2025 IFoS Maths Optional Paper II Solutions | Ramana Sri IAS

2025 IFoS Maths Optional Paper II Solutions

2025 IFoS Maths Optional Paper II Solutions

Ramana Sri IAS provides complete and updated solutions for the 2025 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 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 IFoS Maths Optional Paper II Solutions

These 2025 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 2025 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 question papers page.

These 2025 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 2025 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 2025 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

1Question

If \(H\) and \(K\) are finite subgroups of a group and \(HK=\{hk\mid h\in H,\;k\in K\}\), prove that \(|HK|=\dfrac{|H|\,|K|}{|H\cap K|}\).

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 1(b)

Real Analysis – Bounds

1Question

Let \(S\) be a non-empty subset of \(\mathbb R\), bounded below, and \(T=\{-x:x\in S\}\). Prove that the set \(T\) is bounded above and \(\sup T=-\inf S\).

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 1(c)

Infinite Series

1Question

Prove that the series \(1+\dfrac13-\dfrac12+\dfrac15+\dfrac17-\dfrac14+\cdots\) converges to \(\dfrac32\log2\).

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 1(d)

Complex Analysis – Bilinear Transformation

1Question

Find the image of \(|z|<1\) under the bilinear transformation which maps \(z=1,i,-1\) onto \(w=i,0,-i\) respectively.

2Diagram

Question 1(d): Bilinear Transformation of the Unit Disk

2025 IFoS Maths Optional Paper II Solutions diagram showing the unit disk in the z-plane and its image under a bilinear transformation, with boundary points 1, i, and -1 mapped respectively to i, 0, and -i in the w-plane.

3Concept Related to the Question

This question belongs to Complex Analysis – Bilinear Transformation. The textbook method is to identify the formula or theorem first, write all given data clearly, and then simplify step by step without skipping algebra.

4Detailed Solution

Let \(w=\dfrac{az+b}{cz+d}\). Using \(z=i\mapsto w=0\), we get \(ai+b=0\). Solving the three point correspondences \(1\mapsto i\), \(i\mapsto0\), \(-1\mapsto -i\), one convenient form is \(\displaystyle w=\frac{iz+1}{1-iz}\).

The circle \(|z|=1\) passes through \(1,i,-1\). Their images \(i,0,-i\) lie on the imaginary axis. Hence the boundary maps to \(\operatorname{Re}w=0\). To decide the side, take the interior point \(z=0\). Then \(w=1\), which lies in the right half-plane.

5Final Answer

The transformation may be written as \(\displaystyle w=\frac{iz+1}{1-iz}\), and the image of \(|z|<1\) is \(\operatorname{Re}w>0\).

Question 1(e)

Linear Programming – Formulation

1Question

The forest department aims to afforest up to \(100\) hectares with teak and pine to maximize \(\mathrm{CO}_2\) absorption, planting at least \(10\) hectares of each, and no more than \(60\) hectares of teak. The table below provides the \(\mathrm{CO}_2\) absorption and resource requirements:

Tree	\(\mathrm{CO}_2\) Absorption (tons/ha/year)	Labour (hours/ha/week)	Water (litres/ha/week)
Teak	20	40	200
Pine	15	20	150

The available resources are \(3200\) labour hours and \(16000\) litres of water per week. Formulate this as a linear programming problem.

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 2(a)

Group Theory – Elements of Prime Order

1Question

If \(p\) is an odd prime, prove that there is no group that has exactly \(p\) elements of order \(p\).

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 2(b)

Real Analysis – Pointwise and Uniform Convergence

1Question

Let \(f_n(x)=nx(1-x)^n\), \(x\in[0,1]\), \(n\in\mathbb N\). Show that (i) the sequence \(\{f_n\}\) converges to a function \(f\) on \([0,1]\) and (ii) \(f\) is integrable on \([0,1]\) and \(\displaystyle \lim_{n\to\infty}\int_0^1 f_n=\int_0^1 f\), but still the convergence of the sequence \(\{f_n\}\) is not uniform.

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 2(c)

Complex Analysis – Contour Integration

1Question

Evaluate the integral \(\displaystyle \int_0^\infty \sin(x^2)\,dx\) using the method of contour integration.

