Ramana Sri IAS • IFoS Mathematics Optional 2018 IFoS Maths Optional Paper II Solutions Ramana Sri IAS provides complete and updated solutions for the 2018 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 2018 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 2018 IFoS Maths Optional Paper II Solutions These 2018 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 2018 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.
For the official examination source, students may also refer to the UPSC previous year question papers page .
These 2018 IFoS Maths Optional Paper II Solutions are useful for revision, answer-writing practice, and understanding the step-by-step method expected in the IFoS Mathematics optional paper.
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2018 IFoS Maths Optional Paper II Solutions: Table of Contents
Question 1(a) Sylow Theorem 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionProve that a non-commutative group of order \(2n\), where \(n\) is an odd prime, has a subgroup of order \(n\).
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 1(b) Fixed Point 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(f:[0,1]\to[0,1]\) is continuous, prove that \(f(c)=c\) for some \(c\in[0,1]\).
2 DiagramQuestion 1(b): Fixed Point Theorem on [0, 1]
3 Concept Related to the QuestionThis question belongs to Fixed Point . We use the standard theorem/formula and then simplify step by step.
4 Detailed SolutionLet \(g(x)=f(x)-x\). Then \(g(0)=f(0)\ge0\), and \(g(1)=f(1)-1\le0\). By the Intermediate Value Theorem, \(g(c)=0\) for some \(c\in[0,1]\). Hence \(f(c)=c\).
5 Final AnswerThere exists \(c\) such that \(f(c)=c\).
Question 1(c) Analytic Function 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(u=(x-1)^3-3xy^2+3y^2\), find \(v\) such that \(u+iv\) is analytic.
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 1(d) Simplex Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve by simplex method the following Linear Programming Problem: Maximize \(Z=3x_1+2x_2+5x_3\), subject to \(x_1+2x_2+x_3\le 430\), \(3x_1+2x_3\le 460\), \(x_1+4x_2\le 420\), and \(x_1,x_2,x_3\ge 0\).
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 2(a) Homomorphisms 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind all homomorphisms from \((\mathbb Z,+)\) to \((\mathbb Z_4,+)\).
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) Mixed Partials 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionConsider the function \(f\) defined by
\(f(x,y)=\begin{cases}xy\frac{x^2-y^2}{x^2+y^2},&x^2+y^2\ne 0,\\[4pt]0,&x^2+y^2=0.\end{cases}\)
Show that \(f_{xy}\ne f_{yx}\) at \((0,0)\).
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 2(c) Fresnel Integral 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionProve \(\int_0^\infty\cos x^2dx=\int_0^\infty\sin x^2dx=\frac12\sqrt{\frac\pi2}\).
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(d) Prime Ideal 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionProve that an ideal \(P\) of a commutative ring with unity is prime iff \(R/P\) is an integral domain.
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(a) Minimum Distance 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the minimum value of \(x^2+y^2+z^2\) subject to the condition \(ax+by+cz=p\).
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) Finite Ring Example 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow by an example that in a finite commutative ring, every maximal ideal need not be prime.
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) Trigonometric Integral 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionEvaluate the integral \(\int_0^{2\pi}\cos^{2n}\theta\,d\theta\), where \(n\) is a positive integer.
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(d) Improper Integral 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow that the improper integral \(\int_0^1\frac{\sin(1/x)}{\sqrt{x}}\,dx\) is convergent.
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(a) Beta Integral 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow that
\(\iint_R x^{l-1}y^{m-1}(1-x-y)^{n-1}\,dx\,dy=\frac{\Gamma(l)\Gamma(m)\Gamma(n)}{\Gamma(l+m+n)},\quad l,m,n\gt0,\)
where \(R\) is the triangle bounded by \(x=0\), \(y=0\), and \(x+y=1\).
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(b) Uniform Convergence 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionLet \(f_n(x)=\frac{x}{n+x^2}\), \(x\in[0,1]\). Show that the sequence \(\{f_n\}\) is uniformly convergent on \([0,1]\).
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) Normality 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionLet \(H\) be a cyclic subgroup of a group \(G\). If \(H\) is a normal subgroup of \(G\), prove that every subgroup of \(H\) is a normal subgroup of \(G\).
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 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionThe capacities of three production facilities \(S_1,S_2,S_3\), the requirements of four destinations \(D_1,D_2,D_3,D_4\), and the transportation costs in rupees are given in the following table. Find the minimum transportation cost using Vogel's Approximation Method (VAM).
\(D_1\) \(D_2\) \(D_3\) \(D_4\) Capacity \(S_1\) 19 30 50 10 7 \(S_2\) 70 30 40 60 9 \(S_3\) 40 8 70 20 18 Demand 5 8 7 14 34
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) PDE of Planes 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the partial differential equation of all planes which are at a constant distance \(a\) from 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 5(b) Weddle Rule 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionA solid of revolution is formed by rotating about the \(x\)-axis the area between the \(x\)-axis, the line \(x=0\), and a curve through the points with the following coordinates. Estimate the volume of the solid formed using Weddle's rule.
\(x\) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 \(y\) 1.0000 0.9896 0.9589 0.9089 0.8415 0.8029 0.7635
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 5(c) BASIC Program 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionWrite a program in BASIC to multiply two matrices. Checking for consistency for multiplication is required.
