# P Kalika

## A Site for Higher Mathematics # PhD & JRF Position Interview Asked Questions & Experiences(Collections)

Here we have shared some of Interview Questions asked in PhD Exams, Experience of Candidates and their Suggestions. Hopefully this page will be helpful for you in your preparation of PhD Interviews/Exams.

We request you, if you have recently appeared in PhDs Interview or Assistant Prof. Interview or JRF Position Interview, if possible share your Interview Que. /Experience /Suggestions to us for others benefit, Your little contribution can help PhD Aspirants to Prepare themselves.

Page Editor: P Kalika & K Munesh

❤️ We hearty thanks to those persons who shared their Interview Questions & Experiences for others help. We wish you all the best for your future.

🎯 Some memory based ques from today’s DUET Mphil/PhD exam. (Shared by Kirandeep Kaur)

🎯 Hello sir recently I experienced NBHM PhD interview and sending my interview questions here. (Shared by Parna Saha)

First they asked my preference and I said abstract algebra.
Then they asked two questions from the nbhm phd written test. One is prove the formula ” number of idempotent elements in Zn = 2^d”
And another one from linear algebra and I answered the question in written exam wrong but still they said to try there…

1. What is class equation?
2. What is the cardinality of center of a group of order 27? Explain it by the help of class equation.
3. Number of Homomorphism from S3 to Z6? Explain.
4. Let A,B are closed sets in R. Consider the set dist(A,B) defined by {|x-y|:x in A,y in B} = 0 then what can you say about the intersection of the sets A and B.
5. One question from topology which I didn’t remember.

Suggestions:
If you are appearing nbhm interview try to read the proofs of the formula we generally use and prepare your nbhm exam questions..all of them
And whatever your preference can be…they ask question from real analysis and topology.

Thank you

🎯 I am attaching here pdf of questions asked in PhD interviews at IIT, IIIT, and IISER. These are collection of questions based on my experience of one year (Shared by Divya).

🎯 Hello here is the question of PhD interview University of Calcutta (Ballygunge Science College) pure Mathematics (Shared by Amitabha Bhattacharya)

🎯 IISER Mohali PhD Interview 2021 Questions (Anonymous Share)

1. Give some examples of non- commutative rings.
2. What are the ideals in the ring M_2(R)?
3. Consider a ring R and an ideal I, and if it contains some invertible element. What is special about this ideal?
4. Let f: C to C is an analytic function. Let P(z) = a_0 + a_1 z + … + a_n z^n is a n degree polynomial. Also P(f) = 0. What can we say about this function?
5. A function function in real is always continuous? If yes then prove. If not then give a counter-example.
6. Which type of discontinuities does an increasing function have?
7. Do we have any homomorphisms from S_3 to Z_6? Justify.
8. Is there a continuous map from [0,1] to (0,1)? Why/Why-not?
9. Construct a surjective map from (0,1) to [0,1].
10. Construct a bijective map from (0,1) to [0,1].

🎯 IIIT Delhi PhD Interview 2021 Questions (Anonymous Share)

1. What is Quotient group? How to define it. What is the equivalence relation here?
2. What is a Factor ring?
3. Let G/H be a factor group. Why do we require H to be a normal subgroup.
4. Give an example of a group G and its non-normal subgroup H such that G/H is not a group.
5. What is a Field Extension?
6. When do we say a field extension is finite?
7. Is every finite field extension algebraic?
8. What about the converse? Give a counter-example. Prove it.
9. What is an Algebraically Closed Field?
10. What is the difference between Algebraic Closure and Algebraically Closed Field?
11. Give an example of an Algebraically Closed Field?
12. Give an example of a non-Algebraically Closed Field?
13. Let p be prime and consider the polynomial 1+x+x^2+…+x^(p-1) over Q. Is it reducible or irreducible. Why
14. What is the splitting field of the above polynomial?
15. What is a general linear group?
16. What is the order of GL(2,Z_p)? How.

🎯 NIT Patna PhD Interview 2021 Questions (Shared by Soubhik Paramanik)

🎯 NIT Allahabad PhD Interview 2021 Questions (Anonymously Shared)

1. State existence and uniqueness theorem
2. Define Leibnitz condition
3. y’=siny is linear or non linear? Why?
4. y’=√y what is it’s degree
5. Define regular singular points.
6. Is it possible to find series solution using regular singular point.
7. Define indical equation
8. Value of integration of exp(x²) , limit is from 0 to infinity
9. f(x)=x|x| differentiable at x=0 or not , if yes what is the value of f'(0)
10. How to find tangent plane of f(x,y,z)=0
11. What is equation of plane

🎯 IIT INDORE PhD Interview 2021 Questions (Anonymously Shared)

