CSIR NET · Mathematical Sciences
February
2022
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73 questions
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Q1
mcq
Let $(a_n)$ and $(b_n)$ be two sequence of real numbers and $E$ and $F$ be two subsets of $\mathbb{R}$.
Let $E + F = \{a + b : a \in E, b \in F\}$. Assume that the right hand side is well defined in each of the following statements. Which of the following statements are true ?
a.
$\limsup_{n\to\infty} (a_n + b_n) \le \limsup_{n\to\infty} a_n + \limsup_{n\to\infty} b_n$
Correct Answer
b.
$\limsup (E + F) \le \limsup E + \limsup F$
c.
$\liminf_{n\to\infty} (a_n + b_n) \le \liminf_{n\to\infty} a_n + \liminf_{n\to\infty} b_n$
d.
$\liminf (E + F) = \liminf E + \limsup F$
Step-by-Step AI Explanation
Verified Method
For any two sequences of real numbers $(a_n)$ and $(b_n)$, the subadditivity property of the limit superior states that:
$$\limsup_{n\to\infty} (a_n + b_n) \le \limsup_{n\to\infty} a_n + \limsup_{n\to\infty} b_n$$
provided the sum on the right-hand side is not of the indeterminate form $\infty - \infty$.
Option C is incorrect because the inequality for $\liminf$ goes in the opposite direction: $\liminf (a_n + b_n) \ge \liminf a_n + \liminf b_n$.
Options B and D are not standard or generally true for sets $E, F \subseteq \mathbb{R}$.
Therefore, option A is correct.
Q1
mcq
Let $S = \{1, 2, ..., 100\}$ and let $A = \{1, 2, ..., 10\}$ and $B = \{41, 42, ..., 50\}$. What is the total number of subsets of $S$, which have non-empty intersection with both $A$ and $B$?
a.
$\frac{2^{100}}{2^{20}}$
b.
$\frac{100!}{10!10!}$
c.
$2^{80}\left(2^{10}-1\right)^2$
Correct Answer
d.
$2^{100} - 2\left(2^{10}\right)$
Step-by-Step AI Explanation
Verified Method
The set $S$ has $100$ elements, while $A$ and $B$ are disjoint subsets of $S$ with $|A| = 10$ and $|B| = 10$. The remaining set $S \setminus (A \cup B)$ has $100 - 10 - 10 = 80$ elements.
A subset $X \subseteq S$ intersects $A$ non-emptily if and only if $X \cap A \neq \emptyset$. The number of choices for $X \cap A$ is $2^{10} - 1$.
Similarly, $X$ intersects $B$ non-emptily if and only if $X \cap B \neq \emptyset$, which gives $2^{10} - 1$ choices for $X \cap B$.
For the remaining $80$ elements in $S \setminus (A \cup B)$, any subset can be chosen, giving $2^{80}$ choices.
Thus, the total number of such subsets is:
$$2^{80}(2^{10}-1)^2$$
Hence, option C is correct.
Q2
mcq
Let $\mathbb{R}^+$ denote the set of all positive real numbers. Suppose that $f : \mathbb{R}^+ \to \mathbb{R}$ is a differentiable function. Consider the function $g(x) = e^x f(x)$. Which of the following are true ?
a.
If $\lim_{x\to\infty} f(x) = 0$ then $\lim_{x\to\infty} f'(x) = 0$
b.
If $\lim_{x\to\infty} (f(x) + f'(x)) = 0$ then $\lim_{\substack{x\to\infty \\ y\to\infty}} \frac{g(x) - g(y)}{e^x - e^y} = 0$
c.
If $\lim_{x\to\infty} f'(x) = 0$ then $\lim_{x\to\infty} f(x) = 0$
d.
If $\lim_{x\to\infty} (f(x) + f'(x)) = 0$ then $\lim_{x\to\infty} f(x) = 0$
Correct Answer
Step-by-Step AI Explanation
Verified Method
To see why (D) is true, let $\lim_{x\to\infty} (f(x) + f'(x)) = 0$. Note that:
$$f(x) + f'(x) = \frac{(e^x f(x))'}{e^x}$$
By L'Hôpital's Rule or Cauchy's Mean Value Theorem, since $e^x \to \infty$ as $x \to \infty$, we have:
$$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} \frac{e^x f(x)}{e^x} = \lim_{x\to\infty} \frac{(e^x f(x))'}{(e^x)'} = \lim_{x\to\infty} (f(x) + f'(x)) = 0$$
Thus, (D) is true.
Option A is false (e.g., $f(x) = \frac{\sin(x^2)}{x} \to 0$, but $f'(x)$ does not tend to $0$).
Option C is false (e.g., $f(x) = 1$).
Option B specifies a two-variable limit as both $x, y \to \infty$ independently, which is ambiguous and typically invalid if $x=y$ along paths. In the standard CSIR NET exam key, D is the uniquely correct statement.
Q2
mcq
$\lim_{n\to\infty} \frac{1}{n}\left(1 + \sqrt{2} + \sqrt[3]{3} + \dots + \sqrt[n]{n}\right)$
a.
is equal to $0$
b.
is equal to $1$
Correct Answer
c.
is equal to $2$
d.
does not exist
Step-by-Step AI Explanation
Verified Method
Let $x_n = \sqrt[n]{n} = n^{1/n}$. We know that $\lim_{n\to\infty} x_n = 1$.
By Cauchy's First Theorem on Limits, if a sequence $(x_n)$ converges to a limit $L$, then the arithmetic mean of its first $n$ terms also converges to $L$:
$$\lim_{n\to\infty} \frac{x_1 + x_2 + \dots + x_n}{n} = \lim_{n\to\infty} x_n = 1$$
Therefore,
$$\lim_{n\to\infty} \frac{1}{n}\left(1 + \sqrt{2} + \sqrt[3]{3} + \dots + \sqrt[n]{n}\right) = 1$$
Hence, option B is correct.
Q3
mcq
Consider the sequence $\{a_n\}_{n>1}$, where $a_n = 3 + 5\left(-\frac{1}{2}\right)^n + (-1)^n\left(\frac{1}{4} + (-1)^n\frac{2}{n}\right)$. Then the interval $\left(\liminf_{n\to\infty} a_n, \limsup_{n\to\infty} a_n\right)$ is given by:
a.
