cmi-entrance

Q1 4 marks Groups Group Order and Structure Theorems View

For a field $F$, $F^{\times}$ denotes the multiplicative group ($F \backslash \{0\}, \times$). Choose the correct statement(s) from below:
(A) Every finite subgroup of $\mathbb{R}^{\times}$ is cyclic;
(B) The order of every non-trivial finite subgroup of $\mathbb{R}^{\times}$ is a prime number;
(C) There are infinitely many non-isomorphic non-trivial finite subgroups of $\mathbb{R}^{\times}$;
(D) The order of every non-trivial finite subgroup of $\mathbb{C}^{\times}$ is a prime number.

Q2 4 marks Proof True/False Justification View

Let $R$ be a commutative ring with 1 and $I$ and $J$ ideals of $R$. Choose the correct statement(s) from below:
(A) If $I$ or $J$ is maximal then $IJ = I \cap J$;
(B) If $IJ = I \cap J$, then $I$ or $J$ is maximal;
(C) If $IJ = I \cap J$, then $1 \in I + J$;
(D) If $1 \in I + J$ then $IJ = I \cap J$.

Q3 4 marks Proof True/False Justification View

Let $(X, d)$ and $(Y, \rho)$ be metric spaces and $f : X \longrightarrow Y$ a homeomorphism. Choose the correct statement(s) from below:
(A) If $B \subseteq Y$ is compact, then $f^{-1}(B)$ is compact;
(B) If $B \subseteq Y$ is bounded, then $f^{-1}(B)$ is bounded;
(C) If $B \subseteq Y$ is connected, then $f^{-1}(B)$ is connected;
(D) If $\{y_n\}$ is Cauchy in $Y$, then $\{f^{-1}(y_n)\}$ is Cauchy in $X$.

Q4 4 marks Matrices Linear Transformation and Endomorphism Properties View

Let $a, b \in \mathbb{R}$, and consider the $\mathbb{R}$-linear map $f : \mathbb{C} \longrightarrow \mathbb{C},\ z \mapsto az + b\bar{z}$. Choose the correct statement(s) from below:
(A) $f$ is onto (i.e., surjective) if $ab \neq 0$;
(B) $f$ is one-one (i.e., injective) if $ab \neq 0$;
(C) $f$ is onto if $a^2 \neq b^2$;
(D) if $a^2 = b^2$, $f$ is not one-one.

Q5 4 marks Implicit equations and differentiation Differentiability proof and derivative formula for abstract/matrix-valued functions View

Let $$f(x,y) = \begin{cases} \frac{x^3 y^3}{x^2 + y^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$$ Choose the correct statement(s) from below:
(A) $f$ is continuous on $\mathbb{R}^2$;
(B) $f$ is continuous at every point of $\mathbb{R}^2 \backslash \{(0,0)\}$;
(C) $f$ is differentiable at every point of $\mathbb{R}^2 \backslash \{(0,0)\}$;
(D) $f$ is not differentiable at $(0,0)$.

Q6 4 marks Groups Ring and Field Structure View

Let $K$ be the smallest subfield of $\mathbb{C}$ containing all the roots of unity. Choose the correct statement(s) from below:
(A) $\mathbb{C}$ is algebraic over $K$;
(B) $K$ has countably many elements;
(C) Irreducible polynomials in $K[X]$ do not have multiple roots;
(D) The characteristic of $K$ is zero.

Q7 4 marks Taylor series Identify a closed-form function from its Taylor series View

The power series $$\sum_{n=1}^{\infty} \frac{n^2 x^n}{n!}$$ equals
(A) $x^2 e^x$;
(B) $x e^x$;
(C) $(x^2 + x) e^x$;
(D) $(x^2 - x) e^x$;

Q8 4 marks Differential equations Qualitative Analysis of DE Solutions View

Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be twice continuously differentiable. Suppose further that $f''(x) \geq 0$ for every $x \in \mathbb{R}$. Choose the correct statement(s) from below:
(A) $f$ is bounded;
(B) $f$ is constant;
(C) If $f$ is bounded, then it is infinitely differentiable;
(D) $\int_0^x f(t)\,\mathrm{d}t$ is infinitely differentiable with respect to $x$.

