Ramana Sri IAS - 2016 UPSC Maths Optional Paper II Solutions
2016 UPSC Maths Optional Paper II Solutions
Ramana Sri IAS provides complete and updated solutions for the 2016 UPSC Maths Optional Paper II. Aspirants preparing for the UPSC Mains Examination with Mathematics as their optional subject should solve all these questions carefully at least 5 to 10 times before the mains examination.
Ramana Sri IAS presents complete solutions for UPSC/IAS/CSE-Civil Service Examination 2016 Mathematics Optional Paper II. Each answer follows the same format: Question, Diagram where needed, Concept Related to the Question, Detailed Solution, and Final Answer.
About 2016 UPSC Maths Optional Paper II Solutions
These 2016 UPSC Maths Optional Paper II Solutions are prepared by Ramana Sri IAS for aspirants who want question-wise clarity before the UPSC Mains examination. This public page gives one full sample solution, while the complete paper-wise solutions are available in the full PYQ course.
Students can use these 2016 UPSC Maths Optional Paper II Solutions to understand the expected answer-writing method, diagram presentation, concept application, and final-answer format used by Ramana Sri IAS.
These 2016 UPSC Maths Optional Paper II Solutions are useful for revision, answer-writing practice, and understanding the step-by-step method expected in the UPSC Mathematics optional paper.
Sample Full Solution
We are giving one question as a free sample solution below:
Question 1(e). This free sample includes all five sections:
Question, Diagram,
Concept Related to the Question,
Detailed Solution, and Final Answer.
Complete solutions for all questions are available in the full PYQ course. To purchase the full solutions,
please fill out the admission form first. Our Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
2016 UPSC Maths Optional Paper II Solutions: Table of Contents
Let \(K\) be a field and \(K[X]\) be the ring of polynomials over \(K\) in a single variable \(X\). For a polynomial \(f\in K[X]\), let \((f)\) denote the ideal in \(K[X]\) generated by \(f\). Show that \((f)\) is a maximal ideal in \(K[X]\) if and only if \(f\) is an irreducible polynomial over \(K\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Prove that \(x_{n-1}<x_n<y_n<y_{n-1}\), \(n=2,3,4,\ldots\), and deduce that both the sequences converge to the same limit \(l\), where \(\frac{1}{2}<l<1\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Is \(v(x,y)=x^3-3xy^2+2y\) a harmonic function? Prove your claim. If yes, find its conjugate harmonic function \(u(x,y)\) and hence obtain the analytic function whose real and imaginary parts are \(u\) and \(v\) respectively.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
with constraints \(x+2y\geq1\), \(2x+y\leq1\), \(x\geq0\) and \(y\geq0\) by graphical method.
2Diagram
Question 1(e): Graphical method for linear programming
3Concept Related to the Question
In graphical linear programming, the maximum or minimum of a linear objective function over a polygonal feasible region occurs at a corner point of the feasible region.
4Detailed Solution
The feasible region lies in the first quadrant, below the line \(2x+y=1\), and above the line \(x+2y=1\). The corner points are found by intersecting the boundary lines. The points are \((0,\frac{1}{2})\), \((0,1)\), and the intersection of \(x+2y=1\) and \(2x+y=1\). Solving these two equations gives \(x=y=\frac{1}{3}\).
The largest value is \(\frac{7}{3}\), attained at \((\frac{1}{3},\frac{1}{3})\).
5Final Answer
The maximum value of \(5x+2y\) is \(\frac{7}{3}\), attained at \(x=y=\frac{1}{3}\).
Question 2(a)
Conditional convergence of a series
1Question
Show that the series \(\displaystyle \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n+1}\) is conditionally convergent. (If you use any theorem(s) to show it, then you must give a proof of that theorem(s).)