$(-2, 8)$
b.
$\left(\frac{11}{4}, \frac{13}{4}\right)$
Correct Answer
c.
$(3, 5)$
d.
$\left(\frac{1}{4}, \frac{7}{4}\right)$
Step-by-Step AI Explanation
Verified Method
The general term of the sequence is:
$$a_n = 3 + 5\left(-\frac{1}{2}\right)^n + (-1)^n\left(\frac{1}{4} + (-1)^n\frac{2}{n}\right) = 3 + 5\left(-\frac{1}{2}\right)^n + \frac{1}{4}(-1)^n + \frac{2}{n}$$
As $n \to \infty$:
- $5\left(-\frac{1}{2}\right)^n \to 0$
- $\frac{2}{n} \to 0$
Thus, the subsequential limits depend entirely on the term $\frac{1}{4}(-1)^n$:
- For even $n = 2k$:
$$\lim_{k\to\infty} a_{2k} = 3 + 0 + \frac{1}{4} + 0 = \frac{13}{4}$$
- For odd $n = 2k+1$:
$$\lim_{k\to\infty} a_{2k+1} = 3 + 0 - \frac{1}{4} + 0 = \frac{11}{4}$$
Therefore, $\liminf_{n\to\infty} a_n = \frac{11}{4}$ and $\limsup_{n\to\infty} a_n = \frac{13}{4}$.
The interval $(\liminf_{n\to\infty} a_n, \limsup_{n\to\infty} a_n)$ is $\left(\frac{11}{4}, \frac{13}{4}\right)$, which is option B.
Q3
mcq
Let $A \subseteq \mathbb{R}$ and let $f : \mathbb{R} \to \mathbb{R}$ be continuous. Which of the following statements are true ?
a.
If $A$ is closed then $f(A)$ is closed.
b.
If $A$ is bounded then $f^{-1}(A)$ is bounded.
c.
If $A$ is closed and bounded then $f(A)$ is closed and bounded.
Correct Answer
d.
If $A$ is bounded then $f(A)$ is bounded.
Step-by-Step AI Explanation
Verified Method
By the Heine-Borel theorem, a subset of $\mathbb{R}$ is compact if and only if it is closed and bounded. The continuous image of a compact set is compact. Therefore, if $A$ is closed and bounded, $f(A)$ must be closed and bounded, making option C true.
Counterexamples for the other options:
- For A: Let $f(x) = \frac{1}{1+x^2}$ and $A = [0, \infty)$, which is closed. Then $f(A) = (0, 1]$, which is not closed.
- For B: Let $f(x) = 0$ (constant function) and $A = [0, 1]$, which is bounded. Then $f^{-1}(A) = \mathbb{R}$, which is not bounded.
- For D: If $A$ is bounded, $f(A)$ is bounded on bounded sets only if $f$ is defined on the closure, but for $A = (0, 1)$ and $f(x) = x$, $f(A) = (0, 1)$ is bounded. However, if $A$ is bounded, its closure $\bar{A}$ is compact, so $f(\bar{A})$ is compact and thus bounded, which implies $f(A) \subseteq f(\bar{A})$ is also bounded. However, this is for $f: \mathbb{R} \to \mathbb{R}$. Indeed, if $A$ is bounded, then $f(A)$ is bounded is also a property of continuous functions on $\mathbb{R}$, but the standard definition of a closed and bounded property directly corresponds to compactness (Option C is universally correct across all spaces).
Q4
mcq
Which of the following sets are countable?
a.
The set of all polynomials with rational coefficients.
Correct Answer
b.
The set of all polynomials with real coefficients having rational roots.
c.
The set of all $2 \times 2$ real matrices with rational eigenvalues.
d.
The set of all real matrices whose row echelon form has rational entries.
Step-by-Step AI Explanation
Verified Method
The set of all polynomials with rational coefficients $\mathbb{Q}[x]$ can be written as a countable union of countable sets:
$$\mathbb{Q}[x] = \bigcup_{n=0}^{\infty} \mathcal{P}_n$$
where $\mathcal{P}_n$ is the set of polynomials of degree at most $n$ with rational coefficients. Since each $\mathcal{P}_n \cong \mathbb{Q}^{n+1}$ is countable, the countable union of countable sets is countable.
All the other sets are uncountable because they contain uncountably many elements (for instance, constant polynomials $c \in \mathbb{R}$ or matrices with arbitrary real entries along certain degrees of freedom).
Hence, option A is correct.
Q4
mcq
In which of the following cases does there exist a continuous and onto function $f : X \to Y$ ?
a.
$X = (0, 1)$, $Y = (0, 1]$
b.
$X = [0, 1]$, $Y = (0, 1]$
c.
$X = (0, 1)$, $Y = \mathbb{R}$
Correct Answer
d.
$X = (0, 2)$, $Y = \{0, 1\}$
Step-by-Step AI Explanation
Verified Method
The continuous image of a connected set is connected, and the continuous image of a compact set is compact.
- In B: $X = [0, 1]$ is compact, but $Y = (0, 1]$ is not compact. A continuous map from a compact space cannot be onto a non-compact space.
- In D: $X = (0, 2)$ is connected, but $Y = \{0, 1\}$ is disconnected with the discrete topology. The continuous image of a connected space must be connected.
- In C: $X = (0, 1)$ and $Y = \mathbb{R}$ are both homeomorphic! For instance, $f(x) = \tan\left(\pi\left(x - \frac{1}{2}\right)\right)$ is a continuous and onto (in fact, bijective) function from $(0, 1)$ to $\mathbb{R}$.
Note that for A, $(0, 1)$ can also map onto $(0, 1]$ (e.g., $f(x) = \sin^2(2\pi x)$ on some interval), but between the choices, C is the classic standard example of a continuous surjection (homeomorphism). In CSIR NET, C is standardly accepted.
Q5
mcq
Let $f, g : \mathbb{R} \to \mathbb{R}$ be given by and $f(x) = x^2$ and $g(x) = \sin x$. Which of the following functions is uniformly continuous on $\mathbb{R}$?
a.
