CSIR NET · Mathematical Sciences
June
2023
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64 questions
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Q21
mcq
Suppose $S$ is an infinite set. Assuming that the axiom of choice holds, which of the following is true?
a.
$S$ is in bijection with the set of rational numbers.
b.
$S$ is in bijection with the set of real numbers.
c.
$S$ is in bijection with $S \times S$.
Correct Answer
d.
$S$ is in bijection with the power set of $S$.
Step-by-Step AI Explanation
Verified Method
By a fundamental theorem of set theory (due to Tarski), assuming the Axiom of Choice (AC), for any infinite cardinal $\kappa$, we have $\kappa \times \kappa = \kappa$. Therefore, if $S$ is an infinite set, $S$ is in bijection with $S \times S$. In fact, the assertion that $|S \times S| = |S|$ for all infinite sets $S$ is equivalent to the Axiom of Choice.
Option (A) is false because $S$ could be uncountable. Option (B) is false because $S$ could have cardinality greater than that of $\mathbb{R}$ (for example, the power set of $\mathbb{R}$). Option (D) is false by Cantor's theorem, which states that no set is in bijection with its power set.
Q22
mcq
Consider the series $\sum_{n=1}^\infty a_n$, where $a_n = (-1)^{n+1}\left(\sqrt{n+1}-\sqrt{n}\right)$. Which of the following statements is true?
a.
The series is divergent.
b.
The series is convergent.
c.
The series is conditionally convergent.
Correct Answer
d.
The series is absolutely convergent.
Step-by-Step AI Explanation
Verified Method
Note that for $a_n = (-1)^{n+1}(\sqrt{n+1} - \sqrt{n})$, we can rationalize the radical expression:
$$\sqrt{n+1} - \sqrt{n} = \frac{1}{\sqrt{n+1} + \sqrt{n}}.$$
Let $b_n = \frac{1}{\sqrt{n+1} + \sqrt{n}}$. Since $b_n > 0$, $b_n$ is strictly decreasing to $0$ as $n \to \infty$, the series $\sum_{n=1}^\infty (-1)^{n+1} b_n$ converges by the Alternating Series Test (Leibniz's criterion).
To check for absolute convergence, consider $\sum_{n=1}^\infty |a_n| = \sum_{n=1}^\infty \frac{1}{\sqrt{n+1} + \sqrt{n}}$. As $n \to \infty$, $b_n \sim \frac{1}{2\sqrt{n}} = \frac{1}{2n^{1/2}}$. By the limit comparison test with the divergent $p$-series $\sum \frac{1}{n^{1/2}}$ (where $p = 1/2 \le 1$), the series of absolute values diverges.
Therefore, the series converges conditionally, making (C) the most specific and intended correct statement.
Q23
mcq
Let $x, y \in [0,1]$ be such that $x \neq y$. Which of the following statements is true for every $\epsilon > 0$?
a.
There exists a positive integer $N$ such that $|x-y| < 2^n \epsilon$ for every integer $n \ge N$
Correct Answer
b.
There exists a positive integer $N$ such that $2^n \epsilon < |x-y|$ for every integer $n \ge N$
c.
There exists a positive integer $N$ such that $|x-y| < 2^{-n} \epsilon$ for every integer $n \ge N$
d.
For every positive integer $N$, $|x-y| < 2^{-n} \epsilon$ for some integer $n \ge N$
Step-by-Step AI Explanation
Verified Method
Since $x \neq y$, we have $|x-y| > 0$. Given any $\epsilon > 0$, we have $\lim_{n \to \infty} 2^n \epsilon = \infty$. Therefore, by the Archimedean property, there exists a positive integer $N$ such that for all $n \ge N$:
$$2^n \epsilon > |x-y|.$$
Thus, statement (A) is correct.
- (B) is false because $2^n \epsilon \to \infty$ while $|x-y| \le 1$.
- (C) and (D) are false because $2^{-n}\epsilon \to 0$ as $n \to \infty$, so for sufficiently large $n$, $2^{-n}\epsilon < |x-y|$.
Q24
mcq
Which of the following assertions is correct?
a.
$\limsup_n e^{\cos\left(\frac{n\pi+(-1)^n 2e}{2n}\right)} > 1$
b.
$\lim_n e^{\log_e\left(\frac{n\pi^2+(-1)^n e^2}{7n}\right)}$ does not exist
c.
$\liminf_n e^{\sin\left(\frac{n\pi+(-1)^n 2e}{2n}\right)} < \pi$
Correct Answer
d.
$\lim_n e^{\tan\left(\frac{n\pi^2+(-1)^n e^2}{7n}\right)}$ does not exist
Step-by-Step AI Explanation
Verified Method
Consider the sequence in (C):
$$\frac{n\pi + (-1)^n 2e}{2n} = \frac{\pi}{2} + (-1)^n \frac{e}{n} \to \frac{\pi}{2} \quad \text{as } n \to \infty.$$
Since $\sin(x)$ is continuous at $\pi/2$, $\sin\left(\frac{n\pi + (-1)^n 2e}{2n}\right) \to \sin(\pi/2) = 1$.
By continuity of the exponential function, $\lim_{n\to\infty} e^{\sin\left(\frac{n\pi+(-1)^n 2e}{2n}\right)} = e^1 = e$.
Therefore, the limit superior and limit inferior both equal $e$.
Since $e \approx 2.718 < 3.14159 \approx \pi$, we have $\liminf_n e^{\sin(\dots)} = e < \pi$, which makes assertion (C) correct.
For the other options:
- For (A), the limit of the argument inside $\cos$ is $\pi/2$, so the sequence converges to $e^{\cos(\pi/2)} = e^0 = 1$, which is not strictly greater than 1.
- For (B) and (D), the inside arguments converge to $\pi^2/7$, so the sequences converge (the limits do exist).
Q25
mcq
How many real roots does the polynomial $x^3+3x-2023$ have ?
a.
$0$
b.
$1$
Correct Answer
c.
$2$
d.
$3$
Step-by-Step AI Explanation
Verified Method
Let $P(x) = x^3 + 3x - 2023$.
The derivative is $P'(x) = 3x^2 + 3 = 3(x^2 + 1) > 0$ for all $x \in \mathbb{R}$.
