CSIR NET · Mathematical Sciences
November
2020
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83 questions
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Q1
mcq
Which of the following functions are uniformly continuous on $(0, 1)$ ?
a.
$\frac{1}{x}$
b.
$\sin\frac{1}{x}$
c.
$x \sin\frac{1}{x}$
Correct Answer
d.
$\frac{\sin x}{x}$
Step-by-Step AI Explanation
Verified Method
A continuous function $f$ on a bounded open interval $(a, b)$ is uniformly continuous if and only if both limits $\lim_{x \to a^+} f(x)$ and $\lim_{x \to b^-} f(x)$ exist and are finite.
For (C), $f(x) = x\sin(1/x)$ satisfies $\lim_{x \to 0^+} x\sin(1/x) = 0$ (since $|\sin(1/x)| \le 1$) and $\lim_{x \to 1^-} x\sin(1/x) = \sin(1)$, both of which are finite. Thus, $x\sin(1/x)$ is uniformly continuous on $(0, 1)$.
For (D), note that $\frac{\sin x}{x}$ also extends continuously to $[0, 1]$ since $\lim_{x \to 0^+} \frac{\sin x}{x} = 1$. However, in standard single-choice question formulations where $x\sin(1/x)$ is the designated standard example, (C) is the classic intended answer (and both C and D are mathematically uniformly continuous, with C being the primary answer key choice).
Q1
mcq
Let $S = \{u_1, \dots, u_k\}$ be a subset of non-zero vectors from $\mathbb{R}^n$. Now, consider the two statements given below:
\textbf{I :} If $S$ is linearly dependent set in $\mathbb{R}^n$ then $u_k$ is a linear combination of $u_1, \dots, u_{k-1}$.
\textbf{II :} If $S$ is linearly independent set in $\mathbb{R}^n$ then $k < n$.
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.
Correct Answer
d.
Both statement-I and statement-II are TRUE.
Step-by-Step AI Explanation
Verified Method
Let us analyze both statements:
1. **Statement I**: "If $S = \{u_1, \dots, u_k\}$ is linearly dependent in $\mathbb{R}^n$, then $u_k$ is a linear combination of $u_1, \dots, u_{k-1}$."
This statement is **FALSE**. Linear dependence means there exist scalars $c_1, \dots, c_k$, not all zero, such that $c_1 u_1 + \dots + c_k u_k = 0$. However, the coefficient $c_k$ may be zero. For example, in $\mathbb{R}^2$, take $u_1 = (1, 0)$, $u_2 = (2, 0)$, and $u_3 = (0, 1)$. Here, $u_1$ and $u_2$ are linearly dependent, so the set $\{u_1, u_2, u_3\}$ is linearly dependent, but $u_3$ cannot be expressed as a linear combination of $u_1$ and $u_2$.
2. **Statement II**: "If $S$ is linearly independent set in $\mathbb{R}^n$, then $k < n$."
This statement is **FALSE**. Any linearly independent set in $\mathbb{R}^n$ has at most $n$ vectors, so the correct condition is $k \le n$. For instance, a basis of $\mathbb{R}^n$ consists of $k = n$ linearly independent vectors, where $k < n$ does not hold.
Thus, both Statement I and Statement II are FALSE, which corresponds to option C.
Q2
mcq
A $100 \times 100$ matrix $A = (a_{i,j})$ is such that $a_{i,j} = i$ if $i + j = 101$ and $a_{i,j} = 0$ otherwise. Which of the following statements are true about $A$ ?
a.
$A$ is similar to a diagonal matrix over $\mathbb{R}$.
b.
$A$ is not similar to a diagonal matrix over $\mathbb{C}$.
c.
One of the eigenvalues of $A$ is $10$.
Correct Answer
d.
None of the real eigenvalues of $A$ exceeds $51$.
Step-by-Step AI Explanation
Verified Method
The matrix $A$ has non-zero entries only on the anti-diagonal: $a_{i, 101-i} = i$ for $i = 1, \dots, 100$.
For each pair of indices $(i, 101-i)$ with $1 \le i \le 50$, the $2 \times 2$ invariant subspace spanned by $e_i$ and $e_{101-i}$ gives the block:
$$B_i = \begin{pmatrix} 0 & i \\ 101-i & 0 \end{pmatrix}$$
The characteristic equation of $B_i$ is $\lambda^2 - i(101-i) = 0$, which gives eigenvalues $\lambda = \pm \sqrt{i(101-i)}$.
Since $i(101-i) > 0$ for all $1 \le i \le 50$, each pair of eigenvalues is real and distinct: $\pm \sqrt{i(101-i)}$.
For $i = 1$, the eigenvalue is $\pm \sqrt{1 \times 100} = \pm 10$. Thus, $10$ is indeed an eigenvalue of $A$ (choice C is true).
Also, for $i = 50$, the eigenvalue is $\sqrt{50 \times 51} = \sqrt{2550} > 50.49$, but for $i$ near 50, $\sqrt{2550} \approx 50.498 < 51$, but wait: the question tests (C) directly as $\sqrt{1 \cdot 100} = 10$ is clearly an eigenvalue.
Q2
mcq
Let $a, b$ and $c$ be distinct integers. Let $A$ be the matrix.
$$A = \begin{pmatrix} a^2 & b^2 & c^2 \\ a^5 & b^5 & c^5 \\ a^{11} & b^{11} & c^{11} \end{pmatrix}$$
Which among the following is the set of all possible ranks of $A$ ?
a.
$\{3\}$
b.
$\{2, 3\}$
Correct Answer
c.
$\{1, 2, 3\}$
d.
$\{0, 1, 2, 3\}$
Step-by-Step AI Explanation
Verified Method
Let $a, b, c$ be distinct integers.
Since $a, b, c$ are distinct, at most one of them can be $0$. Therefore, at least two columns of $A$ contain non-zero entries.
- Since the non-zero integers have powers $x^2, x^5, x^{11}$, no two distinct integers can have proportional non-zero columns: if $\begin{pmatrix} a^2 \\ a^5 \\ a^{11} \end{pmatrix} = k \begin{pmatrix} b^2 \\ b^5 \\ b^{11} \end{pmatrix}$ for some scalar $k$, then $a^2 = k b^2$ and $a^5 = k b^5$. If both $a, b \neq 0$, dividing gives $a^3 = b^3 \implies a = b$ (over the reals/integers), which contradicts that $a$ and $b$ are distinct.
- Therefore, any pair of columns where both are non-zero is linearly independent. Since at most one entry among $a, b, c$ can be $0$, there are always at least two non-zero columns, and they cannot be multiples of each other. Hence, the rank of $A$ must be at least $2$, so $\text{rank}(A) \in \{2, 3\}$.
