Ramana Sri IAS • IFoS Mathematics Optional 2016 IFoS Maths Optional Paper II Solutions Ramana Sri IAS provides complete and updated solutions for the 2016 IFoS Maths Optional Paper II. Aspirants preparing for the IFoS Mains Examination with Mathematics as their optional subject should solve all these questions carefully before the mains examination.
Ramana Sri IAS presents complete solutions for Indian Forest 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 IFoS Maths Optional Paper II Solutions These 2016 IFoS Maths Optional Paper II Solutions are prepared by Ramana Sri IAS for aspirants who want question-wise clarity before the IFoS 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 IFoS 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.
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2016 IFoS Maths Optional Paper II Solutions: Table of Contents
Question 1(a) Group of Bijective Functions
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionProve that the set of all bijective functions from a non-empty set \(X\) onto itself is a group with respect to usual composition of functions.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2016 IFoS Maths Optional Paper II PYQ Solutions course.
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3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
5 Final Answer
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Question 1(b) Uniform Convergence
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionExamine the uniform convergence of \(f_n(x)=\dfrac{\sin(nx+n)}{n}\), for \(x\in\mathbb R\), \(n=1,2,3,\ldots\).
2 Diagram
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3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
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5 Final Answer
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Question 1(c) Maxima and Minima of Two-Variable Function
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the maxima and minima of \(f(x,y)=x^3+y^3-3x-12y+20\).
2 Diagram
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3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2016 IFoS 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.
4 Detailed Solution
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
5 Final Answer
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Question 1(d) Analytic Function from Real Part
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the analytic function whose real part is \(e^{-x}\{(x^2-y^2)\cos y+2xy\sin y\}\).
2 Diagram
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3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
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5 Final Answer
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Question 1(e) Convexity of Feasible Region in LPP
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionProve that the set of all feasible solutions of a linear programming problem is a convex set.
2 Diagram
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3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
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5 Final Answer
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Question 2(a) Non-Abelian Group of Order 6
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionShow that any non-abelian group of order \(6\) is isomorphic to the symmetric group \(S_3\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 2(b) Group of Order pq
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(G\) be a group of order \(pq\), where \(p\) and \(q\) are primes such that \(p>q\) and \(q\nmid(p-1)\). Prove that \(G\) is cyclic.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 2(c) Gaussian-Type Integer Ring Example
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionIn the ring \(R=\{a+b\sqrt{-5}:a,b\in\mathbb Z\}\), show that \(\alpha=3\) and \(\beta=1+2\sqrt{-5}\) are relatively prime, but \(\alpha\gamma\) and \(\beta\gamma\) have no greatest common divisor in \(R\), where \(\gamma=7(1+2\sqrt{-5})\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
5 Final Answer
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Question 3(a) Uniform Convergence on a Closed Interval
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionIf \(f_n(x)=\dfrac3{x+n}\), \(0\le x\le2\), state with reasons whether \(\{f_n\}\) converges uniformly on \([0,2]\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 3(b) Continuity at the Origin
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionExamine the continuity at \((0,0)\) of \(f(x,y)=\dfrac{\sin^{-1}(x+2y)}{\tan^{-1}(2x+4y)}\) for \((x,y)\ne(0,0)\), and \(f(0,0)=\dfrac12\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 3(c) Euler’s Homogeneous Function Theorem
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionIf \(u(x,y)=\cos^{-1}\left(\dfrac{x+y}{\sqrt{x}+\sqrt{y}}\right)\), \(0<x<1\), \(0<y<1\), then find the value of \(x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y}\).
2 Diagram
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3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
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5 Final Answer
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Question 3(d) Double Integral by Changing Order
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionEvaluate \(\int_0^2\int_0^{y^2/2}\dfrac{y}{(x^2+y^2+1)^{1/2}}\,dx\,dy\).
2 DiagramQuestion 3(d): Region for changing the order of integration
3 Concept Related to the QuestionThe given region is \(0\le y\le2\), \(0\le x\le y^2/2\). Changing the order gives \(0\le x\le2\), \(\sqrt{2x}\le y\le2\).
4 Detailed SolutionChanging order, the integral becomes \(\int_0^2\int_{\sqrt{2x}}^2 \dfrac{y}{\sqrt{x^2+y^2+1}}\,dy\,dx\). The inner integral is \(\sqrt{x^2+y^2+1}\).
Thus the integral is \(\int_0^2\{\sqrt{x^2+5}-\sqrt{x^2+2x+1}\}\,dx\). Since \(x\ge0\), \(\sqrt{x^2+2x+1}=x+1\).
Therefore the integral equals \(\int_0^2\sqrt{x^2+5}\,dx-\int_0^2(x+1)dx\). Using \(\int\sqrt{x^2+a^2}dx=\dfrac x2\sqrt{x^2+a^2}+\dfrac{a^2}{2}\log(x+\sqrt{x^2+a^2})\), the value becomes \(\dfrac54\log5-1\).