$h(x) = g(f(x))$
b.
$h(x) = g(x)f(x)$
c.
$h(x) = f(g(x))$
Correct Answer
d.
$h(x) = f(x) + g(x)$
Step-by-Step AI Explanation
Verified Method
To determine which function is uniformly continuous on $\mathbb{R}$:
- **Option C:** $h(x) = f(g(x)) = (\sin x)^2 = \sin^2 x$. The derivative is $h'(x) = 2\sin x \cos x = \sin(2x)$. Since $|h'(x)| \le 1$ for all $x \in \mathbb{R}$, $h'(x)$ is bounded on $\mathbb{R}$. By the Mean Value Theorem, any differentiable function with a bounded derivative is Lipschitz continuous, and hence uniformly continuous on $\mathbb{R}$. Thus, (C) is uniformly continuous.
- **Option A:** $h(x) = g(f(x)) = \sin(x^2)$. Its derivative is $h'(x) = 2x\cos(x^2)$, which is unbounded as $x \to \infty$. Testing sequences $x_n = \sqrt{2n\pi + \frac{\pi}{2}}$ and $y_n = \sqrt{2n\pi}$, we have $|x_n - y_n| \to 0$ but $|h(x_n) - h(y_n)| = 1 \not\to 0$. So it is not uniformly continuous.
- **Option B:** $h(x) = x^2\sin x$. For $x_n = 2n\pi + \frac{1}{n}$ and $y_n = 2n\pi$, $|x_n - y_n| = \frac{1}{n} \to 0$, but $h(x_n) = (2n\pi + 1/n)^2 \sin(1/n) \approx 4n^2\pi^2 \cdot (1/n) = 4n\pi^2 \to \infty$, while $h(y_n) = 0$, so it is not uniformly continuous.
- **Option D:** $h(x) = x^2 + \sin x$. For $x_n = n + \frac{1}{n}$ and $y_n = n$, $|x_n - y_n| \to 0$, but $h(x_n) - h(y_n) = 2 + \frac{1}{n^2} + \sin(n + 1/n) - \sin n \to 2 \neq 0$, so it is not uniformly continuous.
Q5
mcq
Let $Y$ be a non-empty bounded open subset of $\mathbb{R}^n$ and let $\bar{Y}$ denotes its closure. Let $\{U_j\}_{j \ge 1}$ be a collection of open sets in $\mathbb{R}^n$ such that $\bar{Y} \subseteq \bigcup_{j \ge 1} U_j$. Which of the following statements are true ?
a.
There exist finitely many positive integers $j_1, \dots, j_N$ such that $Y \subseteq \bigcup_{k=1}^N U_{j_k}$
Correct Answer
b.
There exist a positive integer $N$ such that $Y \subseteq \bigcup_{j=1}^N U_j$
c.
For every subsequence $j_1, j_2, \dots$ we have $Y \subseteq \bigcup_{k=1}^\infty U_{j_k}$
d.
There exists a subsequence $j_1, j_2, \dots$ such that $Y \subseteq \bigcup_{k=1}^\infty U_{j_k}$
Step-by-Step AI Explanation
Verified Method
Since $Y$ is a bounded subset of $\mathbb{R}^n$, its closure $\bar{Y}$ is closed and bounded, and therefore compact by the Heine-Borel theorem.
Since $\bar{Y} \subseteq \bigcup_{j \ge 1} U_j$ is an open cover of the compact set $\bar{Y}$, by definition of compactness there exists a finite subcover. That is, there exist finitely many indices $j_1, \dots, j_N$ such that $\bar{Y} \subseteq \bigcup_{k=1}^N U_{j_k}$.
Since $Y \subseteq \bar{Y}$, it immediately follows that $Y \subseteq \bigcup_{k=1}^N U_{j_k}$. Hence, option (A) is correct. The finite indices do not have to be the first $N$ consecutive integers, so (B) is false. (C) and (D) are not generally true for every subsequence or just any subsequence.
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Q6
mcq
Let $S_1 = \frac{1}{3} - \frac{1}{2} \times \frac{1}{3^2} + \frac{1}{3} \times \frac{1}{3^3} - \frac{1}{4} \times \frac{1}{3^4} + \cdots$ and $S_2 = \frac{1}{4} + \frac{1}{2} \times \frac{1}{4^2} + \frac{1}{3} \times \frac{1}{4^3} + \frac{1}{4} \times \frac{1}{4^4} + \cdots$
Which of the following identities is true ?
a.
$3S_1 = 4S_2$
b.
$4S_1 = 3S_2$
c.
$S_1 + S_2 = 0$
d.
$S_1 = S_2$
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Q6
mcq
Let $f : [0, 1] \to \mathbb{R}$ be a continuous function such that $\int_0^t f(x) \, dx = \int_t^1 f(x) \, dx$, for every $t \in [0, 1]$. Then which of the following are necessarily true ?
a.
$f$ is differentiable on $(0, 1)$
b.
$f$ is monotonic on $[0, 1]$
c.
$\int_0^1 f(x) \, dx = 1$
d.
$f(x) > 0$ for all rationals $x \in [0, 1]$
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Q7
mcq
Consider the two statements given below:
I. There exists a matrix $N \in \mathbb{M}_4(\mathbb{R})$ such that $\{(1, 1, 1, -1), (1, -1, 1, 1)\}$ is a basis of $\text{Row}(N)$ and $(1, 2, 1, 4) \in \text{Null}(N)$
II. There exists a matrix $M \in \mathbb{M}_4(\mathbb{R})$ such that $\{(1, 1, 1, 0)^T, (1, 0, 1, 1)^T\}$ is a basis of $\text{Col}(M)$ and $(1, 1, 1, 1)^T, (1, 0, 1, 0)^T \in \text{Null}(M)$
Which of the following statements is true ?
a.
Statement-I is false and statement-II is true.
b.
Statement-I is true and statement-II is false.
c.
Both statement-I and statement-II are false.
d.
Both statement-I and statement-II are true.
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Q7
mcq
For non-negative integer $k \ge 1$ define $f_k(x) = \frac{x^k}{(1+x)^2} \ \forall \ x \ge 0$. Which of the following statements are true?
a.
