For $n$ a positive whole number, and for $x \neq 0$, let $p _ { n } ( x ) = x ^ { n } + x ^ { - n }$. [0pt] (i) [3 marks] Sketch the graph of $y = p _ { 1 } ( x )$. Label any turning points on your sketch. (ii) $[ 1$ mark $]$ Show that $p _ { 2 } ( x ) = p _ { 1 } ( x ) ^ { 2 } - 2$. (iii) $[ 1$ mark $]$ Find an expression for $p _ { 3 } ( x )$ in terms of $p _ { 1 } ( x )$. [0pt] (iv) [5 marks] Find all real solutions $x$ to the equation $$x ^ { 4 } + x ^ { 3 } - 10 x ^ { 2 } + x + 1 = 0$$ (v) [5 marks] Find all real solutions $x$ to the equation $$x ^ { 7 } + 2 x ^ { 6 } - 5 x ^ { 5 } - 7 x ^ { 4 } + 7 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 1 = 0 .$$
Note that the arguments of all trigonometric functions in this question are given in terms of degrees. You are not expected to differentiate such a function. The notation $\cos ^ { n } x$ means $( \cos x ) ^ { n }$ throughout. (i) $[ 1$ mark $]$ Without differentiating, write down the maximum value of $\cos \left( 2 x + 30 ^ { \circ } \right)$. [0pt] (ii) [4 marks] Again without differentiating, find the maximum value of $$\cos \left( 2 x + 30 ^ { \circ } \right) \left( 1 - \cos \left( 2 x + 30 ^ { \circ } \right) \right)$$ (iii) [4 marks] Hence write down the maximum value of $$\cos ^ { 5 } \left( 2 x + 30 ^ { \circ } \right) \left( 1 - \cos \left( 2 x + 30 ^ { \circ } \right) \right) ^ { 5 }$$ (iv) [6 marks] Find the maximum value of $$\left( 1 - \cos ^ { 2 } \left( 3 x - 60 ^ { \circ } \right) \right) ^ { 4 } \left( 3 - \cos \left( 150 ^ { \circ } - 3 x \right) \right) ^ { 8 }$$
4. For Oxford applicants in Mathematics / Mathematics \& Statistics / Mathematics \& Philosophy, OR those not applying to Oxford, ONLY. Point $A$ is on the parabola $y = \frac { 1 } { 2 } x ^ { 2 }$ at ( $a , \frac { 1 } { 2 } a ^ { 2 }$ ) with $a > 0$. The line $L$ is normal to the parabola at $A$, and point $B$ lies on $L$ such that the distance $| A B |$ is a fixed positive number $d$, with $B$ above and to the left of $A$. [0pt] (i) [6 marks] Find the coordinates of $B$ in terms of $a$ and $d$. [0pt] (ii) [4 marks] Show that in order for $B$ to lie on the parabola, we must have $$a ^ { 2 } d = 2 \left( 1 + a ^ { 2 } \right) ^ { 3 / 2 }$$ (iii) [2 marks] Let $t = a ^ { 2 }$ and express the equality ( $*$ ) in the form $d ^ { 2 / 3 } = f ( t )$ for some function $f$ which you should determine explicitly. [0pt] (iv) [3 marks] Find the minimum value of $f ( t )$. Hence show that the equality ( $*$ ) holds for some real value of $a$ if and only if $d$ is greater than or equal to some value, which you should identify.