5 Final AnswerThe value of the integral is \(\dfrac54\log5-1\).
Question 4(a) Improper Integral
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionEvaluate the integral \(\int_0^\infty \dfrac{dx}{\sqrt{x}(1+x)}\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 4(b) Laurent Series
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the Laurent series for \(f(z)=\dfrac1{1-z^2}\) with centre \(z=1\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 4(c) Contour Integration Integral
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionEvaluate by contour integration \(\int_0^\pi\dfrac{d\theta}{(1+\frac12\cos\theta)^2}\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 4(d) Transportation Problem: Profit Maximization
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionA company manufacturing air-coolers has two plants located at Bengaluru and Mumbai with a weekly capacity of \(200\) units and \(100\) units respectively. The company supplies air-coolers to its \(4\) showrooms situated at Mangalore, Bengaluru, Delhi and Goa which have a demand of \(75,100,100\) and \(25\) units respectively. Due to the differences in local taxes, showroom charges, transportation cost and others, the profits differ. The profits, in Rs., are shown in the following table:
From / To Mangalore Bengaluru Delhi Goa Bengaluru 90 90 100 100 Mumbai 50 70 130 85
Plan the production program so as to maximize the profit. The company may have its production capacity at both plants partially or wholly unused.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(a) Partial Differential Equation by Eliminating Arbitrary Function
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionObtain the partial differential equation governing \(\phi(u,v)=0\), where \(u=xyz\) and \(v=x+y+z\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(b) Lagrange’s Linear PDE
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the general solution of \(xy^2\dfrac{\partial z}{\partial x}+y^3\dfrac{\partial z}{\partial y}=zxy^2-4x^3\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(c) Newton-Raphson Algorithm
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionDevelop an algorithm for Newton-Raphson method to solve \(\phi(x)=0\), starting with initial iterate \(x_0\). Let \(n\) be the number of iterations allowed, \(eps\) the prescribed relative error, and \(delta\) the prescribed lower bound for \(\phi^{\prime}(x)\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(d) Lagrange Interpolation
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionApply Lagrange’s interpolation formula to find \(f(5)\) and \(f(6)\), given \(f(1)=2\), \(f(2)=4\), \(f(3)=8\), and \(f(7)=128\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(e) Moment of Inertia of Ellipse
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionCalculate the moment of inertia of the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\), (i) relative to the \(x\)-axis, (ii) relative to the \(y\)-axis, and (iii) relative to the origin.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 6(a) Lagrange’s PDE
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the general solution of \(x y^2 p+y^3 q=zxy^2-4x^3\), where \(p=\dfrac{\partial z}{\partial x}\) and \(q=\dfrac{\partial z}{\partial y}\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 6(b) Particular Integral of PDE
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the particular integral of \(\dfrac{\partial^2 z}{\partial x^2}-2\dfrac{\partial^2z}{\partial x\partial y}+\dfrac{\partial^2z}{\partial y^2}=2x\cos y\).
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 6(c) Heat Equation in an Insulated Rod
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionA uniform rod of length \(L\), whose surface is thermally insulated, is initially at temperature \(\theta_0\). At time \(t=0\), one end is suddenly cooled to \(\theta=0\) and maintained at this temperature; the other end remains thermally insulated. Find the temperature distribution \(\theta(x,t)\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 7(a) Simpson One-Third Rule
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionEvaluate \(\int_0^{0.6}\dfrac{dx}{\sqrt{1-x^2}}\) by Simpson’s \(\frac13\) rule using \(12\) equal subintervals.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 7(b) Newton-Raphson Cube Root
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the cube root of \(10\) up to \(5\) significant figures by Newton-Raphson method.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 7(c) Fourth-Order Runge-Kutta Method
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionUse the classical fourth-order Runge-Kutta method with \(h=0.2\) to calculate a solution at \(x=0.4\) for \(\dfrac{dy}{dx}=x+y^2\), with \(y=1\) when \(x=0\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 8(a) Moment of Inertia of a Solid Cone
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the moment of inertia of a right solid cone of mass \(M\), height \(h\), and base radius \(a\), about its axis.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 8(b) Lagrange Equation for Bead on Cycloid
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionA bead slides on a wire in the shape of a cycloid described by the equations \(x=a(\theta-\sin\theta)\), \(y=a(1+\cos\theta)\), where \(0\le\theta\le2\pi\), and the friction between the bead and the wire is negligible. Deduce Lagrange’s equation of motion.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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Question 8(c) Pressure on a Pulsating Sphere in Liquid
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionA sphere is at rest in an infinite mass of homogeneous liquid of density \(\rho\), the pressure at infinity being \(P\). If the radius \(R\) of the sphere varies in such a way that \(R=a+b\cos nt\), where \(b<a\), then find the pressure at the surface of the sphere at any time.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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2016 IFoS Maths Optional Paper II Solutions FAQs
Are these 2016 IFoS Maths Optional Paper II Solutions complete? This public page gives one full sample solution. Complete question-wise solutions for the full 2016 IFoS Maths Optional Paper II are available in the full PYQ course.
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