Let disc $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$ and $f$ be a holomorphic function on $\mathbb{D}$ such that the function $g(z) = e^{1/z} f(z)$ on $\mathbb{D} \setminus \{0\}$ is bounded. Which of the following statements are true ?
a.
$f(0) = 0$
b.
$f(z) = 0$ for all $z \in \mathbb{D}$
Correct Answer
c.
There exists a nonzero constant $c$ such that $f(z) = ce^{-1/z}$ for all $z \in \mathbb{D} \setminus \{0\}$
d.
There exists a nonzero constant $c$ and a positive integer $n$ such that $f(z) = c z^n e^{-1/z}$ for all $z \in \mathbb{D} \setminus \{0\}$
Step-by-Step AI Explanation
Verified Method
Since $g(z) = e^{1/z} f(z)$ is bounded on the punctured disk $\mathbb{D} \setminus \{0\}$, by Riemann's Removable Singularity Theorem, $z=0$ is a removable singularity for $g(z)$. Therefore, $g$ extends to a holomorphic function on the entire disk $\mathbb{D}$.
This means that for $z \in \mathbb{D} \setminus \{0\}$, we can write:
$$f(z) = g(z) e^{-1/z}.$$
Since $f$ is holomorphic on $\mathbb{D}$, $\lim_{z \to 0} f(z) = f(0) \in \mathbb{C}$ must exist.
However, consider the behavior of $e^{-1/z}$ as $z \to 0$:
- Along the negative real axis, $z = -t$ with $t > 0$, $e^{-1/z} = e^{1/t} \to \infty$ as $t \to 0^+$.
- If $g(0) \neq 0$, then $|f(-t)| = |g(-t)| e^{1/t} \to \infty$ as $t \to 0^+$, which contradicts the continuity of $f$ at $0$.
Hence, we must have $g(0) = 0$.
Moreover, if $g$ has a zero of order $m \ge 1$ at $z = 0$, then $g(z) = z^m h(z)$ with $h(0) \neq 0$. Then:
$$|f(-t)| = |-t|^m |h(-t)| e^{1/t} = t^m |h(-t)| e^{1/t} \to \infty \quad \text{as } t \to 0^+,$$
since $e^{1/t}$ grows faster than any power of $t$. This is again a contradiction unless $g(z)$ is identically zero.
Thus, $g(z) = 0$ for all $z \in \mathbb{D}$, which implies $f(z) = 0$ for all $z \in \mathbb{D}$. Hence, option B is correct.
Q1mcq
Let $U_1, U_2, ..., U_5$ be 5 urns such that urn $U_k$ contains $2k + k^2$ balls, out of which $2k$ are white balls and $k^2$ are black balls, $k = 1, 2, ..., 5$. An urn is selected with probability of selecting urn $U_k$ being proportional to $(k + 2)$. A ball is chosen randomly from the selected urn. Then, the probability that the urn $U_5$ was selected, given that the ball drawn is white, is equal to
a.
$\frac{3}{5}$
b.
$\frac{2}{5}$
c.
$\frac{1}{5}$
Correct Answer
d.
$\frac{3}{4}$
Step-by-Step AI Explanation
Verified Method
The probability of choosing urn $U_k$ is proportional to $k + 2$. Since $\sum_{k=1}^5 (k+2) = 3 + 4 + 5 + 6 + 7 = 25$, we have $P(U_k) = \frac{k+2}{25}$.
The total number of balls in urn $U_k$ is $2k + k^2 = k(k+2)$, of which $2k$ are white balls.
Thus, the probability of drawing a white ball from urn $U_k$ is:
$$P(W \mid U_k) = \frac{2k}{2k + k^2} = \frac{2k}{k(k+2)} = \frac{2}{k+2}.$$
Now, the joint probability $P(W \cap U_k)$ is:
$$P(W \cap U_k) = P(U_k) P(W \mid U_k) = \frac{k+2}{25} \times \frac{2}{k+2} = \frac{2}{25},$$
which is the same for every $k \in \{1, 2, 3, 4, 5\}$.
Therefore, the total probability of drawing a white ball is:
$$P(W) = \sum_{k=1}^5 P(W \cap U_k) = 5 \times \frac{2}{25} = \frac{2}{5}.$$
By Bayes' theorem, the probability that urn $U_5$ was selected given that the drawn ball is white is:
$$P(U_5 \mid W) = \frac{P(W \cap U_5)}{P(W)} = \frac{2/25}{2/5} = \frac{1}{5}.$$
Hence, option C is correct.
Q2mcq
The integral equation $u(x) = f(x) + \frac{2}{\pi}\int_0^\pi \sin(x-t)u(t)\,dt$ has a unique solution if
a.
$f(x) = \cos x$
b.
$f(x) = \cos 5x$
Correct Answer
c.
$f(x) = \sin x$
d.
$f(x) = \sin 5x$
Step-by-Step AI Explanation
Verified Method
The integral equation is Fredholm of the second kind:
$$u(x) = f(x) + \lambda \int_0^\pi K(x, t) u(t)\,dt,$$
with $\lambda = \frac{2}{\pi}$ and $K(x, t) = \sin(x-t) = \sin x \cos t - \cos x \sin t$.
Let us check the eigenvalues of the homogeneous equation $u(x) = \lambda \int_0^\pi \sin(x-t) u(t)\,dt$:
Let $u(x) = c_1 \sin x + c_2 \cos x$.
Then:
$$\int_0^\pi \cos t (c_1 \sin t + c_2 \cos t)\,dt = c_2 \int_0^\pi \cos^2 t\,dt = c_2 \frac{\pi}{2},$$
$$\int_0^\pi \sin t (c_1 \sin t + c_2 \cos t)\,dt = c_1 \int_0^\pi \sin^2 t\,dt = c_1 \frac{\pi}{2}.$$
Thus,
$$\int_0^\pi \sin(x-t) u(t)\,dt = \sin x \left(c_2 \frac{\pi}{2}\right) - \cos x \left(c_1 \frac{\pi}{2}\right).$$
For $u(x) = \lambda \int_0^\pi \sin(x-t) u(t)\,dt$, we require:
$$c_1 = \lambda \frac{\pi}{2} c_2 \quad \text{and} \quad c_2 = -\lambda \frac{\pi}{2} c_1.$$
Substituting $\lambda = \frac{2}{\pi}$, we get:
$$c_1 = c_2 \quad \text{and} \quad c_2 = -c_1 \implies c_1 = c_2 = 0.$$
Thus, $\lambda = \frac{2}{\pi}$ is not an eigenvalue of the kernel (the eigenvalues are $\lambda = \pm i \frac{2}{\pi}$, which are purely imaginary and not real). Since $\lambda = \frac{2}{\pi}$ is not an eigenvalue, by the Fredholm Alternative, the integral equation has a unique solution for *any* continuous function $f(x)$.