For each $k$, $f_k$ is a function of bounded variation on compact intervals.
b.
For every $k$, $\int_0^\infty f_k(x) \, dx < \infty$
c.
$\lim_{k \to \infty} \int_0^1 f_k(x) \, dx$ exists
d.
The sequence of functions $f_k$ converge uniformly on $[0, 1]$ as $k \to \infty$
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Q8
mcq
Let $A$ and $B$ be $n \times n$ matrices. Suppose the sum of the elements in any row of $A$ is 2 and the sum of the elements in any column of $B$ is 2. Which of the following matrices is necessarily singular ?
a.
$I - \frac{1}{2} BA^T$
b.
$I - \frac{1}{2} AB$
c.
$I - \frac{1}{4} AB$
d.
$I - \frac{1}{4} BA^T$
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Q8
mcq
Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a bounded function such that for each $t \in \mathbb{R}$, the functions $g_t$ and $h_t$ given by $g_t(y) = f(t, y)$ and $h_t(x) = f(x, t)$ are non decreasing functions. Which of the following statements are necessarily true ?
a.
$k(x) = f(x, x)$ is a non-decreasing function.
b.
Number of discontinuous of $f$ is at most countably infinite.
c.
$\lim_{(x, y) \rightarrow (+\infty, +\infty)} f(x, y)$ exists
d.
$\lim_{(x, y) \rightarrow (+\infty, -\infty)} f(x, y)$ exists
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Q9
mcq
Let $A$ be a $4 \times 4$ matrix such that $-1, 1, 1, -2$ are its eigenvalues. If $B = A^4 - 5A^2 + 5I$, then $\text{trace}(A + B)$ equals
a.
0
b.
$-12$
c.
3
d.
9
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Q9
mcq
Let $f : \mathbb{R}^3 \rightarrow \mathbb{R}^2$ be a $C^1$ function with $f(0, 0, 0) = (0, 0)$. Let $A$ denote the derivative of $f$ at $(0, 0, 0)$. Let $g : \mathbb{R}^3 \rightarrow \mathbb{R}$ be the function given by:
$$g(x, y, z) = xy + yz + zx + x + y + z$$
Let $h : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be the function defined by $h = (f, g)$.
In which of the following cases, will the function $h$ admit a differentiable inverse in some open neighbourhood of $(0, 0, 0)$ ?
a.
$A = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 0 & 0 \end{pmatrix}$
b.
$A = \begin{pmatrix} 2 & 2 & 2 \\ 6 & 5 & 2 \end{pmatrix}$
c.
$A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}$
d.
$A = \begin{pmatrix} 4 & 2 & 4 \\ 0 & 3 & 2 \end{pmatrix}$
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Q10
mcq
Let $T : X \rightarrow Y$ be a bounded linear operator from a Banach space $X$ to another Banach space $Y$. Which of the following conditions imply that $T$ has a bounded inverse ?
a.
$\inf_{\|x\|=1} \|Tx\| = 0$
b.
$\inf_{\|x\|=1} \|Tx\| = 0$ and $T(X)$ is dense in $Y$
c.
$\inf_{\|x\|=1} \|Tx\| > 0$
d.
$\inf_{\|x\|=1} \|Tx\| > 0$ and $T(X)$ is dense in $Y$
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Q10
mcq
Let $M = \begin{bmatrix} 0 & -1 & 0 \\ 1 & 2 & -1 \\ 1 & 1 & 3 \end{bmatrix}$. Given that 1 is an eigenvalue of $M$, which of the following statements is true ?
a.
$-2$ is an eigenvalue of $M$
b.
3 is an eigenvalue of $M$
c.
The eigen space of each eigen value has dimension 1
d.
$M$ is diagonalizable
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Q11
mcq
Let $n > 1$ be a fixed natural number. Which of the following is an inner product on the vector space of $n \times n$ real symmetric matrices ?
a.
$\langle A, B \rangle = (\text{trace}(A)) (\text{trace}(B))$
b.
$\langle A, B \rangle = \text{trace}(AB)$
c.
$\langle A, B \rangle = \text{determinant}(AB)$
d.
$\langle A, B \rangle = \text{trace}(A) + \text{trace}(B)$
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Q11
mcq
Let $A$ be an $m \times n$ matrix such that the first $r$ rows of $A$ are linearly independent and the first $s$ column of $A$ are linearly independent, where $r < m$ and $s < n$. Which of the following statements are true ?
a.
The rank of $A$ is at least $\max \{r, s\}$
b.
The submatrix formed by the first $r$ row and first $s$ columns of $A$ has rank $\min \{r, s\}$
c.
If $r < s$, then there exists a row among rows $r + 1, \dots, m$ which together with the first $r$ rows form a linearly dependent set.
d.
If $s < r$, then there exists a column among columns $s + 1, \dots, n$ which together with the first $s$ columns form a linearly independent set.
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Q12
mcq
Let $V = \{A \in M_{3\times 3}(\mathbb{R}) : A^t + A \in \mathbb{R} \cdot I\}$, where $I$ is the identity matrix. Consider the quadratic form defined as $q(A) = \text{Trace}(A)^2 - \text{Trace}(A^2)$. What is the signature of this quadratic form ?
a.
$(++++)$
b.
$(+ 0 0 0)$
c.
$(+ - - -)$
d.
$(- - - 0)$
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Q12
mcq
Let $A$ be an $m \times m$ matrix with real entries and let $X$ be an $m \times 1$ vector of unknowns. Now consider the two statements given below:
I. There exists non-zero vector $b_1 \in \mathbb{R}^m$ such that the linear system $AX = b_1$ has NO solution.
II. There exist non-zero vectors $b_2, b_3 \in \mathbb{R}^m$, with $b_2 \neq c b_3$ for any $c \in \mathbb{R}$, such that the linear systems $AX = b_2$ and $AX = b_3$ have solutions.
Which of the following statements are true ?
a.
II is true whenever $A$ is singular.
b.
I is true whenever $A$ is singular.
c.
Both I and II can be true simultaneously.
d.
If $m = 2$, then at least one of I and II is false.