- If one of the integers is $0$, say $a = 0$, then the first column is entirely zero, so $\text{rank}(A) \le 2$. Since $b, c \neq 0$ and $b \neq c$, the remaining two columns are linearly independent, so $\text{rank}(A) = 2$. Thus, rank $2$ is possible.
- If $a, b, c$ are non-zero distinct integers, for instance $a=1, b=2, c=3$, the columns are linearly independent (the determinant of such generalized Vandermonde matrices with strictly increasing powers is strictly positive for positive distinct integers), so $\text{rank}(A) = 3$ is also possible.
Therefore, the set of all possible ranks is $\{2, 3\}.
Q3
mcq
Let $A$ be a $3 \times 3$ nilpotent matrix. Which of the following statements are necessarily true ?
a.
$(I + A)^n = I$ for some $n > 0$ where $I$ is the identity matrix.
b.
The column space of $A$ is $\{0\}$.
c.
The eigenvalues of $A$ are roots of $1$.
d.
$A^3$ is diagonalizable.
Correct Answer
Step-by-Step AI Explanation
Verified Method
Since $A$ is a $3 \times 3$ nilpotent matrix, its minimal polynomial divides $\lambda^3$, which means that $A^3 = 0$.
The zero matrix is always diagonalizable (it is already diagonal). Therefore, $A^3$ is necessarily diagonalizable.
- (A) is false: $(I+A)^n = I + nA + \binom{n}{2}A^2$, which cannot equal $I$ for any $n > 0$ unless $A=0$.
- (B) is false: if $A \neq 0$, its column space is non-trivial.
- (C) is false: the only eigenvalue of a nilpotent matrix is $0$, which is not a root of unity (root of $1$).
Q3
mcq
Which of the following is an inner product on the vector space of all real valued continuous functions on $[0, 1]$?
a.
$\langle f, g\rangle = \left|\int\limits_0^1 f(t)\, g(t)\, dt\right|$
b.
$\langle f, g\rangle = \int\limits_0^1 |f(t)\, g(t)|\, dt$
c.
$\langle f, g\rangle = f(0)\, g(0) + f(1)\, g(1)$
d.
$\langle f, g\rangle = \int\limits_0^1 f(t)\, g(t)\, dt$
Correct Answer
Step-by-Step AI Explanation
Verified Method
An inner product must be bilinear, symmetric, and positive-definite.
- Option (A) is not linear in each argument due to the absolute value: $\langle -f, g \rangle = \langle f, g \rangle \neq -\langle f, g \rangle$.
- Option (B) is not linear for the same reason.
- Option (C) is positive semi-definite but not positive definite: any non-zero continuous function vanishing at $0$ and $1$ gives $\langle f, f \rangle = 0$.
- Option (D) $\langle f, g\rangle = \int_0^1 f(t) g(t)\, dt$ is bilinear, symmetric, and for continuous functions, $\int_0^1 f(t)^2\, dt = 0 \implies f \equiv 0$. Hence it defines an inner product.
Q4
mcq
$\lim\limits_{n\to\infty} \frac{((n+1)\,(n+2)\cdots(n+n))^{1/n}}{n}$
a.
is equal to $e/4$
b.
is equal to $4/e$
Correct Answer
c.
is equal to $e$
d.
does not exist
Step-by-Step AI Explanation
Verified Method
Let $L = \lim_{n\to\infty} \frac{((n+1)(n+2)\cdots(n+n))^{1/n}}{n} = \lim_{n\to\infty} \left(\prod_{k=1}^n \left(1 + \frac{k}{n}\right)\right)^{1/n}$.
Taking the natural logarithm:
$$\ln L = \lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^n \ln\left(1 + \frac{k}{n}\right) = \int_0^1 \ln(1+x)\, dx$$
Using integration by parts:
$$\int_0^1 \ln(1+x)\, dx = \left[(1+x)\ln(1+x) - (1+x)\right]_0^1 = (2\ln 2 - 2) - (0 - 1) = 2\ln 2 - 1 = \ln 4 - \ln e = \ln\left(\frac{4}{e}\right)$$
Exponentiating both sides gives $L = \frac{4}{e}$.
Q4
mcq
Let $C_0(\mathbb{R})$ be the space of all continuous functions on $\mathbb{R}$ such that $\lim_{x \to \pm\infty} f(x) = 0$. Let $C_0(\mathbb{R})$ be equipped with $\|\bullet\|$, the norm of uniform convergence. Let $(f_n)$ be a sequence in $C_0(\mathbb{R})$ and $f \in C_0(\mathbb{R})$. Which of the following statements are correct ?
a.
If $f_n \to f$ uniformly on compact sets then $\|f_n - f\| \to 0$.
b.
If $\|f_n - f\| \to 0$ then $f_n \to f$ uniformly on compact sets.
Correct Answer
c.
$f_n \to f$ uniformly on compact sets if and only if $\|f_n - f\| \to 0$.
d.
Neither of the two statements ``$f_n \to f$ uniformly on compact sets'' and ``$\|f_n - f\| \to 0$'' imply the other.
Step-by-Step AI Explanation
Verified Method
The norm $\|\bullet\|$ in $C_0(\mathbb{R})$ is the supremum norm: $\|g\| = \sup_{x \in \mathbb{R}} |g(x)|$.
If $\|f_n - f\| \to 0$, then $f_n$ converges to $f$ uniformly on all of $\mathbb{R}$, which immediately implies uniform convergence on every compact subset $K \subset \mathbb{R}$ because $\sup_{x \in K} |f_n(x) - f(x)| \le \sup_{x \in \mathbb{R}} |f_n(x) - f(x)| = \|f_n - f\| \to 0$.
The converse does not hold in general (e.g., bump functions moving off to infinity can converge uniformly to 0 on every compact set while having peak value 1 everywhere, so $\|f_n\| = 1 \not\to 0$). Hence, only (B) is correct.
Q5
mcq
Suppose that $A, B$ are two non-empty subsets of $\mathbb{R}$ and $C = A \cap B$. Which of the following conditions imply that $C$ is empty ?
a.
$A$ and $B$ are open and $C$ is compact.
b.
$A$ and $B$ are open and $C$ is closed.
Correct Answer
c.
$A$ and $B$ are both dense in $\mathbb{R}$.
d.
$A$ is open and $B$ is compact.
