Paper I PYQs-2019

Section A

1.(a) Let $f : [0, \frac{π}{2} \to R]$ be a continuous function such that

f (x) = \frac{c o s^{2} x}{4 x^{2} - π^{2}}, 0 \leq x \leq \frac{π}{2}

Find the value of $f (\frac{π}{2})$ .

[10M]

Some sample ans

1.(b) Let $f : D (\subseteq R^{2}) \to R$ be a function and $(a, b) \in D$ . If $f (x, y)$ is continuous at $(a, b)$ , then show that the function $f (x, b)$ and $f (a, y)$ are continuous at $x = a$ and at $y = b$ respectively.

[10M]

1.(c) Let $T : R^{2} \to R^{2}$ be a linear map such that that $T (2, 1) = (5, 7)$ and $T (1, 2) + (3, 3)$ . If $A$ is the matrix corresponding to $T$ with repect to the standard bases $e_{1}$ , $e_{2}$ , then find $R a n k (A)$ .

[10M]

1.(d) If $A = [\begin{matrix} 1 & 2 & 1 \\ 1 & - 4 & 1 \\ 3 & 0 & - 3 \end{matrix}]$ and $B = [\begin{matrix} 2 & 1 & 1 \\ 1 & - 1 & 0 \\ 2 & 1 & - 1 \end{matrix}]$ , then show that $A B = 6 I_{3}$ . Use this result to solve the following system of equations:

$2 x + y + z = 5$
$x - y = 0$
$2 x + y - z = 1$

[10M]

1.(e) Show that the lines $\frac{x + 1}{- 3} = \frac{y - 3}{2} = \frac{z + 2}{1}$ and $\frac{x}{1} = \frac{y - 7}{- 3} = \frac{z + 7}{2}$ intersect. Find the coordinates of the point of intersection and the equation of the plane containing them.

[10M]

2.(a) Is $f (x) = | \cos x | + | \sin x |$ is differentiable at $x = \frac{π}{2}$ ? If yes, then find its derivative at $x = \frac{π}{2}$ . If no, then give a proof of it.

[15M]

2.(b) Let $A$ and $B$ be two orthogonal matrices of same order and $d e t$ $A$ + $d e t$ $B$ =0.Show that $A + B$ is a singular matrix.

[15M]

2.(c)(i) The plane $x + 2 y + 3 z = 12$ cuts the axes of coordinates in $A$ , $B$ , $C$ . Find the equations of the circle circumscribing the triangle $A B C$ .

[10M]

2.(c)(ii) Prove that the plane $z = 0$ cuts the enveloping cone of the sphere $x^{2} + y^{2} + z^{2} = 11$ which has the vertex at $(2, 4, 1)$ in a rectangulat hyperbola.

[10M]

3.(a) Find the maximum and the minimum value of the function $f (x) = 2 x^{3} - 9 x^{2} + 12 x + 6$ on the interval $(2, 3)$ .

[15M]

3.(b) Prove that, in general, three normals can be drawn from a given point to the paraboloid $x^{2} + y^{2} = 2 a z$ , but if the point lies on the surface

27 a (x^{2} + y^{2}) + 8 (a - z)^{3} = 0

then two of the three normal coincide.

[10M]

3.(c) Let $A = [\begin{matrix} 5 & 7 & 2 & 1 \\ 1 & 1 & - 8 & 1 \\ 2 & 3 & 5 & 0 \\ 3 & 4 & - 3 & 1 \end{matrix}]$ .

(i) Find the rank of matrix $A$ .

[15M]

(ii) Find the dimension of the subspace

V = {(x_{1}, x_{2}, x_{3}, x_{4}) \in R^{4} | A (\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}) = 0}

[5M]

4.(a) State the Cayley-Hamilton theorem. Use this theorem to find $A^{100}$ , where

A = [\begin{matrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]

[15M]

4.(b) Find the length of the normal chord through a point P of the ellipsoid

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1

and prove that if it is equal to $4 P G_{3}$ , where $G_{3}$ is the point where the normal chord through $P$ meets the $x y - p l a n e$ , then $P$ lies on the cone

\frac{x^{6}}{a^{6}} (2 c^{2} - a^{2}) + \frac{y^{6}}{b^{6}} (2 c^{2} - b^{2}) + \frac{z^{4}}{c^{4}} = 0

[15M]

4.(c)(i) If

u = s i n^{- 1} \sqrt{\frac{x^{1 / 3} + y^{1 / 3}}{x^{1 / 2} + y^{1 / 2}}},

then show that $\sin^{2} u$ is homogeneous function of $x$ and $y$ of degree $- \frac{1}{6}$ . Hence show that

$x^{2} \frac{\partial^{2} u}{\partial x^{2}} + 2 x y \frac{\partial^{2} u}{\partial x \partial y} + y^{2} \frac{\partial^{2} u}{\partial y^{2}}$ = $\frac{\tan u}{12} (\frac{13}{12} + \frac{\tan^{2} u}{12})$

[12M]

4.(c)(ii) Using the Jacobian method, show that if $f^{'} (x) = \frac{1}{1 + x^{2}}$ and $f (0) = 0$ , then

f (x) + f (y) = f (\frac{x + y}{1 - x y})