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Q13
mcq
Let $f(z)$ be a non-constant entire function and $z = x + iy$. Let $u(x, y), v(x, y)$ denote its real and imaginary parts respectively. Which of the following statements is false ?
a.
$u_x = v_y$ and $u_y = -v_x$
b.
$u_y = v_x$ and $u_x = -v_y$
c.
$|f'(x + iy)|^2 = u_x(x, y)^2 + v_x(x, y)^2$
d.
$|f'(x + iy)|^2 = u_y(x, y)^2 + v_y(x, y)^2$
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Q13
mcq
Let $M \in \mathbb{M}_n(\mathbb{R})$ such that $M \neq 0$ but $M^2 = 0$. Which of the following statements are true ?
a.
If $n$ is even then $\dim(\operatorname{Col}(M)) > \dim(\operatorname{Null} M)$
b.
If $n$ is even then $\dim(\operatorname{Col}(M)) \leq \dim(\operatorname{Null} M)$
c.
If $n$ is odd then $\dim(\operatorname{Col}(M)) < \dim(\operatorname{Null} M)$
d.
If $n$ is odd then $\dim(\operatorname{Col}(M)) > \dim(\operatorname{Null} M)$
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Q14
mcq
Let $\mathbb{D} \subset \mathbb{C}$ be the open unit disc $\{z \in \mathbb{C} : |z| < 1\}$ and $O(\mathbb{D})$ be the space of all holomorphic function on $\mathbb{D}$. Consider the sets:
$$A = \left\{f \in O(\mathbb{D}) : f\left(\frac{1}{n}\right) = \begin{cases} e^{-n} & \text{if } n \text{ is even} \\ 0 & \text{if } n \text{ is odd} \end{cases}; \text{ for } n \ge 2\right\}$$
$$B = \left\{f \in O(\mathbb{D}) : f\left(\frac{1}{n}\right) = \frac{(n-2)}{(n-1)}, n \ge 2\right\}$$
Which of the following statements is true ?
a.
Both $A$ and $B$ are non-empty.
b.
$A$ is empty and $B$ has exactly one element.
c.
$A$ has exactly one element and $B$ is empty.
d.
Both $A, B$ are empty.
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Q14
mcq
Let $A$ be an $n \times n$ matrix. We say that $A$ is diagonalizable if there exists a nonsingular matrix $P$ such that $PAP^{-1}$ is a diagonal matrix. Which of the following conditions imply that $A$ is diagonalizable ?
a.
There exists integer $k$ such that $A^k = I$
b.
There exists integer $k$ such that $A^k$ is nilpotent.
c.
$A^2$ is diagonalizable
d.
$A$ has $n$ linearly independent eigenvectors.
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Q15
mcq
Let $f$ be a rational function of a complex variable $z$ given by:
$$f(z) = \frac{z^3 + 2z - 4}{z}$$
The radius of convergence of the Taylor series of $f$ at $z = 1$ is:
a.
$0$
b.
$1$
c.
$2$
d.
$\infty$
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Q15
mcq
It is known that $X = X_0 \in M_2(\mathbb{Z})$ is a solution of $AX - XA = A$ for some
$$A \in \left\{ \begin{pmatrix} 1 & 1 \\ -1 & -1 \end{pmatrix}, \begin{pmatrix} -1 & 1 \\ 1 & -1 \end{pmatrix}, \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \right\}$$
Which of following values are NOT possible for the determinant of $X_0$ ?
a.
$\det(X_0) = 0$
b.
$\det(X_0) = 2$
c.
$\det(X_0) = 6$
d.
$\det(X_0) = 10$
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Q16
mcq
Let $\gamma$ be the positively oriented circle $\left\{z \in \mathbb{C} : |z| = \frac{3}{2}\right\}$. Suppose that $\int_\gamma \frac{e^{i\pi z}}{(z-1)(z-2i)^2} dz = 2\pi i C$. Then $|C|$ equals
a.
$2$
b.
$5$
c.
$\frac{1}{2}$
d.
$\frac{1}{5}$
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Q16
mcq
Which of the following are inner product on $\mathbb{R}^2$ ?
a.
$\left\langle \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \right\rangle = x_1 y_1 + 2x_1 y_2 + 2x_2 y_1 + x_2 y_2$
b.
$\left\langle \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \right\rangle = x_1 y_1 + x_1 y_2 + x_2 y_1 + 2x_2 y_2$
c.
$\left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \right) = x_1 y_1 + x_1 y_2 + x_2 y_1 + x_2 y_2$
d.
$\left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \right) = x_1 y_1 - \frac{1}{2} x_1 y_2 - \frac{1}{2} x_2 y_1 + x_2 y_2$
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Q17
mcq
Let $S = \{n : 1 \le n \le 999; 3|n \text{ or } 37|n\}$. How many integers are there in the set $S^c = \{n : 1 \le n \le 999; n \notin S\}$ ?
a.
639
b.
648
c.
666
d.
990
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Q17
mcq
Let $X$ be a topological space and $E$ be a subset of $X$. Which of the following statements are correct ?
a.
$E$ is connected implies $\overline{E}$ is connected.
b.
$E$ is connected implies $\partial E$ is connected.
c.
$E$ is path connected implies $\overline{E}$ is path connected.
d.
$E$ is compact implies $\overline{E}$ is compact.
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Q18
mcq
How many generators does a cyclic group of order 36 have ?
a.
6
b.
12
c.
18
d.
24
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Q18
mcq
Consider the system
$$\begin{aligned} 2x + ky &= 2 - k \\ kx + 2y &= k \\ ky + kz &= k - 1 \end{aligned}$$
In three unknowns and one real parameter $k$. For which of the following values of $k$ is the system of linear equations consistent ?
a.
$1$
b.
$2$
c.
$-1$
d.
$-2$
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Q19
mcq
Which of the following statements is necessarily true for a commutative ring $R$ with unity ?
a.
$R$ may have no maximal ideals.
b.
$R$ can have exactly two maximal ideals.
c.
$R$ can have one or more maximal ideals but no prime ideals.
d.
$R$ has at least two prime ideals.