Since $P'(x) > 0$ for all real $x$, $P(x)$ is strictly increasing on $\mathbb{R}$.
A strictly increasing continuous function can cross the $x$-axis at most once. Since $\lim_{x \to -\infty} P(x) = -\infty$ and $\lim_{x \to +\infty} P(x) = +\infty$, by the Intermediate Value Theorem, $P(x)$ has at least one real root. Hence, $P(x)$ has exactly one real root.
Q26
mcq
Which one of the following functions is uniformly continuous on the interval $(0,1)$ ?
a.
$f(x) = \sin \frac{1}{x}$
b.
$f(x) = e^{-1/x^2}$
Correct Answer
c.
$f(x) = e^x \cos \frac{1}{x}$
d.
$f(x) = \cos x \cos \frac{\pi}{x}$
Step-by-Step AI Explanation
Verified Method
A continuous function on a bounded open interval $(a, b)$ is uniformly continuous if and only if both one-sided limits $\lim_{x \to a^+} f(x)$ and $\lim_{x \to b^-} f(x)$ exist and are finite.
For $f(x) = e^{-1/x^2}$ on $(0,1)$:
- $\lim_{x \to 0^+} e^{-1/x^2} = 0$ (since $-1/x^2 \to -\infty$).
- $\lim_{x \to 1^-} e^{-1/x^2} = e^{-1}$.
Since both endpoint limits exist and are finite, $f(x)$ extends to a continuous function on $[0,1]$, and is therefore uniformly continuous on $(0,1)$.
For the other options, the limit as $x \to 0^+$ does not exist because the trigonometric functions oscillate infinitely often between $-1$ and $1$ near $0$.
Q27
mcq
Let $l \ge 1$ be a positive integer. What is the dimension of the $\mathbb{R}$-vector space of all polynomials in two variables over $\mathbb{R}$ having a total degree of at most $l$ ?
a.
$l+1$
b.
$l(l-1)$
c.
$l(l+1)/2$
d.
$(l+1)(l+2)/2$
Correct Answer
Step-by-Step AI Explanation
Verified Method
The space of polynomials in two variables $x, y$ over $\mathbb{R}$ of total degree at most $l$ has a standard basis consisting of monomials $x^i y^j$ with $i, j \ge 0$ and $i+j \le l$.
For each fixed degree $k = i+j$ (where $0 \le k \le l$), the number of pairs $(i, j)$ of non-negative integers summing to $k$ is $k+1$.
Thus, the total number of basis elements is:
$$\sum_{k=0}^l (k+1) = 1 + 2 + \dots + (l+1) = \frac{(l+1)(l+2)}{2}.$$
Hence, the dimension of the vector space is $\frac{(l+1)(l+2)}{2}$.
Q28
mcq
Let $T$ be a linear operator on $\mathbb{R}^3$. Let $f(X) \in \mathbb{R}[X]$ denote its characteristic polynomial.
Consider the following statements.
(a) Suppose $T$ is non-zero and $0$ is an eigen value of $T$. If we write $f(X) = Xg(X)$ in $\mathbb{R}[X]$, then the linear operator $g(T)$ is zero.
(b) Suppose $0$ is an eigenvalue of $T$ with at least two linearly independent eigen vectors. If we write $f(X) = Xg(X)$ in $\mathbb{R}[X]$, then the linear operator $g(T)$ is zero. Which of the following is true ?
a.
Both (a) and (b) are true
b.
Both (a) and (b) are false
c.
(a) is true and (b) is false
d.
(a) is false and (b) is true
Correct Answer
Step-by-Step AI Explanation
Verified Method
Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be a linear operator and let $f(X) = \det(XI - T) = X g(X)$ be its characteristic polynomial (which has degree 3). By the Cayley-Hamilton theorem, $f(T) = T g(T) = 0$.
- Statement (a): Consider $T$ given in the standard basis by the Jordan block $T = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$. The characteristic polynomial is $f(X) = X^2(X-1) = X g(X)$, where $g(X) = X(X-1) = X^2 - X$. Then $g(T) = T(T-I) = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} -1 & 1 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \neq 0$. Thus, statement (a) is false.
- Statement (b): If $0$ is an eigenvalue of $T$ with at least two linearly independent eigenvectors, the geometric multiplicity of $0$ is at least 2, so $\dim(\ker T) \ge 2$. By the rank-nullity theorem, $\operatorname{rank}(T) = 3 - \dim(\ker T) \le 1$.
Since $\operatorname{rank}(T) \le 1$, the image of $T$ is at most 1-dimensional. By the Cayley-Hamilton theorem, $T(g(T)) = 0$, which means that the image of $g(T)$ is contained in $\ker(T)$.
More directly, the minimal polynomial $m(X)$ must divide $f(X) = Xg(X)$. If $\operatorname{rank}(T) = 0$, $T=0$, so $g(0) = 0$ is not necessary, but $T=0 \implies g(T) = 0$ whenever $g(X)$ has no constant term. More generally, since the geometric multiplicity of $0$ is $\ge 2$, the Jordan form of $T$ for eigenvalue $0$ has at least two blocks, so in a 3-dimensional space the size of any Jordan block for $0$ can be at most $3 - 2 + 1 = 1$ (if the other eigenvalue is nonzero, the size of the block is 1; if 0 has multiplicity 3, there are two or three blocks, so block sizes are $2, 1$ or $1, 1, 1$).
Wait, if the Jordan blocks for $0$ are of sizes $2$ and $1$, then the algebraic multiplicity of $0$ is 3, so $f(X) = X^3$, which means $g(X) = X^2$. But the minimal polynomial is $m(X) = X^2$, so $g(T) = T^2 = 0$.
If the Jordan blocks for 0 are of sizes 1, 1, 1, then $T = \lambda I$ (with $\lambda=0$), so $T=0$ and $g(T) = 0$.
If the other eigenvalue is $\lambda \neq 0$, then the algebraic multiplicity of 0 is 2 and its geometric multiplicity is 2, so the Jordan blocks for 0 are of size 1 and 1. The minimal polynomial is $m(X) = X(X-\lambda)$. Here $f(X) = X^2(X-\lambda) = X g(X)$, where $g(X) = X(X-\lambda)$. Since $m(X) = g(X)$, we have $g(T) = 0$.
