2021 IFoS Maths Optional Paper II Solutions
2021 IFoS Maths Optional Paper II Solutions Ramana Sri IAS provides complete and updated solutions for the 2021 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 2021 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 2021 IFoS Maths Optional Paper II Solutions
These 2021 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 2021 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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UPSC previous year question papers page .
These 2021 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
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Question 1(c) . This free sample includes all five sections:
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2021 IFoS Maths Optional Paper II Solutions: Table of Contents
Question 1(a) Group Theory – Power Automorphism 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionLet \(G\) be a finite commutative group. Let \(n\in\mathbb Z\) be such that \(n\) and the order of \(G\) are relatively prime. Show that the function \(\phi:G\to G\) defined by \(\phi(a)=a^n\) , for all \(a\in G\) , is an isomorphism of \(G\) onto \(G\) .
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 1(b) Real Analysis – Cauchy Criterion 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionApply Cauchy's Principle of Convergence to prove that the sequence \(\langle f_n\rangle\) defined by \(f_n=1+\frac14+\frac17+\cdots+\frac1{3n-2}\) is not 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 1(c) Implicit Differentiation 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind \(\frac{dy}{dx}\) , when \(f(x,y)=\log(x^2+y^2)+\tan^{-1}\left(\frac yx\right)=0\) , on using derivatives of implicit functions.
2 DiagramQuestion 1(c): Implicit Curve as Logarithmic Spiral
3 Concept Related to the QuestionFor an implicit equation \(F(x,y)=0\) , the derivative is \(\frac{dy}{dx}=-\frac{F_x}{F_y}\) , provided \(F_y\ne0\) .
4 Detailed SolutionLet \(F(x,y)=\log(x^2+y^2)+\tan^{-1}(y/x)\) . Then \(F_x=\frac{2x}{x^2+y^2}-\frac{y}{x^2+y^2}=\frac{2x-y}{x^2+y^2}\) .
Also \(F_y=\frac{2y}{x^2+y^2}+\frac{x}{x^2+y^2}=\frac{x+2y}{x^2+y^2}\) . Therefore \(\frac{dy}{dx}=-\frac{F_x}{F_y}=-\frac{2x-y}{x+2y}=\frac{y-2x}{x+2y}\) .
5 Final Answer\(\displaystyle \frac{dy}{dx}=\frac{y-2x}{x+2y}\) .
Question 1(d) Assignment Problem – LPP Formulation 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionAn automobile dealer wishes to put four repairmen \(R_1,R_2,R_3,R_4\) to four different jobs \(J_1,J_2,J_3,J_4\) . But \(R_3\) cannot do the job \(J_2\) . The estimated man-hours are:
Job R1 R2 R3 R4 J1 6 2 3 4 J2 9 7 – 5 J3 6 4 7 5 J4 6 8 8 9
Formulate the above as a Linear Programming Problem.
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(e) Complex Analysis – Analytic Functions 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(f(z)=u+iv\) is any analytic function of the complex variable \(z\) and \(u-v=e^x(\cos y-\sin y)\) , find \(f(z)\) in terms of \(z\) .
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 2(a) Group Theory – Cayley Theorem 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionProve that every group is isomorphic to a permutation group.
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) Improper Integrals 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionExamine the convergence of \(\displaystyle \int_0^\infty\frac{dx}{(1+x)\sqrt{x}}\) and find its value, if possible.
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) Complex Analysis – Taylor Series 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the Taylor's series expansion of the function of the complex variable \(f(z)=\frac1{(z-1)(z-3)}\) about the point \(z=4\) . Find its radius of convergence.
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) Functions of Two Variables – Maxima and Minima 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionExamine the existence of maxima and minima of the function \(u(x,y)=xy+\frac8x+\frac8y\) .
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) Ring Theory – Prime Ideals and Boolean Rings 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 Question(i) Let \(R\) be a non-zero commutative ring with unity. If every ideal of \(R\) is prime, prove that \(R\) is a field.
(ii) Let \(R\) be a commutative ring with unity such that \(a^2=a\) , for all \(a\in R\) . If \(I\) is any prime ideal of \(R\) , find all the elements of \(R/I\) .
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 3(c) Linear Programming – Duality 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionConsider the following Linear Programming Problem as primal: minimize \(z=30x_1+20x_2\) , subject to \(3x_1+5x_2\ge100\) , \(2x_1+x_2\ge120\) , \(5x_1+3x_2\ge90\) , \(x_1,x_2\ge0\) . Then, using the principle of duality, find the optimal solution of the primal.
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 4(a) Complex Analysis – Definite Integral 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow that \(\displaystyle \int_0^{2\pi}\frac{\cos3\theta}{5-4\cos\theta}\,d\theta=\frac\pi{12}\) .
