cm}\).
[10M]
3.(c) If \(2 C\) is the shortest distance between the lines
\[\dfrac{x}{l}-\dfrac{z}{n}=1, y=0\]and \(\dfrac{y}{m}+\dfrac{z}{n}=1, x=0\)
then show that
\[\dfrac{1}{l^{2}}+\dfrac{1}{m^{2}}+\dfrac{1}{n^{2}}=\dfrac{1}{c^{2}}\][10M]
3.(d) Show that the function defined as
\[f(x)=\left\{\begin{array}{cc}\dfrac{\sin 2 x}{x} & \text { when } x \neq 0 \\ 1 & \text { when } x=0\end{array}\right.\]has removable discontinuity at the origin.
[10M]
4.(a) Find by triple Integration the volume cut off from the cylinder \(x^{2}+y^{2}=a x\) by the planes \(z=m x\) and \(z=n x\).
[10M]
4.(b) Show that afl the spheres, that can be drawn through the origin and each set of points where planes parallel to the plane \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=0\) cut the co-ordinate axes, form a system of spheres which are cut orthogonally by the sphere
\[x^{2}+y^{2}+2 f x+2 g y+2 h z=0\]if \(a f+b g+c h=0\)
[10M]
4.(c) A plane makes equal intercepts on the positive parts of the axes and touches the ellipsoid \(x^{2}+4 y^{2}+9 z^{2}=36\). Find its equation.
[10M]
4.(d) Evaluate the following in terms of Gamma function:
\[\int_{0}^{a} \sqrt{\left(\dfrac{x^{3}}{a^{3}-x^{3}}\right) d x}\][10M]
Section B
5.(a) Solve \(\dfrac{d y}{d x}-\dfrac{\tan y}{1+x}=(1+x) e^{x} \sec y\)
[8M]
5.(b) Solve and find the singular solution of \(x^{3} p^{2}+x^{2} p y+a^{3}=0\).
[8M]
5.(c) A particle is projected vertically upwards from the earth’s surface with a velocity just sufficient to carry it to infinity. Prove that the time it takes to reach a height \(h\) is \(\dfrac{1}{3} \sqrt{\left(\dfrac{2 a}{g}\right)}\left[\left(1+\dfrac{h}{a}\right)^{3 / 2}-1\right]\).
[8M]
5.(d) A triangle \(A B C\) is immersed in a liquid with the vertex \(C\) in the surface and the sides \(A C\), \(B C\) equally inclined to the surface. Show that the vertical \(C\) divides the triangle into two others, the fluid pressures on which are as \(b^{3}+3 a b^{2}: a^{3}+3 a^{2} b\) where \(a\) and \(b\) are the sides \(B C\) & \(AC\) respectively.
[8M]
5.(e) If \(u=x+y+z\), \(v=x^{2}+y^{2}+z^{2}\), \(w=y z+z x+x y\), prove that grad \(u\), grad \(v\) and grad \(w\) are coplanar.
[8M]
6.(a) Solve:
\[x^{2} y \dfrac{d^{2} y}{d x^{2}}+\left(x \dfrac{d y}{d x}-y\right)^{2}=0\][10M]
6.(b) Find the value of \(\iint_{s}(\vec{\nabla} \times \vec{F}) \cdot \overrightarrow{d s}\) 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
\[\vec{F}=\left(y^{2}+z^{2}-x\right) \vec{i}+\left(z^{2}+x^{2}-y^{2}\right) \vec{j}+\left(x^{2}+y^{2}-z^{2}\right) \vec{k}\][10M]
6.(c) A particle is projected with a velocity \(u\) and strikes at right angle on a plane through the plane of projection inclined at an angle \(\beta\) to the horizon. Show that the time of flight is \(2u\) \(\overline{g \sqrt{\left(1+3 \sin ^{2} \beta\right)}}\) range on the plane is \(\dfrac{2 u^{2}}{g} \cdot \dfrac{\sin \beta}{1+3 \sin ^{2} \beta}\) and the vertical height of the point struck is \(\dfrac{2 u^{2} \sin ^{2} \beta}{g\left(1+3 \sin ^{2} \beta\right)}\) above the point of projection.
[10M]
6.(d) Solve \(\dfrac{d^{4} y}{d x^{4}}+2 \dfrac{d^{2} y}{d x^{2}}+y=x^{2} \mathrm{c}\).
[10M]
7.(a) A particle is moving with central acceleration \(\mu\left[r^{5}-c^{4} r\right]\) being projected from an apse at. a distance \(c\) with velocity \(\sqrt{\left(\dfrac{2 \mu}{3}\right) c^{3}},\) show that its path is a curve, \(x^{4}+y^{4}=c^{4}\).
[13M]
7.(b) A thin equilateral rectangular plate of uniform thickness and density rests with one end of its base on a rough horizontal plane and the other against a small vertical wall. Show that the least angle, its base can make with the horizontal plane is given by \(\cot \theta=2 \mu+\dfrac{1}{\sqrt{3}}\) \(\mu\), being the coefficient of friction.
[14M]
7.(c) A semicircular area of radius \(a\) is immersed vertically with its diameter horizontal at a depth \(b .\) If the circumference be below the centre, prove that the depth of centre of pressure is \(\dfrac{1}{4} \dfrac{3 \pi\left(a^{2}+4 b^{2}\right)+32 a b}{4 a+3 \pi b}\)
[13M]
8.(a) Solve \(x=y \dfrac{d y}{d x}-\left(\dfrac{d y}{d x}\right)^{2}\).
[10M]
8.(b) Find the value of the line integral aver a circular path given by \(x^{2}+y^{2}=a^{2}, z=0\) where the vector field, \(\vec{F}=(\sin y) \vec{i}+x(1+\cos y) \vec{j}\).
[10M]
8.(c) A heavy elastic string, whose natural length is \(2 \pi a,\) is placed round a smooth cone whose axis is vertical and whose semi vertical angle is \(\alpha\). If \(W\) be the weight and \(\lambda\) the modulus of elasticity of the string, prove that it will be in equilibrium when in the form of a circle whose radius is
\[a\left(1+\dfrac{W}{2 \pi \lambda} \cot \alpha\right)\][10M]
8.(d) Solve \(x^{2} \dfrac{d^{2} y}{d x^{2}}+3 x \dfrac{d y}{d x}+y=(1-x)^{-2}\).
[10M]