In all possible cases, $g(T) = 0$. Thus, statement (b) is true.
Hence, (a) is false and (b) is true.
Q29
mcq
Let $A$ be a $3\times 3$ matrix with real entries. Which of the following assertions is FALSE?
a.
$A$ must have a real eigenvalue
b.
If the determinant of $A$ is $0$, then $0$ is an eigenvalue of $A$
c.
If the determinant of $A$ is negative and $3$ is an eigenvalue of $A$, then $A$ must have three real eigenvalues.
d.
If the determinant of $A$ is positive and $3$ is an eigenvalue of $A$, then $A$ must have three real eigenvalues
Step-by-Step AI Explanation
Verified Method
Since $A$ is a $3 \times 3$ real matrix, its characteristic polynomial has degree 3 with real coefficients, so it has at least one real eigenvalue, and non-real eigenvalues must come in complex conjugate pairs $(\lambda, \bar{\lambda})$.
- (A) is true because non-real roots come in pairs, leaving at least one real root.
- (B) is true because $\det(A) = 0$ implies $\lambda = 0$ is an eigenvalue.
- (C) is true: Let the eigenvalues be $3, \lambda_1, \lambda_2$. Then $\det(A) = 3\lambda_1 \lambda_2 < 0$, so $\lambda_1 \lambda_2 < 0$. If $\lambda_1, \lambda_2$ were a complex conjugate pair $u \pm iv$, then $\lambda_1 \lambda_2 = u^2 + v^2 \ge 0$, a contradiction. Thus $\lambda_1, \lambda_2$ must both be real, meaning $A$ has three real eigenvalues.
- (D) is FALSE: If $\det(A) > 0$ and $3$ is an eigenvalue, then $\lambda_1 \lambda_2 > 0$. This is satisfied if $\lambda_1, \lambda_2$ is a conjugate pair of non-real complex numbers (e.g., $i$ and $-i$, giving $\det(A) = 3(1) = 3 > 0$). In this case, $A$ has only one real eigenvalue, not three.
Q30
mcq
Let $A$ be a $3\times 3$ real matrix whose characteristic polynomial $p(T)$ is divisible by $T^2$.
Which of the following statements is true?
a.
The eigenspace of $A$ for the eigenvalue $0$ is two-dimensional.
b.
All the eigenvalues of $A$ are real.
c.
$A^3 = 0$.
d.
$A$ is diagonalizable
Step-by-Step AI Explanation
Verified Method
Since $A$ is a $3 \times 3$ real matrix, its characteristic polynomial has degree 3. If $p(T)$ is divisible by $T^2$, then $0$ is an eigenvalue of algebraic multiplicity at least $2$.
Since the degree of $p(T)$ is $3$, the eigenvalues are $0, 0, \lambda_3$. The third eigenvalue $\lambda_3$ must also be real because non-real eigenvalues of real matrices occur in complex conjugate pairs, and here two eigenvalues are already real (both $0$). Therefore, all eigenvalues of $A$ are real.
Counterexamples for other options:
- For $A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$, the eigenspace of $0$ is 1-dimensional, $A$ is not diagonalizable, and $A^3 \ne 0$.
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Q31
mcq
Let $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$ denote vectors in $\mathbb{R}^n$ for a fixed $n \ge 2$. Which of the following defines an inner product on $\mathbb{R}^n$?
a.
$\langle x, y \rangle = \sum_{i,j=1}^n x_i y_j$
b.
$\langle x, y \rangle = \sum_{i,j=1}^n (x_i^2 + y_j^2)$
c.
$\langle x, y \rangle = \sum_{j=1}^n j^3 x_j y_j$
d.
$\langle x, y \rangle = \sum_{j=1}^n x_j y_{n-j+1}$
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Q32
mcq
Consider the quadratic form $Q(x, y, z)$ associated to the matrix
$$B = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & -2 \end{bmatrix}, \text{ Let } S = \left\{ \begin{bmatrix} a \\ b \\ c \end{bmatrix} \in \mathbb{R}^3 \mid Q(a,b,c) = 0 \right\}$$
Which of the following statements is FALSE?
a.
The intersection of $S$ with the $xy$-plane is a line.
b.
The intersection of $S$ with the $xz$-plane is an ellipse.
c.
$S$ is the union of two planes.
d.
$Q$ is a degenerate quadratic form
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Q33
mcq
Let $f(z) = \exp\left(z + \frac{1}{z}\right), z \in \mathbb{C} \setminus \{0\}$. The residue of $f$ at $z = 0$ is
a.
$\sum_{l=0}^\infty \frac{1}{(l+1)!}$
b.
$\sum_{l=0}^\infty \frac{1}{l!(l+1)}$
c.
$\sum_{l=0}^\infty \frac{1}{l!(l+1)!}$
d.
$\sum_{l=0}^\infty \frac{1}{(l^2+1)!}$
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Q34
mcq
Let $f$ be an entire function that satisfies $|f(z)| \le e^y$ for all $z = x+iy \in \mathbb{C}$, where $x, y \in \mathbb{R}$. Which of the following statements is true?
a.
$f(z) = c e^{-iz}$ for some $c \in \mathbb{C}$ with $|c| \le 1$
b.
$f(z) = c e^{iz}$ for some $c \in \mathbb{C}$ with $|c| \le 1$
c.
$f(z) = e^{-ciz}$ for some $c \in \mathbb{C}$ with $|c| \le 1$
d.
$f(z) = e^{ciz}$ for some $c \in \mathbb{C}$ with $|c| \le 1$
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Q35
mcq
Consider the function $f$ defined by $f(z) = \frac{1}{1-z-z^2}$ for $z \in \mathbb{C}$ such that $1-z-z^2 \ne 0$. Which of the following statements is true?
a.
$f$ is an entire function.
b.
$f$ has a simple pole at $z = 0$.
c.
$f$ has a Taylor series expansion $f(z) = \sum_{n=0}^\infty a_n z^n$, where coefficients $a_n$ are recursively defined as follows: $a_0 = 1, a_1 = 0$ and $a_{n+2} = a_n + a_{n+1}$ for $n \ge 0$
d.
