Edexcel AEA (Advanced Extension Award) 2002 June

1．Solve the following equation，for \(0 \leq x \leq \pi\) ，giving your answers in terms of \(\pi\) ． $$\sin 5 x - \cos 5 x = \cos x - \sin x$$

2．In the binomial expansion of $$( 1 - 4 x ) ^ { p } , \quad | x | < \frac { 1 } { 4 }$$ the coefficient of \(x ^ { 2 }\) is equal to the coefficient of \(x ^ { 4 }\) and the coefficient of \(x ^ { 3 }\) is positive．
Find the value of \(p\) ．

3．The curve \(C\) has parametric equations $$x = 15 t - t ^ { 3 } , \quad y = 3 - 2 t ^ { 2 }$$ Find the values of \(t\) at the points where the normal to \(C\) at \(( 14,1 )\) cuts \(C\) again．

4．Find the coordinates of the stationary points of the curve with equation $$x ^ { 3 } + y ^ { 3 } - 3 x y = 48$$ and determine their nature．
\includegraphics[max width=\textwidth, alt={}, center]{7f1bc552-3850-43c5-b435-abc87b264f0a-3_553_749_401_618} Figure 1 shows a sketch of part of the curve with equation $$y = \sin ( \cos x ) .$$ The curve cuts the \(x\)-axis at the points \(A\) and \(C\) and the \(y\)-axis at the point \(B\).

1. Find the coordinates of the points \(A , B\) and \(C\).
2. Prove that \(B\) is a stationary point. Given that the region \(O C B\) is convex,
3. show that, for \(0 \leq x \leq \frac { \pi } { 2 }\), $$\sin ( \cos x ) \leq \cos x$$ and $$\left( 1 - \frac { 2 } { \pi } x \right) \sin 1 \leq \sin ( \cos x )$$ and state in each case the value or values of \(x\) for which equality is achieved.
4. Hence show that $$\frac { \pi } { 4 } \sin 1 < \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ( \cos x ) d x < 1$$
   \includegraphics[max width=\textwidth, alt={}]{7f1bc552-3850-43c5-b435-abc87b264f0a-4_682_824_399_704}
   Figure 2 shows a sketch of part of two curves \(C _ { 1 }\) and \(C _ { 2 }\) for \(y \geq 0\).
   The equation of \(C _ { 1 }\) is \(y = m _ { 1 } - x ^ { n _ { 1 } }\) and the equation of \(C _ { 2 }\) is \(y = m _ { 2 } - x ^ { n _ { 2 } }\), where \(m _ { 1 }\), \(m _ { 2 } , n _ { 1 }\) and \(n _ { 2 }\) are positive integers with \(m _ { 2 } > m _ { 1 }\). Both \(C _ { 1 }\) and \(C _ { 2 }\) are symmetric about the line \(x = 0\) and they both pass through the points \(( 3,0 )\) and \(( - 3,0 )\). Given that \(n _ { 1 } + n _ { 2 } = 12\), find

1. the possible values of \(n _ { 1 }\) and \(n _ { 2 }\),
2. the exact value of the smallest possible area between \(C _ { 1 }\) and \(C _ { 2 }\), simplifying your answer,
   (8)
3. the largest value of \(x\) for which the gradients of the two curves can be the same. Leave your answer in surd form.

7．A student was attempting to prove that \(x = \frac { 1 } { 2 }\) is the only real root of $$x ^ { 3 } + \frac { 3 } { 4 } x - \frac { 1 } { 2 } = 0$$ The attempted solution was as follows． $$\begin{array} { r l r } & x ^ { 3 } + \frac { 3 } { 4 } x & = \frac { 1 } { 2 }
\therefore & x \left( x ^ { 2 } + \frac { 3 } { 4 } \right) & = \frac { 1 } { 2 }
\therefore & x & = \frac { 1 } { 2 }
\text { or } & x ^ { 2 } + \frac { 3 } { 4 } & = \frac { 1 } { 2 }
\text { i.e. } & x ^ { 2 } & = - \frac { 1 } { 4 } \quad \text { no solution }
\therefore & \text { only real root is } x & = \frac { 1 } { 2 } \end{array}$$ （a）Explain clearly the error in the above attempt．
（b）Give a correct proof that \(x = \frac { 1 } { 2 }\) is the only real root of \(x ^ { 3 } + \frac { 3 } { 4 } x - \frac { 1 } { 2 } = 0\) ． The equation $$x ^ { 3 } + \beta x - \alpha = 0$$ where \(\alpha , \beta\) are real，\(\alpha \neq 0\) ，has a real root at \(x = \alpha\) ．
（c）Find and simplify an expression for \(\beta\) in terms of \(\alpha\) and prove that \(\alpha\) is the only real root provided \(| \alpha | < 2\) ．
（6）
An examiner chooses a positive number \(\alpha\) so that \(\alpha\) is the only real root of equation（I） but the incorrect method used by the student produces 3 distinct real＂roots＂．
（d）Find the range of possible values for \(\alpha\) ． Marks for style，clarity and presentation： 7