Ramana Sri IAS - 2025 UPSC Maths Optional Paper II Solutions
2025 UPSC Maths Optional Paper II Solutions
Ramana Sri IAS provides complete and updated solutions for the 2025 UPSC Maths Optional Paper II. Aspirants preparing for the UPSC 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 UPSC/IAS/CSE-Civil Service Examination 2025 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 2025 UPSC Maths Optional Paper II Solutions
These 2025 UPSC Maths Optional Paper II Solutions are prepared by Ramana Sri IAS for aspirants who want question-wise clarity before the UPSC 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 2025 UPSC 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 2025 UPSC Maths Optional Paper II Solutions are useful for revision, answer-writing practice, and understanding the step-by-step method expected in the UPSC Mathematics optional paper.
Sample Full Solution
We are giving one question from 2025 UPSC 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 2025 UPSC 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.
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2025 UPSC Maths Optional Paper II Solutions: Table of Contents
Let \(H\) and \(K\) be two subgroups of a group \(G\) such that \(o(H)>\sqrt{o(G)}\) and \(o(K)>\sqrt{o(G)}\). Show that \(H\cap K\neq(e)\), where \(e\) is the identity element. Here \(o(H)\), \(o(K)\) and \(o(G)\) denote the order of \(H\), \(K\) and \(G\) respectively.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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 \(G=\{e,x,x^2,y,yx,yx^2\}\) be a non-Abelian group with \(o(x)=3\) and \(o(y)=2\). Show that \(xy=yx^2\) (where \(e\) is the identity element of \(G\) and \(o(x)\), \(o(y)\) denote the order of the elements \(x\), \(y\) respectively).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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.
Expand \(f(z)=\dfrac1{(z+1)(z+3)}\) in a Laurent series valid for \(1<\lvert z\rvert<3\).
2Diagram
Question 1(d): Laurent series in the annulus
3Concept Related to the Question
Use partial fractions and geometric-series expansion in the annulus \(1<|z|<3\).
4Detailed Solution
We have \(\dfrac1{(z+1)(z+3)}=\dfrac12\left(\dfrac1{z+1}-\dfrac1{z+3}\right)\). For \(|z|>1\), \(\dfrac1{z+1}=\dfrac1z\dfrac1{1+1/z}=\sum_{n=0}^\infty(-1)^n z^{-n-1}\). For \(|z|<3\), \(\dfrac1{z+3}=\dfrac13\dfrac1{1+z/3}=\sum_{n=0}^\infty(-1)^n z^n/3^{n+1}\).
5Final Answer
The required Laurent series is \(\dfrac12\sum_{n=0}^\infty(-1)^n z^{-n-1}-\dfrac12\sum_{n=0}^\infty\dfrac{(-1)^n z^n}{3^{n+1}}\).
Question 1(e)
Basic solutions of linear equations
1Question
How many basic solutions are there for the following system equations?
\[2x_1-x_2+3x_3+x_4=6,\]
\[4x_1-2x_2-x_3+2x_4=10.\]
Find all of them. Further, find the number of basic solutions, which are feasible/non-feasible/non-degenerate.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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 the volume of the greatest rectangular parallelopiped that can be inscribed in the ellipsoid \(\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\) is \(\displaystyle \frac{8abc}{3\sqrt3}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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.
Examine whether the mapping \(\phi:Z[x]\to Z\) defined by \(\phi(f(x))=f(0)\), for \(f(x)\in Z[x]\), is a homomorphism. Deduce that the ideal \((x)\) is a prime ideal in \(Z[x]\), but not a maximal ideal in \(Z[x]\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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.
The following table shows all the necessary information on the available supply to each warehouse, the requirement of each market and the unit transportation cost from each warehouse to each market:
Market \(I\)
Market \(II\)
Market \(III\)
Market \(IV\)
Supply
Warehouse \(A\)
\(5\)
\(2\)
\(4\)
\(3\)
\(22\)
Warehouse \(B\)
\(4\)
\(8\)
\(1\)
\(6\)
\(15\)
Warehouse \(C\)
\(4\)
\(6\)
\(7\)
\(5\)
\(8\)
Requirement
\(7\)
\(12\)
\(17\)
\(9\)
The shipping clerk has worked out the following schedule from experience: \(12\) units from \(A\) to \(II\), \(1\) unit from \(A\) to \(III\), \(9\) units from \(A\) to \(IV\), \(15\) units from \(B\) to \(III\), \(7\) units from \(C\) to \(I\), and \(1\) unit from \(C\) to \(III\). Find the optimal schedule and minimum total shipping cost.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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.