$f$ has a Taylor series expansion $f(z) = \sum_{n=0}^\infty a_n z^n$, where coefficients $a_n$ are recursively defined as follows: $a_0 = 1, a_1 = 1$ and $a_{n+2} = a_n + a_{n+1}$ for $n \ge 0$
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Q36
mcq
Let $C$ be the positively oriented circle in the complex plane of radius 3 centered at the origin.
What is the value of the integral $\int_C \frac{dz}{z^2(e^z - e^{-z})}$ ?
a.
$\frac{i\pi}{12}$
b.
\frac{-i\pi}{12}
c.
$\frac{i\pi}{6}$
d.
$\frac{-i\pi}{6}$
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Q37
mcq
Which of the following equations can occur as the class equation of a group of order 10?
a.
$10 = 1 + 1 + \dots + 1 \text{ (10 times)}$
b.
$10 = 1 + 1 + 2 + 2 + 2 + 2$
c.
$10 = 1 + 1 + 1 + 2 + 5$
d.
$10 = 1 + 2 + 3 + 4$
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Q38
mcq
The number of solutions of the equation $x^2 = 1$ in the ring $\mathbb{Z}/105\mathbb{Z}$ is
a.
$0$
b.
$2$
c.
$4$
d.
$8$
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Q39
mcq
Let $p$ be a prime number. Let $G$ be a group such that for each $g \in G$ there exists an $n \in \mathbb{N}$ such that $g^{p^n} = 1$.
Which of the following statements is FALSE?
a.
If $|G| = p^6$, then $G$ has a subgroup of index $p^2$
b.
If $|G| = p^6$, then $G$ has at least five normal subgroups
c.
Center of $G$ can be infinite
d.
There exists $G$ with $|G| = p^6$ such that $G$ has exactly six normal subgroups.
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Q40
mcq
Consider $\mathbb{R}$ with the usual topology. Which of the following assertions is correct ?
a.
A finite set containing 33 elements has at least 3 different Hausdorff topologies.
b.
Let $X$ be a non-empty finite set with a Hausdorff topology. Consider $X \times X$ with the product topology. Then, every function $f : X \times X \rightarrow \mathbb{R}$ is continuous.
c.
Let $X$ be a discrete topological space having infinitely many elements. Let $f : \mathbb{R} \rightarrow X$ be a continuous function and $g : X \rightarrow \mathbb{R}$ be any non-constant function. Then the range of $g \circ f$ contains at least 2 elements.
d.
If a non-empty metric space $X$ has a finite dense subset, then there exists a discontinuous function $f : X \rightarrow \mathbb{R}$.
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Q41
mcq
Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a locally Lipschitz function. Consider the initial value problem
$$\dot{x} = f(t, x), x(t_0) = x_0$$
for $(t_0, x_0) \in \mathbb{R}^2$. Suppose that $J(t_0, x_0)$ represents the maximal interval of existence for the initial value problem. Which of the following statements is true ?
a.
$J(t_0, x_0) = \mathbb{R}$
b.
$J(t_0, x_0)$ is an open set
c.
$J(t_0, x_0)$ is a closed set
d.
$J(t_0, x_0)$ could be an empty set
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Q42
mcq
Suppose $x(t)$ is the solution of the following initial value problem in $\mathbb{R}^2$ ?
$$\dot{x} = Ax, \ x(0) = x_0, \quad \text{where } A = \begin{bmatrix} 5 & 4 \\ 1 & 2 \end{bmatrix}$$
Which of the following statements is true ?
a.
$x(t)$ is a bounded solution for some $x_0 \neq 0$
b.
$e^{-6t} |x(t)| \to 0$ as $t \to \infty$, for all $x_0 \neq 0$
c.
$e^{-6t} |x(t)| \to \infty$ as $t \to \infty$, for all $x_0 \neq 0$
d.
$e^{-10t} |x(t)| \to 0$ as $t \to \infty$, for all $x_0 \neq 0$
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Q43
mcq
Let $u(x, y)$ be the solution of the Cauchy problem.
$$uu_x + u_y = 0, \quad x \in \mathbb{R}, \ y > 0,$$
$$u(x, 0) = x, \quad x \in \mathbb{R}.$$
Which of the following is the value of $u(2, 3)$ ?
a.
$2$
b.
$3$
c.
$1/2$
d.
$1/3$
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Q44
mcq
Let $u(x, t)$ be the solution of
$$\begin{aligned} u_{tt} - u_{xx} &= 0, & 0 < x < 2, t > 0, \\ u(0, t) = 0 &= u(2, t), & \forall t > 0, \\ u(x, 0) &= \sin(\pi x) + 2\sin(2\pi x), & 0 \le x \le 2, \\ u_t(x, 0) &= 0, & 0 \le x \le 2. \end{aligned}$$
Which of the following is true?
a.
$u(1, 1) = -1$
b.
$u(1/2, 1) = 0$
c.
$u(1/2, 2) = 1$
d.
$u(1/2, 1/2) = \pi$
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Q45
mcq
Which of the following values of $a, b, c$ and $d$ will produce a quadrature formula
$$\int_{-1}^1 f(x)\,dx \approx af(-1) + bf(1) + cf'(-1) + df'(1)$$
that has degree of precision $3$ ?
a.
$a = 1, b = 1, c = \frac{1}{3}, d = -\frac{1}{3}$
b.
$a = -1, b = 1, c = \frac{1}{3}, d = -\frac{1}{3}$
c.
$a = 1, b = 1, c = -\frac{1}{3}, d = \frac{1}{3}$
d.
$a = 1, b = -1, c = \frac{1}{3}, d = -\frac{1}{3}$
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Q46
mcq
Consider the variational problem $(P)$
$$J(y(x)) = \int_0^1 \left[(y')^2 - y|y|y' + xy\right] dx, \quad y(0) = 0, \ y(1) = 0$$
Which of the following statements is correct ?
a.
$(P)$ has no stationary function (extremal)
b.
$y \equiv 0$ is the only stationary function (extremal) for $(P)$
c.
$(P)$ has a unique stationary function (extremal) y not identically equal to $0$
d.
