[Studyplan] Mathematics Optional for UPSC CSE and State PSC Exams

Mathematics Part - I

Before starting with the strategy I would like to answer a common question: What is the final score vis-à-vis correct attempt? It is true that there is heavy moderation of marks in science subjects especially in mathematics. But despite that it is possible to score very high. I got 390/600 in CSE-2008 and 427/600 in CSE-2009.  My final score and correct attempt paper wise is:

Year 2008:

Paper I
: I got 184/300.

Correct attempt was 261/300.

Paper II
: I got 206/300.

Correct attempt was around 250/300.

Year 2009:

Paper I
: I got 209/300.
Correct attempt was 300/300.

Paper II
: I got 218/300.
Correct attempt was 285/300.

In the correct attempt I have counted only those answers which are absolutely correct. Paper I is relatively easy so the reduction is huge. Also do not confuse the term scaling down as only multiplication by a constant factor (some says scaling is 0.6 or 0.7). The statistical formulas are more complex and the subtraction part has larger weightage I guess (seeing the mark trends of all the students in last 2 years).

General tips for preparation:

1. While preparing one should not be bothered about the moderation/scaling. Target should be to score maximum possible marks.
2. In Paper I try to cover all the topics. If time does not permit then cover dynamics and statics from 12 markers point of view.
3. In Paper II focus on the topics: Complex Analysis, Linear Programming, Partial Differential Equations and Numerical Analysis. Rest of the topics should be covered from 12 markers point of view.
4. Make a fair copy and practice all the previous year questions in it.
5. Practice with pen and paper(Many aspirants take this point casually and just read the solved examples)
6. Do at least 3-4 papers in exam like conditions.

General tips while attempting the paper:

1. Ordering of questions does not matter. Attempt the questions which you know the best at the start.
2. Try to finish each question in the 75-80% of the allotted time. For example, a 15 marks question has allotted time of 9 minutes. Try to finish it in 7 minutes. Rest 2 minutes must be spent on revising the answer. Rechecking entire paper at the end may not be possible. Try to work out the answer quickly in rough space. All this is important because wrong answer may carry heavy penalty.
3. All the trivial calculations (like simple integration, algebraic and trigonometric manipulations) can be done in rough space to save time (I believe that showing them in answer might not carry any weightage).
I will also upload some of the answers which I wrote in the paper. In the next articles I will be sharing the detailed strategy for both the papers.

Mathematics Part - II

A major question that comes in the mind of an aspirant is whether she should go for selective or exhaustive preparation. I went for the exhaustive preparation, as most part of the syllabus was part of my graduation curriculum. But if time does not permit one can leave some topics from point of view of full 60 marks question. In this post I am giving topic wise preparation strategy.

Paper I

1. Linear Algebra Focus should on understanding the definition of various terms mentioned in the syllabus (like vector spaces, subspaces, linear dependence and so on). While answering questions explain all the steps.

2. Calculus Single variable calculus is not a problem for candidates. Refer to chapters (book I mentioned in the booklist) that deal with continuity and differentiability for 2 variables. For double and triple integration also refer to that book. The solved examples from that book are sufficient. The relevant chapters are:12,13,15,16,17,18. Theory is not important , just understand the techniques through solved examples.

3. Analytic Geometry Refer book by PN Chaterjee(Rajhans Publications). This book contains solved examples and all the questions are taken from this book.

4. Ordinary Differential Equations In this topic it is better to make a sheet of all the formulas and techniques. Practice all the solved questions of the book I have mentioned.

5. Dynamics and Statics Completing the Krishna Series books(do only topics mentioned in the syllabus) will be sufficient for attempting 60 marks question. Common Catenary, central orbits, constrained motion should be covered in depth(that is do all the solved examples).  For rest of the topics covering only few examples is sufficient.

6. Vector Analysis In this topic make the formula sheet. And practice previous year questions. Regularly practice the derivations asked in the exam.