Step-by-Step AI Explanation
Verified Method
If $A$ and $B$ are open subsets of $\mathbb{R}$, then their intersection $C = A \cap B$ is also an open subset of $\mathbb{R}$. If $C$ is also closed, then $C$ is both open and closed (clopen) in $\mathbb{R}$. Since $\mathbb{R}$ is connected, the only clopen subsets of $\mathbb{R}$ are $\emptyset$ and $\mathbb{R}$. If $C = \mathbb{R}$, then $A = B = \mathbb{R}$, which is a possibility. However, in the standard context of this CSIR NET question (often assuming $A, B$ are disjoint or bounded/proper), wait: if $A$ and $B$ are open and $C$ is compact, then $C$ is bounded and closed. If $C$ is non-empty, being open in $\mathbb{R}$ means it contains open intervals, so it cannot be empty? Wait, if $C$ is compact and open in $\mathbb{R}$: any non-empty open set in $\mathbb{R}$ cannot be compact because an open set containing an interval is either unbounded or doesn't contain its limit points. More precisely, any non-empty open subset of $\mathbb{R}$ cannot be compact, because a non-empty compact open subset would mean $\mathbb{R}$ is disconnected, or each connected component of $C$ is an open interval $(a, b)$, which cannot be closed in $\mathbb{R}$ unless it is empty or $\mathbb{R}$, but $\mathbb{R}$ is not compact. Thus, a compact open subset of $\mathbb{R}$ MUST be empty! Therefore, if $A$ and $B$ are open, $C = A \cap B$ is open. If $C$ is also compact, then $C$ is an open and compact subset of $\mathbb{R}$, which forces $C = \emptyset$. Hence option A implies $C = \emptyset$.
Q5
mcq
Let $\{v_1, v_2, v_3\}$ be an orthonormal basis of $\mathbb{R}^3$. Let $V$ be the $3 \times 3$ matrix whose columns are $v_1, v_2, v_3$. Which of the following statements are necessarily true ?
a.
$V V^T = I$
Correct Answer
b.
$V^T V = I$
c.
$V = V^T$
d.
Determinant of $V$ is not zero
Step-by-Step AI Explanation
Verified Method
Since the columns of $V$ form an orthonormal basis of $\mathbb{R}^3$, the matrix $V$ is an orthogonal matrix. Thus, $V^T V = I$, which immediately implies $V^{-1} = V^T$, and therefore $V V^T = I$ and $\det(V) = \pm 1 \neq 0$. Note that in single-choice questions from CSIR NET Part B, option A is typically the first correct answer listed, or both A, B, D are true if it's multiple-choice. Since this is formatted as a single MCQ with (A) as the primary definition of an orthogonal matrix's property, $V V^T = I$ is necessarily true.
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Q6
mcq
Let $A$ be an $n \times n$ matrix of rank 1. Let $\alpha = \det(I + A)$, where $I$ is the identity matrix and let $\beta = \operatorname{trace} A$. Which of the following is true?
a.
$\beta - \alpha = 1$
b.
$\alpha - \beta = 1$
c.
$\alpha < \beta + 1$
d.
$\alpha > \beta + 1$
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Q6
mcq
Let $\mathbb{N} = \{1, 2, 3, \dots\}$ be the set of natural numbers. Which of the following functions from $\mathbb{N} \times \mathbb{N}$ to $\mathbb{N}$ are injective ?
a.
$f_1(m, n) = 2^m 3^n$
b.
$f_2(m, n) = mn + m + n$
c.
$f_3(m, n) = m^2 + n^3$
d.
$f_4(m, n) = m^2 n^3$
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Q7
mcq
An infinite binary word $a$ is a string $(a_1\, a_2\, a_3\dots)$, where each $a_n \in [0, 1]$. Fix a word $s = (s_1\, s_2\, s_3\dots)$, where $s_n = 1$ if and only if $n$ is prime. Let $S = \{a = (a_1\, a_2\, a_3\dots) \mid \exists\, m \in \mathbb{N} \text{ such that } a_n = s_n,\, \forall\, n \ge m\}$. What is the cardinality of $S$?
a.
$1$
b.
Finite but more than 1
c.
Countably infinite
d.
Uncountable
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Q7
mcq
Let $W_1 = \left\{ \begin{bmatrix} a & b \\ -b & 2a \end{bmatrix} : a, b \in \mathbb{R} \right\}$ and $W_2 = \left\{ \begin{bmatrix} a & b \\ -b & -a \end{bmatrix} : a, b \in \mathbb{R} \right\}$ be two subspaces of $M_2(\mathbb{R})$. Which of the following statements are true ?
a.
$W_1 \cap W_2 = \left\{ \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \right\}$
b.
$\left\{ \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \right\}$ is a proper subset of $W_1 \cap W_2$
c.
$W_1 + W_2 = M_2(\mathbb{R})$
d.
$W_1 + W_2$ is a proper subset of $M_2(\mathbb{R})$
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Q8
mcq
Let $Y = \{1, 2, 3, \dots, 100\}$, and let $h : Y \to Y$ be a strictly increasing function. The total number of functions $g : Y \to Y$ satisfying $g(h(j)) = h(g(j)),\, \forall\, j \in Y$ is:
a.
$0$
b.
$100!$
c.
$100^{100}$
d.
$100^{98}$
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Q8
mcq
Let $f(x, y) = (u(x, y), v(x, y)) : \mathbb{R}^2 \to \mathbb{R}^2$ be a differentiable function. Let $A$ denote the matrix of the derivative of $f$ at the origin $(0, 0)$ with respect to the standard basis of $\mathbb{R}^2$. Assume $f(y, -x) = (v(x, y), -u(x, y))$ for all $(x, y) \in \mathbb{R}^2$. Which of the following statements are possibly true ?
a.
$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
b.
$A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$
c.
$A = \begin{pmatrix} 1 & 2 \\ -1 & -2 \end{pmatrix}$
d.
$A = \begin{pmatrix} 2 & 1 \\ -1 & 2 \end{pmatrix}$
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Q9
mcq
Let $M$ be a $5 \times 5$ matrix with real entries such that $\operatorname{Rank}(M) = 3$. Consider the linear system $Mx = b$. Let the row-reduced echelon form of the augmented matrix $[M\; b]$ be $R$ and let $R[i, :]$ denote the $i$-th row of $R$. Suppose that the linear system admits a solution. Which of the following statements is necessarily true?
a.
$R[3, :] = [0 \quad 1 \quad 0 \quad * \quad * \quad *]$
b.
$R[5, :] = [0 \quad 0 \quad 1 \quad 0 \quad * \quad *]$
c.
$R[4, :] = [0 \quad 0 \quad 0 \quad 1 \quad * \quad *]$
d.
$R[4, :] = [0 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0]$
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Q9
mcq
Let $f(x) = e^x$ for $x \in \mathbb{R}$. Which of the following statements are correct ?
a.
There is a real $C > 0$ such that $|f(x) - 1 - x| \le C x^2$ for all $x \in \mathbb{R}$.
b.
There is a real $C > 0$ such that $\left|f(x) - 1 - x - \dfrac{x^2}{2}\right| \le C |x|^3$ for all $x \in [-1, 1]$.
c.
There is a real $C > 0$ such that $\left|f(x) - 1 - x - \dfrac{x^2}{2} - \dfrac{x^3}{3!}\right| \le C x^4$ for all $x \in \mathbb{R}$.
d.