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Let \(p\) be a prime number and \(\mathbb{Z}_p\) denote the additive group of integers modulo \(p\). Show that every non-zero element of \(\mathbb{Z}_p\) generates \(\mathbb{Z}_p\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Let \(K\) be an extension of a field \(F\). Prove that the elements of \(K\), which are algebraic over \(F\), form a subfield of \(K\). Further, if \(F\subset K\subset L\) are fields, \(L\) is algebraic over \(K\) and \(K\) is algebraic over \(F\), then prove that \(L\) is algebraic over \(F\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Let \(\gamma:[0,1]\to\mathbb{C}\) be the curve \(\gamma(t)=e^{2\pi it}\), \(0\leq t\leq1\). Find, giving justifications, the value of the contour integral \(\displaystyle \int_\gamma \frac{dz}{4z^2-1}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Let \(f:\mathbb{R}\to\mathbb{R}\) be a continuous function such that \(\displaystyle \lim_{x\to+\infty}f(x)\) and \(\displaystyle \lim_{x\to-\infty}f(x)\) exist and are finite. Prove that \(f\) is uniformly continuous on \(\mathbb{R}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
possess vorticity, where \(\vec q=(u,v,w)\) is the velocity in the Cartesian frame, \(\vec r=(x,y,z)\) and \(r^2=x^2+y^2+z^2\)? What is the circulation in the circle \(x^2+y^2=9,\ z=0\)?
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Hamilton characteristic function of a free particle
1Question
Consider a single free particle of mass \(m\), moving in space under no forces. If the particle starts from the origin at \(t=0\) and reaches the position \((x,y,z)\) at time \(\tau\), find the Hamilton's characteristic function \(S\) as a function of \(x,y,z,\tau\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
A simple source of strength \(m\) is fixed at the origin \(O\) in a uniform stream of incompressible fluid moving with velocity \(U\vec i\). Show that velocity potential \(\phi\) at any point \(P\) of the stream is \(\displaystyle \frac{m}{r}-Ur\cos\theta\), where \(OP=r\) and \(\theta\) is the angle which \(OP\) makes with the direction \(\vec i\). Find the differential equation of the streamlines and show that they lie on the surfaces \(Ur^2\sin^2\theta-2m\cos\theta=\text{constants}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Let \(f(x)=e^{2x}\cos3x\), for \(x\in[0,1]\). Estimate the value of \(f(0.5)\) using Lagrange interpolating polynomial of degree \(3\) over the nodes \(x=0\), \(x=0.3\), \(x=0.6\) and \(x=1\). Also, compute the error bound over the interval \([0,1]\) and the actual error \(E(0.5)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
The space between two concentric spherical shells of radii \(a,b\), \((a<b)\), is filled with a liquid of density \(\rho\). If the shells are set in motion, the inner one with velocity \(U\) in the \(x\)-direction and the outer one with velocity \(V\) in the \(y\)-direction, then show that the initial motion of the liquid is given by velocity potential
Find the temperature \(u(x,t)\) in a bar of silver of length \(10\) cm and constant cross-section of area \(1\text{ cm}^2\). Let density \(\rho=10\cdot6\text{ g/cm}^3\), thermal conductivity \(K=1\cdot04\text{ cal}/(\text{cm sec }{}^\circ\text{C})\) and specific heat \(\sigma=0\cdot056\text{ cal/g }{}^\circ\text{C}\). The bar is perfectly isolated laterally, with ends kept at \(0^\circ\text{C}\) and initial temperature \(f(x)=\sin(0\cdot1\pi x)^\circ\text{C}\). Note that \(u(x,t)\) follows the heat equation \(u_t=c^2u_{xx}\), where \(c^2=K/(\rho\sigma)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
A hoop with radius \(r\) is rolling, without slipping, down an inclined plane of length \(l\) and with angle of inclination \(\phi\). Assign appropriate generalized coordinates to the system. Determine the constraints, if any. Write down the Lagrangian equations for the system. Hence or otherwise determine the velocity of the hoop at the bottom of the inclined plane.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 UPSC Maths Optional Paper II PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Let \(A,B,C\) be Boolean variables, \(\bar A\) denote complement of \(A\), \(A+B\) is an expression for \(A\) OR \(B\) and \(A\cdot B\) is an expression for \(A\) AND \(B\). Then simplify the following expression and draw a block diagram of the simplified expression, using AND and OR gates.
Are these 2016 UPSC Maths Optional Paper II Solutions complete?
This public page gives one complete sample solution. Complete solutions for all questions are available in the full PYQ course.
Which question is given as a free sample solution on this page?
Question 1(e) is given as the free sample solution on this page.
How should I use these 2016 UPSC Maths Optional Paper II Solutions for preparation?
Students should first read the question carefully, understand the concept, study the diagram where required, and then practise writing the detailed solution and final answer.
Do these solutions include diagrams and detailed solutions?
Yes. The full PYQ course includes diagrams where needed, concept explanation, detailed solution, and final answer for each question.
How can I get complete solutions for all questions in 2016 UPSC Maths Optional Paper II?
To get complete solutions for all questions, please fill out the admission form first. Our Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
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