Let $A = \begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} \in M_2 (\mathbb{R})$ and $\phi : \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$ be the bilinear map defined by $\phi(v, w) = v^T Aw$. Choose the correct statement from below:
a.
$\phi(v, w) = \phi(w, v)$ for all $v, w \in \mathbb{R}^2$
b.
there exists nonzero $v \in \mathbb{R}^2$ such that $\phi(v, w) = 0$ for all $w \in \mathbb{R}^2$
c.
there exists a $2\times 2$ symmetric matrix $B$ such that $\phi(v, v) = v^T Bv$ for all $v \in \mathbb{R}^2$
Correct Answer
d.
the $\psi : \mathbb{R}^4 \to \mathbb{R}$ map defined by $\psi \left( \begin{bmatrix} v_1 \\ v_2 \\ w_1 \\ w_2 \end{bmatrix} \right) = \phi \left( \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}, \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} \right)$ is linear
Step-by-Step AI Explanation
Verified Method
For any square matrix $A$, the quadratic form $\phi(v, v) = v^T A v$ can always be written as $v^T B v$, where $B = \frac{1}{2}(A + A^T)$ is symmetric. Here, $B = \frac{1}{2}\left(\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} + \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix}\right) = \begin{bmatrix} 1 & 3 \\ 3 & 3 \end{bmatrix}$, which is a symmetric matrix such that $\phi(v, v) = v^T B v$ for all $v \in \mathbb{R}^2$. Therefore, (C) is correct.
Option (A) is false because $A$ is not symmetric ($A \neq A^T$). Option (B) is false because $\det(A) = 1(3) - 2(4) = -5 \neq 0$, so $A$ is invertible, which means if $\phi(v, w) = v^T A w = 0$ for all $w$, then $v^T A = 0 \implies v = 0$. Option (D) is false because $\psi$ is a quadratic (bilinear) map in the components of $v$ and $w$, not a linear map (e.g., $\psi(2u) = 4\psi(u) \neq 2\psi(u)$).
Q2mcq
Let $A = \begin{bmatrix} 1 & -1 & 1 \\ 1 & 1 & 1 \\ 2 & 3 & \alpha \end{bmatrix}$ and $b = \begin{bmatrix} 1 \\ 3 \\ \beta \end{bmatrix}$. Then the system $AX = b$ over the real numbers has
a.
no solution whenever $\beta \neq 7$
b.
an infinite number of solutions whenever $\alpha \neq 2$
c.
an infinite number of solutions if $\alpha = 2$ and $\beta \neq 7$
d.
a unique solution if $\alpha \neq 2$
Correct Answer
Step-by-Step AI Explanation
Verified Method
Let us compute the determinant of the coefficient matrix $A$:
$$\det(A) = \begin{vmatrix} 1 & -1 & 1 \\ 1 & 1 & 1 \\ 2 & 3 & \alpha \end{vmatrix}$$
Subtract row 1 from row 2:
$$\det(A) = \begin{vmatrix} 1 & -1 & 1 \\ 0 & 2 & 0 \\ 2 & 3 & \alpha \end{vmatrix} = 2 \cdot (1 \cdot \alpha - 1 \cdot 2) = 2(\alpha - 2)$$
Thus, $\det(A) \neq 0$ if and only if $\alpha \neq 2$. When $\det(A) \neq 0$, the system $AX = b$ has a unique solution for any vector $b$. Therefore, (D) is correct.
Q3mcq
Let $A = \begin{bmatrix} 0 & 1 \\ -1 & 1 \end{bmatrix}$. Then the smallest positive integer $n$ such that $A^n = \text{I}$ is
a.
$1$
b.
$2$
c.
$4$
d.
$6$
Correct Answer
Step-by-Step AI Explanation
Verified Method
The characteristic polynomial of $A$ is:
$$\det(A - \lambda I) = \begin{vmatrix} -\lambda & 1 \\ -1 & 1 - \lambda \end{vmatrix} = -\lambda(1 - \lambda) + 1 = \lambda^2 - \lambda + 1 = 0$$
The roots are the primitive 6th roots of unity, $\lambda = e^{\pm i\pi/3}$. By the Cayley-Hamilton theorem, $A^2 - A + I = 0$, so multiplying by $(A + I)$ gives $A^3 + I = 0$, which implies $A^3 = -I$. Therefore, $A^6 = (-I)^2 = I$. The powers are $A \neq I$, $A^2 = A - I \neq I$, $A^3 = -I \neq I$, $A^4 = -A \neq I$, $A^5 = -A^2 \neq I$, and $A^6 = I$. Thus, the smallest positive integer $n$ is $6$.
