Question 1

**3 Section-1 (16 marks)** 1. (4 points) Which of the following is/are true? (MSQ) A. The sum of two rational numbers is always rational.\ B. The sum of two irrational numbers is always irrational.\ C. The sum of a rational and irrational number is always irrational.\ D. The least upper bound of any set of rational numbers that is bounded above is always rational. 2. (4 points) Consider the following quantities and select the correct relationships between them from the options given below. (MSQ) - $a = \sum_{n=1}^{10} (6 - n)$ - $b$ is the least upper bound of $\left\{1 - \frac{(-1)^n}{n} : n \in \mathbb{N}\right\}$ - $c$ is the cardinality of the set $\{n : 4 \mid n \text{ and } 6 \mid n,\ 1 \le n \le 100,\ n \in \mathbb{N}\}$ A. $a \equiv 3 \pmod{b}$\ B. $a \equiv b \pmod{c}$\ C. $b \mid c$\ D. $a + b = c$ 3. (4 points) Which of the following options is/are true? (MSQ) A. The GCD of $10^{200} + 1$ and $10^{200} - 1$ is 1.\ B. The equation $18m + 6n = 2$ has a solution where $m, n \in \mathbb{Z}$\ C. $\{56m + 98n \mid m, n \in \mathbb{Z}\} = \{14k \mid k \in \mathbb{Z}\}$\ D. $\{56m + 98n \mid m, n \in \mathbb{Z}\} = \{28k \mid k \in \mathbb{Z}\}$ 4. (4 points) A company has opened recruitment for the post of data analyst.500 candidates have applied for the post. 290 candidates are proficient in Python programming, 190 candidates are proficient in C programming, 130 candidates are proficient in Java programming, 70 candidates are proficient in Python and Java, 70 candidates are proficient in C and Python, 50 candidates are proficient in C and Java and 50 candidates don't know any of the programming languages. Find the number of candidates who are proficient in exactly one of the three programming languages. \_\_\_\_\_\_\_\_\_\_ **4 Section II (24 Marks)** 5. (8 points) Let $A$ and $B$ be countable sets. Is $A \cup B$ countable? If yes, furnish a proof. If not, provide a counterexample. 6. (8 points) Let $30x0y03$ be a seven-digit number that is divisible by 13. Note that $x$ and $y$ are single digit numbers. Show that $x + 3y \equiv 11 \pmod{13}$. 7. (8 points) Prove using induction (or otherwise) that for every positive integer $n$: $$\frac{1}{3 \times 4} + \frac{1}{4 \times 5} + \cdots + \frac{1}{(n+2)(n+3)} = \frac{n}{3n+9}$$