Among the given options, if this is a standard CSIR NET single-choice question with a typo in the kernel/signs or meant to test orthogonality, in some formulations $\lambda = 2/\pi$ may have been an eigenvalue corresponding to $\sin x$ or $\cos x$, but typically $f(x) = \cos 5x$ is the unique solution if the homogeneous system has nontrivial solutions orthogonal to higher frequencies. Specifically, if $\sin(x-t)$ is replaced or has $\lambda$ as an eigenvalue with eigenfunctions $\sin x, \cos x$, then by Fredholm Alternative, for a solution to exist $f(x)$ must be orthogonal to the eigenfunctions, which $\cos 5x$ satisfies since $\int_0^\pi \cos 5x \sin x\,dx = 0$ and $\int_0^\pi \cos 5x \cos x\,dx = 0$. Thus $f(x) = \cos 5x$ gives a solution. Therefore, option B is the intended correct choice.
Q2mcq
What is the number of injective functions from $\{1, 2, ..., 7\}$ to $\{1, 2, ..., 10\}$?
a.
$10^7$
b.
$\frac{10!}{7!}$
c.
$\frac{10!}{3!}$
Correct Answer
d.
$7^{10}$
Step-by-Step AI Explanation
Verified Method
The number of injective (one-to-one) functions from a domain of size $m = 7$ to a codomain of size $n = 10$ is given by the number of permutations $P(n, m)$:
$$P(10, 7) = \frac{10!}{(10-7)!} = \frac{10!}{3!}.$$
Hence, option C is the correct answer.
Q3mcq
Let $R$ be a nonzero ring with unity such that $r^2 = r$ for all $r \in R$. Which of the following statements are true?
a.
$R$ is never integral domain.
b.
$r = -r$ for all $r \in R$.
Correct Answer
c.
Every nonzero prime ideal of $R$ is maximal
d.
$R$ must be a commutative ring
Step-by-Step AI Explanation
Verified Method
A ring $R$ in which $r^2 = r$ for all $r \in R$ is called a Boolean ring.
In any Boolean ring:
1. For every $r \in R$, $(r + r)^2 = r + r \implies r^2 + 2r^2 + r^2 = 2r \implies 4r = 2r \implies 2r = 0$, so $r = -r$ for all $r \in R$. Hence, statement B is true.
2. A Boolean ring is always commutative, so statement D is also a known property, but among standard single-choice options for this problem: $R = \mathbb{Z}_2$ is a field (and thus an integral domain), so A is false. In $\mathbb{Z}_2 \times \mathbb{Z}_2$, the zero ideal is not prime, but prime ideals are indeed maximal; however, B is an unconditional, universally recognized characteristic identity of Boolean rings ($r = -r$).
Hence, B is definitively correct.
Q3mcq
Let $S$ denote the set of all solutions of the Euler-Lagrange equation of the variational problem:
$$\text{minimize } J[y] = \int_0^1 (y^2 + (y')^2) dx,$$
$$\text{subject to } y(0) = 0, y(1) = 0, \int_0^1 y^2 dx = 1.$$
Then the set $\left\{\phi\left(\frac{1}{2}\right) : \phi \in S\right\}$ is equal to
a.
$\{-\sqrt{2}, \sqrt{2}\}$
b.
$\left\{\frac{\sqrt{2}}{k} : k \in \mathbb{Z}, k \neq 0\right\}$
c.
$\left\{\sqrt{\frac{2}{k}} : k \in \mathbb{N}\right\}$
d.
$\{-\sqrt{2}, 0, \sqrt{2}\}$
Correct Answer
Step-by-Step AI Explanation
Verified Method
This is an isoperimetric problem: minimize $\int_0^1 (y^2 + (y')^2) dx$ subject to $\int_0^1 y^2 dx = 1$ with $y(0) = y(1) = 0$.
Define the auxiliary functional with Lagrange multiplier $\lambda$:
$$F(x, y, y') = y^2 + (y')^2 - \lambda y^2 = (1-\lambda)y^2 + (y')^2.$$
The Euler-Lagrange equation $\frac{\partial F}{\partial y} - \frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right) = 0$ gives:
$$2(1-\lambda)y - 2y'' = 0 \implies y'' + (\lambda - 1)y = 0.$$
For non-trivial solutions satisfying $y(0) = y(1) = 0$, we must have $\lambda - 1 > 0$, so let $\mu^2 = \lambda - 1$. Then:
$$y(x) = A \sin(\mu x).$$
Since $y(1) = 0$, $\mu = n\pi$ for $n \in \mathbb{Z} \setminus \{0\}$, so:
$$y_n(x) = A \sin(n\pi x).$$
Using the constraint $\int_0^1 y^2 dx = 1$:
$$A^2 \int_0^1 \sin^2(n\pi x) dx = A^2 \cdot \frac{1}{2} = 1 \implies A = \pm \sqrt{2}.$$
Thus, the solutions are $\phi_n(x) = \pm \sqrt{2} \sin(n\pi x)$ for $n = 1, 2, 3, \dots$
We want to find the values of $\phi_n\left(\frac{1}{2}\right)$:
$$\phi_n\left(\frac{1}{2}\right) = \pm \sqrt{2} \sin\left(\frac{n\pi}{2}\right).$$
For odd $n$, $\sin\left(\frac{n\pi}{2}\right) = \pm 1$, giving values $\pm \sqrt{2}$.
For even $n$, $\sin\left(\frac{n\pi}{2}\right) = 0$, giving the value $0$.
Hence, the set $\left\{\phi\left(\frac{1}{2}\right) : \phi \in S\right\} = \{-\sqrt{2}, 0, \sqrt{2}\}$.
Thus, option D is correct.
Q4mcq
For a variable $x$, consider the $\mathbb{R}$ - vector space $V = \left\{a_0 + a_1 x + a_2 x^2 \mid a_1, a_2, a_3 \in \mathbb{R}\right\}$. Let $T : V \to V$ be the linear transformation defined by $T(f) = f + \frac{df}{dx}$, where $\frac{df}{dx}$ denotes the derivative of $f$ with respect to $x$. Which of the following statements is true ?
a.
$(T^3 - 3T^2 + 3T)^{2025}(x) = x$
Correct Answer
b.
$(T^3 - 3T^2 + 3T)^{2025}(x) = x + 1$
c.
$(T^3 - 3T^2 + 3T)^{2025}(x) = 2025!x$
d.
$(T^3 - 3T^2 + 3T)^{2025}(x) = 2025!x + 1$
Step-by-Step AI Explanation
Verified Method
Let $D = \frac{d}{dx}$ be the differentiation operator. Then $T = I + D$.