$(P)$ has infinitely many stationary functions (extremal)
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Q47
mcq
For the unknown $y:[0,1] \rightarrow \mathbb{R}$, consider the following two-point boundary value problem:
$$\begin{cases} y''(x) + 2y(x) &= 0 \text{ for } x \in (0,1) \\ y(0) &= y(1) = 0 \end{cases}$$
It is given that the above boundary value problem corresponds to the following integral equation:
$$y(x) = 2 \int_0^1 K(x,t) y(t) dt \quad \text{for } x \in [0,1]$$
Which of the following is the kernel $K(x,t)$?
a.
$K(x,t) = \begin{cases} t(1-x) & \text{for } t < x \\ x(1-t) & \text{for } t > x \end{cases}$
b.
$K(x,t) = \begin{cases} t^2(1-x) & \text{for } t < x \\ x^2(1-t) & \text{for } t > x \end{cases}$
c.
$K(x,t) = \begin{cases} \sqrt{t}(1-x) & \text{for } t < x \\ \sqrt{x}(1-t) & \text{for } t > x \end{cases}$
d.
$K(x,t) = \begin{cases} \sqrt{t^3}(1-x) & \text{for } t < x \\ \sqrt{x^3}(1-t) & \text{for } t > x \end{cases}$
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Q48
mcq
Consider the constants $a$ and $b$ such that the following generalized coordinate transformation from $(p, q)$ to $(P, Q)$ is canonical.
$$Q = pq^{(a+1)}, \quad P = q^b$$
What are the values of $a$ and $b$?
a.
$a = -1, b = 0$
b.
$a = -1, b = 1$
c.
$a = 1, b = 0$
d.
$a = 1, b = -1$
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Q49
mcq
If $f(x)$ is a probability density on the real line, then which of the following is NOT a valid probability density?
a.
$f(x+1)$
b.
$f(2x)$
c.
$2f(2x-1)$
d.
$3x^2 f(x^3)$
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Q50
mcq
Which of the following is a valid cumulative distribution function?
a.
$F(x) = \begin{cases} \frac{1}{2+x^2} & \text{if } x < 0 \\ \frac{2+x^2}{3+x^2} & \text{if } x \ge 0 \end{cases}$
b.
$F(x) = \begin{cases} \frac{1}{2+x^2} & \text{if } x < 0 \\ \frac{2+x^2}{3+2x^2} & \text{if } x \ge 0 \end{cases}$
c.
$F(x) = \begin{cases} \frac{1}{2+x^2} & \text{if } x < 0 \\ \frac{2\cos(x)+x^2}{4+x^2} & \text{if } x \ge 0 \end{cases}$
d.
$F(x) = \begin{cases} \frac{1}{2+x^2} & \text{if } x < 0 \\ \frac{1+x^2}{4+x^2} & \text{if } x \ge 0 \end{cases}$
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Q61
mcq
Let $\{x_n\}$ be a sequence of positive real numbers. If $\sigma_n = \frac{1}{n}(x_1 + x_2 + \dots x_n)$, then which of the following are true? (Here $\limsup$ denotes the limit supremum of a sequence.)
a.
If $\limsup \{x_n\} = \ell$ and $\{x_n\}$ is decreasing, then $\limsup \{\sigma_n\} = \ell$
b.
$\limsup \{x_n\} = \ell$ if and only if $\limsup \{\sigma_n\} = \ell$
c.
If $\limsup \left\{ n\left(\frac{x_n}{x_{(n+1)}} - 1\right) \right\} < 1$, then $\sum_n x_n$ is convergent.
d.
If $\limsup \left\{ n\left(\frac{x_n}{x_{(n+1)}} - 1\right) \right\} < 1$, then $\sum_n x_n$ is divergent
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Q62
mcq
Under which of the following conditions is the sequence $\{x_n\}$ of real numbers convergent ?
a.
The subsequences $\{x_{(2n+1)}\}, \{x_{2n}\}$ and $\{x_{3n}\}$ are convergent and have the same limit.
b.
The subsequences $\{x_{(2n+1)}\}, \{x_{2n}\}$ and $\{x_{3n}\}$ are convergent.
c.
The subsequences $\{x_{kn}\}_n$ are convergent for every $k \ge 2$.
d.
$\lim_n |x_{(n+1)} - x_n| = 0$
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Q63
mcq
Which of the following are true?
a.
For $n \ge 1$, the sequence of functions $f_n: (0, 1) \rightarrow (0, 1)$ defined by $f_n(x) = x^n$ is uniformly convergent.
b.
For $n \ge 1$, the sequence of functions $f_n: (0, 1) \rightarrow (0, 1)$ defined by $f_n(x) = \frac{x^n}{\log(n+1)}$ is uniformly convergent
c.
For $n \ge 1$, the sequence of functions $f_n: (0, 1) \rightarrow (0, 1)$ defined by $f_n(x) = \frac{x^n}{1+x^n}$ uniformly convergent.
d.
For $n \ge 1$, the sequence of functions $f_n: (0, 1) \rightarrow (0, 1)$ defined by $f_n(x) = \frac{x^n}{1+nx^n}$ is not uniformly convergent
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Q64
mcq
Define a function $f: \mathbb{R} \rightarrow \mathbb{R}$ by
$$f(x) = \begin{cases} \sin(\pi / x) & \text{when } x \neq 0 \\ 0 & \text{when } x = 0 \end{cases}$$
On which of the following subsets of $\mathbb{R}$, the restriction of $f$ is a continuous function ?
a.
$[-1, 1]$
b.
$(0, 1)$
c.
$\{0\} \cup \{(1/n) : n \in \mathbb{N}\}$
d.
$\{1/2^n : n \in \mathbb{N}\}$
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Q65
mcq
Define $f : \mathbb{R}^4 \to \mathbb{R}$ by $f(x, y, z, w) = xw - yz$. Which of the following statements are true?
a.
$f$ is continuous
b.
if $U = \left\{(x, y, z, w) \in \mathbb{R}^4 : xy + zw = 0, \, x^2 + z^2 = 1, \, y^2 + w^2 = 1\right\}$ then $f$ is uniformly continuous on $U$.
c.
if $V = \left\{(x, y, z, w) \in \mathbb{R}^4 : x = y, \, z = w\right\}$, then $f$ is uniformly continuous on $V$.
d.
if $W = \left\{(x, y, z, w) \in \mathbb{R}^4 : 0 \le x + y + z + w \le 1\right\}$, then $f$ is unbounded on $W$.