[8M]

Section B

5.(a) Solve the differential equation

(2 y \sin x + 3 y^{4} \sin x \cos x) d x - (4 y^{3} \cos^{2} x + \cos x) d y = 0

[10M]

5.(b) Determine the complete solution of the differential equation

\frac{d^{2} y}{d x^{2}} - 4 \frac{d y}{d x} + 4 y = 3 x^{2} e^{2 x} \sin x

[10M]

5.(c) One end of a heavy uniform rod AB can slide along a rough horizontal rod $A C$ , to which it is attracted by a ring. $B$ and $C$ are joined by a string. When the rod is on the point of sliding, then $A C^{2} - A B^{2} = B C^{2}$ . If $θ$ is the angle between $A B$ and the horizontal line, then pove that the coefficient of friction is $\frac{c o t θ}{2 + c o t^{2} θ}$ .

[10M]

5.(d) The force of attraction of a particle by the earth in inversely proportional to the square of its distance from the earth’s centre. A particle, whose weight on the surface of the earth is $W$ , falls to the surface of the earth from a height $3 h$ above it. Show that the magnitude of work done by the earth’s attraction force is $\frac{3}{4} h W$ , where $h$ is the radius of the earth.

[10M]

5.(e) Find the directional derivative of the function $x y^{2} + y z^{2} + z x^{2}$ along the tangent to the curve $x = t$ , $y = t^{2}$ , $z = t^{2}$ at the point $(1, 1, 1)$ .

[10M]

6.(a) A body consists of a cone and underlying hemisphere. The base of the cone and the top of the hemisphere have same radius $a$ . The whole body rests on a rough horizontal table with hemisphere in contact with the table. Show that the greatest height of the cone, so that the equilibrium may be stable, is $\sqrt{3} a$ .

[15M]

6.(b) Find the circulation of $\vec{F}$ round the curve $C$ , where $\vec{F} = (2 x + y^{2}) \hat{i} + (3 y - 4 x) \hat{j}$ and $C$ is the curve $y = x^{2}$ from $(0, 0)$ to $(1, 1)$ and the curve $y^{2} = x$ from $(1, 1)$ to $(0, 0)$ .

[15M]

6.(c)(i) Solve the differential equation

\frac{d^{2} y}{d x^{2}} + (3 s i n x - c o t x) \frac{d y}{d x} + 2 y s i n^{2} x = e^{- c o s x} s i n^{2} x

[10M]

6.(c)(ii) Find the Laplace transforms of $t^{- 1 / 2}$ and $t^{- 1 / 2}$ . Prove that the Laplace transform of $t^{n + 1 / 2}$ , where $n \in N$ , is

\frac{Γ (n + 1 + \frac{1}{2})}{s^{n + 1 + \frac{1}{2}}}

[10M]

7.(a) Find the linearly independent solutions of the corresponding homogeneous differential equation of the equation $x^{2} y^{″} - 2 x y^{'} + 2 y = x^{3} \sin x$ and then find the general solution of the given equation by the method of variation of parameters.

[15M]

7.(b) Find the radius of curvature and radius of torsion of the helix $x = a \cos u$ , $y = a \sin u$ , $z = a u \tan α$ .

[15M]

7.(c) A particle moving along the $y - a x i s$ has an acceleration $F_{y}$ towards the origin, where $F$ is a psoitive and even function of $y$ . The periodic time, when the particle vibrates between $y = - a$ and $y = a$ , is $T$ . Show that

\frac{2 π}{\sqrt{F_{1}}} < T < \frac{2 π}{\sqrt{F_{2}}}

where $F_{1}$ and $F_{2}$ are the greatest and the least values of $F$ within the range $[- a, a]$ . Further, show that when a simple pendulum of length $l$ oscillates through $30^{\circ}$ on either side of the vertical line, $T$ lies between $2 π \sqrt{l / g}$ and $2 π \sqrt{l / g} \sqrt{π / 3}$ .

[20M]

8.(a) Obtain the singular solution of the differential equation

(\frac{d y}{d x})^{2} (\frac{y}{x})^{2} c o t^{2} α - 2 (\frac{d y}{d x}) (\frac{y}{x}) + (\frac{d y}{d x})^{2} c o s e c^{2} α = 1

Also find the complete primitive of the given differential equation. Give the geometrical interpretations of the complete primitive and singular solution.

[15M]

8.(b) Prove that the path of a planet, which is moving so that its acceleration is always directed to a fixed point (star) and is equal to $\frac{μ}{(d i s t a n c e)^{2}}$ is a conic section. Find the conditions under which the path becomes

(i) ellipse,
(ii) parabola and (iii) hyperbola

[15M]

8.(c)(i) State Gauss divergence theorem. Verify this theorem for $\vec{F} = 4 x \hat{i} - 2 y^{2} \hat{j} + z^{2} \hat{k}$ , taken over the region bounded by $x^{2} + y^{2} = 4$ , $z = 0$ and $z = 3$ .

[15M]

8.(c)(ii) Evalute by Stokes’ theorem $\oint_{C} e^{x} d x + 2 y d y - d z$ , where $C$ is the curve $x^{2} + y^{2} = 4$ , $z = 2$ .

[5M]

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