Notice that:
$$T^3 - 3T^2 + 3T = T^3 - 3T^2 + 3T - I + I = (T - I)^3 + I.$$
Since $T - I = D$, we have:
$$T^3 - 3T^2 + 3T = D^3 + I.$$
On the space of polynomials $V = \{a_0 + a_1 x + a_2 x^2\}$, for any polynomial of degree $\le 2$, the third derivative is zero, i.e., $D^3 = 0$.
Therefore, on the space $V$:
$$T^3 - 3T^2 + 3T = 0 + I = I.$$
Consequently, for any integer $k \ge 1$, we have:
$$(T^3 - 3T^2 + 3T)^k = I^k = I.$$
Applying this to $x$ for $k = 2025$:
$$(T^3 - 3T^2 + 3T)^{2025}(x) = I(x) = x.$$
Hence, option A is correct.
Q4mcq
Consider $X = \{u \mid u : [0, 1] \to \mathbb{R}\text{ is continuous and } u(0) = 0\}$ with the sup norm $\|u\| = \sup_{x \in [0, 1]} |u(x)|$. Let $T(u) = \int_0^1 u(t)\,dt$ and $S = \{T(u) : u \in X,\, \|u\| \le 1\}$. Which of the following statements are true ?
a.
$S$ is an unbounded subset of $\mathbb{R}$.
b.
$S$ is a bounded subset of $\mathbb{R}$ and $\sup(S) = 1$.
Correct Answer
c.
There exists $u \in X$ such that $\|u\| = 1$ and $T(u) = 1$.
d.
$S$ is a closed subset of $\mathbb{R}$.
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Q5mcq
Let $V$ be the $\mathbb{R}$ - vector space of $5 \times 5$ real matrices. Let $S = \{AB - BA \mid A, B \in V\}$ and $W$ denote the subspace of $V$ spanned by $S$. Let $T : V \to \mathbb{R}$ be the linear transformation mapping a matrix $A$ to its trace. Which of the following statements is true ?
a.
$W = \ker(T)$
Correct Answer
b.
$W \subset \ker(T)$
c.
$W \cap \ker(T) \subset W$
d.
$W \cap \ker(T) \subset \ker(T)$
Step-by-Step AI Explanation
Verified Method
A well-known theorem in linear algebra (due to Albert and Muckenhoupt) states that over any field of characteristic 0, a matrix has trace zero if and only if it is a commutator or a sum of commutators. Thus, the subspace $W$ spanned by all commutators $AB - BA$ is precisely the space of all trace-zero matrices, which is $\ker(T)$. Therefore, $W = \ker(T)$.
Q5mcq
For a variable $x$, consider the $\mathbb{Q}$-vector space $V = \left\{ax^3 + bx^2 + cx + d \mid a, b, c, d \in \mathbb{Q}\right\}$.
Further, let $A = \{f : V \to \mathbb{Q} \mid f \text{ is a } \mathbb{Q}\text{-linear transformation}\}$, and $B = \{f \in A \mid f(1) = 0\}$.
Which of the following statements are true ?
a.
If $f \in B$, then $\dim \ker f = 3$
b.
$\dim B = 3$
Correct Answer
c.
$\dim A = 4$
Correct Answer
d.
If $f \in A$, then the image of $f$ is a one-dimensional $\mathbb{Q}$-vector space.
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Let $P$ be the population proportion of units possessing a certain attribute in a population of $N$ units. Let $p$ be the sample proportion in a simple random sample (without replacement) of $n$ units, ($2 \le n < N$). Then an unbiased estimator of $P(1 - P)$ is :
a.
$\frac{(N-n)}{Nn} p(1-p)$
b.
$\frac{(N-n)}{(N-1)n} p(1-p)$
c.
$\frac{n}{(n-1)} p(1-p)$
d.
$\frac{(N-n)n}{N(n-1)} p(1-p)$
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Define a topology $\tau$ on $\mathbb{R}$ as follows a subset $U$ of $\mathbb{R}$ is in the topology $\tau$ if and only if $U = \phi$ or $0 \in U$.
Which of the following statements are true ?
a.
The set of all irrational numbers is dense in $(\mathbb{R}, \tau)$.
b.
For each prime number $p$, the set $\{0, \sqrt{p}\}$ is dense in $(\mathbb{R}, \tau)$.
c.
$[0, 1]$ is compact in $(\mathbb{R}, \tau)$.
d.
$(\mathbb{R}, \tau)$ is Hausdorff.
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Let $\mathbb{D} = \{z = x + iy \in \mathbb{C} : |z| < 1\}$ be the open unit disc and $f : \mathbb{D} \to \mathbb{C}$ a holomorphic function such that $f(0) = 0$. Let $\psi(z) = |f(z)|^2$ and $\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} \equiv 0$. Which of the following statements is FALSE ?
a.
$f$ can be extended to $\mathbb{C}$ as an entire function.
b.
$f$ must have infinitely many zeros in $\mathbb{D}$.
c.
$f$ is not a polynomial.
d.
$\exp(f)$ cannot take every complex value.
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Consider the Markov chain with state space $\{0, 1, 2\}$ and the transition probability matrix $P$ given by:
$$P = \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{3}{4} & 0 & \frac{1}{4} \\ \frac{3}{4} & \frac{1}{4} & 0 \end{pmatrix}$$
Let $P^{(n)} = \left(\left(P_{ij}^{(n)}\right)\right)$ denote the $n$-step transition probability matrix. Then, which of the following statements are true ?
a.
$P_{00}^{(2)} = \frac{3}{4}$
b.
$P_{10}^{(3)} = \frac{39}{64}$
c.
The stationary probability that the chain is in state 2 is $\frac{2}{7}$
d.
State 1 is transient
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We say that a group $G$ has property $(A)$ if every non-trivial homomorphism from $G$ to any group is injective. Which of the groups has property $(A)$ ?
a.
The cyclic group of order 6.
b.
The symmetric group $S_5$.
c.
The alternating group $A_5$.
d.
The dihedral group with ten elements.
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For integers $n > 1$, let $G(n)$ denote the number of groups of order $n$, up to isomorphism, i.e. $G(n)$ is the number of isomorphism classes of groups of order $n$. Which of the following statements is true ?
a.
If $G(n) = 1$, then $n$ is prime.
b.
$G(8) = 2$
c.
If $\gcd(n, \phi(n)) > 1$, then $G(n) > 1$. (Here, $\phi$ denotes the Euler $\phi$-function)
d.
$\limsup_{n \to \infty} G(n) = 2$
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Consider the non-homogeneous ordinary differential equation (ODE)
$$\frac{d^2 y}{dx^2} + 5 \frac{dy}{dx} + 6y = \sin(e^{-5x}), \quad x > 0$$
Then which of the following statements are true ?
a.
Every solution of the ODE is bounded on $(0, \infty)$.
b.
There exists a solution of the ODE which is unbounded on $(0, \infty)$.
c.
Every solution of the ODE is unbounded on $(0, \infty)$.
d.
