UPSC Civil Services (Main) Examination 2026 — Mathematics Optional Paper II Analysis, Trends & Comparison | OurEducation
Last Updated: Sep 8, 2026
An analysis of Mathematics Optional Paper II of UPSC Civil Services (Main) Examination 2026 (held 2026-08-30) — difficulty, topic spread, and how it compares with previous years.
Official paper: official source.
“`html
UPSC Civil Services (Main) Examination 2026 — Mathematics Optional Paper II: Analysis
1. Overall Difficulty & Balance
The 2026 Mathematics Optional Paper II maintained a high standard of rigor while preserving a balanced distribution across core topics. The paper tested advanced concepts in Group Theory, Real & Complex Analysis, Linear Programming, PDEs, and Fluid Dynamics, with a noticeable emphasis on problem-solving over rote application. The difficulty curve was steep in sections requiring multi-step derivations (e.g., PDEs, contour integration), but the weightage remained fair, avoiding excessive specialization in niche areas. Candidates with a strong foundation in classical analysis and applied mathematics had a clear advantage.
The paper’s balance leaned toward theoretical depth (40%) and computational/analytical problem-solving (60%), with a slight tilt toward the latter. Questions on dihedral groups, Euclidean domains, and assignment problems provided accessible entry points, while improper integrals, Newton-Raphson iterations, and Milne-Thomson’s theorem demanded higher-order thinking. The inclusion of a logic circuit problem (Question 5c) was a welcome but unexpected deviation from traditional topics, testing interdisciplinary adaptability.
—
2. Section/Topic Distribution
The paper followed a structured yet varied distribution, with the following dominant areas:
- Abstract Algebra (Group Theory): 25% (Questions 1a, 2a, 4a)
- Focused on finite groups, dihedral groups, and integral domains/fields.
- Question 1a (group of order pq) and 2a (two generators of order 2) were classic but required precise reasoning.
- Question 4a (equivalence of integral domain/field/prime n) was a direct test of foundational understanding.
- Real & Complex Analysis: 30% (Questions 1c, 1d, 3b, 4b, 7b)
- Discontinuity analysis (1c) and non-analyticity despite CR equations (1d) were subtle, testing conceptual clarity.
- Improper integral divergence (4b) and Simpson’s 3/8 rule (7b) required computational finesse.
- Question 3b (upper/lower integrals for a piecewise function) was a standout, blending measure theory with classical analysis.
- Linear Programming & Optimization: 15% (Questions 1e, 4c)
- Dual problem (1e) and simplex method (4c) were standard but demanded meticulous constraint handling.
- The assignment problem (4c) introduced a practical twist, testing adaptability to applied scenarios.
- Partial Differential Equations (PDEs) & Integral Transforms: 20% (Questions 6a, 7a, 8a)
- Reduction to canonical form (6a) and harmonic functions in polar coordinates (8a) were core PDE topics.
- Question 7a (non-homogeneous PDE with sin(y−x)) combined method of undetermined coefficients with operator techniques.
- Numerical Methods & Fluid Dynamics: 10% (Questions 5b, 5d, 8b, 8c)
- Newton-Raphson for cube roots (5b) and Runge-Kutta/Euler methods (8b) tested numerical agility.
- Fluid dynamics (5d, 8c) was a niche but recurring theme, emphasizing physical intuition alongside mathematics.
Notable Absences: Probability/Statistics and Number Theory were absent, continuing a trend from recent years. Graph Theory and Combinatorics also saw no representation, aligning with the paper’s focus on continuous mathematics.
—
3. Comparison with 2025 and Multi-Year Trends
Shifts from 2025:
- Increased Weight on PDEs & Integral Transforms: 2025 had a heavier emphasis on Complex Analysis (e.g., residue theorems), while 2026 balanced PDEs (6a, 8a) and harmonic functions more prominently.
- More Applied Problems: Questions like the assignment problem (4c) and fluid dynamics (5d, 8c) reflect a growing trend toward applied mathematics, possibly in response to UPSC’s evolving syllabus priorities.
- Reduced Focus on Number Theory: 2025 included a question on quadratic residues; 2026 omitted it entirely, suggesting a shift toward algebra and analysis.
Repeated Themes:
- Group Theory: Questions on finite groups (order pq, dihedral groups) have appeared in 3 of the last 4 years, indicating a stable area of focus.
- Complex Analysis: Non-analyticity despite CR equations (2026: 1d; 2025: similar) and contour integration (2026: 2c) remain perennial favorites.
- Linear Programming: Dual problems and simplex method have been consistent since 2023, reflecting UPSC’s emphasis on optimization techniques.
Emerging Trends:
- Interdisciplinary Questions: The logic circuit problem (5c) and fluid dynamics (5d, 8c) suggest a move toward applied/interdisciplinary mathematics, possibly aligning with UPSC’s broader syllabus goals.
