Paper II PYQs-2009

1.(a) If \(R\) is the set of real numbers and \(R_{t}\) is the set of positive real numbers, show that \(R\) under addition \((R,+)\) and \(R_{+}\) under multiplication \((R_{+},.)\) are isomorphic. Similarly if \(Q\) is set of rational numbers and \(Q_{+}\) is the set of positive rational numbers, are \((Q,+)\) and \((Q, .)\) isomorphic? Justify your answer.

[12M]

1.(b) Determine the number of homomorphisms from the additive group \(Z_{15}\) to the additive group \(Z_{10}\). ( \(Z_{n}\) is the cyclic group of order \(n\)).

[4+8=12M]

1.(c) State Rolle’s theorem. Use it to prove that between two roots of \(e^{x} \cos x=1\), there will be a root of \(e^{x} \sin x=1\).

[12M]

1.(d) Let \(f(x) = \left\{\begin{array}{ll}{\dfrac{\vert x \vert }{2}+1,} & {\text { if } x< 1} \\ {\dfrac{x}{2}+1,} & {\text { if } 1 \leq x< 2} \\ {-\dfrac{\vert x \vert }{2}+1,} & {\text { if } 2 \leq x}\end{array}\right.\).

What are the points of discontinuity of \(f\), if any? What are the points where \(f\) is not differentiable, if any? Justify your answers.

[12M]

1.(e) Let \(f(z)=\dfrac{a_{0}+a_{1} \ldots \ldots+a_{n-1} z^{n-1}}{b_{0}+b_{1} z+\ldots \ldots \ldots+b_{n} z^{n}}\), \(b_{n} \neq 0\). Assume that the zeros of the denominator are simple. Show that the sum of the residues of \(f(z)\) at its poles is equal to \(\dfrac{a_{n}-1}{b_{n}}\).

[12M]

1.(f) A paint factory produces both interior and exterior paint from materials $M_{1}$ and $M_{2}$ . The basic data is as follows:
\(\begin{array}{|c|c|c|c|}\hline {} & \text {Exterior Paint} & \text {Interior Paint} & \text {Max. Daily Availability}\\ \hline \text { Raw Material } M_{1} & {6} & {4} & {24} \\ \hline \text { Raw Material } M_{2} & {1} & {2} & {2} \\ \hline \text { Profit per ton (Rs. 1000) } & {5} & {4} & { } \\ \hline\end{array}\)

A market survey indicates that the daily demand interior paint cannot exceed that of exterior paint by more than 1 ton. The maximum daily demand of interior paint is 2 tons. The factory wants to determine the optimum product mix of interior and exterior paint that maximizes daily profits. Formulate the LP problem for this situation.

[12M]

2.(a) How many proper, non-zero ideals does the ring \(Z_{12}\) have? Justify your answer. How many ideals does the ring \(Z_{12} \oplus Z_{12}\) have? Why?

[2+3+4+6=15M]

2.(b) Show that the alternating group of four letters \(A_{4}\) has no subgroup of order 6.

[15M]

2.(c) Show that the series \(\left(\dfrac{1}{3}\right)^{2}\)+\(\left(\dfrac{1.4}{3.6}\right)^{2}\)+\(\ldots .+\left(\dfrac{1.4 .7 \ldots . .(3 n-2)}{3.6 .9 \ldots \ldots \ldots . .3 n}\right)^{2}\) till infinity, converges.

[15M]

2.(d) Show that if \(f:[a, b] \rightarrow R\) is a continuous function, then \(f([a, b])=[c, d]\) for some real numbers \(c\) and \(d\), \(c \leq d\).

[15M]

3.(a) Show that \(Z[X]\) is a unique factorization domain that is not a principal ideal domain (\(Z\) is the ring of integers). Is it possible to give an example of principal ideal domain that is not a unique factorization domain? (\(Z[X]\) is the ring of polynomials in the variable \(X\) with integer.

[15M]

3.(b) How many elements does the quotient ring \(\dfrac{Z_{5}[X]}{X^{2}+1}\) have? Is it an integral domain? Justify your answers.

[15M]

3.(c) Show that:

\(\Lt_{x \rightarrow 1} \sum_{n=1}^{\infty} \dfrac{n^2x^2}{n^4+x^4}\)= \(\sum_{n=1}^{\infty} \dfrac{n^2}{n^4+1}\)

Justify all steps of your answer by quoting the theorems you are using.

[15M]

3.(d) Show that a bounded infinite subset of \(R\) must have a limit point.

[15M]

4.(a) If \(\alpha\), \(\beta\), \(\gamma\) are real numbers such that \(\alpha^{2}>\beta^{2}+\gamma^{2}\) show that: \(\int_{0}^{2 \pi} \dfrac{d \theta}{\alpha+\beta \cos \theta+\gamma \sin \theta}\)=\(\dfrac{2 \pi}{\sqrt{\alpha^{2}-\beta^{2}-\gamma^{2}}}\).

[30M]

4.(b) Maximize: \(Z=3x_1+5x_2+4x_3\) subject to:

\(2x_1+3x_2 \leq 8\),
\(3x_1+2x_2+4x_3 \leq 15\),
\(2x_2+5x_3 \leq 10\),
\(x_i \geq 0\)

[30M]

5.(a) Show that the differential equation of all cones which have their vertex at the origin is \(p x+q y=z\). Verify that this equation is satisfied by the surface \(y z+z x+x y=0\).