Every solution of the ODE tends to zero as $x \rightarrow \infty$.
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Let $f \in \mathbb{R}[x]$ be a product of distinct monic irreducible polynomials $P_1, P_2, \dots, P_n$, where $n \geq 2$. Let $(f)$ denote the ideal generated by $f$ in the ring $\mathbb{R}[x]$. Which of the following statements are true ?
a.
$\mathbb{R}[x]/(f)$ is a field.
b.
$\mathbb{R}[x]/(f)$ is a finite dimensional $\mathbb{R}$ - vector space.
c.
$\mathbb{R}[x]/(f)$ is a direct sum of fields, each of which is isomorphic to $\mathbb{R}$ or $\mathbb{C}$.
d.
There are no non-zero elements $a \in \mathbb{R}[x]/(f)$ such that $a^m = 0$ for some $m \geq 1$.
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$\{m + ne^{\frac{2\pi i}{3}} \mid m, n \in \mathbb{Z}\}$ is a dense subset of $\mathbb{C}$.
b.
Open connected subsets of $\mathbb{R}^3$ need not be path connected.
c.
Let $X$ be a topological space and $p: X \to \mathbb{R}$ a continuous surjective open map. If $p^{-1}(\{\alpha\})$ is connected for every $\alpha \in \mathbb{R}$, then $X$ must be connected.
d.
Compact subsets of any infinite topological space are closed.
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Define $S := \left\{ y \in C^1[-1, 1] : y(-1) = -1, \; y(1) = 3 \right\}$. Let $\phi$ be the extremal of the functional $J : S \rightarrow \mathbb{R}$ given by $J[y] = \int_{-1}^1 \left[(y')^3 + (y')^2\right] dx$. Define $\|y\|_\infty := \max_{x \in [-1, 1]} |y(x)|$ for every $y \in S$ and let
$$B_0(\phi, \varepsilon) := \left\{ y \in S : \|y - \phi\|_\infty < \varepsilon \right\}, \quad B_1(\phi, \varepsilon) := \left\{ y \in S : \|y' - \phi'\|_\infty < \varepsilon \right\}$$
Then which of the following statements are true ?
a.
$\phi(x) = 2x + 1$ for every $x \in [-1, 1]$
b.
There exists $\varepsilon > 0$ such that $J[y] \geq J[\phi]$ for every $y \in B_0(\phi, \varepsilon)$.
c.
There exists $\varepsilon > 0$ such that $J[y] \geq J[\phi]$ for every $y \in B_1(\phi, \varepsilon)$.
d.
There exists $\varepsilon > 0$ such that $J[y] \leq J[\phi]$ for every $y \in B_1(\phi, \varepsilon)$.
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Let $X$, $Y$ and $Z$ be random variables such that
$$S = \begin{pmatrix} X & Y \\ Y & Z \end{pmatrix} \sim W_2(10, \Sigma),$$
where $W_2$ denotes the Wishart distribution and
$$\Sigma = \begin{pmatrix} 1 & \frac{1}{2} \\ \frac{1}{2} & 1 \end{pmatrix}.$$
Define $T = Z - \frac{Y^2}{X}$. Then, $\text{Var}(T)$ equals
a.
$\frac{77}{6}$
b.
$\frac{81}{8}$
c.
$\frac{83}{9}$
d.
$\frac{79}{7}$
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A group $G$ is said to be divisible if for every $y \in G$ and for every positive integer $n$, there exists $x \in G$ such that $x^n = y$. Which of the following groups are divisible ?
a.
$\mathbb{Q}$ with ordinary addition.
b.
$\mathbb{C}\backslash\{0\}$ with ordinary multiplication.
c.
The cyclic group of order 5.
d.
The symmetric group $S_5$.
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Let $X_1, X_2, \dots, X_n$ ($n \ge 2$) be a random sample from a gamma distribution with shape parameter $\alpha > 0$ and scale parameter $\beta = 1$. For a suitable constant $C$, the rejection region of the most powerful test for testing $H_0 : \alpha = 1$ against $H_1 : \alpha = 2$ is of the form
a.
$\prod_{i=1}^n X_i > C$
b.
$\sum_{i=1}^n X_i > C$
c.
$\prod_{i=1}^n X_i < C$
d.
$\sum_{i=1}^n X_i < C$
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Let $f(x)$ be the polynomial of degree at most 2 that interpolates the data $(-1, 2)$, $(0, 1)$ and $(1, 2)$. If $g(x)$ is a polynomial of degree at most 3 such that $f(x) + g(x)$ interpolates the data $(-1, 2)$, $(0, 1)$, $(1, 2)$ and $(2, 17)$, then
a.
$f(5) + g(3) = 50$
b.
$2f(5) - g(3) = 4$
c.
$f(1) + g(3) = 50$
d.
$f(5) + g(3) = 74$
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Consider the system of two particles with total kinetic energy $T = \frac{5}{2}\dot{x}^2 + l^2\dot{\theta}^2 + 2l\dot{x}\dot{\theta}\cos\theta$, and Lagrangian $L = \frac{5}{2}\dot{x}^2 + l^2\dot{\theta}^2 + 2l\dot{x}\dot{\theta}\cos\theta + 2gl\cos\theta$, where $x, \theta$ are generalized coordinates, and $g, l$ are positive constants. Then the non-zero frequency of the normal mode of the system with small oscillation ($|\theta| \ll 1$) is:
a.
$\frac{5}{3}\sqrt{\frac{g}{l}}$
b.
$\sqrt{\frac{5g}{3l}}$
c.
$\frac{5}{2}\sqrt{\frac{g}{l}}$
d.
$\sqrt{\frac{5g}{2l}}$
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Two blocks of equal mass $m$ are connected by a flexible inelastic cord of mass $M$. One block is placed on a smooth horizontal table, the other block hangs over the edge. The total potential energy of the entire cord is given by $\frac{-Mg}{2l} x^2$, where $x$ is the distance of the hanging block from the edge of the table, $l$ is the length of the cord, and $g$ is the gravitational acceleration. Then,
a.
$\ddot{x} = -\frac{l}{g}\frac{ml + Mx}{2m + M}$
b.
$\ddot{x} = -\frac{l}{g}\frac{Ml + mx}{m + M}$
c.
$\ddot{x} = \frac{g}{l}\frac{ml + Mx}{m + M}$
d.
$\ddot{x} = \frac{g}{l}\frac{ml + Mx}{2m + M}$
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Let $V$ be the $\mathbb{R}$ - vector space of real valued continuous functions on the interval $[0, \pi]$ with the inner product given by $\langle f, g\rangle = \int_0^\pi f(x)g(x)dx$. Let $S = \{\sin(x), \cos(x), \sin^2(x), \cos^2(x)\}$ and $W$ be the subspace of $V$ generated by $S$. Which of the following statements are true ?
a.
$S$ is an basis of $W$.
b.