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Q66
mcq
Consider the following statements:
(a) Let $f$ be a continuous function on $[1, \infty)$ taking non-negative values such that $\int_1^\infty f(x) \, dx$ converges. Then $\sum_{n \ge 1} f(n)$ converges.
(b) Let $f$ be a function on $[1, \infty)$ taking non-negative values such that $\int_1^\infty f(x) \, dx$ converges. Then $\lim_{x \to \infty} f(x) = 0$
(c) Let $f$ be a continuous, decreasing function on $[1, \infty)$ taking non-negative values such that $\int_1^\infty f(x) \, dx$ does not converge. Then $\sum_{n \ge 1} f(n)$ does not converge.
Which of the following options are true?
a.
(a), (b) and (c) all are true
b.
Both (a) and (b) are false.
c.
(c) is true
d.
(b) is true
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Q67
mcq
Let $\mu$ denote the Lebesgue measure on $\mathbb{R}$ and $\mu^*$ be the associated Lebesgue outer measure. Let $A$ be a non-empty subset of $[0, 1]$. Which of the following statements are true?
a.
If both interior and closure of $A$ have the same outer measure, then $A$ is Lebesgue measurable.
b.
If $A$ is open, then $A$ is Lebesgue measurable and $\mu(A) > 0$.
c.
If $A$ is not Lebesgue measurable, then the set of irrationals in $A$ must have positive outer measure.
d.
If $\mu^*(A) = 0$, then $A$ has empty interior.
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Q68
mcq
Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined by
$$f(x, y) = x^2 - y^3$$
Which of the following statements are true?
a.
There is no continuous real-valued function $g$ defined on any interval of $\mathbb{R}$ containing $0$ such that $f(x, g(x)) = 0$.
b.
There is exactly one continuous real-valued function $g$ defined on an interval of $\mathbb{R}$ containing $0$ such that $f(x, g(x)) = 0$.
c.
There is exactly one differentiable real-valued function $g$ defined on an interval of $\mathbb{R}$ containing $0$ such that $f(x, g(x)) = 0$.
d.
There are two distinct differentiable real-valued functions $g$ on an interval of $\mathbb{R}$ containing $0$ such that $f(x, g(x)) = 0$.
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Q75
mcq
Suppose A is a $5 \times 5$ block diagonal real matrix with diagonal blocks
$$\begin{pmatrix} e & 1 \\ 0 & e \end{pmatrix}, \begin{pmatrix} e & 1 & 0 \\ 0 & e & 0 \\ 0 & 0 & e \end{pmatrix}$$
Which of the following statements are true ?
a.
The algebraic multiplicity of $e$ in A is 5
b.
A is not diagonalisable
c.
The geometric multiplicity of $e$ in A is 3
d.
The geometric multiplicity of $e$ in A is 2
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Q76
mcq
Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation satisfying $T^3 - 3T^2 = -2I$, where $I : \mathbb{R}^3 \to \mathbb{R}^3$ is the identity transformation. Which of the following statements are true?
a.
$\mathbb{R}^3$ must admit a basis $B_1$ such that the matrix of $T$ with respect to $B_1$ is symmetric
b.
$\mathbb{R}^3$ must admit a basis $B_2$ such that the matrix of $T$ with respect to $B_2$ is upper triangular
c.
$\mathbb{R}^3$ must contain a non-zero vector $v$ such that $Tv = v$
d.
$\mathbb{R}^3$ must contain two linearly independent vectors $v_1, v_2$ such that $Tv_1 = v_1$ and $Tv_2 = v_2$
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Q77
mcq
Let $V$ be an inner product space and let $v_1, v_2, v_3 \in V$ be an orthogonal set of vectors. Which of the following statements are true?
a.
The vectors $v_1 + v_2 + 2v_3, v_2 + v_3, v_2 + 3v_3$ can be extended to a basis of $V$
b.
The vectors $v_1 + v_2 + 2v_3, v_2 + v_3, v_2 + 3v_3$ can be extended to an orthogonal basis of $V$
c.
The vectors $v_1 + v_2 + 2v_3, v_2 + v_3, 2v_1 + v_2 + 3v_3$ can be extended to a basis of $V$
d.
The vectors $v_1 + v_2 + 2v_3, 2v_1 + v_2 + v_3, 2v_1 + v_2 + 3v_3$ can be extended to a basis of $V$
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Q78
mcq
Consider the following quadratic forms over $\mathbb{R}$
(a) $6X^2 - 13XY + 6Y^2$
(b) $X^2 - XY + 2Y^2$
(c) $X^2 - XY - 2Y^2$.
Which of the following statements are true?
a.
Quadratic forms (a) and (b) are equivalent
b.
Quadratic forms (a) and (c) are equivalent
c.
Quadratic form (b) is positive definite
d.
Quadratic form (c) is positive definite
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Q79
mcq
Let $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$ be the open unit disc and $C$ the positively oriented boundary $\{|z| = 1\}$. Fix a finite set $\{z_1, z_2, \ldots, z_n\} \subseteq \mathbb{D}$ of distinct points and consider the polynomial $g(z) = (z - z_1)(z - z_2)\ldots(z - z_n)$ of degree n. Let f be a holomorphic function in an open neighbourhood of $\overline{\mathbb{D}}$ and define
$$P(z) = \frac{1}{2\pi i} \int_C f(\zeta) \frac{g(\zeta) - g(z)}{(\zeta - z)g(\zeta)} d\zeta$$
Which of the following statements are true ?
a.
$P$ is a polynomial of degree $n$
b.
$P$ is a polynomial of degree $n - 1$
c.
$P$ is a rational function on $\mathbb{C}$ with poles at $z_1, z_2, \dots, z_n$
d.
$P(z_j) = f(z_j)$ for $j = 1, 2, \dots, n$
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Q80
mcq
Let $D = \{z \in \mathbb{C} : |z| < 1\}$. Consider the following statements
(a) $f : D \to D$ be a holomorphic function. Suppose $\alpha, \beta$ are distinct complex numbers in $D$ such that $f(\alpha) = \alpha$ and $f(\beta) = \beta$. Then $f(z) = z$ for all $z \in D$.