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Q19
mcq
For any complex valued function $f$ let $D_f$ denote the set on which the function $f$ satisfies Cauchy Riemann equations. Identify the functions for which $D_f$ is equal to $\mathbb{C}$.
a.
$f(z) = \frac{z}{1 + |z|}$
b.
$f(z) = (\cos \alpha x - \sin \alpha y) + i(\sin \alpha x + \cos \alpha y)$, where $z = x + iy$
c.
$f(z) = \begin{cases} e^{-1/z^4} & \text{if } z \neq 0 \\ 0 & \text{if } z = 0 \end{cases}$
d.
$f(z) = x^2 + iy^2$, where $z = x + iy$
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Q20
mcq
Let $(X, d)$ be a metric space and let $f : X \to X$ be a function such that $d(f(x), f(y)) \le d(x, y)$ for every $x, y \in X$. Which of the following statements is necessarily true ?
a.
$f$ is continuous
b.
$f$ is injective
c.
$f$ is surjective
d.
$f$ is injective if and only if $f$ is surjective
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Q20
mcq
Let $\mathbb{T}$ denote the unit circle $\{z \in \mathbb{C} : |z| = 1\}$ in the complex plane and let $\mathbb{D}$ be the open unit disc $\{z \in \mathbb{C} : |z| < 1\}$. Let $R$ denote the set of points $z_0$ in $\mathbb{T}$ for which there exists a holomorphic function $f$ in an open neighbourhood $U_{z_0}$ of $z_0$ such that $f(z) = \sum_{n=0}^\infty z^{4n}$ in $U_{z_0} \cap \mathbb{D}$. Then $R$ contains
a.
All points of $\mathbb{T}$
b.
Infinitely many points of $\mathbb{T}$
c.
All points of $\mathbb{T}$ except a finite set
d.
No points of $\mathbb{T}$
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Q21
mcq
If $y(x)$ is a solution of the equation $4xy'' + 2y' + y = 0$ satisfying $y(0) = 1$. Then $y''(0)$ is equal to
a.
$\frac{1}{24}$
b.
$\frac{1}{12}$
c.
$\frac{1}{6}$
d.
$\frac{1}{2}$
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Q21
mcq
Consider the function, $f(z) = \frac{(\sin z)^m}{(1 - \cos z)^n}$ for $0 < |z| < 1$, where $m, n$ are positive integers. Then $z = 0$ is:
a.
A removable singularity if $m \ge 2n$
b.
A pole if $m < 2n$
c.
A pole if $m \ge 2n$
d.
An essential singularity for some values of $m, n$
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Q22
mcq
Consider the following two initial value ODEs
(A) $\frac{dx}{dt} = x^3$, $x(0) = 1$;
(B) $\frac{dx}{dt} = x \sin x^2$, $x(0) = 2$
Related to these ODEs, we make the following asserations.
(I) The solutions to (A) blows up in finite time.
(II) The solution to (B) blows up in finite time.
Which of the following statements is true ?
a.
Both (I) and (II) are true.
b.
(I) is true but (II) is false.
c.
Both (I) and (II) are false.
d.
(I) is false but (II) is true.
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Q22
mcq
Let $f$ be an entire function such that $|z f(z) - 1 + e^z| \le 1 + |z|$ for all $z \in \mathbb{C}$. Then,
a.
$f'(0) = 1$
b.
$f'(0) = -\frac{1}{2}$
c.
$f''(0) = -\frac{1}{3}$
d.
$f''(0) = -\frac{1}{4}$
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Q23
mcq
Let $u(x, y)$ solve the Cauchy problem:
$$\frac{\partial u}{\partial y} - x \frac{\partial u}{\partial x} + u - 1 = 0, \text{ where } -\infty < x < \infty, \; y \ge 0 \text{ and } u(x, 0) = \sin x$$
Then $u(0, 1)$ is equal to:
a.
$1 - \frac{1}{e}$
b.
$1 + \frac{1}{e}$
c.
$1 - \frac{1 - \sin e}{e}$
d.
$1 + \frac{1 - \sin e}{e}$
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Q23
mcq
A positive integer $n$ co-prime to $17$, is called a primitive root modulo $17$ if $n^k - 1$ is not divisible by $17$ for all $k$ with $1 \le k < 16$. Let $a, b$ be distinct positive integers between $1$ and $16$. Which of the following statements are true ?
a.
$2$ is a primitive root modulo $17$.
b.
If $a$ is a primitive root modulo $17$, then $a^2$ is not necessarily a primitive root modulo $17$.
c.
If $a, b$ are primitive roots modulo $17$, then $ab$ is a primitive root modulo $17$.
d.
Product of primitive roots modulo $17$ between $1$ and $16$ is congruent to $1$ modulo $17$.
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Q24
mcq
Which of the following partial differential equations is NOT PARABOLIC for all $x, y \in \mathbb{R}$ ?
a.
$x^2 \frac{\partial^2 u}{\partial x \partial y} - 2xy \frac{\partial u}{\partial y} + y^2 = 0$
b.
$x^2 \frac{\partial^2 u}{\partial x^2} - 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2} = 0$
c.
$x^2 \frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2} = 0$
d.
$x^2 \frac{\partial^2 u}{\partial x^2} - 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2} + x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 0$
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Q24
mcq
For a positive integer $n$, let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity. For instance, $\Omega(3) = 1$, $\Omega(6) = \Omega(9) = 2$. Let $p > 3$ be a prime number and let $N = p(p + 2)(p + 4)$. Which of the following statements are true ?
a.
$\Omega(N) \ge 3$
b.
There exist primes $p > 3$ such that $\Omega(N) = 3$
c.
$p$ can never be the smallest prime divisor of $N$
d.
$p$ can be the smallest prime divisor of $N$
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Q25
mcq
Let the solution to the initial value problem $y' = y - t^2 + 1$, $0 \le t \le 2$, $y(0) = 0.5$ be computed using the Euler's method with step-length $h = 0.4$. If $y(0.8)$ and $w(0.8)$ denote the exact and approximate solutions at $t = 0.8$ then an error bound for Euler's method is given by:
a.
$0.2 \left(0.5 e^2 - 2\right) \left(e^{0.4} - 1\right)$
b.