Q4mcq
Let $A$ be a real symmetric matrix and $B = I + iA$, where $i^2 = -1$. Then
a.
$B$ is invertible if and only if $A$ is invertible
b.
all eigenvalues of are $B$ necessarily real
c.
$B - I$ is necessarily invertible
d.
$B$ is necessarily invertible
Correct Answer
Step-by-Step AI Explanation
Verified Method
Since $A$ is a real symmetric matrix, all its eigenvalues $\lambda$ are real. The eigenvalues of $B = I + iA$ are of the form $1 + i\lambda$ where $\lambda$ is an eigenvalue of $A$. Since $\lambda \in \mathbb{R}$, $1 + i\lambda \neq 0$ for all $\lambda$ (the real part is always $1$). Since no eigenvalue of $B$ is zero, $B$ is necessarily invertible for any real symmetric matrix $A$. Hence, (D) is correct.
Q5mcq
Let $S = \{ x \in [-1, 4] \mid \sin(x) > 0 \}$. Which of the following is true?
a.
$\inf(S) < 0$
b.
$\sup(S)$ does not exist
c.
$\sup(S) = \pi$
Correct Answer
d.
$\inf(S) = \pi/2$
Step-by-Step AI Explanation
Verified Method
We are given $S = \{ x \in [-1, 4] \mid \sin(x) > 0 \}$. Since $\sin(x) > 0$ when $x \in (2k\pi, (2k+1)\pi)$, for $x \in [-1, 4]$:
- For $x \in [-1, 0]$, $\sin(x) \le 0$.
- For $x \in (0, \pi)$, $\sin(x) > 0$.
- For $x \in [\pi, 4]$, $\sin(x) \le 0$ (since $\pi \approx 3.1416$ and $4 < 2\pi$).
Thus, $S = (0, \pi)$. Therefore, $\inf(S) = 0$ and $\sup(S) = \pi$. Hence, statement (C) is true.
Q6mcq
Let $k$ be a positive integer and let $S_k = \{ x \in [0, 1] \mid \text{a decimal expansion of } x \text{ has a prime digit at its } k^{\text{th}} \text{ place} \}$. Then the Lebesgue measure of $S_k$ is
a.
$0$
b.
$4/10$
Correct Answer
c.
$(4/10)^k$
d.
$1$
Step-by-Step AI Explanation
Verified Method
The prime digits in base 10 are $2, 3, 5, 7$ (a total of 4 digits out of 10 possible digits $\{0, 1, \dots, 9\}$). For each fixed position $k$, the choice of the $k$-th decimal digit partitions $[0, 1)$ into $10^k$ subintervals of length $10^{-k}$, grouped into $10^{k-1}$ blocks of 10 consecutive subintervals. In each block, exactly 4 of the subintervals correspond to the $k$-th digit being one of the prime digits $\{2, 3, 5, 7\}$. The total length (Lebesgue measure) is therefore $10^{k-1} \times 4 \times 10^{-k} = \frac{4}{10}$. Ambiguity of expansions with trailing 9s affects only a countable set of points, which has Lebesgue measure $0$. Thus, the Lebesgue measure of $S_k$ is $4/10$.
Q7mcq
Which of the following is necessarily true for a function $f : X \to Y$?
a.
if $f$ is injective, then there exists $g : Y \to X$ such that $f(g(y)) = y$ for all $y \in Y$
b.
if $f$ is surjective, then there exists $g : Y \to X$ such that $f(g(y)) = y$ for all $y \in Y$
Correct Answer
c.
if $f$ is injective and $Y$ is countable then $X$ is finite
d.
if $f$ is surjective and $X$ is uncountable then $Y$ is countably infinite
Step-by-Step AI Explanation
Verified Method
A map $f : X \to Y$ is surjective if and only if it has a right inverse $g : Y \to X$ such that $f(g(y)) = y$ for all $y \in Y$ (this is the definition of a right inverse, guaranteed by the Axiom of Choice). Hence, (B) is necessarily true.
Option (A) defines a right inverse, which corresponds to surjectivity, not injectivity (an injection has a left inverse). Option (C) is false because $X$ can be countably infinite (e.g., $X = Y = \mathbb{N}$). Option (D) is false because $Y$ can be uncountable (e.g., identity on $\mathbb{R}$).
Q8mcq
Let $S = \{ f : \mathbb{R} \to \mathbb{R} \mid \exists \epsilon > 0 \text{ such that } \forall \delta > 0, |x - y| < \delta \Rightarrow |f(x) - f(y)| < \epsilon \}$. Then
a.
$S = \{ f : \mathbb{R} \to \mathbb{R} \mid f \text{ is continuous} \}$
b.
$S = \{ f : \mathbb{R} \to \mathbb{R} \mid f \text{ is uniformly continuous} \}$
c.
$S = \{ f : \mathbb{R} \to \mathbb{R} \mid f \text{ is bounded} \}$
Correct Answer
d.
$S = \{ f : \mathbb{R} \to \mathbb{R} \mid f \text{ is constant} \}$
Step-by-Step AI Explanation
Verified Method
The condition states that there exists $\epsilon > 0$ such that for every $\delta > 0$, whenever $|x - y| < \delta$, we have $|f(x) - f(y)| < \epsilon$. Since this must hold for *all* $\delta > 0$, taking any two arbitrary points $x, y \in \mathbb{R}$, we can choose $\delta > |x - y|$. Then by the condition, $|f(x) - f(y)| < \epsilon$. Fix an arbitrary point $y = 0$; then for every $x \in \mathbb{R}$, $|f(x) - f(0)| < \epsilon$, which implies $|f(x)| < |f(0)| + \epsilon$. Thus $f$ is bounded. Conversely, if $f$ is bounded, say $|f(x)| \le M$ for all $x$, then $|f(x) - f(y)| \le 2M$. Choosing $\epsilon = 2M + 1$, the condition holds trivially for all $\delta > 0$. Thus, $S$ is precisely the set of all bounded functions from $\mathbb{R}$ to $\mathbb{R}$.
Q9mcq
Let $f : (0, \infty) \to \mathbb{R}$ be uniformly continuous. Then
a.
$\lim_{x \to 0^+} f(x)$ and $\lim_{x \to \infty} f(x)$ exist
b.
$\lim_{x \to 0^+} f(x)$ exist but $\lim_{x \to \infty} f(x)$ need not exist
Correct Answer
c.
$\lim_{x \to 0^+} f(x)$ need not exist but $\lim_{x \to \infty} f(x)$ exists
d.
neither $\lim_{x \to 0^+} f(x)$ nor $\lim_{x \to \infty} f(x)$ need exist
Step-by-Step AI Explanation
Verified Method
If a function $f$ is uniformly continuous on a bounded interval $(0, 1)$, it can be continuously extended to $[0, 1]$, which implies that $\lim_{x \to 0^+} f(x)$ must exist. On the other hand, for the limit at infinity, consider $f(x) = \sin(x)$ or $f(x) = x$. Both are uniformly continuous on $(0, \infty)$, but $\lim_{x \to \infty} \sin(x)$ does not exist. Thus, $\lim_{x \to 0^+} f(x)$ exists but $\lim_{x \to \infty} f(x)$ need not exist.
Q10mcq
Let $D$ be a subset of the real line. Consider the assertion: "Every infinite sequence in $D$ has a subsequence which converges in $D$". This assertion is true if
a.
$D = [0, \infty)$
b.
$D = [0, 1] \cup [3, 4]$
Correct Answer
c.
$D = [-1, 1) \cup (1, 2]$
d.
$D = (-1, 1]$
Step-by-Step AI Explanation
Verified Method
The Bolzano–Weierstrass theorem asserts that the property 'every sequence in $D$ has a subsequence converging to a point in $D$' is equivalent to $D$ being sequentially compact, which in $\mathbb{R}$ is equivalent to $D$ being closed and bounded (compact) by the Heine–Borel theorem.
- $D = [0, \infty)$ is unbounded (not compact).
- $D = [0, 1] \cup [3, 4]$ is a union of two closed and bounded intervals, hence it is closed and bounded (compact).
- $D = [-1, 1) \cup (1, 2]$ is not closed (e.g., $1 \notin D$).
- $D = (-1, 1]$ is not closed ($-1 \notin D$).
Therefore, the assertion holds only for option (B).
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Let $\{a_n\}_{n \ge 1}$ be a sequence of real numbers satisfying $a_1 \ge 1$ and $a_{n+1} \ge a_n + 1$ for all $n \ge 1$. Then which of the following is necessarily true?
a.
The series $\sum_{n=1}^\infty \frac{1}{a_n^2}$ diverges
b.
The sequence $\{a_n\}_{n \ge 1}$ is bounded
c.
The series $\sum_{n=1}^\infty \frac{1}{a_n^2}$ converges
d.
The series $\sum_{n=1}^\infty \frac{1}{a_n}$ converges
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Let $\mathbb{Z}$ denote the set of integers and $\mathbb{Z}_{\ge 0}$ denote the set $\{0, 1, 2, 3, .....\}$ . Consider the map $f : \mathbb{Z}_{\ge 0} \times \mathbb{Z} \rightarrow \mathbb{Z}$ given by $f(m, n) = 2^m \cdot (2n+1)$. Then the map $f$ is
a.
onto (surjective) but not one--one (injective)
b.
one--one (injective) but not onto (surjective)
c.
both one--one and onto
d.
neither one--one nor onto
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Let $A$ be a connected open subset of $\mathbb{R}^2$. The number of continuous surjective functions from $\overline{A}$ (the closure of $A$ in $\mathbb{R}^2$) to $\mathbb{Q}$ is:
a.
$1$
b.
$0$
c.
$2$
d.
not finite
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The group $S_3$ of permutations of $\{1, 2, 3\}$ acts on the three dimensional vector space over the finite field $\mathbb{F}_3$ of three elements, by permuting the vectors in basis $\{e_1, e_2, e_3\}$ by $\sigma \cdot e_i = e_{\sigma(i)}$, for all $\sigma \in S_3$. The cardinality of the set of vectors fixed under the above action is
a.
$0$
b.
$3$
c.
$9$
d.
$27$
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Let $\mathbb{D}$ be the open unit disc in the complex plane and $U = \mathbb{D} \setminus \left\{-\frac{1}{2}, \frac{1}{2}\right\}$. Also, let $H_1 = \{f : \mathbb{D} \rightarrow \mathbb{C} \mid f \text{ holomorphic and bounded}\}$ and $H_2 = \{f : U \rightarrow \mathbb{C} \mid f \text{ is holomorphic and bounded}\}$.
Then the map $r : H_1 \rightarrow H_2$ given by $r(f) = f|_U$, the restriction of $f$ to $U$, is
a.
injective but not surjective
b.
surjective but not injective
c.
injective and surjective
d.
neither injective nor surjective
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Let $C$ be the circle of radius $2$ with centre at the origin in the complex plane, oriented in the anti-clockwise direction. Then the integral $\oint_C \frac{dz}{(z-1)^2}$ is equal to
a.
$\frac{1}{2\pi i}$
b.
$2\pi i$
c.
$1$
d.
$0$
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Let $f$ be a holomorphic function in the open unit disc such that $\lim_{z \to 1} f(z)$ does not exist. Let $\sum_{n=0}^{\infty} a_n z^n$ be the Taylor series of $f$ about $z=0$ and let $\text{R}$ be its radius of convergence. Then
a.
$\text{R} = 0$
b.
$0 < \text{R} < 1$
c.
$\text{R} = 1$
d.
$\text{R} > 1$
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Let $I(m)$, denote the moment of inertia of a regular solid tetrahedron about an axis $m$ passing through its centre of gravity. Which of the following is true ?
a.
if the axis $\ell$ passes through a vertex and the axis $\ell'$ does not pass through a vertex then $I(\ell) > I(\ell)$
b.
if the axis $\ell$ passes through the mid-point of an edge and $\ell'$ is any other axis then $I(\ell) > I(\ell')$
c.
$I(\ell)$ is the same for all axes $\ell$
d.
if the axis $\ell$ passes through a vertex and the axis $\ell'$ does not pass through a vertex then $I(\ell) < I(\ell')$
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Let $u(x, t)$ be a solution of the heat equation $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$ in a rectangle $[0, \pi] \times [0, T]$ subject to the boundary conditions $u(0, t) = u(\pi, t) = 0, 0 \le t \le T$ and the initial condition $u(x, 0) = \varphi(x), 0 \le x \le \pi$.
If $f(x) = u(x, T)$, then which of the following is true for a suitable kernel $k(x, y)$?
a.
$\int_0^\pi k(x, y)\varphi(y)dy = f(x), 0 \le x \le \pi$
The iterative method $x_{n+1} = g(x_n)$ for the solution of $x^2 - x - 2 = 0$ converges quadratically in a neighbourhood of the root $x = 2$ if $g(x)$ equals
a.
$x^2 - 2$
b.
$(x-2)^2 - 6$
c.
$1 + \frac{2}{x}$
d.
$\frac{x^2 + 2}{2x - 1}$
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The set of real numbers $\lambda$ for which the boundary value problem
$$\frac{d^2 y}{dx^2} + \lambda y = 0, \quad y(0) = 0, \quad y(\pi) = 0$$
has nontrivial solutions is
a.
$(-\infty, 0)$
b.
$\{\sqrt{n} \mid n \text{ is a possitive integer}\}$
c.
$\{n^2 \mid n \text{ is a possitive integer}\}$
d.