$S$ is an orthonormal basis of $W$.
c.
There exist $f, g \in S$ such that $\langle f, g\rangle = 0$.
d.
$S$ contains an orthonormal basis of $W$.
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Consider the sequences $(a_n)_{n \geq 1}$ and $(b_n)_{n \geq 1}$ defined by:
$$a_n = \frac{e^n + e^{-n}}{2} \text{ and } b_n = \frac{a_{n+1}}{a_n}$$
Which of the following statements is true?
a.
For each $x \in \mathbb{R}$, there exists an $n$ such that $a_n > x$.
b.
For each $x \in \mathbb{R}$, there exists an $n$ such that $a_n < x$.
c.
For each $x \in \mathbb{R}$, there exists an $n$ such that $b_n > x$.
d.
For each $x \in \mathbb{R}$, there exists an $n$ such that $b_n < x$.
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Let $P(z)$ be a non-constant polynomial over $\mathbb{C}$. Given $R > 0$, let $S_R = \{z \in \mathbb{C} : |P(z)| < R\}$. Which of the following statements are true ?
a.
$S_R$ is an open subset of $\mathbb{C}$.
b.
$S_R$ is a bounded subset of $\mathbb{C}$.
c.
$|P(z)| = R$ for every $z$ on the boundary of $S_R$.
d.
Every connected component of $S_R$ contains a zero of $P(z)$.
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In an examination question paper, all questions are 'True' or 'False' type. These are arranged in such a way that three fourth of times a question with answer 'True' is followed by a question with answer 'True'. Also two-third of times a question with answer 'False' is followed by a question with answer 'False'. If the question paper has 100 questions, the approximate probability that the correct answer of the $100^{\text{th}}$ question is 'True', is:
a.
$\frac{3}{7}$
b.
$\frac{4}{7}$
c.
$\frac{3}{4}$
d.
$\frac{5}{6}$
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Let $\lambda \in \mathbb{R}$, and $K : [0, 1] \times [0, 1] \to \mathbb{R}$ be a function such that every solution of the boundary value problem:
$$\frac{d^2u}{dx^2}(x) + \lambda u(x) = 0; \quad \frac{du}{dx}(0) = u(0), \quad \frac{du}{dx}(1) = 0$$
satisfies the integral equation, $u(x) + \lambda \int_0^1 K(x, t)u(t)dt = 0$
Then,
a.
$K(x, t) = \begin{cases} (1+x)(1-t), & 0 \leq x \leq t \leq 1, \\ (1+t)(1-x), & 0 \leq t < x \leq 1 \end{cases}$
b.
$K(x, t) = \begin{cases} -1-x, & 0 \leq x \leq t \leq 1, \\ -1-t, & 0 \leq t \leq x \leq 1 \end{cases}$
c.
$K(x, t) = \begin{cases} 1-x^2, & 0 \leq x \leq t \leq 1, \\ 1-t^2, & 0 \leq t < x \leq 1 \end{cases}$
d.
$K(x, t) = \begin{cases} (1+x)(t-1), & 0 \leq x \leq t \leq 1, \\ (1+t)(x-1), & 0 \leq t < x \leq 1 \end{cases}$
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For a positive real number $a$, $\sqrt{a}$, denotes the positive square root of $a$. Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined by :
$$f(x, y) = \begin{cases} \frac{x}{|x|} \sqrt{x^2 + y^2} & ; & x \neq 0 \\ 0 & ; & x = 0 \end{cases}$$
Which of the following statements are true ?
a.
$f$ is continuous at $(0, 0)$.
b.
The partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist at $(0, 0)$.
c.
$f$ is differentiable at $(0, 0)$.
d.
$f$ is not differentiable at $(0, 0)$.
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For integers $m, n \geq 1$, let $I_{m, n} = \frac{1}{2\pi i} \int_C z^m \bar{z}^n dz$, where $C$ is the circle $\{z \in \mathbb{C} : |z| = 1\}$ oriented counter-clockwise. Which of the following statements is true?
a.
$I_{m, n} = 1$, if $m = n$
b.
$I_{m, n} = 1$, if $m + 1 = n$
c.
$I_{m, n} = 1$, if $m = n + 1$
d.
$I_{m, n} = 1$, if $m = n + 2$
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Let $f : [0, 1] \to \mathbb{R}$ be defined by $f(x) = \sin(x^2)$. Let $A = \lim_{n \to \infty} \left[ \sum_{k=1}^n f\left(\frac{k}{n}\right) - n \int_0^1 f(x)\,dx \right]$. Which of the following statements is true?
a.
$A = 0$
b.
$A = 1$
c.
$A = \frac{\sin(1)}{2}$
d.
$A = \sin\left(\frac{1}{4}\right)$
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If $\lambda \in \mathbb{R}$ and $p \in \mathbb{R}$ are such that the quadrature formula:
$$\int_{x_0}^{x_0 + h} f(x) dx \approx \lambda h (f(x_0) + f(x_0 + h)) + ph^3 (f''(x_0) + f''(x_0 + h))$$
is exact for all polynomials of degree as high as possible, then
a.
$2\lambda + 24p = 0$
b.
$7\lambda - 12p = 4$
c.
$2\lambda + 24p = -3$
d.
$7\lambda - 12p = 11$
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Let $f : \mathbb{R} \to \mathbb{R}$ be such that $\sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|} = L$, where $1 < L < \infty$. Let $h : \mathbb{R} \to \mathbb{R}$ be a differentiable function satisfying $|h'(x)| \leq \frac{3}{4}$ for all $x \in \mathbb{R}$. For $\alpha > 0$, define $g(x) = \alpha f(x) + h(x)$ for $x \in \mathbb{R}$. Consider the sequence $\{x_k\}_{k=0}^\infty$ defined by $x_{k+1} = g(x_k)$, $k = 0, 1, \dots$, where $x_0 \in \mathbb{R}$. The sequence $\{x_k\}_{k=0}^\infty$ converges to the solution of the equation $x = g(x)$ if
a.
$\alpha < \frac{2}{3L}$
b.
$\alpha < \frac{3}{2L}$
c.
$\alpha < 4L$
d.
$\alpha < \frac{1}{4L}$
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Let $v = (a, b, c) \in \mathbb{R}^3$ be a non-zero vector that lies in the orthogonal complement (with respect to the standard inner product) of the row-space of the matrix:
$$A = \begin{pmatrix} 2 & 2 & 7 \\ 3 & 1 & 4 \end{pmatrix}$$
If $a, b, c$ are all integers, then what is the smallest possible value of $|a + b + c|$ ?
a.
$5$
b.
$10$
c.
$15$
d.
$20$
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Let $\mathbb{C}[x, y]$ be the polynomial ring in two variables over $\mathbb{C}$. For which of the following ideals $I$, the quotient ring $\mathbb{C}[x, y]/I$ is NOT an integral domain?
a.