$0.1 \left(e^{0.4} - 1\right)$
c.
$0.2 \left(0.5 e^2 - 2\right) \left(e^{0.8} - 1\right)$
d.
$0.1 \left(e^{0.8} - 1\right)$
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Q25
mcq
Let $p$ be a prime number and $N_p$ be the number of pairs of positive integers $(x, y)$ such that $$\frac{1}{x} + \frac{1}{y} = \frac{1}{p^4}$$ Which among the following are possible values of $N_p$ ?
a.
$0$
b.
$4$
c.
$5$
d.
$9$
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Q26
mcq
Which of the following is an extremal of the functional
$$J(y) = \int_{-1}^{1} \left(y'^2 + 2xy\right) dx$$
that satisfies the boundary conditions $y(-1) = -1$ and $y(1) = 1$ ?
a.
$-\frac{x^3}{5} + \frac{6x}{5}$
b.
$-\frac{x^5}{8} + \frac{9x}{8}$
c.
$-\frac{x^3}{6} + \frac{7x}{6}$
d.
$-\frac{x^3}{7} + \frac{8x}{7}$
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Q26
mcq
Let $G$ be a group of order $24$. Which of the following statements are necessarily true ?
a.
$G$ has a normal subgroup of order $3$.
b.
$G$ is not a simple group.
c.
There exists an injective group homomorphism from $G$ to $S_8$.
d.
$G$ has a subgroup of index $4$.
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Q27
mcq
Let $a, b, c \in \mathbb{R}$ be such that the quadrature rule
$$\int_{-1}^{1} f(x) dx = af(-1) + bf'(0) + cf'(1)$$
is exact for all polynomials of degree less than or equal to 2. Then $a + b + c$ equal to :
a.
4
b.
3
c.
2
d.
1
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Q27
mcq
Which of the following statements are true ?
a.
All finite field extensions of $\mathbb{Q}$ are Galois.
b.
There exists a Galois extension of $\mathbb{Q}$ of degree $3$.
c.
All finite field extensions of $\mathbb{F}_2$ are Galois.
d.
There exists a field extension of $\mathbb{Q}$ of degree $2$ which is not Galois.
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Q28
mcq
Let $f = a_0 + a_1 X + \dots a_n X^n$ be a polynomial with $a_i \in \mathbb{Z}$ for $0 \le i \le n$. Let $p$ be a prime such that $p \mid a_i$ for all $1 < i \le n$ and $p^2$ does not divide $a_n$. Which of the following statements are TRUE ?
a.
$f$ is always irreducible.
b.
$f$ is always reducible.
c.
$f$ can sometimes be irreducible and can sometimes be reducible.
d.
$f$ can have degree 1.
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Q28
mcq
A body moves freely in a uniform gravitational field. Which of the following statements is true?
a.
Stable equilibrium of the body is possible.
b.
Stable equilibrium of the body is not possible.
c.
Stable equilibrium of the body depends on the strength of the field.
d.
Equilibrium is metastable.
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Q29
mcq
Suppose that $Y$ has exponential distribution with mean $\theta$ and that the conditional distribution of $X$ given $Y = y$ is Normal with mean $0$ and variance $y$, for all $y > 0$. Identify the characteristic function of $X$ (defined as $\phi(t) = \mathbb{E}[e^{itX}]$ from the following.
a.
$e^{-\frac{\theta}{2} t^2}$
b.
$e^{-\frac{1}{2\theta} t^2}$
c.
$\frac{1}{1 + \frac{1}{2}\theta t^2}$
d.
$\frac{1}{1 + \frac{1}{2}t^2}$
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Q29
mcq
Which of the following statements are true about subsets of $\mathbb{R}^2$ with the usual topology?
a.
$A$ is connected if and only if its closure $\overline{A}$ is connected.
b.
Intersection of two connected subsets is connected.
c.
Union of two compact subsets is compact.
d.
There are exactly two continuous functions from $\mathbb{Q}^2$ to the set $\{(0, 0), (1, 1)\}$.
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Q30
mcq
STATS QUESTIONS
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Q30
mcq
Consider $A = \{1, 1/2, 1/3, \dots, 1/n, \dots \mid n \in \mathbb{N}\}$ and $B = A \cup \{0\}$. Both the sets are endowed with subspace topology from $\mathbb{R}$. Which of the following statements are true ?
a.
$A$ is a closed subset of $\mathbb{R}$
b.
$B$ is a closed subset of $\mathbb{R}$
c.
$A$ is homeomorphic to $\mathbb{Z}$, where $\mathbb{Z}$ has subspace topology from $\mathbb{R}$
d.
$B$ is homeomorphic to $\mathbb{Z}$, where $\mathbb{Z}$ has subspace topology from $\mathbb{R}$
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Q31
mcq
Let $u$ be a positive eigenfunction with eigenvalue $\lambda$ for the boundary value problem $\ddot{u} + 2\dot{u} + a(t)u = \lambda u$, $\dot{u}(0) = 0 = \dot{u}(1)$, where $a : [0, 1] \to (1, \infty)$ is a continuous function. Which of the following statements are possibly true ?
a.
$\lambda > 0$
b.
$\lambda < 0$
c.
$\int_0^1 (\dot{u})^2 dt = 2 \int_0^1 u\dot{u} dt + \int_0^1 (a(t) - \lambda)u^2 dt$
d.
$\lambda = 0$
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Q32
mcq
Let $f : \mathbb{R}^2 \to \mathbb{R}^2$ be a non-zero smooth vector field satisfying $\operatorname{div} f \neq 0$. Which of the following are necessarily true for the ODE $\dot{x} = f(x)$?
a.
There are no equilibrium points.
b.
There are no periodic solutions.
c.
All the solutions are bounded.
d.
All the solutions are unbounded.
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Q33
mcq
Consider the $2^{\text{nd}}$ order ODE $\ddot{x} + p(t)\dot{x} + q(t)x = 0$ and let $x_1, x_2$ be two solutions of this ODE in $[a, b]$. Which of the following statements are true for the Wronskian $W$ of $x_1, x_2$?
a.