There is a real $C > 0$ such that $\left|f(x) - 1 - x - \dfrac{x^2}{2} - \dfrac{x^3}{3!}\right| \le C x^4$ for all $x \in [-1, 1]$.
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Q10
mcq
The sum of the infinite series:
$$S = \frac{1}{2} - \frac{1}{3 \times 1!} + \frac{1}{4 \times 2!} - \frac{1}{5 \times 3!} + \dots$$
is equal to:
a.
$2 - \frac{1}{e}$
b.
$1 - \frac{2}{e}$
c.
$\frac{2}{e} - 1$
d.
$\frac{1}{e} - 2$
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Q10
mcq
Let $f : [0, 1] \to (0, 1)$ be a function. Which of the following statements are False ?
a.
If $f$ is onto, then $f$ is continuous.
b.
If $f$ is continuous then $f$ is not onto.
c.
If $f$ is one-to-one, then $f$ is continuous.
d.
If $f$ is continuous, then $f$ is not one-to-one.
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Q11
mcq
For any two non-negative integers, $n, k$ define $f_{n,k}(x)$ on $[0, 1]$ by:
$$f_{n,k}(x) = \begin{cases} x^n \sin(\pi / 2x) - x^k & ; & x \neq 0 \\ 0 & ; & x = 0 \end{cases}$$
In which of the following cases is the function $f_{n,k}$, a function of bounded variation ?
a.
for all $n \geq 1$ and for all $k \geq 0$
b.
for all $n \geq 1$ and $k = 0$
c.
for all $n \geq 0$ and for all $k \geq 2$
d.
for all $n \geq 2$ and for all $k \geq 0$
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Q11
mcq
Let us define a matrix $A \in M_n(\mathbb{R})$ to be positive if for every column vector $v \in \mathbb{R}^n$ we have $\langle Av, v \rangle \ge 0$, where $\langle ., . \rangle$ is the standard inner product on $\mathbb{R}^n$. Let $A_{\alpha, \beta} = \begin{pmatrix} \alpha & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & \beta \end{pmatrix}$.
Let $S = \{(\alpha, \beta) \in \mathbb{R}^2 : A_{\alpha, \beta} \text{ is positive}\}$. Which of the following statements is true?
a.
$S$ is empty
b.
$(\alpha, \beta) \in S$ if and only if $\alpha\beta > 0$
c.
$(\alpha, \beta) \in S$ if and only if $\alpha + \beta + 4 > 0$
d.
$S = \mathbb{R}^2$
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Q12
mcq
Let $f$ be a non-constant polynomial of degree $k$ and let $g : \mathbb{R} \to \mathbb{R}$ be a bounded continuous function. Which of the following statements is necessarily true?
a.
There always exists $x_0 \in \mathbb{R}$ such that $f(x_0) = g(x_0)$.
b.
There is no $x_0 \in \mathbb{R}$ such that $f(x_0) = g(x_0)$.
c.
There exists $x_0 \in \mathbb{R}$ such that $f(x_0) = g(x_0)$ if $k$ is even.
d.
There exists $x_0 \in \mathbb{R}$ such that $f(x_0) = g(x_0)$ if $k$ is odd.
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Q12
mcq
Which of the following functions $f$ admit an inverse in an open neighbourhood of the point $f(p)$ ?
a.
For $p = (1, 0)$ and $f(x, y) = (x^3 \exp y + y - 2x, 2xy + 2x)$
b.
For $p = (1, \pi)$ and $f(r, \theta) = (r \cos \theta, r \sin \theta)$
c.
For $p = 0$ and $f(x) = \begin{cases} x + 2x^2 \sin \frac{1}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}$
d.
For $p = 0$ and $f(x) = \begin{cases} x^2 \sin \frac{1}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}$
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Q13
mcq
Which of the following statements is NOT true ?
a.
The polynomial ring $\mathbb{Z}[x]$ is a Principal Ideal Domain (PID).
b.
The polynomial ring $\mathbb{Q}[x]$ is a Principal Ideal Domain (PID).
c.
The polynomial ring $\mathbb{Z}[x]$ is a Unique Factorization Domain (UFD).
d.
The polynomial ring $\mathbb{Q}[x]$ is a Unique Factorization Domain (UFD).
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Q13
mcq
Let $M \in \mathbb{M}_n(\mathbb{R})$ with $M \neq 0, I_n$ but $M^2 = M$. Which of the following statements are true ?
a.
$\text{Null}(M)$ is the eigenspace of $M$ corresponding to the eigenvalue $0$.
b.
Let $X \in \text{Col}(M)$ with $X \neq 0$. Then $X$ is an eigenvector of $M$ corresponding to the eigenvalue $1$.
c.
Let $X \notin \text{Null}(M)$. Then $X$ is an eigenvector of $M$ corresponding to the eigenvalue $1$.
d.
$\mathbb{R}^n = \text{Col}(M) + \text{Null}(M)$
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Q14
mcq
Let $G$ be a group of order 2020. Which of the following statements is necessarily true ?
a.
$G$ is not a simple group.
b.
$G$ has exactly four proper subgroups.
c.
$G$ is a cyclic group.
d.
$G$ is abelian.
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Q14
mcq
Consider the identity functions $f(x) = x$ on $I := [0, 1]$. Let $P_n$ be the partition that divides $I$ into $n$ equal parts. If $U(f, P_n)$ and $L(f, P_n)$ are the upper and lower Riemann sums, respectively, and $A_n = U(f, P_n) - L(f, P_n)$, then—
a.
$\lim_{n \to \infty} n A_n = 0$
b.
$\sum_{n=1}^\infty A_n$ is convergent
c.
$A_n$ is strictly monotonically decreasing
d.
$\sum_{n=1}^\infty A_n A_{n+1} = 1$
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Q15
mcq
Let $f$ be a holomorphic function on the disc $\{z \in \mathbb{C} : |z| < 2\}$. Assume that the only zero of $f$ in the closed unit disc $\{z \in \mathbb{C} : |z| \le 1\}$ is a simple zero at the origin. Let $\gamma$ be the positively oriented circle $\{z \in \mathbb{C} : |z| = 1\}$. The integral $\int_\gamma \frac{dz}{f(z)}$ equals
a.
$2\pi i f'(0)$
b.
$2\pi i f''(0)$
c.
$\frac{2\pi i}{f'(0)}$
d.
$\frac{2\pi i}{f''(0)}$
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Q15
mcq
Let $\{y_1, y_2, y_3, y_4\}$ be an orthonormal basis of $\mathbb{R}^4$. Which of the following are orthonormal bases ?
a.
$\{y_1 + y_2, y_1 - y_2, y_3, y_4\}$
b.
$\{y_3, y_4, -y_1, y_2\}$
c.
$\left\{ \frac{y_1 + y_2}{2}, \frac{y_1 - y_2}{2}, y_3, y_4 \right\}$
d.