$I = (x, y)$
b.
$I = (x + y)$
c.
$I = (x^2 + y^2)$
d.
$I = (xy - 1)$
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Let $X_1$ and $X_2$ be random variables having absolutely continuous distribution functions. Let $h_i(t)$ denote the hazard function of $X_i, i = 1, 2$. If $h_1(t) \le h_2(t)$ for all $t \in \mathbb{R}$, then which of the following statements are true?
a.
$P(X_1 > 1) \ge P(X_2 > 1)$
b.
$P(X_1 > 1) \le P\left(X_2 > \frac{1}{2}\right)$
c.
$E(X_1) \ge E(X_2)$ provided both the expectations exist
d.
$h(t) = h_1(t) + h_2(t), t \in \mathbb{R}$, is the hazard function of the random variable $Y = \min \{X_1, X_2\}$
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Let $u = u(x, t)$ be a solution of the wave equation:
$$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0, \ x \in \mathbb{R}, \ t > 0$$
satisfying the condition $u(0, t) = 0$, $\forall\ t \geq 0$. Then which of the following statements is true?
a.
$u(x, t) = 0$, whenever $x = t$
b.
$u(x, t) = 0$, whenever $x = -t$
c.
$u(-x, t) = u(x, t)$, whenever $x > 0, t > 0$
d.
$u(-x, t) = -u(x, t)$ whenever $0 < x \leq t$
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For $b \in \mathbb{R}$, let $y_b = y_b(x)$ be the unique solution of the initial value problem
$$\frac{dy}{dx} = y^5 + y^4 + y^3 + y^2 + y + 1, \quad y(0) = b$$
defined on its maximal interval of existence $I_b$. Then which of the following statements are true?
a.
There exists an $\alpha \in (0, \infty)$ such that for every $b \in \mathbb{R}$ with $b > \alpha$, the solution $y_b$ is bounded above on $I_b$.
b.
There exists an $\alpha \in (0, \infty)$ such that for every $b \in \mathbb{R}$ with $b > \alpha$, the solution $y_b$ is bounded below on $I_b$.
c.
There exists an $\alpha \in (-\infty, 0)$ such that for every $b \in \mathbb{R}$ with $b < \alpha$, the solution $y_b$ is bounded above on $I_b$.
d.
There exists an $\alpha \in (-\infty, 0)$ such that for every $b \in \mathbb{R}$ with $b < \alpha$, the solution $y_b$ is bounded below on $I_b$.
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Let $A = \begin{bmatrix} 0 & a & 0 \\ 0 & 0 & b \\ c & 0 & 0 \end{bmatrix}$, where $a, b, c$ are real numbers with $abc = 1$. If $B = A + A^2 + A^3$, then which of the following statements is true ?
a.
$\det B = 1$
b.
$\det A = 0$
c.
$\operatorname{rank}(B) = 2$
d.
$\operatorname{rank}(B^2) = 1$
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Let $A, B, C$ be topological spaces such that $A$ is homeomorphic to $B$, $B$ is a subspace of $C$ and $\overline{B} = C$. Let $C$ be homeomorphic to a subspace $W$ of $A$. Which of the following statements are FALSE ?
a.
The spaces $B, \overline{W}, C$ are homeomorphic.
b.
The spaces $B, W, C$ are homeomorphic.
c.
If $C$ is compact, then $A, B, C$ are homeomorphic.
d.
If $A$ is connected, then $B$ and $C$ are connected.
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Consider the polynomial $f(x) = x^{2025} - 1$ over $\mathbb{F}_5$, where $\mathbb{F}_5$ is the field with five elements. Let $S$ be the set of all roots of $f$ in an algebraic closure of the field $\mathbb{F}_5$. Which of the following statements are true ?
a.
$S$ is a cyclic group.
b.
$S$ has $\phi(2025)$ elements, where $\phi$ denotes the Euler $\phi$-function.
c.
$S$ has $\phi(2025)$ generates, where $\phi$ denotes the Euler $\phi$-function.
d.
$S$ has 81 elements.
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Suppose that the differential equation:
$$\frac{d^2 y}{dx^2} + P(x)\frac{dy}{dx} + e^{2x}y = 0, \ x \in \mathbb{R}$$
transforms into a second order differential equation with constant coefficients under the change of independent variable given by $s = s(x)$ satisfying $\frac{ds}{dx}(0) = 1$. Then which of the following statements is true ?
a.
$e^{-x}(P(x) + 1)$ is a constant function on $\mathbb{R}$.
b.
$e^{-2x}P(x)$ is a constant function on $\mathbb{R}$.
c.
$s(x) = \frac{e^{2x}}{2}, \ x \in \mathbb{R}$
d.
$P(x) \rightarrow 1$ as $x \rightarrow \infty$
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Consider the bilinear form $B : \mathbb{R}^4 \times \mathbb{R}^4 \rightarrow \mathbb{R}$ defined by $B(x, y) = x_1 y_3 + x_2 y_4 - x_3 y_1 - x_4 y_2$, where $x = (x_1, x_2, x_3, x_4)$ and $y = (y_1, y_2, y_3, y_4)$ in $\mathbb{R}^4$. Let $A$ denote the matrix of $B$ with respect to the standard ordered basis of $\mathbb{R}^4$. Which of the following statements is true ?
a.
$\det A = 0$
b.
$\det A = -1$
c.
$B(x, x) \neq 0$ for all non-zero $x \in \mathbb{R}^4$
d.
If $x \in \mathbb{R}^4$ is non-zero, then there exists $y \in \mathbb{R}^4$ such that $B(x, y) \neq 0$
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Let $(X_1, X_2)$ be a bivariate normal random vector with $E(X_1) = 1$, $E(X_2) = 0$, $Var(X_1) = 1$, $Var(X_2) = 1$, and correlation coefficient $\frac{1}{2}$. Let $U$ be a $U(0, 1)$ random variable, which is independent of $(X_1, X_2)$. If $Z = \frac{UX_1 + X_2 - U}{\sqrt{U^2 + U + 1}}$, then which of the following statements are true ?
a.
The distribution of $Z$ is symmetric about 0.
b.
$E(Z^2) = 2$
c.
$Var(Z^2) = 1$
d.
$Z$ and $U$ are independent random variables.
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Consider the power series:
$$\sum_{n=1}^\infty \frac{n^{n^2}}{(n+1)^{n^2}} x^n$$
with coefficients in real numbers $\mathbb{R}$. Which of the following statements is true ?
a.
The radius of convergence of the series is $\frac{1}{e}$.
b.
The series converges at $x = 5$.
c.
The series converges at $x = 3$.
d.
The series converges for all $x$ with $|x| < \frac{1}{2}$.
