, 2023

(a) $x$, $y$ and $z$ are real numbers, with $x \leq y \leq z$ and $x + y = 4$, $y + z = 7$. Let $T = x + y + z$. (i) Show that $x \leq 2$ and that $T = 11 - y$. (ii) Find the minimum possible value of $T$, giving one example of values of $x$, $y$ and $z$ where this occurs. (2 marks) (b) $a$, $b$, $c$, $d$ and $e$ are real numbers, with $a \leq b \leq c \leq d \leq e$ and $a + b + c = 4$, $b + c + d = 5$, $c + d + e = 9$. Let $S = a + b + c + d + e$. Find the minimum possible value of $S$, giving one example of values of $a$, $b$, $c$, $d$ and $e$ where this occurs. Your solution must fully justify why no smaller value of $S$ is possible. (8 marks)