2024 IFoS Maths Optional Paper II Solutions
2024 IFoS Maths Optional Paper II Solutions Ramana Sri IAS provides complete and updated solutions for the 2024 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 2024 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 2024 IFoS Maths Optional Paper II Solutions These 2024 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 2024 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 2024 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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Question 1(d) . This free sample includes all five sections:
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2024 IFoS Maths Optional Paper II Solutions: Table of Contents
Question 1(a) Ring Theory – Matrix Ideals 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionLet \(R\) be the ring of \(n\times n\) matrices over reals. Show that \(R\) has only two ideals, namely \(\{0\}\) and \(R\) .
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) Series – Absolute and Conditional Convergence 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow that the series \(\displaystyle \frac1{(1+a)^p}-\frac1{(2+a)^p}+\frac1{(3+a)^p}-\cdots,\;a>0\) is
(i) absolutely convergent if \(p\gt1\) ,
(ii) conditionally convergent if \(0\lt p\le1\)
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) Maxima and Minima 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(f'(x)=(x-a)^{2n}(x-b)^{2m+1}\) , where \(m,n\) are positive integers, show that \(f\) has neither a maximum nor a minimum at \(a\) and \(f\) has a local minimum at \(b\) .
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) Complex Analysis – Analytic Function 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionLet \(f(z)=u(r,\theta)+iv(r,\theta)\) be an analytic function. If \(u=-r^3\sin3\theta\) , then construct the corresponding analytic function \(f(z)\) in terms of \(z\) .
2 DiagramQuestion 1(d): Analytic function from polar real part
3 Concept Related to the QuestionThis question belongs to Complex Analysis – Analytic Function . The method is to identify the relevant theorem or formula first, substitute the given data, and simplify each step clearly.
4 Detailed SolutionSince \(z=re^{i\theta}\) , we have \(z^3=r^3e^{3i\theta}=r^3(\cos3\theta+i\sin3\theta)\) . Multiplying by \(i\) , \(iz^3=ir^3\cos3\theta-r^3\sin3\theta\) . Thus the real part of \(iz^3\) is \(-r^3\sin3\theta\) .
Therefore the required analytic function is \(iz^3\) , up to an additive purely imaginary constant.
5 Final Answer\(f(z)=iz^3+iC\) , where \(C\) is real.
Question 1(e)(i) Linear Programming – Graphical Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind all optimal solutions of the following linear programming problem graphically:
Maximize \(z=3x_1+6x_2\)
subject to \(x_1+x_2\le8\) , \(x_1-x_2\le4\) , \(2x_1-x_2\ge4\) , and \(x_1,x_2\ge0\) .
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)(ii) Linear Programming – Alternative Optima 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionThe LPP in part (i) with the first constraint \(x_1+x_2\le8\) changed to \(x_1+2x_2\le12\) .
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)(i) Group Theory 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(G\) is a group of even order, then show that there exists an element \(a\) other than the identity element such that \(a^2=e\) .
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)(ii) Ring Theory – Maximal Ideal 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionProve that an ideal \(S\) of the ring \(\mathbb Z\) of all integers is a maximal ideal if \(S\) is generated by some prime 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 2(b) Improper Integral – Beta Function 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionExamine the convergence of the improper integral \(\displaystyle \int_0^\infty \frac{x^{p-1}}{1+x}\,dx\) and hence evaluate it.
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) Residue Theorem 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionDetermine the poles of the function \(\displaystyle f(z)=\frac{z^2}{(z-1)^2(z-2)}\) and the residue at each pole, and hence evaluate \(\displaystyle \oint_C f(z)\,dz\) , where \(C\) is the circle \(|z|=2.5\) .
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) Limits 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionEvaluate \(\displaystyle \lim_{x\to0}\frac{\int_0^{x^2}e^{\sqrt{1+t}}\,dt}{x^2}\) .
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) Unique Factorization Domain 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionProve that in a Unique Factorization Domain \(R\) , an element is prime if and only if it is irreducible.
2 Diagram
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Question 3(c) Simplex Method and Dual 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve the LPP:
Maximize \(z=2x_1+x_2+x_3\)
subject to \(x_1+2x_2-x_3\le3\) , \(x_1-2x_2-5x_3\ge-9\) , and \(x_1,x_2,x_3\ge0\) , by the simplex method. Write its dual problem and from the optimal table of the given problem, obtain the optimal solution of the dual problem.
2 Diagram
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Question 4(a) Group Homomorphism 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionLet \(G\) and \(H\) be finite groups such that \(\gcd(|G|,|H|)=1\) . Show that the trivial homomorphism is the only homomorphism from \(G\) into \(H\) .
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 4(b)(i) Complex Mapping 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the image of \(|z-3i|=3\) under the mapping \(w=\dfrac1z\) .
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 4(b)(ii) Complex Integration 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the value of the integral \(\displaystyle \int_0^{1+i}(x-y+ix^2)\,dz\) along the straight line from \(z=0\) to \(z=1+i\) .
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) Transportation Problem 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the initial basic feasible solution of the following minimum cost transportation problem by Vogel's Approximation Method (VAM). Using it, find the optimal solution and the minimum transportation cost. Is the optimal solution unique? If not, find an alternative optimal solution.