$\mathbb{R}$
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Let $D$ denote the unit disc given by $\{(x, y) \mid x^2 + y^2 \le 1\}$ and let $D^C$ be its complement in the plane. The partial differential equation
$$(x^2 - 1)\frac{\partial^2 u}{\partial x^2} + 2y\frac{\partial^2 u}{\partial x \partial y} - \frac{\partial^2 u}{\partial y^2} = 0$$
a.
parabolic for all $(x, y) \in D^C$
b.
hyperbolic for all $(x, y) \in D$
c.
hyperbolic for all $(x, y) \in D^C$
d.
parabolic for all $(x, y) \in D$
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A parallel system consists of $n$ identical components. The lifetimes of the components are independent identically distributed uniform random variables with mean 30 hours and range 60 hours. If the expected lifetime of the system is 50 hours, then the value of $n$ is
a.
3
b.
4
c.
5
d.
6
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Suppose we draw a random sample of size $n$ from a population of size $N$, where $1 < n < N$, using simple random sampling without replacement scheme. Let $P$ be the population proportion of units possessing a particular attribute and $p$ be the corresponding sample proportion. Which of the following is an unbiased estimator for $P(1-P)$?
a.
$p(1-p)$
b.
$\frac{N-n}{N-1}(1-p)$
c.
$\frac{n(N-1)}{N(n-1)}p(1-p)$
d.
$\frac{N(n-1)}{n(N-1)}p(1-p)$
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Suppose $X_1 \sim N_{p_1}(0, \Sigma_1), X_2 \sim N_{p_2}(0, \Sigma_2)$, where $X_1$ and $X_2$ are independently distributed. If $p_1 > p_2$ and $\Sigma_1, \Sigma_2$ are positive definite then which of the following statements is necessarily true?
Consider the following regression problem
$$Y_i = \alpha + \beta i + \epsilon_i; \quad i = 1, \dots, n$$
Here $\epsilon_i, i = 1, 2, \dots, n$ are i.i.d $N(0, 1)$ random variables. It is assumed that $\alpha \neq 0$ and $\beta$ is known. If $\hat{\alpha}_n$ is the MLE of $\alpha$, which of the following statements is true ?
Let $X_1, X_2, \dots$ be a random sample from uniform $(0, 3\theta), \theta > 0$. Define $T_n = \frac{1}{3} \max \{X_1, X_2, \dots X_n\}$. Which of the following is NOT true ?
a.
$T_n$ is consistent for $\theta$
b.
$T_n$ is unbiased for $\theta$
c.
$T_n$ is a sufficient statistic
d.
$T_n$ is complete
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Let $X$ be a random sample of size 1 from a Cauchy distribution with probability density function
$$f_\theta(x) = \frac{1}{\pi} \left( \frac{1}{1 + (x - \theta)^2} \right), -\infty < x < \infty, \text{ where } \theta \in (-\infty, \infty).$$ For testing $H_0 : \theta = -1$ against $H_1 : \theta = 0$, the following test is suggested. Reject $H_0$ if $\frac{X}{\sqrt{1 + X^2}} > C$, otherwise do not reject $H_0$. What is the value of $C$ so that the power of the test is 0.5?
a.
$\frac{\pi}{4}$
b.
$0$
c.
$\tan^{-1}\left(\frac{1}{2}\right)$
d.
a solution of $\tan^{-1} \sqrt{\frac{C}{1-C}} = \frac{\pi}{3}$
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Let $X_1$ and $X_2$ be a random sample of size two from a distribution with probability density function
$$f_\theta(x) = \theta \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2} + (1-\theta) \frac{1}{2} e^{-|x|}$$
$-\infty < x < \infty$,
where $\theta \in \left\{0, \frac{1}{2}, 1\right\}$. If the observed values of $X_1$ and $X_2$ are $0$ and $2$, respectively, then the maximum likelihood estimate of $\theta$ is
a.
$0$
b.
$1/2$
c.
$1$
d.
not unique
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There are five empty boxes. Balls are placed independently one after another in randomly selected boxes. The probability that the fourth ball is the first to be placed in an occupied box equals
Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be a continuously differentiable map satisfying $\|f(x) - f(y)\| \ge \|x - y\|$, for all $x, y \in \mathbb{R}^n$. Then
a.
$f$ is onto
b.
$f(\mathbb{R}^n)$ is a closed subset of $\mathbb{R}^n$
c.
$f(\mathbb{R}^n)$ is an open subset of $\mathbb{R}^n$
d.
$f(0) = 0$
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Let $f : \mathbb{R}^4 \to \mathbb{R}$ be defined by $f(x) = x^t Ax$, where $A$ is a $4\times 4$ matrix with real entries and $x^t$ denotes the transpose of $x$. The gradient of at a point $x_0$ necessarily is
a.