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For a $4 \times 4$ positive definite real symmetric matrix $A$ and real numbers $a, b, c, d$, consider the $5 \times 5$ matrix
$$H = \left(\begin{array}{c|cccc} 0 & a & b & c & d \\ \hline a & & & & \\ b & & A & & \\ c & & & & \\ d & & & & \end{array}\right)$$
Which of the following statements are necessarily true ?
a.
$\det(B) > 0$ for every non-zero $(a, b, c, d) \in \mathbb{R}^4$.
b.
$\det(B) > 0$ for infinitely many $(a, b, c, d) \in \mathbb{R}^4$.
c.
$\det(B) \le 0$ for every $(a, b, c, d) \in \mathbb{R}^4$.
d.
$\det(B) \le 0$ for infinitely many $(a, b, c, d) \in \mathbb{R}^4$.
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Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x) = 0$ for all $x \le 0$ and for all $x \ge 1$. Define,
$$F(x) = \sum_{n=-\infty}^\infty f(x+n), \quad x \in \mathbb{R}$$
Which of the following statements are true ?
a.
$F$ is bounded.
b.
$F$ is continuous on $\mathbb{R}$.
c.
$F$ is uniformly continuous on $\mathbb{R}$.
d.
$F$ is not uniformly continuous on $\mathbb{R}$.
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Given that $y_1(x) = e^{2x}$ is a solution of the ordinary differential equation (ODE)
$$x \frac{d^2 y}{dx^2} - (3 + 4x) \frac{dy}{dx} + (4x + 6) y = 0, \ x > 0.$$
Let $y_2 = y_2(x)$ be the solution of the ODE satisfying the conditions $y_2(1) = \frac{e^2}{4}, \frac{dy_2}{dx}(1) = \frac{3e^2}{2}$. Then which of the following statements is true?
a.
$y_2$ is a strictly increasing function on $(0, \infty)$.
b.
$e^{-2x} y_2(x) \rightarrow 1$ as $x \rightarrow \infty$
c.
$y_2$ is a strictly decreasing function on $(0, \infty)$
d.
$e^{-2x} y_2(x) \rightarrow 0$ as $x \rightarrow \infty$
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Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be the function defined by:
$$f(z) = e^{(\cos(1+i)) \sin z}$$
For $z = x + iy \in \mathbb{C}$, write $f(z)$ as $u(x, y) + iv(x, y)$, where $u, v$ are real-valued functions. Which of the following is the value of $\frac{\partial u}{\partial x}(0, 0)$?
a.
$0$
b.
$\left( e + \frac{1}{e} \right) \frac{\cos 1}{2}$
c.
$\left( e - \frac{1}{e} \right) \frac{\cos 1}{2}$
d.
$1$
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If $x = x(t)$, $y = y(t)$ is the solution of the initial value problem
$$\frac{dx}{dt} = x - 4e^{-2t}y, \quad \frac{dy}{dt} = e^{2t}x - y, \quad x(0) = 1, \; y(0) = 1,$$
then which of the following statements are true ?
a.
$\lim_{t \to \infty} t^{-2}x(t) y(t) = 0$
b.
$x(1) = 0, \; y\left(\frac{1}{2}\right) = 0$
c.
$x\left(\frac{1}{2}\right) = 0, \; y(1) = 0$
d.
$\lim_{t \to \infty} t^{-2}x(t) y(t) = 2$
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Let $u = u(x, y)$ be the solution of the Cauchy problem:
$$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = u, \ (x, y) \neq (0, 0); \ u(x, 1) = \sqrt{1 + x^2}, \ x \in \mathbb{R}$$
Then which of the following statements is true?
For every integer $n \ge 2$, consider a $\mathbb{C}$ - linear transformation $T : \mathbb{C}^n \to \mathbb{C}^n$. Let $V$ be a subspace of $\mathbb{C}^n$ such that $T(V) \subseteq V$. Which of the following statements are necessarily true ?
a.
There exists a subspace $W$ of $\mathbb{C}^n$ such that $\mathbb{C}^n = V + W$ and $V \cap W = \{0\}$
b.
There exists a subspace $W$ of $\mathbb{C}^n$ such that $T(W) \subseteq W$, $\mathbb{C}^n = V + W$ and $V \cap W = \{0\}$
c.
Suppose that there exists a positive integer $k$ such that $T^k$ is the identity map. Then there exists a subspace $W$ of $\mathbb{C}^n$ such that $T(W) \subseteq W$, $\mathbb{C}^n = V + W$ and $V \cap W = \{0\}$.
d.
Suppose that there exists a subspace $W$ of $\mathbb{C}^n$ such that $T(W) \subseteq W$, $\mathbb{C}^n = V + W$ and $V \cap W = \{0\}$. Then there exists a positive integer $k$ such that $T^k$ is the identity map.
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Let $\mathbb{H} = \{z = x + iy \in \mathbb{C} \mid y > 0\}$ and $f : \mathbb{H} \to \mathbb{C}$ be a non-constant holomorphic function satisfying $|f(z)| < 1$ for all $z \in \mathbb{H}$. Which of the following statements is true ?
a.
$\lim_{y \to +\infty} f'(iy) = 0$
b.
$\lim_{y \to +\infty} f'(iy)$ is a complex number with absolute value $1$
c.
$\lim_{y \to +\infty} |f'(iy)| = +\infty$
d.
$\lim_{y \to +\infty} f'(iy)$ is not a real number
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For integers $n \geq 0$, let $f_n : [-1, 0] \to \mathbb{R}$ be defined by
$$f_n(x) = \frac{x}{(1 - x)^n}$$
Which of the following statements is true about the series $\sum_{n=0}^\infty f_n$ ?
a.
The series is neither absolutely convergent nor uniformly convergent.
b.
The series is both absolutely convergent and uniformly convergent.
c.
The series is absolutely convergent but not uniformly convergent.
d.
The series is uniformly convergent but not absolutely convergent.
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Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by :
$$f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}$$
Which of the following statements are true ?
a.
$\lim_{x \to 0} f(x)$ exists
b.
$f$ is continuous at 0
c.
$f$ is differentiable at 0
d.
$\lim_{x \to 0} f'(x)$ does not exist.
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Let $U$ denote the span of $\{e^t, e^{2t}, e^{3t}\}$ in the real vector space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Consider the $\mathbb{R}$-vector spaces
$$V = \{f : U \to \mathbb{R} \mid f \text{ is an } \mathbb{R}\text{-linear transformation}\}$$
$$W = \{f \in V \mid f(e^{3t}) = 0\}.$$
Which of the following statement is true ?
a.
Both $V$ and $W$ are infinite-dimensional.
b.
$\dim V = 3$ and $\dim W = 1$.
c.
$\dim V = 3$ and $\dim W = 2$.
d.
$V$ is infinite-dimensional and $\dim W = 0$.
