2022 IFoS Maths Optional Paper II Solutions | Ramana Sri IAS
2022 IFoS Maths Optional Paper II Solutions
2022 IFoS Maths Optional Paper II Solutions
Ramana Sri IAS provides complete and updated solutions for the 2022 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 2022 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 2022 IFoS Maths Optional Paper II Solutions
These 2022 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 2022 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.
These 2022 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 2022 IFoS Maths Optional Paper II Solutions as a free sample solution below:
Question 1(d). This free sample includes all five sections:
Question, Diagram,
Concept Related to the Question,
Detailed Solution, and Final Answer.
Complete 2022 IFoS Maths Optional Paper II Solutions for all questions are available in the full PYQ course.
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Question 1(a)
Finite Fields
1Question
Let \(F\) be a finite field of characteristic \(p\), where \(p\) is a prime. Then show that there is an injective homomorphism from \(\mathbb Z_p\) to \(F\). Also show that the number of elements in \(F\) is \(p^n\), for some positive integer \(n\).
2Diagram
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The Diagram section for this question is available in the full 2022 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.
Let \(\mathbb R\) denote the set of real numbers and \(\mathbb Q\) denote the set of rational numbers. If \(x\in\mathbb R,\;x\gt0\) and \(y\in\mathbb R\), then show that there exists a positive integer \(n\) such that \(nx\gt y\). Use it to show that if \(x\lt y\), then there exists \(p\in\mathbb Q\) such that \(x\lt p\lt y\).
2Diagram
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The Diagram section for this question is available in the full 2022 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.
subject to the constraints \(2x_1+x_2\le2\), \(3x_1+4x_2\ge12\), and \(x_1,x_2\ge0\), does not admit an optimum basic feasible solution.
2Diagram
Question 1(d): LPP with No Optimum Basic Feasible Solution
3Concept Related to the Question
This question belongs to Linear Programming – Infeasibility. The method is to first write the definition or standard formula, then substitute the given data, and finally simplify each step without skipping the reason behind the step.
4Detailed Solution
From \(2x_1+x_2\le2\) and non-negativity, the largest possible value of \(3x_1+4x_2\) occurs on the boundary at one of the intercepts. At \((1,0)\), it is \(3\), and at \((0,2)\), it is \(8\). Hence \(3x_1+4x_2\le8\) for all points satisfying the first constraint. This can never be at least \(12\).
5Final Answer
The feasible region is empty. Therefore there is no optimum basic feasible solution.
Question 1(e)
Complex Integration – Residues
1Question
Compute the integral \(\displaystyle \int_C \frac{1+2z+z^2}{(z-1)^2(z+2)}\,dz\), where \(C\) is \(|z|=3\).
2Diagram
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The Diagram section for this question is available in the full 2022 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.
Suppose \(\{f_n\}\) is a sequence of functions defined on \([a,b]\) and \(\displaystyle \lim_{n\to\infty}f_n(x)=f(x),\;x\in[a,b]\). Put \(\displaystyle M_n=\sup_{x\in[a,b]}|f_n(x)-f(x)|\). Then show that:
(i) \(f_n\) converges to \(f\) uniformly on \([a,b]\) if and only if \(M_n\to0\) as \(n\to\infty\).
(ii) If \(|f_n(x)|\le M_n,\;(x\in[a,b],\;n=1,2,\ldots)\), then \(\displaystyle \sum_{n=1}^{\infty}f_n\) converges uniformly on \([a,b]\) if \(\displaystyle \sum_{n=1}^{\infty}M_n\) converges.
2Diagram
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
(i) Prove that every bounded and monotonically increasing sequence is convergent and converges to the lub, that is, the least upper bound of the sequence.
(ii) If \(a_n=1+\frac12+\frac13+\cdots+\frac1n,\;\forall n\in\mathbb N\), then using Cauchy criterion for convergence of the sequence, show that \(\{a_n\}\) is not convergent.
2Diagram
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(i) Let \(P\) be a Sylow \(p\)-subgroup of a group \(G\) and \(H\) be any \(p\)-subgroup of \(G\) such that \(HP=PH\). Then show that \(H\subseteq P\).
(ii) Show that every group of order \(15\) is cyclic.
