1. 1
   Consider the statement $P(n)$: $\displaystyle\sum_{r=1}^{n} r(r+1) = \frac{n(n+1)(n+2)}{3}$, for $n \in \mathbb{Z}^{+}$. A student plans to prove $P(n)$ by mathematical induction.

   (a)

   Find the common value of both sides of $P(n)$ when $n = 4$.

   [1 mark]

   • A$20$
   • B$40$
   • C$60$
   • D$30$

   (b)

   In the inductive step, which expression must be added to the sum for $n = k$ to obtain the sum for $n = k + 1$?

   [1 mark]

   • A$(k+1)(k+2)$
   • B$k(k+1)$
   • C$\frac{(k+1)(k+2)(k+3)}{3}$
   • D$(k+2)(k+3)$

   (c)

   Show that $\frac{k(k+1)(k+2)}{3} + (k+1)(k+2) = \frac{(k+1)(k+2)(k+3)}{3}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A student makes two claims. Claim 1: for every positive integer $n$, the number $n^{2} + n + 41$ is prime. Claim 2: for every real number $x$, $x^{2} \ge x$.

   (a)

   Which value of $n$ is a counterexample to Claim 1?

   [1 mark]

   • A$20$
   • B$10$
   • C$40$
   • D$39$

   (b)

   Which value of $x$ is a counterexample to Claim 2?

   [1 mark]

   • A$0$
   • B$2$
   • C$-1$
   • D$\frac{1}{2}$

   (c)

   Show that $n = 41$ is also a counterexample to Claim 1.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A real number is rational if it can be written as $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$. A real number that is not rational is irrational.

   (a)

   Let $a$ be a rational number and $b$ an irrational number. Prove by contradiction that $a + b$ is irrational.

   [3 marks]

   (b)

   Prove by contradiction that $\sqrt[3]{5}$ is irrational.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x) = xe^{2x}$, $x \in \mathbb{R}$. For $n \in \mathbb{Z}^{+}$, $f^{(n)}(x)$ denotes the $n$th derivative of $f(x)$.

   (a)

   Prove by mathematical induction that $f^{(n)}(x) = 2^{n-1}(2x + n)e^{2x}$ for all $n \in \mathbb{Z}^{+}$.

   [6 marks]

   (b)

   Hence show that, for each $n \in \mathbb{Z}^{+}$, the graph of $y = f^{(n)}(x)$ has exactly one stationary point, that it is a minimum, and that the minimum value is $-2^{n-1}e^{-(n+1)}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions