The IISER Mohali PhD Screening Test in Mathematics is one of the most respected research entrance examinations in India.
It evaluates deep conceptual understanding, analytical reasoning, and problem-solving ability at the postgraduate level.
This exam is especially important for students aiming for doctoral studies or research fellowships in mathematics at
institutes such as IISER, TIFR, ISI, and leading Indian universities.
The test covers a wide range of subjects including real analysis, complex analysis,
linear algebra, abstract algebra, topology, differential equations, and measure theory.
This paper offers an authentic look at the type of questions asked in a PhD mathematics entrance exam
and helps aspirants strengthen their foundation in core mathematical topics.
Each question is designed to test logical depth rather than memorization, making it ideal for those preparing
for CSIR NET Mathematics, NBHM PhD fellowships, or GATE Mathematics.
Working through such problems improves critical thinking, precision in mathematical writing,
and confidence for research interviews and doctoral entrance tests.
On this page, we share the complete question paper of the IISER Mohali PhD Screening Test (Part I, 2019)
formatted clearly for online reading and self-study.
It is a valuable resource for postgraduate students and researchers pursuing excellence in advanced mathematics.
IISER Mohali PhD Screening Test in Mathematics (Part I) - May 16, 2019
Indian Institute of Science Education and Research, Mohali
PhD Screening Test in Mathematics – Part I
May 16, 2019
Instructions
Each question may have multiple correct answers. Zero points will be awarded for unattempted questions.
For a question that has been attempted, the grading scheme is as follows:
+1 for marking a correct option
+1 for not marking an incorrect option
−1 for marking an incorrect option
−1 for not marking a correct option
Notation: For any ring \( R \), \( M_n(R) \) denotes the ring of \( n \times n \) matrices with entries from \( R \).
1). Consider the polynomial ring \( \mathbb{C}[X,Y] \) and the ideal \( I \) generated by \( Y^2 - X \).
How many maximal ideals are there in the quotient ring \( \mathbb{C}[X,Y]/I \)?
a). 0
b). 1
c). 2
d). Infinitely many
2). Let \( R \subset R[\alpha] \) be commutative rings with 1, where \( \alpha \neq 0 \) satisfies a minimal monic polynomial of degree ≥ 2 over \( R \).
Which of the following statements are correct?
a). If \( R[\alpha] \) is a field then \( R \) is a field.
b). If \( R \) is a field then \( R[\alpha] \) is a field.
c). If \( R \) is a domain then \( R[\alpha] \) is a domain.
d). If \( R[\alpha] \) is a domain then \( R \) is a domain.
3). Let \( f_n \) be a sequence of Lebesgue measurable functions. Which of the following statements are correct?
a). If \( \delta_n(A) := \int_A |f_n|\,d\lambda \), then \( \delta_n \) is a measure for each \( n \).
c). \( \{ x \in \mathbb{R} \mid \lim f_n(x) \text{ is a real number} \} \) is a measurable set.
d). \( \{ x \in \mathbb{R} \mid \limsup\limits_{n \to \infty} f_n(x) \text{ is a real number} \}\) is a measurable set.
4). Let \( E \) be a finite Galois extension of \( F \) with Galois group \( G \).
Let \( K_0 = F \), \( K_n = E \), and \( K_1, K_2, \ldots, K_{n-1} \) be the fields intermediate between \( F \) and \( E \). Which of the following are true?
a). If \( K_i \) is not Galois over \( F \) for any \( i \), then \( G \) is a simple group.
b). If \( G \) is a simple group, then for any \( u \in E - F \), the smallest Galois extension containing \( u \) is \( E \).
c). If \( n = 2 \) then \( G \) is cyclic.
d). If \( K_0 \subset K_1 \subset K_2 \subset \cdots \subset K_n \), then \( G \) is cyclic.
5). Identify \( M_n(\mathbb{R}) \) and \( M_n(\mathbb{C}) \) with the standard topological spaces \( \mathbb{R}^{n^2} \) and \( \mathbb{C}^{n^2} \) respectively.
Which of the following statements are correct?
a). \( GL_n(\mathbb{R}) \) is connected.
b). \( GL_n(\mathbb{R}) \) is path connected.
c). \( SL_n(\mathbb{R}) \) is connected.
d). \( SL_n(\mathbb{R}) \) is connected but not path connected.
6). Let \( a_1a_2a_3... \) be the binary expansion of \( a \in (0,1] \) and define
\( N := \{ a \in (0,1] : \lim_{n\to\infty} \displaystyle \frac{\sum_{i=1}^n a_i}{n} = \frac{1}{2}\} \).
