Mathematics is one of the optional subjects offered by the UPSC in the civil services mains exam. Generally considered by many as a tough subject, mathematics ‘the queen of the sciences’ is taken by many aspirants, many of whom have come out with flying colours in the IAS exam. Maths, often vilified as the toughest subject in school, students, barring exceptions, universally dread maths. In the IAS exam, the performance of this subject has been ambiguous. Many toppers have attributed their success and high rank to the maths optional, whereas many candidates have had very low marks in the maths optional papers. In this article, we will try to dispel many myths surrounding the mathematics optional while presenting a realistic picture. We also discuss some strategies and UPSC maths study material relevant for this subject.

## How many take Mathematics optional?

As per the latest available data, in 2015, 258 candidates had taken mathematics as their optional subject out of which 31 cleared the exam giving the subject a success rate of 12% that year. The following table shows the number of candidates who had appeared with the maths optional.

### Table showing maths optional success rate

Year |
No. of candidates appeared |
No. of candidates cleared |
Success rate (%) |

2015 | 258 | 31 | 12 |

2014 | 351 | 35 | 10 |

2013 | 329 | 20 | 6.1 |

2012 | 325 | 23 | 7.1 |

2011 | 337 | 28 | 8.3 |

### IAS toppers with maths optional

Name |
Year |
Success rate (%) |

Atul Prakash | 2017 | 4 |

Anubhav Singh | 2017 | 8 |

Sagar Kumar | 2017 | 13 |

Abhishek Verma | 2017 | 32 |

Prateek Jain | 2017 | 86 |

Utsav Kaushal | 2016 | 14 |

Manish Gurwani | 2016 | 17 |

Yogesh Kumbhejkar | 2015 | 8 |

Ashish Sangwan | 2015 | 12 |

Siddharth Jain | 2015 | 13 |

Pratap Singh | 2015 | 15 |

Nitish K | 2014 | 8 |

Kashish Mittal | 2010 | 58 |

Mutyalaraju Revu | 2006 | 1 |

## Mathematics optional pros and cons

Maths, as an optional subject, is opted for by between 300 and 350 candidates every year. There are benefits and drawbacks associated with this optional. In this section, we discuss some of the advantages and disadvantages of mathematics optional.

### Maths optional pros

**Scoring**– maths is a scoring subject. There is literally no theory here to write about in the paper. You only need to understand the underlying theories and then solve problems accordingly. If you have prepared thoroughly and are able to write answers correctly and with accuracy, you can score maximum marks, more than any humanities subject. For instance, in 2017, the highest optional marks was**375 out of 500**which went to Anubhav Singh for mathematics optional.**Direct**– the questions asked in this paper are generally direct and straight-forward. If you are good in mathematics and enjoy the subject, you should go for mathematics optional as you will find it easy to solve the papers with proper preparation of course.**Static**– maths is a static subject unlike some other humanities subjects. There is no linkage to current affairs here.**Competition**– in this optional, you will face far less competition compared to some popular subjects.**Predictable**– you can assess your performance once you have given the paper. There is no scope for subjectivity and nothing is left to the interpretation of the examiner.

### Maths optional cons

**No overlap**– there is virtually zero overlap with the general studies papers.**Time-consuming**– preparation will take more time since the syllabus is to be done thoroughly and completely if you want to perform well. Moreover, maths can be studied only by practicing the sums. There is no way around it.**No marks for attempting**– here, there is very little margin for error. If your answer is wrong, you can lose marks severely. Only if your attempt is in the right direction, you may be given some grace marks for attempting.**Technical**– maths can be taken only by candidates who are maths or engineering graduates or postgraduates.

## UPSC Mathematics optional syllabus

Let us take a look at the syllabus for Mathematics for the UPSC mains exam.

There are two optional papers in the UPSC exam pattern. Both the papers are for a total of 250 marks making the total optional marks to 500.

Download the Mathematics syllabus PDF.