Origin / Destination D1 D2 D3 D4 Availability O1 3 5 8 2 50 O2 5 7 2 9 40 O3 7 1 3 4 30 Demand 40 35 25 20 120
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 – 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 by eliminating the arbitrary function \(f\) from the equation \(f(x+y+z,\;x^2+y^2+z^2)=0\) .
2 Diagram
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Question 5(b) Numerical Integration – Simpson 3/8 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionObtain the following approximate quadrature formula: \(\displaystyle \int_0^3 f(x)\,dx=\frac38\{f(0)+3f(1)+3f(2)+f(3)\}\) .
2 Diagram
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Question 5(c)(i) Number Systems 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionConvert \((523.0234375)_{10}\) into an equivalent octal number and then convert it to its binary form.
2 Diagram
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Question 5(c)(ii) Hexadecimal Arithmetic 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(x=(1D2.2)_{16}\) and \(y=(52E.02)_{16}\) , then find the value of \(x+y\) in the decimal system.
2 Diagram
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5 Final Answer
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Question 5(d) Fluid Dynamics – Streamlines and Pathlines 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionThe velocity components in an unsteady three dimensional flow are given by \(u=\dfrac{x}{1+t}\) , \(v=\dfrac{y}{1+t}\) , and \(w=\dfrac{z}{1+t}\) . Describe the streamlines and pathlines.
2 Diagram
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Question 5(e) Holonomic Constraint 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIs a system of two particles which are connected by a rod of constant length holonomic? Justify your answer.
2 Diagram
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Question 6(a) PDE – Lagrange Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionUsing Charpit's method, find the complete integral of \(yq+3xp=2(z-y^2p^2)\) , where \(p=\dfrac{\partial z}{\partial x}\) and \(q=\dfrac{\partial z}{\partial y}\) .
2 Diagram
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Question 6(b) Euler Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionWrite down the algorithm for solving the differential equation \(\dfrac{dy}{dx}=f(x,y),\;y(x_0)=y_0\) numerically by Euler's method with step length \(h\) up to \(x=x_n=x_0+nh\) .
Solve the following differential equation for \(x=1\) with step length \(h=0.2\) by Euler's method: \(\dfrac{dy}{dx}=x^2+y,\;y(0)=1\) .
2 Diagram
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Question 6(c) Hamilton Equations 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionDerive the Hamilton equations for holonomic systems and use them to discuss the motion of a simple pendulum.
2 Diagram
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Question 7(a) Newton-Raphson Method 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow that the iteration formula for the Newton-Raphson method for finding the \(K^{\text{th}}\) root of a positive real number \(a\) is \(\displaystyle x_{n+1}=\frac1K\left[(K-1)x_n+\frac{a}{x_n^{K-1}}\right]\) , where \(K\gt0\) . Use this formula to find \(\sqrt[3]{13}\) , correct up to three decimal places.
2 Diagram
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Question 7(b) Linear PDE with Constant Coefficients 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the general solution of the partial differential equation \([D^2-(D')^2-3D+3D']z=(1-x)(1-y)+e^{x+2y}\) , where \(D=\dfrac{\partial}{\partial x}\) and \(D'=\dfrac{\partial}{\partial y}\) .
2 Diagram
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Question 7(c) Fluid Dynamics – Pressure 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionConsider an inviscid incompressible fluid flow with velocity \(\displaystyle \bar q=\left(x,\frac{y}{1+t},\frac{z}{2+t}\right)\) under the body force \(\bar F=-gz\hat k\) , where \(g\) is the gravitational constant. Find the pressure at a point \((x,y,z)\) if \(p(0,0,0)=p_0\) .
2 Diagram
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Question 8(a) Fluid Dynamics – Source and Sink 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionConsider a source and a sink of equal strength at points \((\pm\frac14a,0)\) within a fixed circular boundary \(x^2+y^2=a^2\) . Determine the equation of streamlines.
2 Diagram
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Question 8(b) Heat Equation – Fourier Sine Series 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the solution of the heat equation \(\displaystyle \frac{\partial u}{\partial t}=4\frac{\partial^2u}{\partial x^2},\;0\lt x\lt\pi,\;t\gt0\) under the boundary conditions \(u(0,t)=0=u(\pi,t)\) and the initial condition \(\displaystyle u(x,0)=\begin{cases}x,&0\le x<\frac\pi2,\\ \pi-x,&\frac\pi2\le x\le\pi.\end{cases}\)
2 Diagram
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Question 8(c) Gauss-Jordan Elimination 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionUse Gauss-Jordan elimination method to solve the following system of equations: \(3x_1+x_2+x_3=7\) , \(2x_1+x_2+5x_3=13\) , and \(x_1+4x_2+x_3=9.4\) , correct up to two significant figures.
2 Diagram
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2024 IFoS Maths Optional Paper II Solutions FAQs
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This public page gives one full sample solution for 2024 IFoS Maths Optional Paper II Solutions. Complete question-wise solutions for the full paper are available in the full PYQ course by Ramana Sri IAS.
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Question 1(d) is given as the free sample solution on this page. The sample includes all five sections: Question, Diagram, Concept Related to the Question, Detailed Solution, and Final Answer.
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