[Maximum number: 2]

The following question explores features of a family of curves. The family is then linked to a homogeneous differential equation.
Consider the curve given by $y=\frac{x\left(x^{2}-16\right)}{x^{2}+16}$ .

(a)

(i)

Show that $x-\frac{2 A x}{x^{2}+A} \equiv \frac{x\left(x^{2}-A\right)}{x^{2}+A}$ .

[ 2 ]

[Maximum number: 5]

Consider a geometric sequence with a first term of 4 and a fourth term of -2.916 .

(a)

Find the common ratio of this sequence.

[ 3 ]

(b)

Find the sum to infinity of this sequence.

[ 2 ]

[Maximum number: 5]

Given the sets A and B, use the properties of sets to prove that $A \cup\left(B^{\prime} \cup A\right)^{\prime}=A \cup B$ , justifying each step of the proof.

[Maximum number: 5]

A geometric sequence has $u_{4}=-70$ and $u_{7}=8.75$ . Find the second term of the sequence.

[Maximum number: 4]

In an arithmetic sequence, the sum of the 3rd and 8th terms is 1 .
Given that the sum of the first seven terms is 35 , determine the first term and the common difference.

[Maximum number: 7]

This question considers two possible models for the occurrence of random events in a computer game.
In a new computer game, each time a player performs an action, there is a random chance that the action will be boosted, meaning that it provides a benefit to the player.
The designer of this computer game is considering two possible models for when to boost an action.
In the first model, the probability that an action will be boosted is constant.

(a)

Suppose the probability that an action will be boosted is p, where 0<p<1.

[ 1 ]

(i)

Hence, write down an expression, using sigma notation, for E(X) in terms of x and p.

Consider the sum of an infinite geometric sequence, with first term a and common ratio r(|r|<1),

a+a r+a r^{2}+a r^{3}+\ldots=\frac{a}{1-r}

[ 1 ]

(b)

(i)

By differentiating both sides of the above equation with respect to r, find an expression for $\sum_{n=1}^{\infty} n a r^{n-1}$ in terms of a and r.

[ 4 ]

(ii)

Hence, show that $\mathrm{E}(X)=\frac{1}{p}$ .

It can be shown that $\operatorname{Var}(X)=\frac{1-p}{p^{2}}$ .

[ 2 ]

[Maximum number: 5]

The 7 th term of an arithmetic sequence is 6 .
The sum of the 6th term and the 12th term is 24 .
Find the first term and the common difference.

[Maximum number: 3]

Consider two subsets, A and B, of a universal set U.

(a)

Use the laws of set operations to show that $A \cap(A \cap B)^{\prime}=A \backslash B$ .

[ 3 ]

[Maximum number: 4]

Consider the arithmetic sequence $8,26,44, \ldots$ .

(a)

Find an expression for the $n^{\text {th }}$ term.

[ 1 ]

(b)

Write down the sum of the first n terms using sigma notation.

[ 1 ]

(c)

Calculate the sum of the first 15 terms.

[ 2 ]

(a)

Use the Euclidean algorithm to show that 1463 and 389 are relatively prime.

[ 4 ]

(b)

Find positive integers a and b such that 1463 a-389 b=1.

[ 5 ]

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