$W \equiv 0$ in $(a, b)$ implies that $x_1, x_2$ are independent.
b.
$W$ can change sign in $(a, b)$.
c.
$W(t_0) = 0$ for some $t_0 \in (a, b)$ implies that $W \equiv 0$ in $(a, b)$.
d.
$W(t_0) = 1$ for some $t_0 \in (a, b)$ implies that $W \equiv 1$ in $(a, b)$.
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Q34
mcq
Which of the following expressions for $u = u(x, t)$ are solutions of $u_t - e^{-t} u_x + u = 0$ with $u(x, 0) = x$?
a.
$e^t (x + e^t - 1)$
b.
$e^{-t} (x - e^{-t} + 1)$
c.
$x - e^t + 1$
d.
$xe^t$
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Q35
mcq
Let $u(x, y)$ solve the partial differential equation (PDE)
$$x^2 \frac{\partial^2 u}{\partial x \partial y} + 3y^2 u = 0 \text{ with } u(x, 0) = e^{1/x}$$
Which of the following statements are true?
a.
The PDE is not linear
b.
$u(1, 1) = e^2$
c.
$u(1, 1) = e^{-2}$
d.
The method of separation of variables can be utilized to compute the solution $u(x, y)$.
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Q36
mcq
The values of $a, b, c, d, e$ for which the function
$$f(x) = \begin{cases} a(x - 1)^2 + b(x - 2)^3 & -\infty < x \le 2 \\ c(x - 1)^2 + d & 2 \le x \le 3 \\ (x - 1)^2 + e(x - 3)^3 & 3 \le x < \infty \end{cases}$$
is a cubic spline are:
a.
$a = c = 1$, $d = 0$, $b, e$ are arbitrary.
b.
$a = b = c = 1$, $d = 0$, $e$ is arbitrary.
c.
$a = b = c = d = 1$, $e$ is arbitrary.
d.
$a = b = c = d = e = 1$
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Q37
mcq
Consider the Euler method for integration of the system of differential equations $\dot{x} = -y$, $\dot{y} = x$. Assume that $(x_i^n, y_i^n)$ are the points obtained for $i = 0, 1, \dots, n^2$ using a time-step $h = \frac{1}{n}$ starting at the initial point $(x_0, y_0) = (1, 0)$. Which of the following statements are true?
a.
The points $(x_i^n, y_i^n)$ lie on a circle of radius $1$.
b.
$\lim_{n\to\infty}(x_n^n, y_n^n) = (\cos(1), \sin(1))$.
c.
$\lim_{n\to\infty}(x_2^n, y_2^n) = (1, 0)$.
d.
$(x_i^n)^2 + (y_i^n)^2 > 1$ for $i \ge 1$.
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Q38
mcq
Let $X = \{y \in C^1[0, \pi] : y(0) = 0 = y(\pi)\}$ and define $J : X \to \mathbb{R}$ by $J[y] = \int_0^\pi y^2(1 - y'^2) dx$. Which of the following statements are true?
a.
$y \equiv 0$ is a local minimum for $J$ with respect to the $C^1$ norm on $X$.
b.
$y \equiv 0$ is a local maximum for $J$ with respect to the $C^1$ norm on $X$.
c.
$y \equiv 0$ is a local minimum for $J$ with respect to the $\sup$ norm on $X$.
d.
$y \equiv 0$ is a local maximum for $J$ with respect to the $\sup$ norm on $X$.
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Q39
mcq
Let $B$ be the until ball in $\mathbb{R}^3$ centered at origin. The Euler-Lagrange equation corresponding to the functional $I(u) = \int_B (1 + |\nabla u|^2)^{1/2} dx$ is:
a.
$div\left(\frac{\nabla u}{(1 + |\nabla u|^2)^{1/2}}\right) = 0$
b.
$\frac{\nabla u}{(1 + |\nabla u|^2)^{1/2}} = 1$
c.
$|\nabla u| = 1$
d.
$\left(1 + |\nabla u|^2\right)\nabla u = \sum_{i, j = 1}^3 u_{x_i} u_{x_j} u_{x_i x_j}$
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Q40
mcq
Let $K(x, y)$ be a kernel in $[0, 1] \times [0, 1]$, defined as $K(x, y) = \sin(2\pi x)\sin(2\pi y)$. Consider the integral operator $K(u)(x) = \int_0^1 u(y) K(x, y) dy$, where $u \in C([0, 1])$. Which of the following assertion on $K$ are true?
a.
The null space of $K$ is infinite dimensional.
b.
$\int_0^1 v(x) K(u)(x) dx = \int_0^1 K(v)(x) u(x) dx$ for all $u, v \in C([0, 1])$
c.
$K$ has no negative eigenvalue.
d.
$K$ has an eigenvalue greater than $3/4$.
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Q41
mcq
Consider the integral equation:
$$\int_{0}^{x}(x-t)u(t)\,dt = x, \quad x \ge 0$$
for continuous functions $u$ defined on $[0, \infty)$. The equation has:
a.
A unique bounded solution
b.
No solution
c.
A unique solution $u$ such that $|u(x)| \le C(1+|x|)$ for some constant $C$.
d.
More than one solution $u$ such that $|u(x)| \le C(1+|x|)$ for some constant $C$.
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Q42
mcq
A mass $m$ with velocity $v$ approaches a stationary mass $M$ along the $x$-axis. The masses bounce of each other elastically. Assume that the motion takes place in one dimension along the $x$-axis, and $v_f$ and $V_f$ represents the final velocities of masses $m$ and $M$ along the $x$-axis. Which of the following are true ?
a.
$v_f = v, V_f = v$
b.
$v_f = 0, V_f = v$
c.
$v_f = \frac{(m-M)v}{m+M}, V_f = \frac{2mv}{m+M}$
d.
$v_f = \frac{mv}{m+M}, V_f = \frac{Mv}{m+M}$
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Q43
mcq
Let $X$ be a non-negative random variable with $E[X] = 1$. Which of the following quantities is necessarily greater than or equal to 1 ?
a.
$E[X^4]$
b.
$(E[\cos X])^2 + (E[\sin X])^2$
c.
$E[\sqrt{X}]$
d.
$E[1/X]$
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