## Mathematics optional strategy

**General tips for UPSC maths optional:**

- Go through the maths syllabus in the beginning. Understand the syllabus and have it at the back of your mind all the time.
- Then, make a list of books from where you will prepare each topic in the syllabus. For this, you can follow the specific topic-wise strategy we have given in this article. You can also take guidance from your seniors and mentors for this.
- You should also go through the previous years’ question papers for maths optional so that you are aware of the most important and repeated topics from where questions are asked in the mains exam.
- The way to go about preparing for this optional is simple – first read and understand the concepts and theorems. After that, devote 80% of the time in practicing the sums and the remaining 20% in revising them.
- You must also practice the proofs of the theorems very well. It will help if you are able to prove and derive the theory or theorem from scratch. Sometimes, proofs of theorems are also asked in the exam particularly in Paper II.
- Always prepare and keep a formula sheet ready. This will help you memorise certain formulae that would be needed to solve the problems.
- Always practice maths with a relaxed mind. Panic can cause confusion and you will end up faring badly in the paper.
- Practice is most important in this subject. Practice as much as you can as it will improve your speed.
- It is very important to enrol for a good mains
**test series for maths optional**. This is because exam-like practice will help you gain confidence and also teach you how to remain calm in the exam hall. It is possible that you might start off the paper with a difficult question and you aren’t able to solve it. Panic at this stage will ruin the whole paper for you. It is precisely to handle such situations that candidates are advised to join a test series and gain enough exam experience. - When you are solving the IAS exam papers, don’t worry about scaling and moderation. Your aim should be to score maximum. Always check your answers once after finishing them. Take out time for this as any careless mistakes can be rectified. Practicing test papers can ensure that you learn time management well enough to spare time for checking the answers.

### Maths paper I strategy

- Linear Algebra – this is a topic that you would have done in your XI and XII classes also. There are more theorems here. But, it is also one of the easier topics in maths.

Example questions:

*Q. Show that similar matrices have the same characteristic polynomial. [10 Marks, 2017]*

*Q. Prove that distance non-zero eigenvectors of a matrix are linearly independent. [10 Marks, 2017]*

*Q. Using elementary row operation find the condition that the linear equations have a solution*

*x – 2y + z = a*

*2x + 7y – 3z = b*

*3x + 5y -2z = c [7 marks, 2016]*

*Q. Prove that Eigen values of a unitary matrix have absolute value 1. [7 Marks, 2015]* - Calculus – even though an straight-forward topic, this requires some effort from you if you are to perform well here.

Example questions:

*Q. Integrate the function f(x,y) = xy(x**2**+ y**2**) over the domain R: {-3**<**x**2**– y**2**<**3,**1**<**xy**<**4} [10 marks, 2017]*

*Q. Find the surface area of the plane x + 2y + 2z cut off by x = 0, y = 0 and x**2**+ y**2**= 16 [15 marks, 2016]*

*Q. A conical tent is of given capacity. For the least amount of Canvas required, for it, find the ratio of its height to the radius of its base. [13 marks, 2015]* - Analytic Geometry – in this section, you should complete all the theorems and then study the solved questions that come in the Krishna Series books for IAS maths. This will cover all the types of questions that come in the exam.

Example questions:

*Q. A plane through a fixed point ( a, b, c) and cuts the axes at the points A, B, C respectively. Find the locus of the center of the sphere which passes through the origin O and A, B, C. [15 marks, 2017]*

*Q. Find the surface generated by a line which intersects the line y = a = z, x + 3z = a = y + z [10 marks, 2016]* - Ordinary Differential Equations – even though it looks like a vast topic, this section is relatively simple because once you have done all the types of differential equations, they are done. Even the type of questions that are asked are the same every year. So, this is a scoring section.

Example questions:

*Q. Find the differential equation representing the entire circle in the xy plane. [10 marks, 2017]*

*Q. Find the curve for which the part of the tangent cut-off by the axes is bisected at the point of tangency. [10 marks, 2015]* - Dynamics & Statics – this part is related to physics to a large extent. For this also, refer to Krishna series book for the solved examples. This will cover all the types of questions that are usually asked in the exam.

Example questions:

*Q. Suppose that the streamlines of the fluid flow are given by a family of curves xy = c. Find the equipotential lines, that is, the orthogonal trajectories of the family of curves representing the streamlines. [10 marks, 2017]*

*Q. A particle moves with a central acceleration which varies inversely as the cube of the distance. If it is projected from an apse at a distance a from the origin with a velocity which √2 is times the velocity for a circle or radius a then find the equation to the path. [10 marks, 2016]*

*Q. A rod of 8kg is movable in a vertical plane about a hinge at one end another end is fastened a weight equal to half of the rod, this is fastened by a string of length l to a point at a height to above the hinge vertically. Obtain the tension in the sting. [10 marks, 2015]* - Vector Analysis – read all the theorems properly. You can perform well in this section if you cover all the theorems and their applications completely.