Determine the truth table for the Boolean function \(F(x,y,z)=(x+y+z')(x'+y')\). Also derive the full disjunctive normal form of \(F(x,y,z)\) from the truth table.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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 bead of mass \(m\) slides on a frictionless wire in the shape of a cycloid given by \(x=a(\theta-\sin\theta)\), \(y=a(1+\cos\theta)\), \(0\leq\theta\leq2\pi\). Find the Lagrangian function. Hence show that the equation of motion can be written as \(\frac{d^2u}{dt^2}+\frac{g}{4a}u=0\), where \(u=\cos\left(\frac{\theta}{2}\right)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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 source and a sink of equal strength are placed at points \(\left(\pm\frac a2,0\right)\) within a fixed circular boundary \(x^2+y^2=a^2\). Show that the streamlines are given by \(\left(r^2-\frac{a^2}{4}\right)(r^2-4a^2)-4a^2y^2=k y(r^2-a^2)\), where \(k\) is a constant and \(r^2=x^2+y^2\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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.
Solve \(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0\) for a rectangular plate subject to the boundary conditions \(u(0,y)=0\), \(u(a,y)=0\), \(u(x,0)=0\), and \(u(x,b)=f(x)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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.
Calculate the moment of inertia of a uniform solid cylinder of mass \(M\), radius \(R\) and length \(L\) with respect to a set of axes passing through the centre of the cylinder, where \(z\)-axis is the axis of the cylinder and \(\rho\) is the constant density at any point of the cylinder. Also find \(\frac LR\) for which the moment of inertia about \(x\)- or \(y\)-axis will be minimum for a given mass of the cylinder.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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 for an incompressible steady flow with constant viscosity, the velocity components \(u(y)=\left(\frac Uh\right)y-\frac{hy}{2\mu}\frac{dp}{dx}\left(1-\frac yh\right)\), \(v=0=w\), with \(p=p(x)\), satisfy the equation of motion in the absence of body force. Given that \(U\), \(h\) and \(\frac{dp}{dx}\) are constants.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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 characteristics of the partial differential equation \(p^2+q^2=2;\ p=\frac{\partial z}{\partial x},\ q=\frac{\partial z}{\partial y}\), and determine the integral surface which passes through \(x=0,\ z=y\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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 constant \(p\) and error term for the quadrature formula \(\displaystyle \int_{x_0}^{x_1}f(x)\,dx=\frac h2(f_0+f_1)+ph^2(f'_0-f'_1)\), where \(x_0+h=x_1\), \(f_0=f(x_0)\), \(f_1=f(x_1)\) and prime \((\, '\, )\) represents derivative with respect to \(x\). Hence deduce the composite rule for integrating \(\displaystyle \int_a^b f(x)\,dx\), \(a=x_0\lt x_1\lt\cdots\lt x_N=b\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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 mass \(m\) moves in a force field of potential \(V(r)=\dfrac{k\cos\theta}{r^2}\), \(k\) constant. Find the Hamiltonian and Hamilton’s equations in spherical polar coordinates \((r,\theta,\phi)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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 the Lagrangian \(L=m\dot x\dot y-m\omega_0^2xy\), where \(m\) and \(\omega_0\) are constants. Find the Hamiltonian and Hamilton’s equations of motion. Identify the system.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 UPSC 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 2025 UPSC Maths Optional Paper II Solutions complete?
This public page gives one full sample solution for 2025 UPSC Maths Optional Paper II Solutions. 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. The sample includes all five sections: Question, Diagram, Concept Related to the Question, Detailed Solution, and Final Answer.
How should I use these 2025 UPSC Maths Optional Paper II Solutions for preparation?
Students should first solve the questions independently, then compare their approach with the solution method. These solutions are useful for revision, answer-writing practice, diagram presentation, and concept clarity before the UPSC Mains examination.
Do these solutions include diagrams and detailed explanations?
Yes. The full PYQ course follows a structured format wherever required: Question, Diagram, Concept Related to the Question, Detailed Solution, and Final Answer. This public page shows the format through one free sample solution.
How can I get complete solutions for all questions in 2025 UPSC Maths Optional Paper II?
To access complete solutions for all questions, please fill out the admission form first. Our Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
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