$2Ax_0$
b.
$Ax_0 + A^t x_0$
c.
$2A^t x_0$
d.
$Ax_0$
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Consider the set of rational numbers $\mathbb{Q}$ as a subspace of $\mathbb{R}$ with the usual metric. Suppose $a$ and $b$ are irrational numbers with $a < b$ and let $K = [a, b] \cap \mathbb{Q}$. Then
a.
$K$ is a bounded subset of $\mathbb{Q}$
b.
$K$ is a closed subset of $\mathbb{Q}$
c.
$K$ is a compact subset of $\mathbb{Q}$
d.
$K$ is an open subset of $\mathbb{Q}$
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Let $f : \mathbb{R} \to \mathbb{R}$ be a function satisfying $f(x+y) = f(x)f(y), \forall x, y \in \mathbb{R}$ and $\lim_{x \to 0} f(x) = 1$. Which of the following are necessarily true ?
a.
$f$ is stricly increasing
b.
$f$ is either constant or bounded
c.
$f(rx) = f(x)^r$ for every rational $r \in \mathbb{Q}$
d.
$f(x) \ge 0, \forall x \in \mathbb{R}$
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Let $\mathbb{R}$ denote the set of real numbers and $\mathbb{Q}$ the set of all rational numbers. For $0 \le \epsilon \le \frac{1}{2}$, let $A_\epsilon$ be the open interval $(0, 1 - \epsilon)$. Which of the following are true ?
Let $a_{mn}, m \ge 1, n \ge 1$ be a double array of real numbers. Define
$$P = \liminf_{n \to \infty} \liminf_{m \to \infty} a_{mn}, \quad Q = \liminf_{n \to \infty} \limsup_{m \to \infty} a_{mn}$$
$$R = \limsup_{n \to \infty} \liminf_{m \to \infty} a_{mn}, \quad S = \limsup_{n \to \infty} \limsup_{m \to \infty} a_{mn}$$
Which of the following statements are necessarily true?
a.
$P \le Q$
b.
$Q \le R$
c.
$R \le S$
d.
$P \le S$
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Let $A$ be an $m \times n$ matrix of rank $m$ with $n > m$. If for some non-zero real number $\alpha$, we have $x'AA'x = \alpha x'x$, for all $x \in \mathbb{R}^m$ then $A'A$ has
a.
exactly two distinct eigenvalues
b.
$0$ as an eigenvalue with multiplicity $n - m$
c.
$\alpha$ as a non-zero eigenvalue
d.
exactly two non-zero distinct eigenvalues
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Let $V$ be the vector space over $\mathbb{C}$ of all polynomials in a variable $X$ of degree at most 3. Let $D : V \to V$ be the linear operator given by differentiation with respect to $X$. Let $A$ be the matrix of $D$ with respect to some basis for $V$. Which of the following are true?
a.
A is a nilpotent matrix
b.
A is a diagonalizable matrix
c.
the rank of $A$ is 2
d.
the Jordan canonical form of A is $$\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$
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Let $M = \left\{ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \;\middle|\; a,b,c,d \in \mathbb{Z} \text{ and the eigenvalues of } A \text{ are in } \mathbb{Q} \right\}$. Then
a.
M is empty
b.
$M = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \;\middle|\; a,b,c,d \in \mathbb{Z} \right\}$
c.
if $A \in M$ then the eigenvalues of A are in $\mathbb{Z}$
d.
if $A, B \in M$ are such that $AB = I$ then $\det A \in \{+1, -1\}$
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Let $f : [-1,1] \to \mathbb{R}$ be a function given by $$f(x) = \begin{cases} x^2 \cos\left(\frac{1}{x}\right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}$$ Then
a.
$f$ is of bounded variation on $[-1, 1]$
b.
$f'$ is of bounded variation on $[-1, 1]$
c.
$|f'(x)| \le 1 \quad \forall x \in [-1, 1]$
d.
$|f'(x)| \le 3 \quad \forall x \in [-1, 1]$
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Let $\ell^2 = \left\{ x = (x_n)_{n \ge 1} \mid x_n \in \mathbb{R}, \sum_{n=1}^\infty x_n^2 < \infty \right\}$ be the Hilbert space of square summable sequences and let $e_k$ denote the $k^{\text{th}}$ co-ordinate vector (with $1$ in $k^{\text{th}}$ place, $0$ elsewhere). Which of the following subspaces is NOT dense in $\ell^2$?
Consider $X = \left\{\left(x, \sin\frac{1}{x}\right) \,\middle|\, 0 < x \le 1\right\} \cup \{(0, y) \mid -1 \le y \le 1\}$ as a subspace of $\mathbb{R}^2$ and $Y = [0,1]$ as a subspace of $\mathbb{R}$. Then
a.
