Ramana Sri IAS - 2020 IFoS Maths Optional Paper II Solutions
2020 IFoS Maths Optional Paper II Solutions Ramana Sri IAS provides complete and updated solutions for the 2020 IFoS Maths Optional Paper II. Aspirants preparing for the IFoS 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 Indian Forest Service Examination 2020 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 2020 IFoS Maths Optional Paper II Solutions These 2020 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 2020 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 2020 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.
Sample Full Solution
We are giving one question from 2020 IFoS Maths Optional Paper II Solutions as a free sample solution below:
Question 1(c) . This free sample includes all five sections:
Question , Diagram ,
Concept Related to the Question ,
Detailed Solution , and Final Answer . Complete 2020 IFoS Maths Optional Paper II Solutions for all questions are available in the full PYQ course. To purchase the full solution, please fill out the admission form first. Our Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
2020 IFoS Maths Optional Paper II Solutions: Table of Contents
Question 1(a) Modern Algebra – Wilson Theorem 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionLet \(p\) be prime. Show \((p-1)!+1\equiv0\pmod p\) . Also find the remainder when \(6^{44}(22)!+3\) is divided by \(23\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 1(b)(i) Partial Derivatives – Chain Rule 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(u=u(y-z,z-x,x-y)\) , find \(u_x+u_y+u_z\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 1(b)(ii) Euler Theorem – Homogeneous Functions 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(u=\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}\) , find \(xu_x+yu_y+zu_z\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 1(c) Double Integral – Change of Variables 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionEvaluate \(\iint_R(x-y)^2\cos^2(x+y)\,dxdy\) , where \(R\) is the rhombus with vertices \((\pi,0),(2\pi,\pi),(\pi,2\pi),(0,\pi)\) .
2 DiagramQuestion 1(c): Rhombus transformed into a rectangle
3 Concept Related to the QuestionThis question belongs to Double Integral – Change of Variables . First identify the standard theorem or formula, then substitute the given data carefully.
4 Detailed SolutionPut \(u=x-y\) and \(v=x+y\) . Then \(dxdy=\frac12dudv\) . The rhombus becomes the rectangle \(-\pi\le u\le\pi\) , \(\pi\le v\le3\pi\) . Therefore the integral is \(\frac12\int_{-\pi}^{\pi}u^2du\int_{\pi}^{3\pi}\cos^2v\,dv\) . These are \(2\pi^3/3\) and \(\pi\) .
5 Final AnswerThe value is \(\pi^4/3\) .
Question 1(d) Linear Programming – Graphical Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve graphically: maximize \(z=5x_1-3x_2\) subject to \(3x_1+2x_2\le12\) , \(-x_1+x_2\ge1\) , \(-x_1+x_2\le2\) , and \(x_1,x_2\ge0\) . Then discuss \(z=6x_1+4x_2\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 1(e) Complex Integration 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionEvaluate \(\int_C \operatorname{Re}(z^2)\,dz\) from \(0\) to \(2+4i\) along \(C:y=x^2\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 2(a) Ring Theory – Maximal Ideal 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionLet \(R\) be a non-zero commutative ring with unity. Show that \(M\) is maximal iff \(R/M\) is a field.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 2(b) Real Analysis – Uniform Convergence 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow that \(f_n(x)=nx(1-x)^n\) does not converge uniformly on \([0,1]\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 2(c) Complex Analysis – Residues/Cauchy Formula 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionEvaluate \(\oint_C \frac{e^z}{z^2(z+1)^3}\,dz\) , where \(C:|z|=2\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 3(a) Maxima and Minima with Constraints 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the extreme values of \(f(x,y,z)=2x+3y+z\) subject to \(x^2+y^2=5\) and \(x+z=1\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 3(b) Group Theory – Sylow Theorem 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionLet \(G\) be a finite group and let \(p\) be a prime. If \(p^m\) divides the order of \(G\) , show that \(G\) has a subgroup of order \(p^m\) , where \(m\) is a positive integer.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 3(c) Linear Programming – Simplex Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve by simplex method: maximize \(z=2x_1+x_2\) subject to \(2x_1-2x_2\le1\) , \(2x_1-4x_2\le3\) , \(2x_1+x_2\le2\) , \(x_1,x_2\ge0\) . State whether alternate optima exist.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 4(a) Complex Analysis – Bilinear Transformation 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow that \(w=e^{i\theta}\frac{z-z_0}{z-\overline{z_0}}\) maps the upper half-plane into the unit circle. If \(z=i\) maps to \(w=0\) and infinity maps to \(w=-1\) , find the transformation.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 4(b) Finite Fields 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionLet \(K\) be a finite field. Show that \(|K|=p^n\) . Also prove that \(\mathbb Z_3[X]/(X^2+1)\) is a field and find its number of elements.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 4(c) Transportation Problem – VAM 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the minimum transportation cost using Vogel’s approximation method for the following transportation problem:
Source \(D_1\) \(D_2\) \(D_3\) \(D_4\) Availability \(S_1\) 9 16 15 9 15 \(S_2\) 2 1 3 5 25 \(S_3\) 6 4 7 3 20 Demand 10 15 25 10
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 5(a) Partial Differential Equations 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionConstruct a PDE of all surfaces of revolution having the \(z\) -axis as axis of rotation.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 5(b) Numerical Analysis – Newton-Raphson 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionUsing Newton-Raphson method, find \((37)^{1/3}\) correct to four decimals.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 5(c) Computer Arithmetic – Number Conversion 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionConvert \((14231)_8\) to binary and decimal. Convert \((43503)_{10}\) to binary and hexadecimal.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 5(d) Hamiltonian System 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the condition on real numbers \(a,b,c\) such that \(\dot p=aq-q^2\) , \(\dot q=bp+cq\) is Hamiltonian, and compute the Hamiltonian.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 5(e) PDE – Lagrange Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the general solution of \(p\tan x+q\tan y=\tan z\) , where \(p=z_x\) and \(q=z_y\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 6(a) PDE – Charpit Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the general and singular solution of \(6yz-6pxy-3qy^2+pq=0\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 6(b) Numerical Analysis – Lagrange Interpolation 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the Lagrange interpolating polynomial that fits the following data values, and interpolate at \(x=2.5\) , correct to three decimal places.
\(x\) -1 2 3 5 \(f(x)\) -1 10 25 60
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 6(c) Fluid Dynamics – Streamlines 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFor \(\vec V=2x\hat i+3y\hat j-5z\hat k\) , find the streamline through \((4,8,1)\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 7(a) Numerical Analysis – Euler Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionWrite down the algorithm and flowchart for solving numerically the differential equation \(\frac{dy}{dx}=f(x,y)=1+x\cos y\) with initial condition \(x=x_0\) , \(y=y_0\) and step length \(h\) by Euler’s method up to \(x=x_n=x_0+nh\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 7(b) Fluid Dynamics – Potential and Stream Function 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFor \(u=x-ay\) and \(v=-ax-y\) , show that velocity potential exists and find both potential and stream function.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 7(c) PDE – Linear Equation 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve \(2z_{xx}+5z_{xy}+3z_{yy}=ye^x\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 8(a) Analytical Mechanics – Inverse Cube Force 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionA particle is attracted to a center by a force which varies inversely as the cube of its distance from the center. Identify the generalized coordinates and write down the Lagrangian of the system. Derive the equations of motion and solve them for the orbits. Discuss how the nature of orbits depends on the parameters of the system.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 8(b) Numerical Integration – Trapezoidal Rule 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionEvaluate the integral \(\int_0^2\frac{x}{1+x^3}\,dx\) , using trapezoidal rule with \(h=\frac14\) , correct to three decimal places. Here \(h\) is the length of each subinterval.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
Question 8(c) Numerical Linear Algebra – Gaussian Elimination 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve by Gaussian elimination: \(5x_1+2x_2+x_3=-2\) , \(6x_1+3x_2+2x_3=1\) , \(x_1-x_2+2x_3=0\) .
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2020 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2020 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
Full Solution Access The Detailed Solution section for this question is available in the full 2020 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.
5 Final Answer
Full Solution Access The Final Answer section for this question is available in the full 2020 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.
2020 IFoS Maths Optional Paper II Solutions FAQs
Are these 2020 IFoS Maths Optional Paper II Solutions complete? This public page gives one full sample solution for Question 1(c). Complete question-wise solutions for the full paper are available in the full PYQ course by Ramana Sri IAS.
Which question is given as a free sample solution on this page? Question 1(c) is given as the free sample solution on this page. It includes the Question, Diagram, Concept Related to the Question, Detailed Solution, and Final Answer sections.
How should I use these 2020 IFoS Maths Optional Paper II Solutions for preparation? Use these solutions for revision, answer-writing practice, diagram presentation, and step-by-step understanding of the IFoS Mathematics optional paper.
Do these solutions include diagrams and detailed solutions? Yes. The full PYQ course includes diagrams wherever needed, concept explanations, detailed solutions, and final answers for all questions.
How can I get complete solutions for all questions in 2020 IFoS Maths Optional Paper II Solutions? To get the complete solutions, please fill out the admission form. The Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