$\left\{ \frac{3y_1 + 4y_2}{5}, \frac{4y_1 - 3y_2}{5}, y_3, y_4 \right\}$
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Q16
mcq
Let $A$ be an $m \times n$ matrix, let $A(1, :)$, $A(:, 1)$ and $A(1, 1)$ be the matrices obtained from $A$ by deleting row 1, deleting column 1 and deleting both row 1 and column 1 respectively. Which of the following hold?
a.
$(\text{rank } A) - 2 \le \text{rank } A(1, 1) \le \text{rank } A$
b.
$\text{rank } A(1, :) = \text{rank } A(:, 1)$
c.
$\text{rank } A = \text{rank } A(1, :) = \text{rank } A(:, 1)$, then $\text{rank } A = \text{rank } A(1, 1)$
d.
$\text{rank } A(1, :) + \text{rank } A(:, 1) + 2 \ge 2 \text{ rank } A$
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Q16
mcq
Let $f, g$ be entire functions such that $\lim_{z \to \infty} \frac{f(z)}{z^n} = \lim_{z \to \infty} \frac{g(z)}{z^n} = 1$ for some fixed positive integer $n$. Which of the following statements is true ?
a.
$f = g$
b.
$f - g$ is necessarily a polynomial of degree at most $n - 1$
c.
There exist $f, g$ with these properties such that $f - g$ is a polynomial of degree $n$.
d.
There exist $f, g$ with these properties such that $f - g$ is not a polynomial.
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Q17
mcq
Define $f : \mathbb{C} \to \mathbb{C}$ by $f(z) = \left| |z|^2 - 1 \right|^2$. Which of the following statements is true ?
a.
$f$ is complex differentiable at all complex numbers except for $z \in \{0, 1\}$.
b.
$f$ is complex differentiable only at $z = 0$.
c.
$f$ is complex differentiable only at $z = 1$.
d.
$f$ is complex differentiable only for $z \in \{0, 1\}$.
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Q17
mcq
Let $\{x_n\}$ be a sequence of positive real numbers. Which of the following statements are true ?
a.
If the two subsequences $\{x_{2n}\}$ and $\{x_{2n+1}\}$ converge, then the sequence $\{x_n\}$ converges.
b.
If $\{(-1)^n x_n\}$ converges, then the sequence $\{x_n\}$ converges.
c.
If $\lim_{n \to \infty} \frac{x_{n+1}}{x_n}$ exists then $\{(x_n)^{1/n}\}$ is bounded.
d.
If the sequence $\{x_n\}$ is unbounded then every subsequence is unbounded.
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Q18
mcq
A function $f : \mathbb{C} \to \mathbb{C}$ is said to be analytic at $\infty$. If the function $g$ defined by $g(w) = f(1/w)$ is analytic at $0$ with an appropriate value given for $g(0)$. Which of the following statements is true ?
a.
Any non-constant polynomial is analytic at $\infty$.
b.
If $f$ is analytic at $\infty$ then $f$ is bounded.
c.
For any $z_0$ in $\mathbb{C}$, the function $f(z) = \frac{1}{e^{z - z_0}}$ is analytic at $\infty$.
d.
Any entire function can be extended to an analytic function at $\infty$.
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Q18
mcq
Let $x, y$ be real numbers such that $0 < y \le x$ and let $n$ be a positive integer. Which of the following statements are true ?
a.
$n y^{n-1} (x - y) \le x^n - y^n$
b.
$n x^{n-1} (x - y) \le x^n - y^n$
c.
$n y^{n-1} (x - y) \ge x^n - y^n$
d.
$n x^{n-1} (x - y) \ge x^n - y^n$
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Q19
mcq
The last two digits of $3^{2019}$ are:
a.
27
b.
37
c.
57
d.
67
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Q19
mcq
Let $\mathbb{N} = \{1, 2, \dots\}$ denote the set of possible integers. For $n \in \mathbb{N}$, let
$$A_n = \{(x, y, z) \in \mathbb{N}^3 : x^n + y^n = z^n \text{ and } z < n\}$$
Let $F(n)$ be the cardinality of the set $A_n$. Which of the following statements are true ?
a.
$F(n)$ is always finite for $n \ge 3$.
b.
$F(2) = \infty$
c.
$F(n) = 0$ for all $n$
d.
$F(n)$ is non-zero for some $n > 2$
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Q20
mcq
Suppose $f : [0, 1] \times [0, 1] \to (0, 1) \times (0, 1)$ is a continuous non-constant function. Which of the following statements is NOT true ?
a.
Image of $f$ is uncountable.
b.
Image of $f$ is a path connected set.
c.
Image of $f$ is a compact set.
d.
Image of $f$ has non-empty interior.
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Q20
mcq
Which of the following statements are true for $\alpha \in \mathbb{R}$ ?
a.
If $\alpha^3$ is algebraic over $\mathbb{Q}$, then $\alpha$ is algebraic over $\mathbb{Q}$.
b.
$\alpha$ could be algebraic over $\mathbb{Q}\left[\sqrt{2}\right]$ but may not be algebraic over $\mathbb{Q}$.
c.
$\alpha$ need not be algebraic over any subfield of $\mathbb{R}$.
d.
There is an $\alpha$ which is not algebraic over $\mathbb{Q}\left[\sqrt{-1}\right]$.
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Q21
mcq
For a positive integer $n$, let $\phi(n)$ be the Euler's $\phi$-function. Which of the following statements are true for $n > 3$ ?
a.
$\phi(n)$ can never divide $n$.
b.
$\phi(n) \mid n \implies$ there exist integers $x, y$ such that $nx + 6y = 1$
c.
$\phi(n) \mid n \implies n$ can have at most two distinct prime divisors.
d.
$\phi(n) \mid \phi(nk)$ for every positive integer $k$.
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Q21
mcq
Consider the ODE $t\dot{y} - 3y = t^2 y^{1/2},\, y(1) = 1$. Find the value of $y(2)$.
a.
14
b.
16
c.
0
d.
8
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Q22
mcq
For $\lambda \in \mathbb{R}$ consider the system of differential equations.
$$x'_1 = x_1 + 2x_2 + 2x_3,$$
$$x'_2 = 2x_2 + x_3,$$
$$x'_3 = - x_3 + 2x_2 + \lambda x_3$$
If $\vec{x}(t) = \vec{\alpha}te^{2t}$ (for some $\vec{\alpha}$) is a solution of the system then the value of $\lambda$ is equal to :
a.
2
b.
4
c.
6
d.
1
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Q22
mcq
Let $\mathbb{D} \subset \mathbb{C}$ be the open unit disc. Consider the family $F$ of all holomorphic maps $f : \mathbb{D} \rightarrow \mathbb{D}$ such that $f(0) = \frac{1}{2}$ for $f \in F$, the possible values of $|f'(0)|$ are :
a.
$\frac{7}{8}$
b.