Let \( N^c \) denote the complement of \( N \) in \( (0,1] \). Then:
a). \( N^c \) is countably infinite set.
b). \( N^c \) is uncountable set.
c). \( N^c \) is empty set.
d). \( N^c \) is dense subset in \( (0,1] \).
7). Let \( y_1(z), y_2(z) \) be two linearly independent solutions of the linear differential operator \( L(y(z)) = y(z)'' - 2zy(z)' - y(z) \) defined on a region \( R \) of the complex plane.
If \( W(z) = \det\begin{pmatrix}y_1 & y_2 \\ y_1' & y_2'\end{pmatrix} \), then:
a). \( W(z) = ce^{2z} \) for some nonzero \( c \in \mathbb{C} \).
b). \( W(z) = 0 \).
c). \( W(z) = i \).
d). \( W(z) = ce^{z^2} \) for some nonzero \( c \in \mathbb{C} \).
8). Let \( A \) be a ring with 1 such that \( a^2 = a \) for all \( a \in A \).
Which of the following are correct?
a). \( A \) must be commutative.
b). Every prime ideal of \( A \) is maximal.
c). \( A \) has a unique maximal ideal.
d). \( 2a = 0 \) for all \( a \in A \).
9). Consider \( S := \{ A \in M_3(\mathbb{Q}) : A^3 = I \} \), where \( I \) is the identity matrix in \(M_3(\mathbb{Q})\).
Which statement is true?
a). \( S \) has exactly one element.
b). \( S \) is infinite but \( S \cap M_3(\mathbb{Z}) \) has exactly one element.
c). For every \(A \in S\), there exists a vector \(v \in \mathbb{Q}^{3}\), such that \(Av = v\).
d). \( S \cap M_3(\mathbb{Z}) \) is infinite.
10). Let \( f:\mathbb{C}\setminus\{0\}\to\mathbb{C} \) be holomorphic function such that \( \int_\gamma f(z)dz=0 \) for any closed curve \( \gamma \) in \(\mathbb{C}\setminus\{0\}\), which of the follwoing is true?
Which are true?
a). \( f \) has a removable singularity at 0.
b). \( f \) cannot have an essential singularity at 0.
c). There exists a holomorphic \( F : \mathbb{C} \setminus\{0\} \to \mathbb{C} \) such that \( F'(z)=f(z)\).
d). There exists an entire function\( F : \mathbb{C} \to \mathbb{C} \) such that \( F'(z)=f(z) \) for \( z\in\mathbb{C}\setminus\{0\} \).
11). Let \( X,Y\) be subsets of \(\mathbb{R}^2 \) and \( f:X\to Y \) be a continuous bijection.
Let \( \Gamma_f=\{(x,f(x)):x\in X\}\subset X\times Y \). Which are true?
a). \( X\) and \(Y \) are homeomorphic if both are closed in \( \mathbb{R}^2 \).
b). \( X\) and \(Y \) are homeomorphic if \( X \) is closed and bounded subset of \( \mathbb{R}^2 \).
c). \( X\) and \(Y \) are homeomorphic if \( Y \) is closed and bounded subset of \( \mathbb{R}^2 \).
d). \( \Gamma_f \) is homeomorphic to \( X \).
12). Let \( n \) be a positive integer. For any prime number \( p \), let
\( \phi_p : M_n(\mathbb{Z}) \rightarrow M_n(\mathbb{Z}/p\mathbb{Z}) \)
be the map which takes the matrix \( (a_{ij})_{i,j} \) to the matrix \( (\bar{a}_{ij})_{i,j} \),
where \( \bar{a}_{ij} \) is the image of \( a_{ij} \) under the quotient map
\( \mathbb{Z} \rightarrow \mathbb{Z}/p\mathbb{Z} \).
Let \( A, B \) denote elements of \( M_n(\mathbb{Z}) \).
Which of the following are true?
a). If \( \phi_p(A)\phi_p(B) = \phi_p(B)\phi_p(A) \) for all prime numbers \( p \), then \( AB = BA \).
b). If \( \phi_p(A)\phi_p(B) = \phi_p(B)\phi_p(A) \) for infinitely many primes \( p \), then \( AB = BA \).
c). There exist matrices \( A, B \) such that \( AB \neq BA \) and \( \phi_p(A)\phi_p(B) = \phi_p(B)\phi_p(A) \) for infinitely many primes \( p \).
d). There exist matrices \( A, B \) such that \( AB \neq BA \) and \( \phi_p(A)\phi_p(B) = \phi_p(B)\phi_p(A) \) for all primes \( p \).
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