[12M]

5.(b)(i) Form the partial differential equation by elimination the arbitrary function \(f\) given by: \(f\left(x^{2}+y^{2}, z-x y\right)=0\).

[6M]

5.(b)(ii) Find the integral surface of: \(x^{2} p+y^{2} p+z^{2}=0\) which passes through the curve: \(x y=x+y\), \(z=1\).

[6M]

5.(c)(i) The equation \(x^{2}+a x+b=0\) has two real roots \(\alpha\) and \(\beta\). Show that the iterative method given by: \(x_{k+1}=-\dfrac{\left(a x_{k}+b\right)}{x_{k}}\), \(k=0,1,2 \ldots\) is convergent near \(x=\alpha\), if \(\vert \alpha \vert > \vert \beta \vert\).

[6M]

5.(c)(ii) Find the values of two valued Boolean variables \(A\), \(B\), \(C\), \(D\) by solving the following simultaneous equations:
\({\quad \overline{A}+A B=0} \\ {\quad A B+A C} \\ {\quad A B+A \overline{C}+C D=\overline{C} D}\)
where \(\overline{x}\) represents the complement of \(x\).

[6M]

5.(d)(i) Realize the following expressions by using NAND gates only:
\(g=(\overline{a}+\overline{b}+c) \overline{d}(\overline{a}+e) f\), where \(\overline{x}\) represents the complement of \(x\).

[6M]

5.(d)(ii) Find the decimal equivalent of \((357.32)_{8}\).

[6M]

5.(e) The flat surface of a hemisphere of radius \(r\) is cemented to one flat surface of a cylinder of the same radius and of the same material. If the length of the cylinder be \(l\) and the total mass be \(m\), show that the moment of inertia of the combination about the axis of the cylinder is given by: \(m r^{2} \dfrac{\left(\dfrac{l}{2}+\dfrac{4}{15} r\right)}{\left(l+\dfrac{2 r}{3}\right)}\).

[12M]

5.(f) Two sources, each of strength \(m\) are placed at the point \((-a, 0)\), \((a, 0)\) and a sink of strength \(2m\) is at the origin. Show that the stream lines are the curves: \(\left(x^{2}+y^{2}\right)^{2}=a^{2}\left(x^{2}-y^{2}+\lambda x y\right)\) where \(\lambda\) is a variable parameter. Show also that the fluid speed at any point is \(\left(2 m a^{2}\right)/\left(r_{1} r_{2} r_{3}\right)\), where \(r_{1} r_{2}\) and \(r_{3}\) are the distance of the points from the source and the sink.

[12M]

6.(a) Find the characteristics of: \(y^{2} r-x^{2} t=0\) where \(r\) and \(t\) have their usual meanings.

[15M]

6.(b) Solve: \(\left(D^{2}-D D^{\prime}-2 D^{\prime 2}\right) z\)=\(\left(2 x^{2}+x y-y^{2}\right) \sin x y-\cos x y\) where \(D\) and \(D^{\prime}\) represent \(\dfrac{\partial}{\partial x}\) and \(\dfrac{\partial}{\partial y}\) and \(\dfrac{\partial}{\partial y}\).

[15M]

6.(c) A tightly stretched string has its ends fixed at \(x=0\) and \(x=1\). At time \(t=0\), the string is given a shape defined by \(f(x)=\mu x(l-x)\), where \(\mu\) is a constant, and then released. Find the displacement of any point \(x\) of the string at time \(t > 0\).

[30M]

7.(a) Develop an algorithm for Regula-Falsi method to find a root of \(f(x)=0\) starting with two initial iterates \(x_{0}\) and \(x_{1}\) to the root such that sign \(\left(f\left(x_{0}\right)\right) \neq \operatorname{sign}\left(f\left(x_{1}\right)\right)\). Take \(n\) as the maximum number of iterations allowed and epsilon be the prescribed error.

[30M]

7.(b) Using Lagrange interpolation formula, calculate the value of \(f(3)\) from the following table of values of \(x\) and \(f(x)\):
\(\begin{array}{|c|c|c|c|c|c|c|}\hline x & {0} & {1} & {2} & {4} & {5} & {6} \\ \hline f(x) & {1} & {14} & {15} & {5} & {6} & {19} \\ \hline \end{array}\)

[15M]

7.(c) Find the value of \(y(1.2)\) using Runge-Kutta fourth order method with step size \(h=0.2\) from the initial value problem: \(y^{\prime}=x y\), \(y(1)=2\).

[15M]

8.(a) A perfectly rough sphere of mass \(m\) and radius \(b\), rests on the lowest point of a fixed spherical cavity a radius \(a\). To the highest point of the movable sphere is attached a particle of mass \(m^{\prime}\) and the system is disturbed. Show that the oscillations are the same as of a simple pendulum of length \((a-b) \dfrac{4 m^{\prime}+\dfrac{7}{5} m}{m+m^{\prime}\left(2-\dfrac{a}{b}\right)}\).

[30M]

8.(b) An infinite mass of fluid is acted on by a force \(\dfrac{\mu}{r^{3 / 2}}\) per unit mass directed to the origin. If initially the fluid is at rest and there is a cavity in the form of the sphere \(r=C\) in it, show the cavity will be filled up after an interval of time \(\left(\dfrac{2}{5 \mu}\right)^{\dfrac{1}{2}} \cdot C^{\dfrac{5}{4}}\).

[30M]