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Memory-Based Questions from CSIR NET Mathematical Science Exam (July 25, 2024)

Mathematics Questions

Part C

Q.1 The function \(x|x|\) is/are?

A) \(f\) is continuous on \(\mathbb{R}\)

B) \(f\) is differentiable on \(\mathbb{R}\)

C) \(f\) is not continuous

D) \(f\) is differentiable only at 0

Q.2 Consider the function \(f(x, y)=\left\{\begin{array}{cl}\frac{y \sqrt{x^2+y^2}}{x} & ,(x, y) \neq(0,0) \\ 0 & ,(x, y)=(0,0)\end{array}\right.\)

A) \(f_x\) exists at \((0,0)\)

B) \(f_y\) exists at \((0,0)\)

C) \(f\) is continuous at \((0,0)\)

D) \(f\) is differentiable at \((0,0)\)

Q.3 Consider the quadratic form \(f(x, y, z)\) for which \(\exists(a, b, c) \neq(0,0,0)\) such that \(f(a, b, c)=0\)

A) \(x^2+2 y^2+3 z^2\)

B) \(x^2+2 y^2+3 z^2-2 x y\)

C) \(x^2+2 y^2+3 z^2-2 x y-2 y z\)

D) \(x^2+y^2-3 z^2\)

Q.4 If \(\sum a_n\) is a convergent series and \(b_n= \begin{cases}a_n, & a_n>0 \\ 0, & a_n<0\end{cases}\) then which of the following is/are correct? and \(c_n=\left\{\begin{array}{cc}a_n, & a_n<0 \\ 0, & a_n>0\end{array}\right.\)

A) \(\sum b_n\) and \(\sum c_n\) both converge to 0

B) If \(\sum a_n\) is absolutely convergent then \(\sum b_n\) and \(\sum c_n\) are absolutely convergent

C) If \(\sum a_n\) is not absolutely convergent then \(\sum b_n \& \sum c_n\) are divergent

Q.5 If \(f(z)\) is an entire function and \(|f(z)| \geqslant 2024\) then

A) \(f(z)=2024\)

B) \(f(z)\) is constant

Q.6 If \(f(z)\) is an entire function then its power series \(\sum a_n z^2\) is

A) convergent for \(\forall z \in \mathbb{C}\)

B) convergent for \(\forall z \in \mathbb{R}\)

C) convergent for all \(z \in\left\{2^n: n \in \mathbb{N}\right\}\)

D) convergent for all \(z \in\left\{\frac{1}{5^n}: n \in \mathbb{N}\right\}\)

Q.7 If \(A\) is a \(4 \times 4\) matrix with real entries where the minimal polynomial is \(x^2+x+1\) and \(B=A+I_4\) for a given \(4 \times 4\) identity matrix, then which of the following are true?

A) Minimal polynomial of \(B\) is \(x^2+x+1\)

B) Minimal polynomial of \(B\) is \(x^2-x+1\)

C) \(B^3=I_4\)

D) \(B^3+I_4=0\)

Q.8 If \(f(z)=\frac{1}{z} \sin \left(\frac{1}{z}\right)\) and \(g(z)=f(z) \cdot \sin z\), then which of the following are/ is true?

A) \(f\) has an essential singularity at \(z=0\)

B) \(f\) has a removable singularity at \(z=0\)

C) \(g\) has a removable singularity at \(z=0\)

D) \(g\) has an essential singularity at \(z=0\)

Part B

Q.9 If \(C\) is the collection of del sets \(S\) such that the power set of those is countably infinite, then the cardinality of \(C\) is

A) Empty

B) Finite

C) Countably infinite

D) Uncountable

Q.10 If \(f_n(x)=\frac{x^2}{\sqrt{x^2+\frac{1}{n}}}\) then

A) \(f(x)\) does not exist for any \(x \in \mathbb{R}\)

B) \(f(x)=x \quad \forall x \in \mathbb{R}\)

C) \(f(x)=|x| \quad \forall x \in \mathbb{R}\)

D) \(f(x)=0 \quad \forall x \in \mathbb{R}\)

Q.11 Let \(T\) be a linear transformation of all \(2 \times 2\) matrices given by \(T(B)=A B\) and \(A=[2, 0; 0, 1]\), then the characteristic polynomial of \(T\) is:

A) \((x-1)(x-2)\)

B) \(\left(x^2-1\right)\left(x^2-2\right)\)

C) \((x-1)^2(x-2)^2\)

D) \((x-1)(x-2)^2\)

Q.12 The number of solutions of the equation \(e^x + x = 1\) is

A) 0

B) 1

C) 2

D) 3

Q.13 The supremum of the set \(\{x: 0 < (\sqrt{2}-1)x < \sqrt{2}+1\}\) is

A) \(2 + 2\sqrt{2}\)

B) \(2 + 3\sqrt{2}\)

C) \(3 + 2\sqrt{2}\)

D) \(3 + \sqrt{2}\)

Q.14 Let \(A=\left[\begin{array}{ll}2 & b \\ a & c\end{array}\right]\) be a given matrix such that ‘6’ is one eigenvalue then which of the following is necessarily true?

A) \(a b+4 c=24\)

B) \(a \cdot b=0\)

C) \(c=12\)

D) \(c=0\)

Q.15 If \(A\) is a \(10 \times 10\) real matrix such that \(\operatorname{rank}(A)=7\), then which of the following options can be true?

A) \(\exists\) some non-zero vector \(u\) such that \(A^2 u=0\)

B) \(\exists\) a non-zero vector \(v\) such that \(A v \neq 0\) but \(A^2 v=0\)

C) \(A\) must have a non-zero eigenvalue

D) \(A^7=0\)

Q.16 The possible orders of an element in the symmetric group \(S_5\) are

A) 2

B) 3

C) 4

D) 6

Q.17 The number of group homomorphisms from \(\mathbb{Z_{150}}\) to \(\mathbb{Z_{90}}\) is

A) 30

B) 60

C) 45

D) 10

Q.18 If \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous one-to-one function, then

A) \(f(x)\) is strictly increasing

B) \(f(x)\) is strictly decreasing

C) \(f(x)\) is either increasing or decreasing

D) \(f\) is onto

Analysis and Insights

These memory-based questions provide a glimpse into the types of problems encountered in the CSIR NET Mathematical Science exam. By reviewing these questions, you can gain valuable insights into the exam pattern and focus on areas that may require more attention.

Practice these questions and review your solutions to build a stronger foundation and improve your problem-solving skills. For more detailed solutions and explanations, consider consulting additional study materials or contacting experts in the field.

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