- Numerical Methods: Questions on Runge-Kutta (8b) and Newton-Raphson (5b) are becoming more frequent, indicating a need for computational skills.
—
4. Notable/Unexpected Questions and Why They Matter
- Question 1c: Discontinuity of a Limit Function
Why it matters: This question tested the ability to analyze pointwise limits of sequences of functions, a topic often glossed over in standard curricula. The function involved a limit of the form (1 + sin(π/x))ⁿ, requiring candidates to consider the behavior of sin(π/x) near its singularities (e.g., x = 1/k for integer k). The discontinuities at x = 1/k (for k ∈ ℕ) were non-trivial to identify, as the limit could converge to different values depending on the subsequence. This question rewarded deep understanding of limits and continuity over mechanical computation.
- Question 3b: Upper and Lower Integrals
Why it matters: The piecewise function defined by f(x) = x² (rational) and f(x) = x³ (irrational) is a classic example in measure theory, testing whether candidates grasp the distinction between Riemann and Lebesgue integration. The question required evaluating upper and lower integrals separately, a skill rarely emphasized in standard UPSC preparation. It highlighted the importance of theoretical foundations in real analysis, which are often sidelined in favor of problem-solving drills.
- Question 5c: Logic Circuit Analysis
Why it matters: The inclusion of a logic circuit problem (Boolean algebra) was unexpected, as UPSC’s Mathematics Optional traditionally avoids discrete mathematics. This question tested the ability to translate a circuit diagram into a Boolean expression and derive a truth table, bridging pure mathematics with computer science. It underscored the need for aspirants to develop interdisciplinary problem-solving skills, possibly reflecting UPSC’s evolving expectations.
- Question 6c: Milne-Thomson’s Circle Theorem
Why it matters: Fluid dynamics questions (e.g., 5d, 8c) have appeared before, but this one combined potential theory with complex analysis via Milne-Thomson’s theorem. The question required deriving the complex velocity potential for a cylinder in a uniform stream and analyzing stagnation points with circulation. This is a high-level topic, typically reserved for advanced fluid dynamics courses, and tested the ability to apply abstract theorems to physical scenarios.
—
5. Key Takeaways for Aspirants
- Master the Fundamentals, Then Specialize:
While the paper included advanced topics (e.g., Milne-Thomson’s theorem, PDE canonical forms), the majority of questions were rooted in core concepts. Aspirants should prioritize:
- Group Theory (finite groups, Sylow theorems,
Questions asked
- Let $G$ be a group of order $pq$, where $p$ and $q$ are primes. Show that either $G$ is an abelian group or no non-identity element commutes with every element of $G$. (b) Show that every non-zero prime ideal in a Euclidean Domain is maximal. 10 (c) $f(x) = \lim_{n \rightarrow \infty} \frac{\left(1 + \sin \frac{\pi}{x}\right)^n – 1}{\left(1 + \sin \frac{\pi}{x}\right)^n + 1}, x \in (0, 1)$ द्वारा परिभाषित फलन $f : (0, 1) \rightarrow \mathbb{R}$ के Find the points of discontinuity of the function $f : (0, 1) \rightarrow \mathbb{R}$ defined by $f(x) = \lim_{n \rightarrow \infty} \frac{\left(1 + \sin \frac{\pi}{x}\right)^n – 1}{\left(1 + \sin \frac{\pi}{x}\right)^n + 1}, x \in (0, 1).$ 10 (d) Show that the function $f(z) = \sqrt{|xy|}$ is not analytic at the origin, although Cauchy-Riemann equations are satisfied at that point. 10 (e) $$y_1 + y_2 + y_3 + y_4 \geq 2$$ $$2y_1 + y_2 – y_3 – 2y_4 \geq 1$$ $$y_1, y_2, y_3, y_4 \geq 0$$ Solve the dual problem of the following linear programming problem by the graphical method : $$\text{Minimize } Y = 10y_1 + 6y_2 + 2y_3 + y_4$$ subject to the constraints : $$y_1 + y_2 + y_3 + y_4 \geq 2$$ $$2y_1 + y_2 – y_3 – 2y_4 \geq 1$$ $$y_1, y_2, y_3, y_4 \geq 0$$