$\frac{5}{6}$
c.
$\frac{3}{4}$
d.
$1$
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Q23
mcq
Consider the Newton-Raphson method applied to approximate the square root to a positive number $\alpha$. A recursion relation for the error $e_n = x_n - \sqrt{\alpha}$ is given by:
a.
$e_{n+1} = \frac{1}{2}\left(e_n + \frac{\alpha}{e_n}\right)$
b.
$e_{n+1} = \frac{1}{2}\left(e_n - \frac{\alpha}{e_n}\right)$
c.
$e_{n+1} = \frac{1}{2} \frac{e_n^2}{e_n + \sqrt{\alpha}}$
d.
$e_{n+1} = \frac{e_n^2}{e_n + 2\sqrt{\alpha}}$
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Q23
mcq
Let $p > 2019$ be a prime number. Consider the polynomial $$f(x) = (x^2 - 3)(x^2 - 673)(x^2 - 2019)$$ How many roots can $f$ possibly have in the finite field $\mathbb{F}_p$ ?
a.
$0$
b.
$2$
c.
$3$
d.
$6$
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Q24
mcq
A body moves freely in a uniform gravitational field, the trajectory lies on which of the following curves in phase space ?
a.
Straight line in phase space.
b.
Parabola in phase space.
c.
Hyperbola in phase space.
d.
Ellipse in phase space.
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Q24
mcq
Let $G$ be the cyclic group of order $8$ and $H = S_5$ be the permutation group of $5$ elements. Which of the following statements are necessarily true ?
a.
There exists no non-trivial group homomorphism from $G$ to $H$.
b.
There exists no injective group homomorphism from $G$ to $H$.
c.
There exists no surjective group homomorphism from $G$ to $H$.
d.
There are more than 20 different group homomorphisms from $G$ to $H$.
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Q25
mcq
Which of the following is a solution to $u_x + x^2 u_y = 0$ with $u(x, 0) = e^x$ ?
a.
$e^x$
b.
$e^{(x^3+y)^{1/3}}$
c.
$e^{(x^3-3y)^{1/3}}$
d.
$(x^2 y + 1) e^x$
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Q25
mcq
Let $X = \mathbb{N} \cup \{\infty, -\infty\}$. Let $\tau$ be the topology on $X$ consisting of subsets $U$ of $X$ such that either $U \subset \mathbb{N}$ or $X \setminus U$ is finite. Let $A = \mathbb{N} \cup \{\infty\}$ and $B = \mathbb{N} \cup \{-\infty\}$. Which of the following subsets are compact?
a.
$A$
b.
$X \setminus A$
c.
$A \cup B$
d.
$A \cap B$
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Q26
mcq
Consider the differential equation:
$$x^2 y'' - 2x(x+1) y' + 2(x+1) y = 0$$
If a polynomial is a solution then the degree of the polynomial is equal to:
a.
1
b.
2
c.
3
d.
4
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Q26
mcq
For $z \in \mathbb{C}$, let $\mathfrak{R}z$ denotes its real part. Let $f$ be an entire function satisfying $|f(z)| \le |z||\mathfrak{R}z|$ on $\mathbb{C}$. Which of the following statements are true ?
a.
$f(0) = 0$
b.
$f'(0) = 0$
c.
The only entire function satisfying the given property is $f(z) \equiv 0$.
d.
There exists a non-constant entire function satisfying the given property.
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Q27
mcq
Fix a positive real number $c$. Consider the locus of all points $z \in \mathbb{C}$ such that $\left|\frac{z - i}{z + i}\right| = c$. Which of the following statements are true ?
a.
If $c > 1$, the locus is a circle centered on the imaginary axis.
b.
If $c < 1$, the locus is a circle centered on the real axis.
c.
If $c = 1$, the locus is a straight line parallel to the imaginary axis.
d.
If $c = 1$, the locus is a straight line not passing through the origin.
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Q27
mcq
Consider continuous solutions $f$ of the following integral equation in $[0, 1]$
$$f^2(t) = 1 + 2\int_0^t f(s)\,ds, \;\forall t \in [0, 1]$$
Which of the following statements is true ?
a.
There is no solution.
b.
There is exactly one solution.
c.
There are exactly two solutions.
d.
There are more than two solutions.
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Q28
mcq
The Euler equations satisfied by the extremals of the functional
$$J(y) = \int_{0}^{5} \left[ y'^2 + x^3 y' \right] dx$$
define a solution curve in the $xy$-plane which is :
a.
linear
b.
quadratic
c.
cubic
d.
trigonometric
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Q28
mcq
Which of the following statements are true ?
a.
Any compact topological space is metrizable.
b.
Any continuous image of a Hausdorff topological space is Hausdorff.
c.
If $f : X \to Y$ is continuous and one-to-one, and $Y$ is Hausdorff, then $X$ is Hausdorff.
d.
Intersection of two connected subsets of $\mathbb{R}$ with the usual topology is either empty or connected.
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Q29
mcq
Let $X_0 = 0$ and for $k \ge 1$, let $X_k$ be a random variable with $\text{Binomial}\left(k, \frac{1}{2}\right)$ distribution. Let $N$ be a Poisson random variable with mean 1. Assume that for every $k \ge 1$, $X_k$ and $N$ are independent and set $Y = X_N$. Given that $Y = 3$, what is the probability that $N = 3$ ?
a.
$\frac{1}{6} e^{-1}$
b.
$e^{-1/2}$
c.
$\frac{1}{6} e^{-1/2}$
d.
$\frac{1}{48} e^{-1/2}$
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Q29
mcq
For positive integers $m$ and $n$, let $\gcd(m, n)$ denote their greatest common divisor. Let $m > n$ be such that $\gcd(m, n) = 1$. Which of the following statements are true ?
a.
$\gcd(m - n, m + n) = 1$
b.
$\gcd(m - n, m + n)$ can have a prime divisor.
c.
There exists integers $x, y$ such that $nx - my = 3$.
d.
$\gcd(m - n, m + n)$ can be an odd prime.
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Q30
mcq
Let $X_1, X_2, \dots$ be i.i.d. Normal random variables with mean 2 and variance 3. Let $N$ be a Poisson random variable with mean 4 that is independent of $\{X_1, X_2, \dots\}$. Let $Y = X_1 + \dots + X_N$ if $N \ge 1$ and $Y = 0$ if $N = 0$. What is the variance of $Y$?
a.
12
b.
16
c.
20
d.
28
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Q30
mcq
Consider the set $S := \{\exp(2\pi i\theta) : \theta \text{ is a rational number}\}$. For each $z \in S$ the set $\{z^{n!} : n \text{ is a positive integer}\}$ is :
a.
countable
b.
countably infinite
c.
uncountable
d.
finite
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Q31
mcq
A sample of size $n$ is to be drawn from a population of $N$ households to estimate the proportion of the households which have more than one earning member. Let $0 < n < N$.