1. What is dim of R(Q)?
2. Why dimension is infinite?
3. Any n dimension vector space over Q is countable or not
4. f(x) = x sin(1/x) for x≠0, f(x)=0 for x=0Is f(x) continuous ?
5. Why limit of sin(1/x) does not exist at x=0 ?
6. The given function in que 4 is differentiable or not?
7. If the function is continuous then its derivative is continuous?
8. f(x) = x² sin(1/x) for x≠0, = 0 for x=0 is continuous. Then it is differentiable. What if we replace x² by x³?
9. In general f(x) = x^alpha sin(1/x^beta) for  x≠0, = 0  for x=0 Condition on alpha beta for differentiability
10. sin(1/x) as x tends to zero range of limit
11. What is Sturm Separation Theorem?
12. y1=x & y2= sinx then can you form second order differential equation
13. V=P5(x) (R) Let set {1,x}Prove it is LI
14. v1 & v2 are any two vector in 2 dim plane the line joining point v1 & v2 passing through origin is LI or LD

🎯 IIT Bhubneswar PhD Interview 2021 Questions (Shared by Radhanandan Mandal).

🎯 IIT Jodhpur PhD Interview 2021 Questions (Anonymously Shared)

1. What is Chinese Remainder Theorem
2. Application of Chinese Remainder Theorem
3. What is Prime Number
4. Chinese remainder theorem in Ring Theory
5. What is an Irreducible Element
6. What you know about set of Integers
7. What is relation in Prime element & Irreducible element

🎯 IISc Bangalore PhD Interview 2021 Questions (Thanks to Lalbahadur Sahu)

🎯 IIT Mandi PhD Interview 2021 Questions (Anonymously Shared)

1. State and explain existence and uniqueness theorem of ODE.
2. What is ODE ? What is PDE ?
3. What is order and degree ?
4. What is characteristics equations ?

🎯 IIT Delhi PhD Stage-1 Interview 2021 Questions (Anonymously Shared)

🎯 Anonymously Shared – 1

1. Definition of Differentiability
2. f(x) = x sin(1/x) for x≠0, = 0 for x=0 is diff. or not. Prove it.
3. Definition of continuous function. Give an example of Discontinuous function.
4. If A & B are 2×2 matrices such that AB =0 then BA=0 ?
5. In above question, if (AB)²=0 what about BA² ?
6. In group G if every non identity element has order 2 then is it Abelian group? If yes, Prove it.
7. Let G be a group of odd order, F: G to G defined as F(x) = x². Is it a homomorphism? If yes, prove it.

🎯 Anonymously Shared – 2

1. Def. Of open ball.
2. Def. Of closed ball.
3. Def. Of boundary point.
4. [0,1] Union [7,9] center 8 and radius 1 boundary point Or isse related kya puch rahe samajh nhi aaya
5. Complex se fundamental theorem ka proof.
6. Difference between differential function and analytic function.
7. Analytic function, singular point, CR equation, types of singularity, examples of isolated singularity.
8. df/dzbaar = 0 wala result proof kro

🎯 Anonymously Shared – 3

1. If a subring of a ring is integral domain, does it imply Ring is also an integral domain?
2. Definition of Integral domain.
3. A given n×n non invertible matrix will be a zero divisor or not?
4. G=Gl(n, C) , S= set of all n×n invertible diagonal matrices, then
1. Is S a subgroup of G?
2. Is S a normal subgroup of G
5. Def of similar matrices
6. does there exist some infinite group having only finite no. Of subgroups?
7. There are infinitely many prime numbers. Proof.
8. If f is non negative continuous function such that [integral (0to1) fdx]=0
9. Then what can u conclude about f?
10. √x is Uc. Why so?

🎯 Anonymously Shared – 4

🎯 Interview Questions for PhD Admission 2021 at IISc Bangalore (Shared By Sayan Roy)

🎯 Interview Questions for PhD Admission 2021 at IIT Hyderabad (Anonymously Shared)

I have given an online PhD Interview for IIT Hyderabad. The questions are as follows:

1. Is the collection of all polynomials of degree at most 2 over the field of reals a vector space?
2. Now the function that p(x) goes to its derivative, is it a Linear Transformation?
3. Give its Matrix Representation.
4. Is the matrix diagonalizable? Why/Why not?
5. Prove that a Nilpotent matrix is diagonalizable if and only if it is a zero matrix.
6. Give examples of 2000 (or infinitely) rings.
7. Give infinite examples of fields of characteristic 0.
8. Give infinite examples of Galois extensions.
9. What is the degree of a field extension?
10. What is the degree of the extension Q(w_n) over Q, where w_n is n-th root of unity.

🎯 Interview Questions for PhD Admission 2021 at IISER Kolkata (Anonymously Shared)

🎯 PhD Admission 2021 IIT Delhi interview question… (Shared by Sandeep)

🎯 PhD Admission 2021 IISER Kolkata telephonic interview question… (Anonymously Shared)

1. How many finite algebraically closed fields are there upto isomorphism?
2. how many fields of order 12 are there upto isomorphism?
3. what is caley hamilton theorem?
4. how can we represent all those primes which are of the form -sum of 2 integer.
5. series (n^1/n -1) is convergent or divergent.
6. series (n/(n+1))^(1/n) convergent or divergent. ?

🎯 PhD Admission 2021 Interview Que. Asked at IIT Mandi (Shared by Dhiman Das)

● sir maine linear algebra and pde mai interview diya

◆ linear algebra se eigen vector defination

◆ null vector eigen vector ho sakta hai kiya ??

◆ diagonalisibility se and A*A’ diagonalisble or not ase question kiyaa

◆ pde se classifiacation se kuch question kiya

◆ and heat wave de almbert se question kiya sir

🎯 PhD Admission 2021 interview Que. Asked at IIT Kharagpur (Anonymously Shared)

1. Expand f(z)=1/z(z-2) by Laurent series expansion in the domain 0<|z|<2 and 2<|z|<infinity.
2. How do these two series differ?
3. What are different iterative methods for solving system of equations?
4. How is Gauss Jacobi method different from Gauss Seidel method?
5. Which of the above has better convergence?
6. When does the Gauss Jacobi method converges?
7. What is meaning of diagonally dominant matrix?
8. Is a diagonally dominant matrix invertibe?
9. Is the function f(x)=x^2-y^2 harmonic?

📚 (Suggestion)👉🏻 Questions asked are easy so one has to be clear about all fundamentals while preparing for interview.

🎯 PhD Admission 2021 interview Que. Asked at IISc Banglore (Shared By Sandip)

One of my friends give interview(today) in iisc bangalore for integrated phd..

🎯 PhD interview questions Asked at IIT Mandi (Shared by Manisha Agarwal)

• Definitions of field, ring, vector space, field extension, linear transformation
• Application of field and field extension
• Collection of all polynomial form a vector space or not?
• Collection of all polynomial of degree greater than or equal to n is subspace or not?
• Collection of all polynomial of deg less than or equal to n form subspace or not?
• If subspace then give any linear transformation on this.

📚 Their Suggestions…

Sir please try to cover latex and mtlab computer applications which we require in phd interview…try to cover as soon as possible.

As they ask that do u know any computer applications like latex and matlab
I think it is also one of the factor so that anyone got selection or not

🎯 PhD interview questions Asked at IIT Guwahati (Topic: Real Analysis) (Shared by Harinand Singh)

1. Sequence of function x^n [0, 1] uniform cgt Or not.
2. Above question on [0, 1)
3. Examples of continus function
4. Range of polynomial
5. Example of a nonconstant poly which
6. Rang is bdd.

📚 Their Suggestions …

This was my first interview,
According to me, jisme intersted h us topic ko ache se taiyar kre aur basic bhi.
Maine unhe ode bola but O bole kisi aur me interested h to maine real bola O bs real se hi Sare question puche aur thodi bahut help bhi krte h

🎯 PhD Interview Questions Asked at IIT Mandi (Topic: PDE ,& ODE) (Shared by Shadhank)

I have appeared in IIT Mandi PhD interview.

The questions were from topic PDE & ODE.
The questions were as follows:

1. Classify second order PDE with examples of each one? Which type of PDE is classified in these categories (Hyperbolic, Parabolic & Elliptic)?
2. From where the Characteristic Equation comes in such PDE and on the basis of roots of which how we decide the above classification?
3. Classify the homogeneous Laplace Equation in two dimension. (which is also a second order PDE)
4. What is degree and order of a differential equation?
5. What is Initial Value Problem? Why it is called as IVP? Is IVP holds for the differential equation having order greater than one? (Discuss)
6. What is Boundary Value Problem?
7. What are the boundary conditions and how many types of boundary conditions are there? How it it different from Initial Condition?
8. What is Sturm-Liouville’s Problem? Is it regular or not?
9. How many types of BVP? What is regular BVP?
10. Is Heat equation in one dimension an IVP or BVP?

Thats all.

🎯 Some of the PhD Interview Questions that were asked to me in the academic year 2020-2021. The interviews were held online (Anonymously Shared)

Interview Questions in IIT Delhi.(Round 1)

1. Give an example of an ode such that f(x,y) does not satisfy the Lipschitz’ condition, but still a solution exists.
2. What is the most important theorem on finite groups? State and sketch the proof.
3. State Sylow’s Second Theorem.
4. If the Correspondence (aH)(bH)=(abH) defines the binary operation on the Set of distinct left cosets of H in G. Is H a normal subgroup in G?
5. What was the need of an orthonormal basis, when we already had a standard ordered basis?

Interview Questions in IISER Kolkata.(Round 1: Telephonic)

1. Prove that R is a field if and only if R[x] is a P.I.D.
2. Does R/p become an integral domain if we have only R as a commutative ring without unity and p a prime ideal in R.?
3. Give an explicit form of the prime ideals in the ring Z*Z.
4. If R is a ring such that every subring in R is an ideal in R, prove that R is either isomorphic to Z or isomorphic to Z/nZ.
5. Does there exist a non-commutative ring of order 15?

Interview Questions in IISER Kolkata.(Round 2: Google Meet)

1. Is every complex matrix diagonalizable?
2. Prove that rank(A*A)=rank(A).
3. Relation between the minimal and characteristic polynomial of similar matrices.
4. If G and H are Abelian groups, M and N are their subgroups respectively, such that G is isomorphic to H, M is isomorphic to N, then is G/M isomorphic to G/N?
5. Let f be an onto ring homomorphism from a PID R to an Integral domain S, such that f is not injective, prove that S is a field.
6. what is the dimension of R[x,y]/<x^3,y^2> as a vector space over R?
7. How many irreducible polynomials are there in Z_2[x]?
8. Show that in a finite group of odd order every element has a unique square root.
9. Prove that the image of a non constant entire function is dense in C.

🎯 This Interview Experience is Shared by Gautam Kaushik (PhD Aspirant 2021 ).

🔢 PhD Interview Questions Shared by ‘Ritika Das’ (PhD Aspirant 2021 ).

Interview Questions Asked at Visva Bharati UniversityKolkata.

🎯 PhD Interview Questions & Suggestions Shared by ‘Prince Bharti’ (PhD Aspirant 2021 ).

I selected subjects for interview namely Differential equations and Fuzzy Theory. Because I do project in MPhil on Fuzzy Theory so it’s beneficial for me. They ask some questions about my project.

And from Differential equations they ask from SLP and BVP , what is the difference between them, and example of these, and knowledge about boundary conditions that’s it.

Thank you sir 🙏

📚 (Our Question) Any Suggestions from your Side…

Yes sir according to me , interview k liye apna ek subject tyyar kr ke jana chahiye, or uske main main topics per command ho and given conditions , related examples ko dhyan se smajhna kafi halpful rhega. Jiska nhi pta use dheere se sorry bol de or apni pasand ka topic unhe bta do bss , ab aap tyyar rhe sb kuch usi me se poocha jane wala h😊😊 that’s it. Thank you sir.

Ye bhi add kr lena, aapke notes kafi useful rhe mere addmission m , thank you very much sir 🙏 , isi tarah se hmari help krte rhiye …

🎯 PhD Interview Questions & Suggestions Shared by ‘Sanket’ (PhD Aspirant 2021 ).

Recently I’ve appeared for VIT Vellore and BITS Pilani interview. Here I’m providing my Interview Questions in handwritten format…

📚 (Our Question) Any Suggestions from your Side…

” Just prepare some basics … Means whatever you are going to tell about research interests. Prepare basics nicely. Prepare one favourite theorem. Lots of example and counter examples”

🎯 PhD Interview Questions & Suggestions Shared by ‘Sanjeev(PhD Aspirant 2021 ).

Hello sir
I was asked few questions in Tezpur Univ phd interview

1. Shift operators
2. Formula for Sequence of rationals that converges to a given irrational c
3. Difference between hamel and schauder bases and benefits or need of schauder basis
4. Idea of Cantor’s set and its prop and proof sketch of some of the prop
5. Basic on group theory

Thank you very much for providing useful tips and updating us on events happening in Mathematical world.

📚 (Our Question) Any Suggestions from your Side…

Hello Sir…my suggestions are
One may clear the competitive exams but clearing interview is completely different. We need to have good knowledge of topics. Must read standard books (including proofs) like Bartle for Analysis, Gallian for Algebra, etc. Avoid random guess because one wrong answer would expose all your weaknesses. If you don’t know the answer them simply say I can try try and after try if could not get answer then say that you don’t know the answer… Logical implications must be sound.

### 5 thoughts on “PhD & JRF Position Interview Asked Questions & Experiences(Collections)”

1. Q Is there collection of the DU PHD Interview questions?

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2. Whether any computer science related questions for phd

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3. Thanks a lot sir. 🙏

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4. very nice

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5. I need to get new updates

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