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 3(a)

Double Integration – Change of Variables

1Question

Evaluate \(\displaystyle \iint_E \frac{\sqrt{a^2b^2-b^2x^2-a^2y^2}}{\sqrt{a^2b^2+b^2x^2+a^2y^2}}\,dx\,dy\), the field of integration \(E\) being the positive quadrant of the ellipse \(\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\).

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 3(b)

Ring Theory – Ideals in Product Rings

1Question

Show that in the ring \(\mathbb Z\times\mathbb Z\), (i) the ideal \(S=\{(a,0):a\in\mathbb Z\}\) is a prime ideal but not maximal and (ii) the ideal \(T=\{(m,n)\in\mathbb Z\times\mathbb Z:3\mid n\}\) is a maximal ideal.

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 3(c)

Linear Programming – Simplex Method

1Question

Solve the following linear programming problem by simplex method:

Maximize \(Z=3x_1+2x_2+3x_3\)

subject to \(x_1+x_2+x_3\le4\), \(2x_1+x_2+3x_3\le10\), \(x_1+2x_2+x_3\le6\), and \(x_1,x_2,x_3\ge0\). Is the solution unique? If not, find all the optimal solutions.

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 4(a)

Abstract Algebra – Eisenstein Criterion

1Question

Let \(f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0\in\mathbb Z[x]\), and suppose that \(p\) is prime such that \(p\nmid a_n\), \(p\mid a_{n-1},\ldots,p\mid a_0\) and \(p^2\nmid a_0\). Prove that \(f(x)\) is irreducible over \(\mathbb Q\).

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 4(b)

Complex Analysis – Laurent Series

1Question

Let \(\displaystyle f(z)=\frac{z}{(z-1)(z+2)}\). Find the Laurent series expansion of \(f(z)\) in the following regions: (i) \(1<|z|<2\), (ii) \(|z|>2\).

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 4(c)

Operations Research – Assignment Problem

1Question

A company has \(5\) workers \(A,B,C,D,E\) and \(4\) jobs \(I,II,III,IV\). The profit in rupees that each worker earns from completing each job is given below:

Worker / Job	I	II	III	IV
A	17	20	18	21
B	19	16	17	20
C	16	15	20	18
D	18	17	15	16
E	14	19	13	15

Assign each job to exactly one worker in such a way that the profit is maximized.

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 5(a)

PDE – Eliminating Arbitrary Function

1Question

Eliminate the arbitrary function \(F\) from the given equation \(\displaystyle F\left(\frac{xy}{z},\frac{x-y}{z}\right)=0\) and find the corresponding partial differential equation.

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 5(b)

Numerical Analysis – Bisection Method

1Question

Find the interval in which the root of the equation \(xe^x=1\) lies between \(0\) and \(1\), obtained by using three iterations of bisection method.

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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Question 5(c)

Computer Arithmetic – Two’s Complement

1Question

Perform the operations (i) \((+42)+(-13)\) and (ii) \((-42)-(-13)\) in binary using signed \(2\)'s complement representation for negative numbers in \(8\)-bit system. Give the final answer in decimal.

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 5(d)

Analytical Mechanics – Cycloid

1Question

Derive Lagrange's equation for a particle of mass \(m\) that slides on a frictionless wire hanging as a cycloid given by the equations \(x=a(\theta-\sin\theta)\), \(y=a(1+\cos\theta)\). Further, if \(\cos\dfrac{\theta}{2}=f\), then transform Lagrange's equation to the form \(\displaystyle \frac{d^2f}{dt^2}+\frac{g}{4a}f=0\), where \(t\) symbolizes time.

2Diagram

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3Concept Related to the Question

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4Detailed Solution

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5Final Answer

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Question 5(e)

Fluid Dynamics – Euler Equation and Vorticity

1Question

Prove that the equation of motion of a homogeneous inviscid liquid moving under forces arising from a potential \(V\) may be written in the form \(\displaystyle \frac{\partial\vec q}{\partial t}-\vec q\times\vec\zeta=-\vec\nabla\left(\frac{p}{\rho}+\frac12\vec q^{\,2}+V\right)\), where \(\vec q,t,\vec\zeta,p,\rho\) respectively stand for velocity vector, time, vorticity vector, pressure, density, and \(\vec\nabla\) the gradient operator.