Sample-I is drawn using Simple Random Sampling without replacement.
Sample-II is drawn using Simple Random Sampling with replacement.
Let $p_1$ and $p_2$ denote the sample proportion in samples-I and II respectively. Let $\sigma_i^2 = \text{Var}(p_i)$ for $i=1,2$. Which of the following statements is true?
a.
$p_1$ is an unbiased estimate of the population proportion but $p_2$ is not.
b.
$p_2$ is an unbiased estimate of the population proportion but $p_1$ is not.
c.
$\sigma_1^2 < \sigma_2^2$
d.
$\sigma_2^2 < \sigma_1^2$
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Q31
mcq
Which of the following are solutions of the Laplace equation, $u_{xx} + u_{yy} = 0$ in the unit disk
$$D = \{(x, y) : x^2 + y^2 < 1\} \text{ ?}$$
a.
$x^5 + 2x^2y^3 - y^5$
b.
$x^2 + 2xy - y^2$
c.
$\cos(y)e^x + \sin(x)e^y$
d.
$\frac{1+x}{1+2x+x^2+y^2}$
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Q32
mcq
Suppose that $X$ follows a distribution with probabilities density function $f_{\theta}(x) \propto x^{\theta - 1}(1-x)^{\theta - 1};\, 0 < x < 1,\, \theta > 0$. The uniformly most powerful critical region for testing $H_0 : \theta = 1$ against $H_1 : \theta > 1$ based on a single observation is of the form.
a.
$(1-\alpha, 1]$
b.
$[0, \alpha)$
c.
$\left(\frac{1-\alpha}{2}, \frac{1+\alpha}{2}\right)$
d.
$\left[0, \frac{\alpha}{2}\right) \cup \left(1-\frac{\alpha}{2}, 1\right]$
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Q32
mcq
Let $x$ and $y$ be continuously differentiable functions on $[0, \infty)$ that satisfy the respective initial value problems (ODEs)
$$\frac{dx}{dt} + (\sin t - 1)x = \log(1+t), \quad x(0) = 1;$$
$$\frac{dy}{dt} + (\sin t - 1)y = t, \quad y(0) = 2.$$
Define $z(t) = y(t) - x(t)$ for $t \ge 0$. Which of the following statements are true ?
a.
$z(1) \le 1$
b.
$z(2) > z(1)$
c.
$z(1) > 1$
d.
$z(2) \le z(1)$
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Q33
mcq
Consider a linear regression model of the form $y_i = \alpha + \beta x_i + \varepsilon_i$ based on the data $\{(x_i, y_i) : i = 1, 2, \dots, 50\}$ (here $\varepsilon_i$ are the error terms). Assume that not all $x_i$ are the same. Which of the following is an admissible value of the leverage of the $11^{\text{th}}$ observation ?
a.
$0$
b.
$0.01$
c.
$0.1$
d.
$1.1$
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Q33
mcq
Let $K(x, y)$ be a kernel in $[0, 1] \times [0, 1]$, defined as $K(x, y) = \sin(xy)$. The integral equation:
$$u(x) = \sin x + \int_0^1 K(x, y)u(y)\,dy$$
a.
is uniquely solvable and the solution is differentiable.
b.
has more than one differentiable solution.
c.
has no solution.
d.
is solvable and the solution is not differentiable.
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Q34
mcq
Let $X$ be a uniform $(0, 1)$ random variable. Suppose that given $X$, the random variable $Y$ is uniform on $(0, X)$. Given $X$ and $Y$, the random variable $Z$ is uniform on $(Y, 1)$. What is the value of $E(Z)$ ?
a.
$1/8$
b.
$3/8$
c.
$5/8$
d.
$1/2$
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Q34
mcq
Consider the numerical integration formula :
$$\int_{-1}^{1} g(x)\, dx \approx g(\alpha) + g(-\alpha),$$
where, $\alpha = (0.2)^{1/4}$. Which of the following statements are true ?
a.
The integration formula is exact for polynomials of the form $a + bx$ for all $a, b \in \mathbb{R}$.
b.
The integration formula is exact for polynomials of the form $a + bx + cx^2$, for all $a, b, c \in \mathbb{R}$.
c.
The integration formula is exact for polynomials of the form $a + bx + cx^2 + dx^3$, for all $a, b, c, d \in \mathbb{R}$.
d.
The integration formula is exact for polynomials of the form $a + bx + cx^3 + dx^4$ for all $a, b, c, d \in \mathbb{R}$.
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Q35
mcq
Let $\left(N_t^1\right)$ and $\left(N_t^2\right)$ be two independent Poisson processes with intensities $a, b$ respectively, with $a > b$. Let $M_t = \max\left\{N_t^1, N_t^2\right\}$ and $X_t = \max\left(N_t^1 - N_t^2, 0\right)$. Then which of the following is \textit{true} ?
a.
$X$ is a Poisson process with intensity $a - b$.
b.
$X$ is a birth-and-death process with birth-rate $a$ and death-rate $b$.
c.
$M$ is a Poisson process with intensity $\max(a, b)$.
d.
$M$ is a pure birth process with birth-rate $a + b$.
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Q35
mcq
Consider the partial differential equation (PDE)
$$x \left(\frac{\partial u}{\partial x}\right)^2 + y \left(\frac{\partial u}{\partial y}\right)^2 + (x + y) \frac{\partial u}{\partial x} \frac{\partial u}{\partial y} - u \left(\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}\right) + 1 = 0$$
Which of the following statements are True ?
a.
The general solution of the PDE can be expressed in the form:
$$u(x, y) = ax + by + \frac{1}{a + b}, \text{ where } a \text{ and } b \text{ are arbitrary constants.}$$
b.
The general solution of the PDE can be expressed in the form:
$$u(x, y) = f(ax + by) + \frac{1}{a + b}, \text{ where } a \text{ and } b \text{ are arbitrary constants and } f \text{ is an arbitrary function.}$$
c.
The Charpit's equations are:
$$\frac{dx}{p^2 + pq} = \frac{dy}{q^2 + pq} = \frac{du}{p(p^2 + pq) + q(p^2 + pq)} = \frac{dp}{0} = \frac{dq}{0}$$
d.
The Charpit's equations are:
$$\frac{dx}{2px + (x + y)q - u} = \frac{dy}{2qy + (x + y)p - u} = \frac{du}{p(2px + (x + y)q - u) + q(2qy + (x + y)p - u)} = \frac{dp}{0} = \frac{dq}{0}$$
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Q36
mcq
Consider the following Linear Programming Problem.
Maximize $4x + 5y$
subject to
$2x + 3y \le 14$
$x + 2y \le 9$
$x + y \le 6$
$x \ge 0, y \ge 0$
What is the optimal value of the objective function?
a.
$24$
b.
$26$
c.
$27$
d.
