2023 IFoS Maths Optional Paper II Solutions | Ramana Sri IAS
Ramana Sri IAS - 2023 IFoS Maths Optional Paper II Solutions
2023 IFoS Maths Optional Paper II Solutions
Ramana Sri IAS provides complete and updated solutions for the 2023 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 2023 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 2023 IFoS Maths Optional Paper II Solutions
These 2023 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 2023 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 2023 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 2023 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 2023 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.
Question 1(a)
Group Theory – Cyclic Groups
1Question
Prove that a subgroup of a cyclic group is cyclic. Let \(G\) be a cyclic group with generator \(a\). If the order of \(G\) is infinite, then prove that \(G\) is isomorphic to \((\mathbb Z,+)\).
2Diagram
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The Diagram section for this question is available in the full 2023 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.
Prove that \(x_1=2,x_2=1,x_3=0\) is a feasible solution to the following set of equations:
\(2x_1-x_2+3x_3=3\), \(-6x_1+3x_2+7x_3=-9\).
Is the solution basic? Justify your answer. If the solution is not basic, reduce it to a basic feasible one.
2Diagram
Question 1(d): Basic Feasible Solution in Linear Programming
3Concept Related to the Question
This question belongs to Linear Programming – Basic Feasible Solution. The solution uses the standard method: write the relevant theorem or formula, substitute the given data carefully, and simplify step by step.
4Detailed Solution
Substituting \((2,1,0)\) gives \(2(2)-1+3(0)=3\) and \(-6(2)+3(1)+7(0)=-9\), so the point is feasible.
The positive variables are \(x_1\) and \(x_2\). Their coefficient columns are \((2,-6)^T\) and \((-1,3)^T\), which are linearly dependent. Hence the feasible solution is not basic. From the equations, \(16x_3=0\), so \(x_3=0\). Then \(2x_1-x_2=3\). Taking \(x_2=0\) gives \(x_1=\frac32\).
5Final Answer
The point is feasible but not basic. A reduced basic feasible solution is \(\left(\frac32,0,0\right)\).
Question 1(e)
Bilinear Transformation
1Question
Find a bilinear transformation which maps the points \(z=0,-i,-1\) into \(w=i,1,0\) respectively.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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) Prove that if \(u_n(x),\;n=1,2,3,\ldots\) are continuous in \([a,b]\) and if \(\sum u_n(x)\) converges uniformly to the sum \(S(x)\) in \([a,b]\), then \(S(x)\) is continuous in \([a,b]\).
(ii) Prove that an absolutely convergent series is convergent. Show that \(1-\frac12+\frac13-\frac14+\cdots\) is conditionally convergent.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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.
Group Theory – Normal Subgroups and Second Isomorphism
1Question
(i) If \(N\) is a normal subgroup of a group \(G\) and if \(H\) is any subgroup of \(G\), then prove that \(H\vee N=HN=NH\), where \(H\vee N\) denotes the join of \(H\) and \(N\).
(ii) State the Second Isomorphism Theorem of groups and apply it to the case \(G=\mathbb Z\times\mathbb Z\times\mathbb Z\), \(H=\mathbb Z\times\mathbb Z\times\{0\}\) and \(N=\{0\}\times\mathbb Z\times\mathbb Z\).
2Diagram
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The Diagram section for this question is available in the full 2023 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 \(x_1+2x_2+2x_3\ge1\), \(x_1-2x_3\ge-1\), \(x_1-x_2+3x_3\ge3\), and \(x_i\ge0,\;i=1,2,3\). Solve the dual of the above LPP and find the minimum value of \(Z\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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.
Five workers perform five jobs and the operating cost is given below, but there is a restriction that the worker \(C\) cannot perform the third job and \(B\) cannot perform the fifth job. Find the optimal assignment and the optimal assignment cost.
Worker / Job
I
II
III
IV
V
A
24
29
18
32
19
B
17
26
34
22
Not allowed
C
27
16
Not allowed
17
25
D
22
18
28
30
24
E
28
16
31
24
27
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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.
Consider a particle of mass \(m\) moving in a plane under attractive force \(\frac{k}{r^2}\) directed towards the origin, where \(k\gt0\). Using the polar coordinates \((r,\theta)\), write the corresponding Lagrangian and obtain the equations of motion. Also show that the angular momentum is conserved.
2Diagram
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The Diagram section for this question is available in the full 2023 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 function \(f\), defined on \([0,1]\), is such that \(f(0)=0\), \(f\left(\frac12\right)=-1\), \(f(1)=0\). Find the quadratic polynomial \(p(x)\) which agrees with \(f(x)\) for \(x=0,\frac12,1\). If \(\left|\frac{d^3f}{dx^3}\right|\le1\) for \(0\le x\le1\), show that \(|f(x)-p(x)|\le\frac1{12}\) for \(0\le x\le1\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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.
In a two-dimensional flow there are sources at \((a,0)\), \((-a,0)\) and sinks at \((0,a)\), \((0,-a)\), all are of equal strength. Determine the stream function and show that the circle through these four points is a streamline.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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 \(u_x-u u_y+u=0\) for the initial values \(x_0(s)=0\), \(y_0(s)=s\), \(u_0(s)=-2s\). Does the solution break down for any finite \(x\)? Is the solution unique?
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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.
For a dynamical system having two degrees of freedom, the Lagrangian is given by \(\displaystyle L=\frac m2(a^2\dot q_1^{\,2}+\dot q_2^{\,2})-\frac k2(a^2+q_2^2)\), where \(q_1\) and \(q_2\) are generalized coordinates. Find the corresponding Hamiltonian and derive the Hamiltonian equations of motion. Show further that the generalized momentum corresponding to \(q_1\) is constant. Show that the system exhibits a simple harmonic motion with respect to the generalized coordinate \(q_2\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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 the flow-chart of Runge-Kutta method of fourth order to find \(y(0.8)\) for \(\frac{dy}{dx}=xy\), \(y(0)=2\), taking \(h=0.2\). Also solve the above IVP to find \(y(0.4)\) by Runge-Kutta method of fourth order.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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.
Consider two-dimensional Navier-Stokes equations of a steady fluid flow. Show that there exists a stream function \(\Psi(x,y)\) for such a flow. Find the equation satisfied by \(\Psi(x,y)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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.
Show that \(f(x,y,z,p,q)=x^2p^2+y^2q^2-4=0\) and \(g(x,y,z,p,q)=qy-a=0\), where \(a\) is a constant, are compatible and hence solve \(f(x,y,z,p,q)=0\). Is it complete integral?
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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.
There is a doublet at \((c,0)\) in a two-dimensional flow. A cylinder of radius \(a\), \(a\lt c\), with \(z\)-axis as axis of the cylinder was introduced into the flow. Find the complex potential and image system for the flow.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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 2023 IFoS Maths Optional Paper II Solutions complete?
This public page gives one full sample solution for Question 1(d). 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(d) 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 2023 IFoS Maths Optional Paper II Solutions for preparation?
Students should first solve the question independently, then compare their method with the given step-by-step solution, concept application, diagram presentation, and final-answer format.
Do these solutions include diagrams and detailed solutions?
Yes. The full PYQ course includes diagrams where needed, concept explanations, detailed solutions, and final answers for the questions.
How can I get complete solutions for all questions in 2023 IFoS Maths Optional Paper II?
To get complete solutions for all questions, fill out the admission form. The Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
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