Question 1

**1 Section I (18 Marks)** 1. True or False? [2 marks] If $a$ is relatively prime to $n$ then $\gcd(n - a, n) = 1$. 2. What is the digit in the units place in the decimal expansion of $2^{100}$? [2 marks] 3. Let $G$ be the graph on five vertices with adjacency matrix $$\begin{bmatrix} 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \end{bmatrix}$$ Which of the following hold for $G$? [3 marks] (a) The degree of every vertex in the graph is 2.\ (b) $G$ has six edges.\ (c) $G$ is three-colourable.\ (d) $G$ is planar. 4. Consider the function $f : \mathbb{R} \longrightarrow \mathbb{R}$ defined by $$f(x) = \begin{cases} \frac{|(x-1)(x-3)|}{x-3}, & \text{if } x \neq 3 \\ 2, & \text{if } x = 3 \end{cases}$$ Which of the following options is/are true? [3 marks] (a) $\lim\limits_{x \to 3^+} f(x) = f(3)$.\ (b) $\lim\limits_{x \to 3^-} f(x) = f(3)$.\ (c) $f(2) = -1$.\ (d) $f$ is continuous at $x = 3$. 5. Evaluate $\lim\limits_{\theta \to 0} \dfrac{\sin(5\theta)}{\sin(4\theta)}$. [2 marks] 6. Consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $(x) = |x - 1| + |x|$. Which of the following options is/are true? [3 marks] (a) The function $f(x)$ is continuous at $x = 0$ and $x = 1$.\ (b) The function $f(x)$ is differentiable at $x = 0$ and $x = 1$.\ (c) The function $f : \mathbb{R} \to \mathbb{R}$ is surjective.\ (d) The function $f : \mathbb{R} \to \mathbb{R}$ is injective. 7. Let $f(x) = 6x^2 + 4x$ be a function defined in the interval $[0, 6]$. Compute the Mid-Riemann sum of the function $f$ by dividing the interval $[0, 6]$ into 3 sub-intervals of equal length and using the midpoints of the sub-intervals for the height of the rectangles. [3 marks] **2 Section II (17 Marks)** 1. Use the Intermediate Value Theorem to prove that the equation $x^5 = 20 + \sqrt{x}$ has a solution in the interval $[1, 4]$. [3 marks] 2. Let $f(x) = \cos(\sin(x))$. Compute the third derivative $f'''(x)$. [4 marks] 3. Discuss the convergence of the sequence $\{x^n\}_{n \geq 1}$, where $x$ is any positive real number, treating each of the cases $x > 1$, $x = 1$, and $0 < x < 1$ separately. [5 marks] 4. Let $\{F_n\}$ be the usual Fibonacci sequence defined by $$F_1 = F_2 = 1 \text{ and } F_n = F_{n-1} + F_{n-2} \text{ for } n > 2.$$ Define a new sequence $\{G_n\}$ by $$G_1 = 4, G_2 = 3 \text{ and } G_n = G_{n-1} + G_{n-2} \text{ for } n > 2.$$ Using the principle of mathematical induction, prove that $$G_n = 3F_{n-1} + 4F_{n-2}$$ for all $n > 2$. [5 marks]