Q9 4 marks Complex numbers 2 Properties of Analytic/Entire Functions View

Let $f(z)$ be a power-series (with complex coefficients) centred at $0 \in \mathbb{C}$ and with a radius of convergence 2. Suppose that $f(0) = 0$. Choose the correct statement(s) from below:
(A) $f^{-1}(0) = \{0\}$;
(B) If $f$ is a non-constant function on $\{|z| < 2\}$, then $f^{-1}(0) = \{0\}$;
(C) If $f$ is a non-constant function, then for all $\zeta \in \mathbb{C}$ with sufficiently small $|\zeta|$, the equation $f(z) = \zeta$ has a solution;
(D) $$\int_{\gamma} f^{(n)}(z)\,\mathrm{d}z = 0$$ for every $n \geq 1$, where $\gamma$ is a unit circle centred at 0, oriented clockwise, and $f^{(n)}$ is the $n$th derivative of $f(z)$.

Q10 4 marks Complex numbers 2 Contour Integration and Residue Calculus View

Let $z$ be a complex variable, and write $x = \Re(z)$ and $y = \Im(z)$ for the real and the imaginary parts, respectively. Let $f(z)$ be a complex polynomial. Let $R > 0$ be a real number and $\gamma$ the circle in $\mathbb{C}$ of radius $R$ and centre at 0, oriented in the counterclockwise direction. What is the value of $$\frac{1}{2\pi \imath R} \int_{\gamma} \left( \Re(f(z))\,\mathrm{d}x + \Im(f(z))\,\mathrm{d}y \right)$$

Q11 10 marks Proof True/False Justification View

Fix a non-negative integer $d$. Let $$\mathcal{A}_d := \{A \subseteq \mathbb{C} : A \text{ is the zero-set of a polynomial of degree } \leq d \text{ in } \mathbb{C}[X]\}.$$ Let $\mathcal{T}$ be the coarsest topology on $\mathbb{C}$ in which $A$ is closed for every $A \in \mathcal{A}_d$.
(A) Determine whether $\mathcal{T}$ is Hausdorff.
(B) Show that for every polynomial $f(X) \in \mathbb{C}[X]$, the function $\mathbb{C} \longrightarrow \mathbb{C}$ defined by $z \mapsto f(z)$ is continuous, where $\mathbb{C}$ (on both the sides) is given the topology $\mathcal{T}$.

Q12 10 marks Matrices Eigenvalue and Characteristic Polynomial Analysis View

Let $a_n,\ n \geq 0$ be complex numbers such that $\lim_n a_n = 0$.
(A) Show that $F(z) := \sum_{n \geq 0} a_n z^n$ is a holomorphic function on $\{z \in \mathbb{C} : |z| < 1\}$.
(B) Let $G(z)$ be a meromorphic function on $\{z \in \mathbb{C} : |z| < 2\}$, with a pole at 1. Show that $G \neq F$ on $\{z \in \mathbb{C} : |z| < 1\}$. (Hint: consider the function $(1-z)F(z)$ as $z \longrightarrow 1$.)

Q13 10 marks Number Theory Congruence Reasoning and Parity Arguments View

Let $|\cdot| : \mathbb{R} \longrightarrow \mathbb{R}_{\geq 0}$ be a function such that for every $x, y \in \mathbb{R}$, (i) $|x| = 0$ if and only if $x = 0$; (ii) $|x + y| \leq |x| + |y|$; (iii) $|xy| = |x||y|$. Show that the following are equivalent:
(A) The set $\{|n| : n \in \mathbb{Z}\}$ is bounded;
(B) $|x + y| \leq \max\{|x|, |y|\}$ for every $x, y \in \mathbb{R}$.