- Let $G$ be a finite group generated by two elements $u$ and $v$ of order 2 each. Show that $G$ is isomorphic to the dihedral group of order $2n$, where $O(uv) = n$. (b) Prove that the sequence $(u_n)$ defined by the recursion formula $u_{n+1} = \sqrt{7 + u_n}$, $u_1 = \sqrt{7}$, converges to the positive root of $x^2 – x – 7 = 0$. (c) $$\text{तो } \int_0^{2\pi} \frac{\sin^2 \theta}{a + b \cos \theta} \, d\theta = \frac{2\pi}{b^2} \left\{ a – \sqrt{a^2 – b^2} \right\} \text{ है।}$$ Use the method of contour integration to prove $$\int_0^{2\pi} \frac{\sin^2 \theta}{a + b \cos \theta} \, d\theta = \frac{2\pi}{b^2} \left\{ a – \sqrt{a^2 – b^2} \right\}, \text{ if } a > b > 0. \quad 20$$
- $$\cosh\left(z + \frac{1}{z}\right) = a_0 + \sum_{n=1}^{\infty} a_n \left( z^n + \frac{1}{z^n} \right) \text{ है,}$$ $$\text{जहाँ } n = 0, 1, 2, \dots \text{ के लिए } a_n = \frac{1}{2\pi} \int_0^{2\pi} \cos n\theta \cosh(2\cos \theta) \, d\theta \text{ है।}$$ Prove that $$\cosh\left(z + \frac{1}{z}\right) = a_0 + \sum_{n=1}^{\infty} a_n \left( z^n + \frac{1}{z^n} \right),$$ where $$a_n = \frac{1}{2\pi} \int_0^{2\pi} \cos n\theta \cosh(2\cos \theta) \, d\theta, \text{ for } n = 0, 1, 2, \dots. \quad 15$$ (b) A function $f$ is defined on $[0, 1]$ by $$f(x) = \begin{cases} x^2, & \text{when } x \text{ is rational} \\ x^3, & \text{when } x \text{ is irrational} \end{cases}$$ Evaluate lower integral $\int_0^1 f$ and upper integral $\int_0^1 f$. Does the integral $$\int_0^1 f \text{ exist? } \quad 20$$ (c) | — | — | — | — | — | — | | | | M_{1} | M_{2} | M_{3} | M_{4} | | | J_{2} | 8 | 13 | 17 | 19 | | | J_{3} | 8 | 15 | 19 | 22 | The following table represents the estimated costs of assigning jobs to machines. Solve this assignment problem to minimize the total cost. | | | Machine | | | | | — | — | — | — | — | — | | | | M_{1} | M_{2} | M_{3} | M_{4} | | Job | J_{1} | 8 | 24 | 28 | 32 | | | J_{2} | 8 | 13 | 17 | 19 | | | J_{3} | 8 | 15 | 19 | 22 | Is the assignment unique for the total minimum cost? Justify your answer and obtain an alternate assignment, if it exists.
- Let $n$ be a positive integer. Show that the following statements are equivalent : (i) $\mathbb{Z}/n\mathbb{Z}$ is an integral domain. (ii) $\mathbb{Z}/n\mathbb{Z}$ is a field. (iii) $n$ is prime. (b) Prove that the improper integral $\int_0^\infty \frac{1}{1+x^2\sin^2 x}\,dx$ is divergent. (c) $$2x_1 + x_2 \leq 10$$ $$2x_1 + 5x_2 \leq 20$$ $$2x_1 + 3x_2 \leq 18$$ $$x_1, x_2 \geq 0$$ Solve the following linear programming problem by the Simplex method : Maximize $Z = 4x_1 + 10x_2$ subject to the constraints $$2x_1 + x_2 \leq 10$$ $$2x_1 + 5x_2 \leq 20$$ $$2x_1 + 3x_2 \leq 18$$ $$x_1, x_2 \geq 0$$ Obtain an alternate optimal solution, if it exists, with explanation.
- Find the integral surface of the equations $$\frac{dx}{xz-y} = \frac{dy}{yz-x} = \frac{dz}{1-z^2}$$ . 10 (b) By using the Newton-Raphson iterative formula, establish the formula $$x_{i+1} = \frac{1}{3} \left( 2x_i + \frac{N}{x_i^2} \right)$$ to find the cube root of N. Use this formula, if established, to find the cube root of 63 correct to four decimal places with the initial approximation 3.9. 10 (c)  Consider the logic circuit L :  (i) Express Y as a Boolean expression. (ii) Write 8-bit special sequences for A, B and C. (iii) Find the truth table of L using 8-bit special sequences. (d) A simple source of strength m is fixed at the origin O in a uniform stream of incompressible fluid moving with velocity $U\hat{i}$. Find out the velocity potential $\phi$ at any point P of the stream where $OP = r$ and $\theta$ is the angle $\overrightarrow{OP}$ makes with the direction $\hat{i}$. Find the differential equation of the stream lines and show that they lie on the surface $Ur^2 \sin^2 \theta – 2m \cos \theta =$ constant. (e) A uniform heavy solid hemisphere of radius ‘a’ is held at rest with its base vertical and its curved surface in contact with a horizontal plane. If the hemisphere is released when the plane is rough enough to prevent slipping, show that the angle θ that the base makes with the horizontal at time t, is such that $$\left(\frac{d\theta}{dt}\right)^2 = \frac{15g\cos\theta}{a(28-15\cos\theta)}$$ .