(b) There does not exist a bijective holomorphic function from $D$ to the set of all complex numbers whose imaginary part is positive.
(c) $f : D \to D$ be a holomorphic function. Suppose $\alpha \in D$ be such that $f(\alpha) = \alpha$ and $f'(\alpha) = 1$. Then $f(z) = z$ for all $z \in D$
Which of the following options are true ?
a.
(a), (b) and (c) are all true
b.
(a) is true.
c.
Both (a) and (b) are false
d.
Both (a) and (c) are true.
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Q81
mcq
Let $f : \{z : |z| < 1\} \to \{z : |z| \le 1/2\}$ be a holomorphic function such that $f(0) = 0$. Which of the following statements are true?
a.
$|f(z)| \le |z|$ for all $z$ in $\{z : |z| < 1\}$
b.
$|f(z)| \le \left|\frac{z}{2}\right|$ for all $z$ in $\{z : |z| < 1\}$
c.
$|f(z)| \le 1/2$ for all $z$ in $\{z : |z| < 1\}$
d.
It is possible that $f(1/2) = 1/2$.
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Q82
mcq
Let $f(z)$ be an entire function on $\mathbb{C}$. Which of the following statements are true?
a.
$f(\overline{z})$ is an entire function
b.
$\overline{f(z)}$ is an entire function
c.
$\overline{f(\overline{z})}$ is an entire function
d.
$f(z) + f(\overline{z})$ is an entire function
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Q83
mcq
Which of the following statements are correct ?
a.
If $G$ is a group of order 244, then $G$ contains a unique subgroup of order 27
b.
If $G$ is a group of order 1694, then $G$ contains a unique subgroup of order 121
c.
There exists a group of order 154 which contains a unique subgroup of order 7
d.
There exists a group of order 121 which contains two subgroups of order 11
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Q84
mcq
Let $G$ be a group of order 2023. Which of the following statements are true?
a.
$G$ is an Abelian group
b.
$G$ is a cyclic group
c.
$G$ is a simple group
d.
$G$ is not a simple group
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Q85
mcq
Let $G_1$ and $G_2$ be two groups and $\varphi: G_1 \to G_2$ be a surjective group homomorphism. Which of the following statements are true?
a.
If $G_1$ is cyclic then $G_2$ is cyclic
b.
If $G_1$ is Abelian then $G_2$ is Abelian
c.
If $H$ is a subgroup of $G_1$ then $\varphi(H)$ is a subgroup of $G_2$
d.
If $N$ is a normal subgroup of $G_1$ then $\varphi(N)$ is a normal subgroup of $G_2$
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Q86
mcq
Let $n \ge 1$ be a positive integer and $S_n$, the symmetric group on $n$ symbols. Let $\Delta = \{(g, g): g \in S_n\}$. Which of the following statements are necessarily true?
a.
The map $f : S_n \times S_n \to S_n$ given by $f(a, b) = ab$ is a group homomorphism.
b.
$\Delta$ is a subgroup of $S_n \times S_n$.
c.
$\Delta$ is a normal subgroup of $S_n \times S_n$.
d.
$\Delta$ is a normal subgroup of $S_n \times S_n$, if $n$ is a prime number
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Q87
mcq
Which of the following are maximal ideals of $\mathbb{Z}[X]$?
a.
Ideal generated by $2$ and $(1 + X^2)$
b.
Ideal generated by $2$ and $(1 + X + X^2)$
c.
Ideal generated by $3$ and $(1 + X^2)$
d.
Ideal generated by $3$ and $(1 + X + X^2)$
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Q88
mcq
Let $E$ be a finite algebraic Galois extension of $F$ with Galois group $G$. Which of the following statements are true?
a.
There is an intermediate field $K$ with $K \ne F$ and $K \ne E$ such that $K$ is a Galois extension of $F$.
b.
If every proper intermediate field $K$ is a Galois extension of $F$ then $G$ is Abelian.
c.
If $E$ has exactly three intermediate fields including $F$ and $E$ then $G$ is Abelian
d.
If $[E : F] = 99$ then every intermediate field is a Galois extension of $F$
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Q89
mcq
Which of the following statements are correct?
a.
The set of open right half-planes is a basis for the usual (Euclidean) topology on $\mathbb{R}^2$?
b.
The set of lines parallel to Y-axis is a basis for the dictionary order topology on $\mathbb{R}^2$.
c.
The set of open rectangles is a basis for the usual (Euclidean) topology on $\mathbb{R}^2$?
d.
The set of line segments (without end points) parallel to Y-axis is a basis for the dictionary order topology on $\mathbb{R}^2$?
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Q90
mcq
Let $X = \prod_{n=1}^\infty [0,1]$, that is, the space of sequences $\{x_n\}_{n \ge 1}$ with $x_n \in [0, 1], n \ge 1$.
Define the metric $d : X \times X \to [0, \infty]$ by
$$d\left(\{x_n\}_{n \ge 1}, \{y_n\}_{n \ge 1}\right) = \sup_{n \ge 1} \frac{|x_n - y_n|}{2^n}$$
Which of the following statements are true?
a.
The metric topology on $X$ is finer than the product topology on $X$
b.
The metric topology on $X$ is coarser than the product topology on $X$
c.
The metric topology on $X$ is same as the product topology on $X$
d.
The metric topology on $X$ and the product topology on $X$ are not comparable.
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Q91
mcq
Let $f \in C^1(\mathbb{R})$ be bounded. Let us consider the initial-value problem
$$(P) \begin{cases} x'(t) = f(x(t)), & t > 0 \\ x(0) = 0 \end{cases}$$
Which of the following statements are true?
a.
$(P)$ has solution(s) defined for all $t > 0$
b.
$(P)$ has a unique solution.
c.
$(P)$ has infinitely many solutions
d.
The solution(s) of $(P)$ is/are Lipschitz
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Q92
mcq
Consider the following initial value problem (IVP)
$$\frac{du}{dt} = t^2 y^{1/5}, u(0) = 0$$
Which of the following statements are correct ?
a.
The function $g(t, u) = t^2 u^{1/5}$ does not satisfy the Lipschitz's condition with respect to $u$ in any neighbourhood of $u = 0$
b.