Paper II

1. Algebra For the people who are totally new to this topic i would suggest that read the books i have mentioned 2-3 times. Then practice all the theorems and questions yourself 2-3 times.

2. Real Analysis Cover chapters: 2- The Real Numbers(till page 24), 3-Neighbourhoods and Limit Point of a set(till page 11),  4 Countability of Sets(first 12 theorems and examples given at the end) 5,6,7(Regarding Sequence and series) 8. Real Functions. Limit and Continuity, 9,10,11,12,13,15,16. For the topics underlined cover the theorems also. For rest of the chapters solved examples are enough.

Complex Analysis Cover this topic from any book. Practice this topic thoroughly.

4. Linear Programming Practice is important. Improve upon the speed and accuracy for this topic.

5. Partial Differential Equations Same as Ordinary Differential Equations.

6. Numerical Analysis and Computer Programming Make the formula sheet for all the methods. Cover error analysis for newton raphson, lagrange interpolation, numerical integration. Draw flow chart and algorithm for all the numerical methods. Practice them regularly. For other topics i had M Morris Mano(this is digital electronics book). One can refer to the internet if she does not find the leftover topics in book.

7. Mechanics and Fluid Dynamics For Mechanics focus on chapters of Lagrangian(cover it from Vol-II, Rigid Dyanamics-Krishna Series), Hamiltonian, Moment Of Inertia, D’Alembert’s principle. For Fluid Dynamics  cover topics mentioned in the syllabus. The book of MD Raisinghania covers syllabus from IAS exam perspective also.
My methodology was to first cover the topic from the books. Then practice previous year questions from that topic. I covered the topics in the order mentioned in the syllabus. However it does not matter much. After covering the entire syllabus I practiced last 3-4 year papers in exam like conditions.

Mathematics Book List :

A lot of aspirants have asked me the best books to refer to for Mathematics Mains preparation. Almost all of the questions asked in the paper are solved examples from standard text-books. So it is important to identify the correct set of books for your preparation. Here I am providing the list of books which I referred to during my preparation. I was able to attempt all the questions during both my attempts.

Paper – I

  1. Calculus – Shanti Narayan – Course on Mathematical Analysis (S. Chand)
  2. Analytic Geometry – Shanti Narayan (S. Chand)
  3. Ordinary Differential Equations – M.D. Raisinghania (S. Chand)
  4. Statics: Statics (Krishna Series) Statics
    Statics & Dynamics

    Dynamics (Krishna Series)
  5. Vector Analysis:
    Vector Calculus (Krishna Series)
    Curves in Space
    Calculus by Thomas & Finney

Paper – II

  1. Algebra - (a) A Course In Algebra - Khanna and Bhambri
                        (b) Topics In Algebra - I. N. Herstein
  2. Real Analysis – MD. Raisinghania – Elements of Real Analysis (S. Chand)
  3. Complex AnalysisFunctions of a Complex Variable (Krishna Series)
  4. Linear Programming –  Krishna Series
  5. Partial Differential Equations- M.D. Raisinghania + (Some portion of book on Boundary Value Problem by S. Chand  or Advanced Engineering Mathematics by Erwin Kreyszig )
  6. Numerical Analysis – Jain and Iynger
  7. Fluid Dynamics – M.D. Raisinghania
  8. Mechanics – Krishna Series (Rigid Dynamics vol.-1 and Vo.-II)
    Rigid Dynamics -I (Dynamics of Rigid Bodies) (Krishna Series)
    Rigid Dynamics - II (Analytical Dynamics) (Krishna Series)

For Linear Algebra and Vector Analysis, I could not find any books relevant to the exam. So I referred to Brilliant Tutorials material to practice questions.

Source :  UPSC Topper Prakash Rajpurohit's Blog

Studyplan by Another Topper:

I chose Mathematics and Physics as optional subjects as I had a considerable exposure to the topics in their syllabi from my study since classes +1, +2 upto college level.