$X$ is connected
b.
$X$ is compact
c.
$X \times Y$ (in product topology) is connected
d.
$X \times Y$ (in product topology) is compact
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Let $X$ and $Y$ be topological spaces where $Y$ is Hausdorff. Let $X \times Y$ be given the product topology. Then for a function $f : X \to Y$ which of the following statements are necessarily true ?
a.
if $f$ is continuous, then $\text{graph}(f) = \{(x, f(x)) \mid x \in X\}$ is closed in $X \times Y$
b.
if $\text{graph}(f)$ is closed in $X \times Y$, then $f$ is continuous
c.
if $\text{graph}(f)$ is closed in $X \times Y$, then $f$ need not be continuous
d.
if $Y$ is finite, then $f$ is continuous
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For any prime number $p$, let $A_p$ be the set of integers $d \in \{1, 2, \ldots, 999\}$ such that the power of $p$ in the prime factorisation of $d$ is odd. Then the cardinality of
a.
$A_3$ is 250
b.
$A_5$ is 160
c.
$A_7$ is 124
d.
$A_{11}$ is 82
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For a set $X$, let $P(X)$ be the set of all subsets of $X$ and let $\Omega(X)$ be the set of all functions $f : X \to \{0,1\}$. Then
a.
if $X$ is finite then $P(X)$ is finite
b.
if $X$ and $Y$ are finite sets and if there is a 1--1 correspondence between $P(X)$ and $P(Y)$, then there is a 1--1 correspondence between $X$ and $Y$
c.
there is no 1--1 correspondence between $X$ and $P(X)$
d.
there is a 1--1 correspondence between $\Omega(X)$ and $P(X)$
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Let $p(z) = z^n + a_{n-1}z^{n-1} + \dots + a_0$, where $a_0, \dots, a_{n-1}$ are complex numbers and let $q(z) = 1 + a_{n-1}z + \dots + a_0 z^n$. If $|p(z)| \le 1$ for all $z$ with $|z| \le 1$ then
a.
$|q(z)| \le 1$ for all $z$ with $|z| \le 1$
b.
$q(z)$ is a constant polynomial
c.
$p(z) = z^n$ for all complex numbers
d.
$p(z)$ is a constant polynomial
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Let $f : \mathbb{C} \to \mathbb{C}$ be a holomorphic function and let $u$ be the real part of $f$ and $v$ the imaginary part of $f$. Then, for $x, y \in \mathbb{R}, |f'(x + iy)|^2$, is equal to
a.
$u_x^2 + u_y^2$
b.
$u_x^2 + v_x^2$
c.
$v_y^2 + u_y^2$
d.
$v_y^2 + v_x^2$
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Consider a non-zero, real-valued polynomial function $p(x) = a_0 + a_1 x + a_2 x^2$ of degree at most 2. Let $y = y(x)$ be a solution of the integral equation
$$y = p(x) + \int_0^x y(t) \sin(x-t)\,dt$$
Which of the following statements are necessarily correct?
a.
$y(x)$ is a polynomial function of degree $\le 2$
b.
$y(x)$ is a polynomial function of degree $\le 4$
c.
If $a_1 \neq 0$ and $a_0 + 2a_2 = 0$ then $y'(0) = 0$
d.
If $a_1 \neq 0$ and $a_0 + 2a_2 = 0$, then $y''(0) = 0$
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Let $I : C^1[0,1] \to \mathbb{R}$ be defined as
$$I(u) := \frac{1}{2} \int_0^1 \left((u'(t))^2 - 4\pi^2 u(t)^2\right) dt$$
Let us set $(P)\, m := \inf \left\{ I(u) : u \in C^1[0,1] : u(0) = u(1) = 0 \right\}$
Let $\bar{u} \in C^1[0,1]$ satisfy the Euler-Lagrange Equation associated with (P). Then
a.
$m = -\infty$ i.e. $I$ is not bounded below
b.
$m \in \mathbb{R}$, with $I(\bar{u}) = m$
c.
$m \in \mathbb{R}$, with $I(\bar{u}) > m$
d.
$m \in \mathbb{R}$, with $I(\bar{u}) < m$
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Let $X = \left\{ u \in C^1[0,1] \mid u(0) = 0 \right\}$ and let $I : X \to \mathbb{R}$ be defined as
$$I(u) = \int_0^1 (u'(t))^2\,dt$$
Which of the following are correct?
a.
$I$ is bounded below
b.
$I$ is not bounded below
c.
$I$ attains its infimum
d.