- Reduce the partial differential equation $$y\frac{\partial^2z}{\partial x^2} + (x+y)\frac{\partial^2z}{\partial x\partial y} + x\frac{\partial^2z}{\partial y^2} = 0$$ to canonical form and hence solve it. (b) Find the decimal equivalent of the following floating point 10-bit numbers with 5 bits as fractional part : 1010111011 and 0100101010 Evaluate $(11001.1011)_{10} + (33.24)_8 + (101101)_2$ and convert the final result in hexadecimal system. (c) Suppose that there is an infinitely long cylinder of radius 'a' placed in a uniform stream having velocity $-V_i^\wedge$. Then using Milne-Thomson's circle theorem, what will be its complex velocity potential? If in addition, a circulation round the cylinder of k is produced, find out the stagnation points. Do they produce a lifting tendency in the vertical direction? Explain how.
- (a) Solve the partial differential equation $$(2D^2 – 5DD' + 2D'^2)z = 24(y – x) + \sin(y – x)$$ where $D \equiv \frac{\partial}{\partial x}, D' \equiv \frac{\partial}{\partial y}, z = z(x, y)$. (b) $$f(x) = 0.2 + 25x – 200x^2 + 675x^3 – 900x^4 + 400x^5 \text{ है।}$$ Evaluate the integral $\int_{0}^{0.8} f(x)dx$, $$\text{where } f(x) = 0.2 + 25x – 200x^2 + 675x^3 – 900x^4 + 400x^5$$ by using single application of Simpson's $\frac{3}{8}$ rule. (c) A smooth uniform rod, say OA, of length 2a and mass m is pivoted at one end to a fixed point O. The rod is inclined at an angle $\theta$ with the downward vertical line OZ and the plane OAZ makes an angle $\phi$ with a fixed vertical plane. A bead of mass $\lambda m$ slides smoothly on the rod and is connected to O by a light elastic string of modulus nmg and natural length a. Show that the kinetic energy T of the system is given by $2T = \frac{4}{3}ma^2(\dot{\theta}^2 + \dot{\phi}^2 \sin \theta) + \lambda m(\dot{x}^2 + x^2\dot{\theta}^2 + x^2\dot{\theta}^2 \sin^2 \theta)$, where x is the stretched length of the string, and derive the equations of motion.
- (a) $$\frac{\partial^2 V}{\partial r^2} + \frac{1}{r} \frac{\partial V}{\partial r} + \frac{1}{r^2} \frac{\partial^2 V}{\partial \theta^2} = 0$$ (ii) r = a पर V = ∑n cn cos(nθ) है Two-dimensional harmonic equation in plane polar coordinates (r, θ) takes the form : $$\frac{\partial^2 V}{\partial r^2} + \frac{1}{r} \frac{\partial V}{\partial r} + \frac{1}{r^2} \frac{\partial^2 V}{\partial \theta^2} = 0$$ Deduce that it has solutions of the form (Arⁿ + Br⁻ⁿ) e^±inθ, where A, B, n are constants. Determine V if it satisfies the two-dimensional harmonic equation in the region 0 ≤ r ≤ a, 0 ≤ θ ≤ 2π and satisfies the conditions : (i) V remains finite as r → 0, (ii) V = ∑n cn cos(nθ) on r = a. (b) Consider the initial value problem : $$\frac{dy}{dx} = 1 + \frac{y}{x}, y(1) = 1, h = 1 \text{ (Step-size)}$$ (i) Apply the classical fourth order Runge-Kutta method to approximate the solution y(2). (ii) Use Euler's method to find y(2) and y(3). (c) A viscous incompressible fluid is filled between two concentric cylinders of radius a, b (b > a). The flow is steady and no body forces are taken into consideration. Discuss and formulate the velocity of the fluid if : (i) the inner cylinder is given angular velocity $\Omega$ while the outer one is held at rest; (ii) the outer cylinder is rotated with angular velocity $\Omega$ while the inner one is held at rest.
- Group Theory (finite groups, Sylow theorems,
Discover more from Our Education | Best Coaching Institutes Colleges Rank
Subscribe to get the latest posts sent to your email.
« UPSC Civil Services (Main) Examination 2026 — Mathematics Optional Paper I Analysis, Trends & Comparison | OurEducation UPSC Civil Services (Main) Examination 2026 — Zoology Optional Paper I Analysis, Trends & Comparison | OurEducation »
Tell us Your Queries, Suggestions and Feedback