The books I referred for Mathematics are as follows:

Mathematics Book list:

Paper I

  1. Linear Algebra:
    Linear Algebra by Vasishtha and Sharma (Krishna Series)

    Matrices by Vasishtha and Vasishtha (Krishna Series)
  2. Calculus: Differential Calculus by Shanti Narayan (S. Chand)/ Or Here
    Integral Calculus by Shanti Narayan (S. Chand)
    A Course of Mathematical Analysis by Shanti Narayan (S. Chand)
  3. Analytic Geometry: Analytical Solid Geometry by Shanti Narayan (S. Chand)
  4. Ordinary Differential Equations: Ordinary and Partial Differential Equations by M.D. Raisinghania (S. Chand)
    Laplace Transforms:

    Brilliant Tutorials
    Advanced Engineering Mathematics by Erwin Kreyszig
  5. Statics: Statics (Krishna Series)

    Statics & Dynamics

    Dynamics (Krishna Series)
  6. Vector Analysis:
    Vector Calculus (Krishna Series)
    Curves in Space

    Calculus by Thomas & Finney

Paper II

  1. Algebra:  A Course in Algebra by Khanna and Bhambri
                       Topics in Algebra by I. N. Herstein
  2. Real Analysis: Elements of Real Analysis by MD Raisinghania (S. Chand)
  3. Complex Analysis: Functions of a Complex Variable (Krishna Series)
  4. Linear Programming: Linear Programming (Krishna Series)
  5. Partial Differential Equations: Ordinary and Partial Differential Equations by M.D. Raisinghania (S. Chand)
    Boundary Value Problems:
    Advanced Engineering Mathematics by Erwin Kreyszig
  6. Numerical Analysis and Computer Programming: Numerical Methods by Jain, Jain and Iynger
  7. Mechanics:
    Rigid Dynamics -I (Dynamics of Rigid Bodies) (Krishna Series)
    Rigid Dynamics - II (Analytical Dynamics) (Krishna Series)
    Fluid Dynamics:
    M.D. Raisinghania (S. Chand)

    [Due to some time constraint, I studied Abstract Algebra and Real Analysis mostly from 12 markers point of view. However, looking back, I think I could (& should) have done them thoroughly as well.]
    While studying these books I basically attempted the solved examples. In the first cycle of my study, I read selective solved examples and tried to register them in my mind, and solved some of them on my own. In the second cycle of the syllabus I properly attempted the solved examples on paper, trying my best to maintain the quality and language of the answers as close to the solutions in the books. The way of writing a solution is very important in the Mathematics paper, and can be best learnt from the solved examples themselves.

    After that I also attempted questions from the past 15-20 years’ papers. The answers/solutions can be verified from the Brilliant Tutorials and the books mentioned above. This exercise helps in giving an idea as to what is the kind and level of questions asked in the exam, and may also prove beneficial in case some questions in the exam come on similar lines as in the past.

    Finally, I also made summary sheets for every unit/topic (~ 2 pages per unit) in which I listed the important results/formulae/theorems/tips which can be gone through a day before the exam. This is helpful as it can give you a holistic & quick revision of the entire syllabus before the exam.

    In the exam, my correct attempt was ~ 258 in Paper I and ~ 260 in Paper II. My score in the exam was 207 in Paper I and 198in Paper II. I think the accuracy in the Mathematics paper is extremely necessary, as candidates are penalised heavily for any errors in the solution/answer.

Source : UPSC Topper Kashish Mittal

Mathematics Civil Services Mains Papers


  1. I work in Intel,Bangalore. So it is not possible for me to join the class in delhi. So is it possible for any aspirants that they can do self preparation which will be sufficient to crack the exam, if the preparation is done with cocnentration and dedication. Please do help in this regard.

    1. for Statics , dynamics ,fluid dynamics and mechanics you can join online course on Neosstencil of a faculty named "vajpeyi" but the course is expensive around 5000 per topic so instead of joining alone ,you should join it in group , and for rest of the topics purchase IMS notes either directly from the institute or from any shop in Old Rajinder Nagar in Delhi

  2. Would you please suggest book for mathematics option in Odisha state psc