Q14 10 marks Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Show that the sequence $$\left[\int_0^1 |f(x)|^n\,\mathrm{d}x\right]^{\frac{1}{n}}$$ is convergent.

Q15 10 marks Matrices Linear Transformation and Endomorphism Properties View

Let $V$ be a subspace of the complex vector space $M_n(\mathbb{C})$. Suppose that every non-zero element of $V$ is an invertible matrix. Show that $\dim_{\mathbb{C}} V \leq 1$.

Q16 10 marks Groups Group Order and Structure Theorems View

Let $n$ be a positive integer such that every group of order $n$ is cyclic. Show the following.
(A) For all prime numbers $p$, $p^2$ does not divide $n$.
(B) If $p$ and $q$ are prime divisors of $n$, then $p$ does not divide $q - 1$. (Hint: Consider $2 \times 2$ matrices $$\left[\begin{array}{ll} x & y \\ 0 & 1 \end{array}\right]$$ with $x, y \in \mathbb{Z}/q\mathbb{Z}$ and $x^p = 1$.)
(C) Show that $(n, \phi(n)) = 1$, where $\phi(n)$ is the number of integers $m$ such that $1 \leq m \leq n$ with $\gcd(n, m) = 1$.

Q17* 10 marks Groups Symmetric Group and Permutation Properties View

Let $F$ be a field and $G = \mathrm{GL}_n(F)$. For $g \in G$, write $C_g = \{hgh^{-1} \mid h \in G\}$. Let $X = \{C_g \mid g \in G,\ \text{the order of } g \text{ is } 2\}$. Determine $|X|$.

Q18* 10 marks Proof Proof That a Map Has a Specific Property View

A compactification of a topological space $X$ is a compact topological space $Y$ which contains a dense subspace homeomorphic to $X$. Let $X = (0,1]$, in the subspace topology of $\mathbb{R}$ and $f : X \longrightarrow \mathbb{R},\ x \mapsto \sin\frac{1}{x}$. Show the following:
(A) $Y := [0,1]$ is a compactification of $X$, but $f$ does not extend to a continuous function $Y \longrightarrow \mathbb{R}$, i.e., there does not exist a continuous function $g : Y \longrightarrow \mathbb{R}$ such that $\left.g\right|_X = f$.
(B) $X$ is homeomorphic to the set $X_1 := \left\{\left.\left(t, \sin\frac{1}{t}\right)\right\rvert\, t \in X\right\} \subseteq \mathbb{R}^2$.
(C) The closure $Y_1$ of $X_1$ in $\mathbb{R}^2$ is a compactification of $X$.
(D) $f$ extends to a continuous function $Y_1 \longrightarrow \mathbb{R}$.

Q19* 10 marks Groups Ring and Field Structure View

Let $f(X) \in \mathbb{Z}[X]$ be a monic polynomial. Suppose that $\alpha \in \mathbb{C}$ and $3\alpha$ are roots of $f$.
(A) Show that $f(0) \neq 1$. (Hint: if $\zeta$ and $\zeta'$ are complex numbers satisfying monic polynomials in $\mathbb{Z}[X]$, then $\zeta\zeta'$ satisfies a monic polynomial in $\mathbb{Z}[X]$.)
(B) Assume that $f$ is irreducible. Let $K$ be the smallest subfield of $\mathbb{C}$ containing all the roots of $f$. Let $\sigma$ be a field automorphism of $K$ such that $\sigma(\alpha) = 3\alpha$. Show that $\sigma$ has finite order and that $\alpha = 0$.

Q20* 10 marks Proof Proof That a Map Has a Specific Property View

Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Define $g(0) = f(0)$ and $g(x) = \max\{f(y) \mid 0 \leq y \leq x\}$ for $0 < x \leq 1$. Show that $g$ is well-defined and that $g$ is a monotone continuous function.

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