Maths 1 End Term: 11 December 2022, Set ETF1 (September 2022 term)

A company has branches in each of the six cities $C_1, C_2, ...., C_6$. Let a relation $R$ be defined as

1. $R = \{(a, b, c) \mid \text{there is a direct flight between } a \text{ and } b, a \neq b, \text{ with a fare of ₹ } c\}$
2. The elements in the relation $R$ are $(C_1, C_2, 4000), (C_2, C_1, 4000), (C_1, C_5, 5000), (C_5, C_1, 5000), (C_2, C_3, 5000), (C_3, C_2, 5000), (C_4, C_5, 4000), (C_5, C_4, 4000), (C_3, C_4, 7000), (C_4, C_3, 7000), (C_5, C_6, 1000), (C_6, C_5, 1000), (C_1, C_6, 2000)$, and $(C_6, C_1, 2000)$.

An employee of that company wanted to travel from the city $C_2$ to the city $C_5$. If he traveled by the cheapest route possible, then find the total fare that he should pay.

The polynomial $p(x) = a_n x^n + a_{n-1}x^{n-1} + \ldots + a_0$ has the following properties:

• $p(x)$ is an odd degree polynomial with at least three distinct roots.
• $p(x)$ has exactly two positive real roots.
• $(x - 5)^2$ is a factor of $p(x)$.
• $p(0) \neq 0$

Choose the best possible representations of $p(x)$.

Edwin plotted a graph on Desmos (an online graphing tool), which was continuous and differentiable at every point. Later he remembered the form of the function $f(x)$ that represents the graph that he plotted but forgot some of the values that appeared in the expression. So he used $m, n$ and $p$ in place of the missing values and the function he wrote down had the following expression:

$f(x) = \begin{cases} e^{nx} + 3, & x < 0 \\ m + 2, & x = 0 \\ 5x + p + 1, & x > 0 \end{cases}$

Can you help Edwin by finding the values of $m, n$, and $p$?

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.