If the velocity \(\vec q\), referred to cylindrical polar coordinates \((r,\theta,z)\), is given by \(\displaystyle \vec q=\begin{cases}(0,\frac12\omega r,0),&0\le r\le a,\\(0,\frac12\omega a^2/r,0),&r\ge a,\end{cases}\) where \(\omega\) is a constant, prove that the vorticity is given by \(\displaystyle \vec\zeta=\begin{cases}(0,0,\omega),&0\le r\le a,\\(0,0,0),&r\ge a.\end{cases}\)

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Question 6(a)

PDE – Complete Integral

1Question

Find a complete integral of the equation \(p^2x+q^2y=z\), where \(p=\dfrac{\partial z}{\partial x}\), \(q=\dfrac{\partial z}{\partial y}\).

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Question 6(b)

Numerical Integration – Simpson Rule

1Question

Write down the algorithm for finding the integral \(\displaystyle I=\int_a^b f(x)\,dx\) using Simpson's \(\dfrac13\) rule. Hence, find \(\displaystyle I=\int_0^1(4x-3x^2)\,dx\) taking \(10\) intervals. Also, find the relative error.

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Question 6(c)

Mechanics – Rotating Rod

1Question

A uniform rod of length \(2a\) and mass \(m\) can rotate freely about a fixed end. What is the least angular velocity required to start with, from the lowest position so as to reach the top in order to make a complete revolution? Also, find out the time consumed.

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Question 7(a)

Statistics – Ogive / Interpolation

1Question

In an examination, the number of candidates who secured marks between certain limits was as follows:

Marks	0–19	20–39	40–59	60–89	90–99
No. of candidates	41	62	65	50	17

Estimate the number of candidates getting marks less than \(50\).

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Question 7(b)

PDE – Linear Partial Differential Equation

1Question

Find the general solution of the partial differential equation \(\displaystyle (4D^2-4DD'+D'^2)z=8\log(x+2y)+\cos2x\cos y\), where \(D=\dfrac{\partial}{\partial x}\) and \(D'=\dfrac{\partial}{\partial y}\).

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Question 7(c)

Fluid Dynamics – Blasius Theorem

1Question

A long infinite cylinder of radius \(a\) is placed in a uniform stream such that its axis lies perpendicular to the stream. Besides, a circulation round the cylinder is produced by a uniform line vortex through the origin. If the uniform stream velocity is \(-U\hat i\) and the circulation is \(2\pi k\), then find out the complex velocity potential. Show, by using the theorem of Blasius, that the cylinder experiences an uplifting force.

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Question 8(a)

Fluid Dynamics – Source in Uniform Stream

1Question

A simple source, of strength \(m\), is fixed at the origin \(O\) in a uniform stream of incompressible fluid moving with velocity \(U\hat i\). Show that the velocity potential \(\phi\) at any point \(P\) of the stream is \(\displaystyle \frac mr-Ur\cos\theta\), where \(OP=r\) and \(\theta\) is the angle \(\overrightarrow{OP}\) makes with the direction \(\hat i\). Find the differential equation of the streamlines and show that they lie on the surfaces \(Ur^2\sin^2\theta-2m\cos\theta=\text{constant}\).

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Question 8(b)

Wave Equation – Fourier Series

1Question

Let \(u(x,t)\) be the solution of the wave equation \(\displaystyle \frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}=0\), \(0<x<\pi,\;t>0\), with the initial conditions \(u(x,0)=\sin x+\sin2x+\sin3x\), \(\displaystyle \frac{\partial u}{\partial t}(x,0)=0\), \(0<x<\pi\), and the boundary condition \(u(0,t)=u(\pi,t)=0\), \(t\ge0\). Find the value of \(u\left(\frac\pi2,\pi\right)\).

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Question 8(c)

Numerical Analysis – Runge-Kutta Method

1Question

Using Runge-Kutta method of fourth order, find \(y\) at \(x=0.2\), given that \(\displaystyle \frac{dy}{dx}=3e^x+2y\), \(y(0)=0\) and \(h=0.1\), correct to four decimal places.

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