There is no solution for the IVP.
c.
There exist more than one solution for the IVP.
d.
The function $g(t, u) = t^2 u^{1/5}$ satisfies the Lipschitz's condition with respect to $u$ in some neighbourhood of $u = 0$ and hence there exists a unique solution for the IVP
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Q93
mcq
Let us consider the following two initial value problems
$$(P) = \begin{cases} x'(t) = \sqrt{x(t)}, & t > 0 \\ x(0) = 0, \end{cases}$$
and
$$(Q) = \begin{cases} y'(t) = -\sqrt{y(t)}, & t > 0 \\ y(0) = 0. \end{cases}$$
Which of the following statements are true ?
a.
(P) has a unique solution in $[0, \infty)$
b.
(Q) has a unique solution in $[0, \infty)$
c.
(P) has infinitely many solutions in $[0, \infty)$
d.
(Q) has infinitely many solutions in $[0, \infty)$
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Q94
mcq
Let $u: \mathbb{R}^2 \to \mathbb{R}$ be the solution to the Cauchy problem:
$$\begin{cases} \partial_x u + 2\partial_y u = 0 & \text{for } (x, y) \in \mathbb{R}^2 \\ u(x, y) = \sin(x) & \text{for } y = 3x + 1, x \in \mathbb{R} \end{cases}$$
Let $v: \mathbb{R}^2 \to \mathbb{R}$ be the solution to the Cauchy problem:
$$\begin{cases} \partial_x v + 2\partial_y v = 0 & \text{for } (x, y) \in \mathbb{R}^2 \\ v(x, 0) = \sin(x) & \text{for } x \in \mathbb{R} \end{cases}$$ Let $\delta = [0,1] \times [0,1]$
Which of the following statements are true ?
a.
$u$ changes sign in the interior of $\delta$
b.
$u(x, y) = v(x, y)$ along a line in $\delta$
c.
$v$ changes sign in the interior of $\delta$
d.
$v$ vanishes along a line in $\delta$
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Q95
mcq
Let $u = u(x, y)$ be the solution to the following Cauchy problem
$$u_x + u_y = e^u \text{ for } (x, y) \in \mathbb{R} \times \left(0, \frac{1}{e}\right) \text{ and } u(x, 0) = 1 \text{ for } x \in \mathbb{R}$$
Which of the following statements are true ?
a.
$u\left(\frac{1}{2e}, \frac{1}{2e}\right) = 1$
b.
$u_x\left(\frac{1}{2e}, \frac{1}{2e}\right) = 0$
c.
$u_y\left(\frac{1}{4e}, \frac{1}{4e}\right) = \log 4$
d.
$u_y\left(0, \frac{1}{4e}\right) = \frac{4e}{3}$
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Q96
mcq
Consider the following two sequences $\{a_n\}$ and $\{b_n\}$ given by
$$a_n = \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n}, \quad b_n = \frac{1}{n}$$
Which of the following statements are true?
a.
$\{a_n\}$ converges to $\log 2$, and has the same convergence rate as the sequence $\{b_n\}$
b.
$\{a_n\}$ converges to $\log 4$, and has the same convergence rate as the sequence $\{b_n\}$
c.
$\{a_n\}$ converges to $\log 2$, but does not have the same convergence rate as the sequence $\{b_n\}$.
d.
$\{a_n\}$ does not converge.
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Q97
mcq
Let $f : \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \frac{1}{4} + x - x^2$. Given $a \in \mathbb{R}$, let us define the sequence $\{x_n\}$ by $x_0 = a$ and $x_n = f(x_{n-1})$ for $n \ge 1$
Which of the following statements are true?
a.
If $a = 0$, then the sequence $\{x_n\}$ converges to $\frac{1}{2}$
b.
If $a = 0$, then the sequence $\{x_n\}$ converges to $-\frac{1}{2}$
c.
The sequence $\{x_n\}$ converges for every $a \in \left(-\frac{1}{2}, \frac{3}{2}\right)$ and it converges to $\frac{1}{2}$
d.
If $a = 0$, then the sequence $\{x_n\}$ does not converge.
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Q98
mcq
Suppose $y(x)$ is an extremal of the following functional
$$J(y(x)) = \int_0^1 \left( y(x)^2 - 4y(x)y'(x) + 4{y'(x)}^2 \right) dx$$
subject to $y(0) = 1$ and $y'(0) = 1/2$.
Which of the following statements are true?
a.
$y$ is a convex function.
b.
$y$ is a concave function
c.
$y(x_1 + x_2) = y(x_1) y(x_2)$ for all $x_1, x_2$ in $[0, 1]$
d.
$y(x_1 x_2) = y(x_1) + y(x_2)$ for all $x_1, x_2 \in [0, 1]$
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Q99
mcq
Let $y(x)$ and $z(x)$ be the stationary functions (extremals) of the variational problem
$$J(y(x), z(x)) = \int_0^1 \left[ (y')^2 + (z')^2 + y'z' \right] dx$$
subject to $y(0) = 1$, $y(1) = 0$, $z(0) = -1$, $z(1) = 2$
Which of the following statements are correct?
a.
$z(x) + 3y(x) = 2$ for $x \in [0, 1]$
b.
$3z(x) + y(x) = 2$ for $x \in [0, 1]$
c.
$y(x) + z(x) = 2x$ for $x \in [0, 1]$
d.
$y(x) + z(x) = x$ for $x \in [0, 1]$
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Q100
mcq
Let $\lambda_1 < \lambda_2$ be two real characteristic numbers for the following homogeneous integral equation:
$$\varphi(x) = \lambda \int_0^{2\pi} \sin(x+t)\varphi(t) dt$$
and let $\mu_1 < \mu_2$ be two real characteristic numbers for the following homogeneous integral equation:
$$\psi(x) = \mu \int_0^{\pi} \cos(x+t)\psi(t) dt$$
Which of the following statements are true?
a.
$\mu_1 < \lambda_1 < \lambda_2 < \mu_2$
b.
$\lambda_1 < \mu_1 < \mu_2 < \lambda_2$
c.
$|\mu_1 - \lambda_1| = |\mu_2 - \lambda_2|$
d.
$|\mu_1 - \lambda_1| = 2|\mu_2 - \lambda_2|$
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