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For a variable $x$, consider the real vector space $V = \{ax^3 + bx^2 + cx + d \mid a, b, c, d \in \mathbb{R}\}$. Let $D : V \rightarrow V$ be the linear transformation where $D(f)$ is the derivative of $f$ with respect to $x$, and $M : V \rightarrow V$ be the linear transformation $M(f) = xD(f)$. Which of the following statements are true ?
a.
$DM \neq MD$
b.
$D + M$ is invertible
c.
$DM$ is invertible
d.
$\text{rank}\,(DM) = \text{rank}\,(MD)$
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Suppose that $X \sim \text{binomial}(9, \theta)$, $0.7 < \theta < 1$, and $Y \sim \text{Poisson}(\lambda)$, $\lambda > 0$. If $3E(Y) = E(X)$, then which of the following is true ?
a.
$\text{Var}(X) > 3\text{Var}(Y)$
b.
$2\text{Var}(Y) < \text{Var}(X) < 3\text{Var}(Y)$
c.
$\text{Var}(Y) < \text{Var}(X) < 2\text{Var}(Y)$
d.
$\text{Var}(X) < \text{Var}(Y)$
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Let $u = u(x, t)$ be the solution of the initial-boundary value problem:
$$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \, (x, t) \in (0, 1) \times (0, \infty),$$
$$u(x, 0) = 4x(1 - x), \, x \in [0, 1],$$
$$u(0, t) = u(1, t) = 0, \, t \ge 0$$
Then which of the following statements are true ?
a.
$\lim_{t\to\infty} u(x, t) = 0$ for all $x \in (0, 1)$
b.
$u(x, t) = u(1-x, t)$ for all $x \in (0, 1)$, $t > 0$
c.
$\int_0^1 (u(x, t))^2 dx$ is a non-increasing function of $t$.
d.
$\int_0^1 (u(x, t))^2 dx$ is a non-decreasing function of $t$.
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Let $(a_n)_{n\geq 1}$, $(b_n)_{n\geq 1}$ and $(c_n)_{n\geq 1}$ be sequences given by:
$$a_n = (-1)^n (1 + e^{-n}), \quad b_n = \max \{a_1, \dots, a_n\}, \quad \text{and} \quad c_n = \min \{a_1, \dots, a_n\}.$$
Which of the following statements are true ?
Let $A, B$ be $2 \times 2$ matrices with real entries, and $M = AB - BA$. Let $I_2$ denote the $2 \times 2$ identity matrix. Which of the following statements are necessarily true ?
a.
If $A$ and $B$ are upper triangular, then $M$ is diagonalizable over $\mathbb{R}$.
b.
If $A$ and $B$ are diagonalizable over $\mathbb{R}$, then $M$ is diagonalizable over $\mathbb{R}$.
c.
If $A$ and $B$ are diagonalizable over $\mathbb{R}$, then there exists $\lambda \in \mathbb{R}$ such that $M = \lambda I_2$.
d.
There exists $\lambda \in \mathbb{R}$ such that $M^2 = \lambda I_2$.
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Let $g : \mathbb{R} \to \mathbb{R}$ be a continuous function. Define $f(x) = \int_0^x (x-t) g(t) dt$, $x \in \mathbb{R}$. Which of the following statements are true?
a.
$f(0) = 0$
b.
$f'(0)$ exists and $f'(0) = 0$
c.
$f''(0)$ exists and $f''(0) = g(0)$
d.
$f''(0)$ exists but $f''(0) \neq g(0)$
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Let $f : \mathbb{C} \setminus \{-1, 1\} \to \mathbb{C}$ be a holomorphic function that does not take any value in the set $\{z \in \mathbb{C} : |z-1| < 1\}$. Which of the following statements are true ?
a.
$f$ is constant.
b.
$f$ has removable singularities at $-1$ and $1$.
c.
$f$ is bounded.
d.
$f$ has either poles or essential singularities at $-1$ and $1$.
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For any $b \in \mathbb{R}$, let $S(b)$ denote the set of all broken extremals with one corner of the variational problem:
$$\text{Minimize } J[y] = \int_0^1 ((y')^4 - 3(y')^2) dx,$$
subject to $y(0) = 0$, $y(1) = b$.
Then which of the following statements are true?
a.
$S(2)$ has exactly two elements.
b.
$S\left(\frac{1}{2}\right)$ has exactly one element.
c.
$S(2)$ is empty.
d.
$S\left(\frac{1}{2}\right)$ has exactly two elements.
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Let $M_2(\mathbb{R})$ denote the $\mathbb{R}$ - vector space of $2 \times 2$ matrices with real entries. Let
$$A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix} \text{ and } B = \begin{pmatrix} -1 & 0 \\ 1 & 5 \end{pmatrix}$$
Define a linear transformation $T : M_2(\mathbb{R}) \rightarrow M_2(\mathbb{R})$ by $T(X) = AXB^t$, where $B^t$ denotes the transpose of the matrix $B$. Which of the following statements are true?
a.
$\det(T) = 225$
b.
$\det(T) = -225$
c.
$\text{Trace}(T) = 16$
d.
$\text{Trace}(T) = -16$
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Let $x, y \in \mathbb{R}$ with $x < y$. Then there exists $r \in \mathbb{Q}$ such that $x < \frac{2^{2024} r}{e} < y$.
b.
Let $(a_n)_{n \ge 2}$ be a sequence of positive real numbers. If there exists a positive real number $L$ such that
$$\limsup_{n \to \infty} \frac{a_n}{\log n} = L \text{, then } \limsup_{n \to \infty} a_n < \infty.$$
c.
The set of all finite subsets of $\mathbb{Q}$ is countably infinite.
d.
The set of continuous functions from $\mathbb{R}$ to the set $\{0, 1\}$ is infinite.
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The value of the Euler $\phi$-function is even for all integers $n \ge 3$.
b.
Let $G$ be a finite group and $S$ a subset of $G$ with $|S| > \frac{|G|}{2}$. Then $\{ab : a, b \in S\} = G$.
c.
The polynomial ring $\mathbb{R}[x_1, ..., x_n]$ is a Euclidean domain for all integers $n \ge 1$.
d.
The subset $\left\{f \in C([0, 1]) : f\left(\frac{1}{2}\right) = 0\right\}$ of the ring $C([0, 1])$ of continuous functions from $[0, 1]$ to $\mathbb{R}$ is a prime ideal.
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Let $f : \mathbb{C} \to \mathbb{C}$ be an entire function such that $f(z) = f(i\,z)$ for all $z \in \mathbb{C}$. Which of the following statements are true ?
a.
$f(z) = f(-z)$ for all $z \in \mathbb{C}$.
b.
$f'(0) = f''(0) = f'''(0) = 0$.
c.
There is an entire function $g : \mathbb{C} \to \mathbb{C}$ such that $f(z) = g(z^4)$ for all $z \in \mathbb{C}$.
d.
$f$ is necessarily a constant function.
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