2Diagram
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The Diagram section for this question is available in the full 2022 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.
(i) Find an upper bound for the absolute value of the integral \(\displaystyle I=\int_C e^z\,dz\), where \(C\) is the line segment joining the points \((0,0)\) and \((1,3)\).
(ii) Find the length of the curve \(C\) defined by \(z(t)=(1-2i)t^3,\;-1\le t\le1\).
2Diagram
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The Diagram section for this question is available in the full 2022 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.
Equation of any cone with vertex at the point \((a,b,c)\) is of the form \(\displaystyle f\left(\frac{x-a}{z-c},\frac{y-b}{z-c}\right)=0\). Find the partial differential equation of the cone.
2Diagram
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The Diagram section for this question is available in the full 2022 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.
A rod of length \(2a\) revolves with uniform angular velocity \(\omega\) about a vertical axis through a smooth joint at one extremity of the rod so that it describes a cone of semi-vertical angle \(\alpha\). Prove that the direction of reaction at the hinge makes with the vertical an angle \(\displaystyle \tan^{-1}\left(\frac34\tan\alpha\right)\).
2Diagram
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The Diagram section for this question is available in the full 2022 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.
Find the system of equations for obtaining the general equation of surfaces orthogonal to the family given by \(x(x^2+y^2+z^2)=Cy^2\), where \(C\) is a parameter.
2Diagram
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The Diagram section for this question is available in the full 2022 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.
Write down an algorithm for Simpson's \(\frac13\) rule. Hence, compute \(\displaystyle \int_0^1x^2(1-x)\,dx\) correct up to three decimal places with step size \(h=0.1\) and compare the result with its exact value.
2Diagram
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The Diagram section for this question is available in the full 2022 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.
If the velocity of an incompressible fluid at the point \((x,y,z)\) is given by \((-Ay,Ax,0)\), then prove that the surfaces intersecting the streamlines orthogonally exist and are the planes through \(z\)-axis, although the velocity potential does not exist. Discuss the nature of the fluid flow.
2Diagram
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The Diagram section for this question is available in the full 2022 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.
Verify that \(\displaystyle w=ik\log\left(\frac{z-ia}{z+ia}\right)\) is the complex potential of a steady fluid flow about a circular cylinder, the plane \(y=0\) being a rigid boundary. Further show that the fluid exerts a downward force of magnitude \(\displaystyle \frac{\pi\rho k^2}{2a}\) per unit length on the cylinder, where \(\rho\) is the fluid density.
2Diagram
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The Diagram section for this question is available in the full 2022 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.
Find the solution of the partial differential equation \(\displaystyle z=\frac12(p^2+q^2)+(p-x)(q-y),\;p=\frac{\partial z}{\partial x},\;q=\frac{\partial z}{\partial y}\), which passes through the \(x\)-axis, using Cauchy's method of characteristics.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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.
A particle of unit mass is projected so that its total energy is \(h\) in a field of force of which the potential energy is \(\phi(r)\) at a distance \(r\) from the origin. By employing the principle of energy and least action, show that the path is given by the following differential equation: \(\displaystyle c^2\left[r^2+\left(\frac{dr}{d\theta}\right)^2\right]=r^4[h-\phi(r)]\), where \(c\) is a constant.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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.
Find a complete integral of the partial differential equation \((p^2+q^2)x=pz,\;p=\frac{\partial z}{\partial x},\;q=\frac{\partial z}{\partial y}\) using Charpit's method and hence deduce the solution which passes through the curve \(x=0,\;z^2=4y\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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.
Are these 2022 IFoS Maths Optional Paper II Solutions complete?
This public page gives one full sample solution for 2022 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(d) is given as the free sample solution on this 2022 IFoS Maths Optional Paper II Solutions page.
How should I use these 2022 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.
Do these solutions include diagrams and detailed solutions?
Yes. The complete PYQ course includes question-wise diagrams where needed, concept related to the question, detailed solutions, and final answers.
How can I get complete solutions for all questions in 2022 IFoS Maths Optional Paper II?
To get complete solutions for all questions, students can fill out the admission form. The Ramana Sri IAS admission team will guide students through WhatsApp, email, or call.
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