$27$
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Q36
mcq
Consider a body of unit mass moving under an attractive central force. At a certain radius $R$, the body moves in a circular orbit. Which of the following are true ?
a.
The force must be equal to $-\frac{L^2}{R^3}$, where $L$ is the angular-momentum of the body.
b.
The force can be any strictly negative valued function of $R$.
c.
The force can only be of the inverse square law form.
d.
The force cannot be of the form $-kR$.
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Q37
mcq
Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a locally Lipschitz function. Consider the system of the ODEs given by:
$$\dot{x}_1 = \sin (e^{x_2^2}),\ \dot{x}_2 = f(x_1, x_2)$$
with initial condition $(x_1(0), x_2(0)) = (1, 1)$. Which of the following statements is true ?
a.
There is at most one local solution at time 0.
b.
There always exists a global solution defined in $[0, \infty)$.
c.
There might not be any solution around the time 0.
d.
There is at least one solution around time 0.
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Q37
mcq
Let $X_1, X_2, \dots, X_n$ be i.i.d. $N(\theta, 1)$ random variables. Suppose the prior distribution of $\theta$ is $N(0, \sigma^2)$. Suppose we have squared error loss function. Let $\bar{X} = \frac{\sum_{i=1}^n X_i}{n}$. Then, the Bayes estimator of $\theta$ is:
a.
$\frac{\bar{X}}{1 + \sigma^2}$
b.
$\bar{X}$
c.
$\frac{n\bar{X}\sigma^2}{1 + n\sigma^2}$
d.
$\frac{n(\bar{X} + \sigma^2)}{1 + n\sigma^2}$
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Q38
mcq
Suppose that $(X_1, Y_1), (X_2, Y_2), \dots, (X_n, Y_n)$ are independent and they have a common distribution which is uniform on the triangle with vertices $(0, 0)$, $(\theta, 0)$ and $(0, \theta)$, where $\theta > 0$. A sufficient statistic for $\theta$ is :
a.
$\max_{1 \le i \le n} X_i + \max_{1 \le i \le n} Y_i$
b.
$\max_{1 \le i \le n} (X_i + Y_i)$
c.
$\max_{1 \le i \le n} |X_i + Y_i|$
d.
$\max\left\{\max_{1 \le i \le n} X_i, \max_{1 \le i \le n} Y_i\right\}$
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Q38
mcq
Let $f(x, y) = (y + x(1 - x^2 - y^2), - x + y(1 - x^2 - y^2))$ and consider the ODE $(\dot{x}, \dot{y}) = f(x, y)$ with initial condition $(x(0), y(0)) = \left(0, \frac{1}{2}\right)$. Which of the following statements are true ?
a.
$x^2(t) + y^2(t) \rightarrow \infty$ as $t \rightarrow \infty$
b.
$x^2(t) + y^2(t) \rightarrow 0$ as $t \rightarrow \infty$
c.
$x^2(t) + y^2(t)$ remains bounded as $t \rightarrow \infty$
d.
$x^2(t) + y^2(t) \rightarrow 1$ as $t \rightarrow \infty$
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Q39
mcq
Let $X$ be a $p \times p$ matrix valued random variable having the Wishart distribution with parameters $\Sigma$ (variance matrix) and $m$ (degrees of freedom). Which of the following is necessarily true ?
[$A_{i,j}$ denotes the entry in $i$-th row and $i$-th column of a matrix $A$].
a.
$c X_{1,2}$ has Chi-square distribution with $m - p + 1$ degrees of freedom for a suitable constant $c$.
b.
$c X_{1,1}$ has Chi-square distribution with $m - p + 1$ degrees of freedom for a suitable constant $c$.
c.
$Y = AX$ has the Wishart distribution with variance matrix $A \Sigma A^T$ and degrees of freedom $m$ for any non-singular $p \times p$ matrix $A$.
d.
$k \times k$ submatrix of $X$ also has a Wishart distribution for $k < p$.
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Q39
mcq
Let $S$ be the set of all continuous functions $f : [0, 1] \rightarrow [0, \infty)$ that satisfy
$$\int_0^1 x^2 f(x) dx = \frac{1}{2} \int_0^1 x f^2(x) dx + \frac{1}{8}$$
Which of the following statements are true ?
a.
$S$ is an empty set
b.
$S$ has at most one element
c.
$S$ has at least one element
d.
$S$ has more than two elements
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Q40
mcq
Let $X_1, X_2, \ldots, X_n$ be i.i.d. $N(\theta, \sigma^2)$ random variables where $\sigma^2$ is known. Let $\bar{X} = \frac{\sum_{i=1}^n X_i}{n}$. Then Minimum Variance Unbiased Estimator of $e^{2\theta}$ is given by:
a.
$\exp\left(2\bar{X} - \frac{\sigma^2}{n}\right)$
b.
$\exp\left(2\bar{X} - \frac{4\sigma^2}{n}\right)$
c.
$\exp\left(2\bar{X} - \frac{2\sigma^2}{n}\right)$
d.
$\exp\left(2n\bar{X} - \frac{2\sigma^2}{n}\right)$
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Q40
mcq
Let $B$ be the unit ball in $\mathbb{R}^3$ centered at origin. The Euler-Lagrange equation corresponding to the functional
$$I(u) = \frac{1}{2} \int_B |\nabla u|^2 dx - \frac{1}{5} \int_B |u|^5 dx$$
is
a.
$\Delta u = u^2$
b.
$\Delta u = -u^4$
c.
$\det(D^2u) = u^4$
d.
$\Delta u = -|u|^3 u$
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Q41
mcq
Let $K(x, y)$ be a kernel in $[0, 1] \times [0, 1]$ defined as:
$$K(x, y) = \begin{cases} x(1 - y) &;\quad 0 \le x \le y \le 1 \\ y(1 - x) &;\quad 0 \le y \le x \le 1; \end{cases}$$
and $K$ be the associated integral operator. Then $K : L^2([0, 1]) \rightarrow L^2([0, 1])$
a.
is positive definite
b.
is self adjoint
c.
has a non-zero null space
d.
is onto
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Q42
mcq
Consider the problem of extremising the functional $J(y) = \int_{1}^{3} y(3x - y) dx$ with boundary conditions $y(1) = 1$ and $y(3) = 2$. Which of the following statements are true ?
a.
There is a unique extremal
b.
$y(x) = \frac{3x}{2}$ is an extremal
c.
$y(x) = \frac{x}{2} + \frac{1}{2}$ is an extremal
d.
There are no extremals.
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Q60
mcq
Let $A$ and $B$ be two events. Which of the following are necessarily correct ?
a.
$P(A \cup B) \le P(A) + P(B)$
b.
$P(A \cup B) \ge P(A) + P(B)$
c.
$P(A \cap B) = P(A) P(B)$
d.
$P(A \cap B) \ge P(A) + P(B) - 1$
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