2. (a) For what values of the constant $k$ does the quadratic equation $$x ^ { 2 } - 2 x - 1 = k$$ have: (i) no real solutions; (ii) one real solution; (iii) two real solutions. (b) Showing your working, express $\left( x ^ { 2 } - 2 x - 1 \right) ^ { 2 }$ as a polynomial of degree 4 in $x$. (c) Show that the quartic equation $$x ^ { 4 } - 4 x ^ { 3 } + 2 x ^ { 2 } + 4 x + 1 = h$$ has exactly two real solutions if either $h = 0$ or $h > 4$. Show that there is no value of $h$ such that the above quartic equation has just one real solution.
3. Let $$f ( x ) = \left\{ \begin{array} { c c }
x + 1 & \text { for } 0 \leq x \leq 1 \\
2 x ^ { 2 } - 6 x + 6 & \text { for } 1 \leq x \leq 2
\end{array} \right.$$ (a) On the axes provided below, sketch a graph of $y = f ( x )$ for $0 \leq x \leq 2$, labelling any turning points and the values attained at $x = 0,1,2$. (b) For $1 \leq t \leq 2$, define $$g ( t ) = \int _ { t - 1 } ^ { t } f ( x ) \mathrm { d } x$$ Express $g ( t )$ as a cubic in $t$. (c) Calculate and factorize $g ^ { \prime } ( t )$. (d) What are the minimum and maximum values of $g ( t )$ for $t$ in the range $1 \leq t \leq 2$ ? [Figure] 4. [Figure][Figure] The triangle $A B C$, drawn above, has sides $B C , C A$ and $A B$ of length $a , b$ and $c$ respectively, and the angles at $A , B$ and $C$ are $\alpha , \beta$ and $\gamma$. (a) Show that the area of $A B C$ equals $\frac { 1 } { 2 } b c \sin \alpha$. Deduce the sine rule $$\frac { a } { \sin \alpha } = \frac { b } { \sin \beta } = \frac { c } { \sin \gamma } .$$ (b) In the triangle above, let $P , Q$ and $R$ respectively be the feet of the perpendiculars from $A$ to $B C , B$ to $C A$, and $C$ to $A B$, as shown. Prove that $$\text { Area of } P Q R = \left( 1 - \cos ^ { 2 } \alpha - \cos ^ { 2 } \beta - \cos ^ { 2 } \gamma \right) \times ( \text { Area of } A B C ) .$$ For what triangles $A B C$, with angles $\alpha , \beta , \gamma$, does the equation $$\cos ^ { 2 } \alpha + \cos ^ { 2 } \beta + \cos ^ { 2 } \gamma = 1$$ hold?
5. The game of Oxflip is for one player and involves circular counters, which are white on one side and black on the other, placed in a grid. During a game, the counters are flipped over (changing between black and white side uppermost) following certain rules. Given a particular size of grid and a set starting pattern of whites and blacks, the aim of the game is to reach a certain target pattern. Each "move" of the game is to flip over either a whole row or a whole column of counters (so one whole row or column has all its blacks swapped to whites and vice versa). For example, in a game played in a three-by-three square grid, if you are given the starting and target patterns [Figure][Figure] a sequences of three moves to achieve the target is: [Figure] There are many other sequences of moves which also have the same result. (a) Consider the two-by-two version of the game with starting pattern [Figure] Draw, in the blank patterns opposite, the eight different target patterns (including the starting pattern) which it is possible to obtain. What are the possible numbers of white counters that may be present in these target patterns? (b) In the four-by-four version of the game, starting with pattern [Figure] explain why it is impossible to reach a pattern with only one white counter. [0pt] [Hint: don't try to write out every possible combination of moves.] (c) In the five-by-five game, explain why any sequence of moves which begins [Figure] and ends with an all-white pattern, must involve an odd number of moves. What is the least number of moves needed? Give reasons for your answer. [Figure]