### Maths paper II strategy

- Modern Algebra – this is a feared section for most students of maths. But with sincere effort and systematic study, this can be covered well. Study repeatedly in order to understand the topic comprehensively. Proofs of theorems are very important here because they help you in solving questions also from this topic.

Example questions:

*Q. Let G be a group of order n. Show that G is isomorphic to a subgroup of the permutation group S**n**. [10 marks, 2017]*

*Q. Show that the groups Z**5**X Z**7**and Z**35**are isomorphic. [15 marks, 2017]*

*Q. Show that every algebraically closed field is infinite. [15 marks, 2016]* - Real Analysis – focus on real analysis and number theory here. This section also requires repeated study and enough revision.

Example questions:

*Q. Find the Supremum and the Infimum of x/sinx on the interval (0, 𝝅/2). [10 marks, 2017]*

*Q. Find the absolute maximum and minimum values of the function f(x, y) = x**2**+ 3y**2**– y over the region x**2**+ 2y**2**<**1. [15 marks, 2015]* - Complex Analysis – this can be one of the easiest sections in this paper is done thoroughly. Here, Cauchy’s theorem is very important.

Example question:

*Q. Determine all entire functions f (z) such that 0 is removable singularity of f(1/z). [10 marks, 2017]*

*Q. Prove that every power series represents an analytic function inside its circle of convergence. [20 marks, 2016]* - Linear Programming – this is again a simple topic and limited kinds of questions are asked from here.

Example question:

*Q. Using graphical method, find the maximum values of 2x + y*

*Subject to*

*4x + 3y**<**12*

*4x + y**<**8*

*4x – y**<**8*

*X,y**>**0 [10 marks, 2017]*

*Q. For each hour per day that Ashok studies mathematics, it yields him 10 marks and for each hour that he studies physics, it yields him 5 marks. He can study at most 14 hours a day and he must get at least 40 marks in each. Determine graphically how many hours a day he should study mathematics and physics each, in order to maximize his marks? [12 marks, 2012]* - Partial Differential Equations – this is analogous to Ordinary Differential Equations in paper I. You need to know the methods of solving questions here. Once you know of the method and the variety of methods, you can easily solve questions from here. Laplace solutions are very important from this section.
- Numerical Analysis and Computer programming – this topic contains some aspects of computer programming. This can be done well if adequate practice is done.

Example questions:

*Q. Explain the main steps of the Gauss-Jordan method and apply this method to find the inverse of the matrix*

*2 6 6*

*2 8 6*

*2 6 8 [10 marks, 2017*] - Mechanics and Fluid Dynamics – this is related to physics and it is very important to understand the concepts first. This is a time consuming section and you must go through the previous year papers and understand which subtopics are more important.

Example question:

*Q. A hoop with radius r is rolling, without slipping, down an inclined plane of length l and with angle of inclination Ф. Assign appropriate generalized coordinate to the system. Determine the constraints, if any. Write down the Lagrangian equation for the system. Hence or otherwise determine the velocity of the hoop at the bottom of the inclined plane. [15 marks, 2016]*

*Q. Find the equation of motion of a compound pendulum using Hamilton’s equations. [10 marks, 2014]*

## IAS Maths study material (Optional)

- Schaum’s Outline of Linear Algebra by Seymour Lipschutz
- Krishna Series on Matrices
- Krishna Series on Differential Calculus
- Krishna Series on Integral Calculus
- Krishna Series on Analytical Geometry
- Krishna Series on Analytical Solid Geometry
- Mathematical Analysis by Malik and Arora
- Ordinary and Partial Differential Equations by MD Raisinghania
- Advanced Differential Equations by MD Raisinghania
- Krishna Series on Statics
- Krishna Series on Dynamics
- Krishna Series on Vector Calculus
- Vector Analysis: Schaum’s Outline Series by Murray Spiegel
- Abstract Algebra, Group Theory by R Kumar
- Abstract Algebra, Ring Theory by R Kumar
- Real Analysis by MD Raisinghania
- Krishna Series on Complex Analysis
- Operations Research by JK Sharma
- Engineering Maths by Grewal
- Numerical Methods by Jain and Iyengar
- Fluid Dynamics by MD Raisinghania
- Krishna Series on Dynamics for Moment of Inertia

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