$I$ does not attain its infimum
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For $f \in C[0,1]$ and $n > 1$, let $T(f) = \frac{1}{n} \left[ \frac{1}{2} f(0) + \frac{1}{2} f(1) + \sum_{j=1}^{n-1} f\left(\frac{j}{n}\right) \right]$ be an approximation of the integral
$$I(f) = \int_0^1 f(x)\,dx$$
For which of the following functions $f$ is $T(f) = I(f)$?
a.
$1 + \sin 2\pi n x$
b.
$1 + \cos 2\pi n x$
c.
$\sin^2 2\pi n x$
d.
$\cos^2 2\pi (n+1) x$
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Consider the linear system $Ax = b$ with
$$A = \begin{bmatrix} 2 & 1 & -3 \\ 1 & 2 & -2 \\ -3 & -2 & 1 \end{bmatrix}$$
Let $x_n$ denote the $n^{\text{th}}$ Gauss-Seidel iteration and $e_n = x_n - x$. Let $M$ be the corresponding matrix such that $e_{n+1} = M e_n$, $n \ge 0$. Which of the following statements are necessarily true?
a.
all eigenvalues of $M$ have absolute value less than 1
b.
there is an eigenvalue of $M$ with absolute value at least 1
c.
$e_n$ converges to 0 as $n \to \infty$ for all $b \in \mathbb{R}^3$ and any $e_0$
d.
$e_n$ does not converge to 0 as $n \to \infty$ for any $b \in \mathbb{R}^3$ unless $e_0 = 0$
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Consider the second order PDE
$$8 \frac{\partial^2 z}{\partial x^2} - 2 \frac{\partial^2 z}{\partial x \partial y} - 3 \frac{\partial^2 z}{\partial y^2} = 0$$
Then which of the following are correct ?
a.
the equation is elliptic
b.
the equation is hyperbolic
c.
the general solution is $z = f\left(y - \frac{x}{2}\right) + g\left(y + \frac{3x}{4}\right)$, for arbitrary differentiable functions $f$ and $g$
d.
the general solution is $z = f\left(y + \frac{x}{2}\right) + g\left(y - \frac{3x}{4}\right)$, for arbitrary differentiable functions $f$ and $g$
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Consider the Lagrange equation $x^2 \frac{\partial z}{\partial x} + y^2 \frac{\partial z}{\partial y} = (x + y)z$. Then the general solution of the given equation is
a.
$F\left(\frac{xy}{z}, \frac{x-y}{z}\right) = 0$ for an arbitrary differentiable function $F$
b.
$F\left(\frac{x-y}{z}, \frac{1}{x} - \frac{1}{y}\right) = 0$ for an arbitrary differentiable function $F$
c.
$z = f\left(\frac{1}{x} - \frac{1}{y}\right)$ for an arbitrary differentiable function $f$
d.
$z = xy f\left(\frac{1}{x} - \frac{1}{y}\right)$ for an arbitrary differentiable function $f$
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Consider a boundary value problem (BVP) $\frac{d^2y}{dx^2} = f(x)$ with boundary conditions $y(0) = y(1) = y'(1)$, where $f$ is a real-valued continuous function on $[0, 1]$. Then which of the following are true?
a.
the given BVP has a unique solution for every $f$
b.
the given BVP does not have a unique solution for some $f$
c.
$y(x) = \int_0^x xt f(t)\,dt + \int_x^1 (t - x + xt) f(t)\,dt$ is a solution of the given BVP
d.
$y(x) = \int_0^x (x - t + xt) f(t)\,dt + \int_x^1 xt f(t)\,dt$ is a solution of the given BVP
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Consider the differential equation $\frac{d^2y}{dx^2} - 2\tan x\frac{dy}{dx} - y = 0$ defined on $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Which among the following are true?
a.
there is exactly one solution $y = y(x)$ with $y(0) = y'(0) = 1$ and $y\left(\frac{\pi}{3}\right) = 2\left(1 + \frac{\pi}{3}\right)$
b.
there is exactly one solution $y = y(x)$ with $y(0) = 1, y'(0) = -1$ and $y\left(-\frac{\pi}{3}\right) = 2\left(1 + \frac{\pi}{3}\right)$
c.
any solution $y = y(x)$ satisfies $y''(0) = y(0)$
d.
if $y_1$ and $y_2$ are any two solutions then $(ax+b)y_1 = (cx+d)y_2$ for some $a,b,c,d \in \mathbb{R}$
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Consider a system of first order differential equations
$$\frac{d}{dt}\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = \begin{bmatrix} x(t) + y(t) \\ -y(t) \end{bmatrix}$$
The solution space is spanned by