Define the sequence, $F _ { n }$, as follows: $F _ { 1 } = 1 , F _ { 2 } = 1$, and for $n \geqslant 3$, $$F _ { n } = F _ { n - 1 } + F _ { n - 2 } .$$ (i) [3 marks] What are the values $F _ { 3 } , F _ { 4 } , F _ { 5 }$ ? [0pt] (ii) [1 mark] Using the equation (*) repeatedly, in terms of $n$, how many additions do you need to calculate $F _ { n }$ ? We now consider sequences of 0 's and 1 's of length $n$, that do not have two consecutive 1 's. So, for $n = 5$, for example, ( $0,1,0,0,1$ ) and ( $1,0,1,0,1$ ) would be valid sequences, but ( $0,1,1,0,0$ ) would not. Let $S _ { n }$ denote the number of valid sequences of length $n$. [0pt] (iii) [1 mark] What are $S _ { 1 }$ and $S _ { 2 }$ ? [0pt] (iv) [3 marks] For $n \geqslant 3$, by considering the first element of the sequence of 0 's and 1's, show that $S _ { n }$ satisfies the same equation (*). Hence conclude that $S _ { n } = F _ { n + 2 }$ for all $n$. [0pt] (v) [2 marks] For $n \geqslant 2$, by considering valid sequences of length $2 n - 3$ and focusing on the element in the $( n - 1 ) ^ { \text {th } }$ position, show that, $$F _ { 2 n - 1 } = F _ { n } ^ { 2 } + F _ { n - 1 } ^ { 2 }$$ (vi) [3 marks] For $n \geqslant 2$, show that, $$F _ { 2 n } = F _ { n } ^ { 2 } + 2 F _ { n } F _ { n - 1 }$$ (vii) [2 marks] Let $k \geqslant 3$ be an integer. By using the equations (O) and (E) repeatedly, how many arithmetic operations do you need to calculate $F _ { 2 ^ { k } }$ ? You should only count additions and multiplications needed to calculate values using the equations (O) and (E) .
6. For Oxford applicants in Computer Science / Mathematics \& Computer Science / Computer Science \& Philosophy ONLY. In an octatree, all the digits 1 to 8 are arranged in a diagram like trees $T _ { 1 }$ and $T _ { 2 }$ shown below. There is a single digit at the root, drawn at the top (so the root is 3 in $T _ { 1 }$ ), and every other digit has another digit as its parent, so that by moving up the tree from parent to parent, each non-root digit has a unique path to the root. The order in which the children of any parent are drawn does not matter, so for simplicity we show them in increasing order from left to right. A leaf is a digit that is not the parent of any other digit: in tree $T _ { 1 }$, the leaves are $2,4,6$ and 7 . $T _ { 1 }$[Figure] $T _ { 2 }$[Figure] The code for an octatree is a sequence of seven digits obtained as follows. We use $T _ { 1 }$ as an example.
Remove the numerically smallest leaf and write down its parent. In $T _ { 1 }$, we remove 2 and write down its parent 8 .
In the tree that remains, remove the smallest leaf and write down its parent. In $T _ { 1 }$, after having removed 2, we remove 4 and write down its parent 5 .
Continue in this way until only the root remains. In $T _ { 1 }$, we would have deleted the digits $2,4,5,6,7,1,8$ in that order and obtained the code 8538183. (i) $[ 1$ mark $]$ Find the code for the octatree $T _ { 2 }$. [0pt] (ii) [1 mark] Draw the octatree that has the code $\mathbf { 8 8 8 8 8 8 8 }$. [0pt] (iii) [2 marks] Draw the octatree that has the code $\mathbf { 3 1 6 5 4 7 2 }$. [0pt] (iv) [3 marks] What are the leaves of the octatree that has the code $\mathbf { 1 6 1 8 3 8 8 }$ ? Justify your answer. [0pt] (v) [2 marks] Find all the digits in the octatree that has the code $\mathbf { 1 6 1 8 3 8 8 }$ that have 1 as their parent. [0pt] (vi) [2 marks] Reconstruct the whole tree that has the code 1618388. [0pt] (vii) [2 marks] Briefly describe a procedure that given a sequence of seven digits from 1 to 8 constructs an octatree with that sequence as its code. [0pt] (viii) [2 marks] Is the number of distinct octatrees greater than or smaller than $2,000,000$ ? Justify your answer. (You may use the fact that $2 ^ { 10 } = 1024$.)