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 4(b) Algebra – Euclidean Domains 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow that an element \(x\) in a Euclidean domain is a unit if and only if \(d(x)=d(1)\) , where the notations have their usual meanings.
2 Diagram
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Question 4(c) Operations Research – Transportation Problem 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionStarting with Least Cost Method, find all the solutions to the following transportation problem:
Plants/Warehouses I II III IV Supply A 8 6 5 3 18 B 6 7 6 8 20 C 10 8 4 5 18 Demand 15 16 12 13 56
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 5(a) Partial Differential Equations – Complete Primitive 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the complete primitive of \(4r-4s+t=16\log_e(x+2y)\) , where \(r,s,t\) bear their usual meanings.
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 5(b) Numerical Analysis – Interpolation 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFrom the following table, estimate the number of students who obtained marks between \(40\) and \(46\) :
Marks 30–40 40–50 50–60 60–70 70–80 No. of students 32 43 55 40 30
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 5(c) Computer Arithmetic – Bitwise Operations 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionConsider the following integers and their 8-bit binary representations: \(13=00001101\) , \(20=00010100\) . Perform the following bitwise operations and express the results in decimal system: (i) \(13\&20\) (Bitwise AND), (ii) \(13|20\) (Bitwise OR), (iii) \(13\,\hat{}\,20\) (Bitwise XOR), (iv) \(\sim20\) (Bitwise Complement).
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 5(d) Analytical Mechanics – Particle on Parabolic Wire 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionExamine the motion of a particle sliding on a parabolic wire given by \(x^2=2y\) .
2 Diagram
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Question 5(e) Differential Equations – Orthogonal Trajectories 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the orthogonal trajectory of the family of curves \(x^2-y^2=a^2\) . Then sketch the two families to demonstrate whether they cut orthogonally.
2 Diagram
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Question 6(a) Partial Differential Equations – Charpit Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve the following by Charpit's method: \(pxy+pq+qy=yz\) , where \(p=\frac{\partial z}{\partial x}\) and \(q=\frac{\partial z}{\partial y}\) .
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Question 6(b) Numerical Analysis – Regula-Falsi Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionUsing Regula-Falsi method, find the fourth root of \(28\) correct to three decimal places.
2 Diagram
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Question 6(c) Fluid Dynamics – Possible Motion and Streamlines 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionVerify whether the motion given by \(\vec q=(3x\hat i-2y\hat j)xy^2\) is a possible fluid motion. If so, is it of the potential kind? Accordingly find out the streamlines and the velocity potential or the angular velocity if the fluid was replaced by a rigid solid.
2 Diagram
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4 Detailed Solution
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Question 7(a) Numerical Analysis – Runge-Kutta Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionWrite down the algorithm and flowchart of Runge-Kutta method of fourth order to find the numerical solution at \(x=0.8\) for \(\frac{dy}{dx}=\sqrt{2(x+y)}\) , \(y(0.4)=0.82\) . Assume the step length \(h=0.2\) .
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 7(b) Fluid Dynamics – Complex Potential 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionDiscuss the flow given by the complex potential \(w=\log_e\left(z-\frac{a^2}{z}\right)\) . Draw sketches of the streamlines and explain the flow directions along the streamlines.
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) Partial Differential Equations – Lagrange Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve the following differential equation: \((y^2+z^2-x^2)p-2xyq+2xz=0\) , where \(p=\frac{\partial z}{\partial x}\) and \(q=\frac{\partial z}{\partial y}\) .
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 8(a) Analytical Mechanics – Lagrange Equations in Spherical Coordinates 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionDerive the Lagrange's equation for a spherical problem.
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) Numerical Analysis – Gauss-Seidel Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve the following system of equations by Gauss-Seidel method: \(20x+y-3z=16\) , \(2x+20y-z=-19\) , \(3x-2y+20z=25\) , starting with the initial solution \(x_0=y_0=z_0=0\) .
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 8(c) Differential Equations – Singular Solution 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the singular solution of \(yp^2-2xp+y=0\) . Also trace the graph.
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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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
2021 IFoS Maths Optional Paper II Solutions FAQs
Are these 2021 IFoS Maths Optional Paper II Solutions complete? This public page gives one full sample solution for 2021 IFoS Maths Optional Paper II Solutions. Complete question-wise solutions for the full paper are available in the full PYQ course.
Which question is given as a free sample solution on this page? Question 1(c) is given as the free sample solution on this 2021 IFoS Maths Optional Paper II Solutions page.
How should I use these 2021 IFoS Maths Optional Paper II Solutions for preparation? Students should first solve the question independently, then compare their method with the solution format, diagram presentation, concept explanation, detailed solution, and final answer.
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