Question 1

**1 Section I (18 Marks)** 1. Which of the following statements is/are true? [Marks: 3] (a) $$1 + 4 + 7 + \cdots + (3n-2) = \frac{(3n-1)n}{2}$$ (b) There is no bijection between the set of natural numbers and the set of integers. (c) $3 \mid n(n+1)(n+2)$ for all $n \in \mathbb{N}$. (d) 3 is the remainder when $3^{60}$ is divided by 7. 2. Consider the sequence $\{s_n\}$ defined by $$s_n = 1 + \frac{1}{4} + \frac{1}{7} + \ldots + \frac{1}{3n-2}$$ for $n \geq 1$. Which of the following options is/are true? [Marks: 3] (a) $\{s_n\}$ is a convergent sequence. (b) $\{s_n\}$ is a Cauchy sequence. (c) $\{s_n\}$ is an increasing sequence. (d) $\{s_n\}$ is a divergent sequence. 3. Which of the following is the infinite continued fraction of the golden ratio $\dfrac{1+\sqrt{5}}{2}$? [Marks 3] (a) $1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{\ldots}}}$ (b) $1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\ldots}}}$ (c) $1 + \cfrac{1}{2 + \cfrac{1}{3 + \cfrac{1}{\ldots}}}$ (d) None of these 4. Consider a function $f(x) = |x-1|$ in the interval $[-3, 4]$. Which of the following options is/are true? [Marks: 3] (a) $f(x)$ is continuous. (b) $f(x)$ is uniformly continuous. (c) $f(x)$ is a differentiable. (d) $f(x)$ is a Riemann Integrable. 5. Which of the following binomial coefficients gives the cardinality of the set [Marks: 3] $$\{(i_1, i_2, i_3, i_4) \mid 1 \leq i_1 \leq i_2 \leq i_3 \leq i_4 \leq 10\}?$$ (a) $\binom{10}{4}$ (b) $\binom{13}{4}$ (c) $\binom{14}{4}$ (d) $\binom{14}{3}$ 6. The number of permutations on six letters that have exactly 14 inversions is equal to which of the following? [Marks: 3] (a) 0 (b) 1 (c) The number of permutations of six letters that have exactly one inversion. (d) The number of permutations of six letters that have exactly two inversions. **2 Section II (22 Marks)** 1. Let $\phi(n)$ denote Euler's totient function, i.e., $\phi(n)$ is equal to the cardinality of the set $$\{1 \leq a \leq n \mid \gcd(a, n) = 1\}$$ Prove that $\phi(n)$ is even if $n > 2$. [Marks: 6] 2. Let $S(1, \sqrt{2})$ be the following subset of the real numbers: $$S(1, \sqrt{2}) = \{a + b\sqrt{2} \mid a, b \in \mathbb{Q}\}$$ where $\mathbb{Q}$ denotes the set of rational numbers. (a) Prove that $S(1, \sqrt{2})$ is closed under addition and multiplication. (b) Let $x = 3 + \sqrt{2}$. Find $y \in S(1, \sqrt{2})$ such that $xy = 1$. [6 marks] 3. Prove that there exists a positive real number $x$ such that $x^3 + x^2 + x = 22$. [6 marks] 4. Let $\{x_n\}$ be a sequence of real numbers such that $\lim_{n \to \infty} x_n = 1$. Prove that $$\lim_{n \to \infty} (x_n^2 - x_{n-1}x_{n+1}) = 0$$ [4 marks]