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 5(d) Fluid Equation 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionAir, obeying Boyle's law, is in motion in a uniform tube of small section. Prove that if \(\rho\) is the density and \(v\) is the velocity at a distance \(x\) from a fixed point at time \(t\), then
\(\frac{\partial^2\rho}{\partial t^2}=\frac{\partial^2}{\partial x^2}\{\rho(v^2+k)\}.\)
2 Diagram
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4 Detailed Solution
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Question 6(a) Charpit PDE 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the complete integral of the partial differential equation \(x(p^2+q^2)=zp\), and deduce the solution which passes through the curve \(x=0\), \(z^2=4y\). Here \(p=\frac{\partial z}{\partial x}\) and \(q=\frac{\partial z}{\partial y}\).
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 6(b) Runge-Kutta 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionApply the fourth-order Runge-Kutta method to compute \(y\) at \(x=0.1\) and \(x=0.2\), given that \(\frac{dy}{dx}=x+y^2\), \(y=1\) at \(x=0\).
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 6(c) Hamiltonian 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFor a particle having charge \(q\) and moving in an electromagnetic field, the potential energy is \(U=q(\phi-\vec v\cdot\vec A)\), where \(\phi\) and \(\vec A\) are, respectively, known as the scalar and vector potentials. Derive the expression for the Hamiltonian for the particle in the electromagnetic field.
2 Diagram
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Question 6(d) BASIC Trapezoidal 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionWrite a program in BASIC to implement the trapezoidal rule to compute \(\int_0^{10}e^{-x^2}\,dx\) with 10 subdivisions.
2 Diagram
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Question 7(a) Lagrange PDE 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve \((z^2-2yz-y^2)p+(xy+zx)q=xy-zx\), where \(p=\frac{\partial z}{\partial x}\), \(q=\frac{\partial z}{\partial y}\). If the solution of the above equation represents a sphere, what will be the coordinates of its centre?
2 Diagram
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Question 7(b) Simpson Rule 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionThe velocity \(v\) in km/min of a moped is given at fixed intervals of time \(t\) in minutes as below. Estimate the distance covered during the time using Simpson's one-third rule.
\(t\) 0.1 0.2 0.3 0.4 0.5 0.6 \(v\) 1.00 1.104987 1.219779 1.34385 1.476122 1.615146 \(t\) 0.7 0.8 0.9 1.0 1.1 \(v\) 1.758819 1.904497 2.049009 2.18874 2.31977
2 Diagram
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Question 7(c) Two Complement 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionAssuming a 16-bit computer representation of signed integers, represent \(-44\) in 2's complement representation.
2 Diagram
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3 Concept Related to the Question
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Question 7(d) Potential Flow 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIn the case of two-dimensional motion of a liquid streaming past a fixed circular disc, the velocity at infinity is \(u\) in a fixed direction, where \(u\) is variable. Show that the maximum value of the velocity at any point of the fluid is \(2u\). Prove that the force necessary to hold the disc is \(2m\dot u\), where \(m\) is the mass of the liquid displaced by the disc.
2 Diagram
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Question 8(a) Poisson Equation 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind a real function \(V\) of \(x\) and \(y\), satisfying \(\frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\partial y^2}=-4\pi(x^2+y^2)\), and reducing to zero when \(y=0\).
2 Diagram
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Question 8(b) Regula Falsi 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionThe equation \(x^6-x^4-x^3-1=0\) has one real root between \(1.4\) and \(1.5\). Find the root to four places of decimal by regula-falsi method.
2 Diagram
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Question 8(c) Lagrange Mechanics 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionA particle of mass \(m\) is constrained to move on the inner surface of a cone of semi-angle \(\alpha\) under the action of gravity. Write the equation of constraint and mention the generalized coordinates. Write down the equation of motion.
2 Diagram
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Question 8(d) Sources and Sink 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionTwo sources, each of strength \(m\), are placed at the points \((-a,0)\), \((a,0)\), and a sink of strength \(2m\) at the origin. Show that the streamlines are the curves
\((x^2+y^2)^2=a^2(x^2-y^2+\lambda xy),\)
where \(\lambda\) is a variable parameter. Show also that the fluid speed at any point is \(\frac{2ma^2}{r_1r_2r_3}\), where \(r_1,r_2,r_3\) are the distances of the point from the sources and the sink.
2 Diagram
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2018 IFoS Maths Optional Paper II Solutions FAQs
Are these 2018 IFoS Maths Optional Paper II Solutions complete? This public page gives one full sample solution. Complete paper-wise 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(b) is given as the free sample solution on this page with Question, Diagram, Concept Related to the Question, Detailed Solution, and Final Answer.
How should I use these 2018 IFoS Maths Optional Paper II Solutions for preparation? Use the solutions for revision, answer-writing practice, diagram presentation, and concept clarity before attempting the full IFoS Mathematics optional paper.
Do these solutions include diagrams and detailed explanations? Yes. The full PYQ course includes diagrams where needed, concepts, detailed explanations, and final answers for all questions.
How can I get complete solutions for all questions in 2018 IFoS Maths Optional Paper II? Complete solutions are available in the full PYQ course. Please fill out the admission form first. Our Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
![2018 IFoS Maths Optional Paper II Solutions diagram for Question 1(b), showing a continuous function from [0,1] to [0,1] intersecting the line y=x at a fixed point.](https://mathsoptional.com/wp-content/uploads/2026/06/2018-ifos-maths-optional-paper-2-question-1-b-diagram.png)