Edexcel AEA (Advanced Extension Award) 2013 June

1．In the binomial expansion of $$\left( 1 + \frac { 12 n } { 5 } x \right) ^ { n }$$ the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) are equal and non－zero．
（a）Find the possible values of \(n\) ．
（4）
（b）State，giving a reason，which value of \(n\) gives a valid expansion when \(x = \frac { 1 } { 2 }\)
（2）

2．（a）Use the formula for \(\sin ( A - B )\) to show that \(\sin \left( 90 ^ { \circ } - x \right) = \cos x\)
（b）Solve for \(0 < \theta < 360 ^ { \circ }\) $$2 \sin \left( \theta + 17 ^ { \circ } \right) = \frac { \cos \left( \theta + 8 ^ { \circ } \right) } { \cos \left( \theta + 17 ^ { \circ } \right) }$$

3．The lines \(L _ { 1 }\) and \(L _ { 2 }\) have equations given by
\(L _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { c } - 7
7
1 \end{array} \right) + \lambda \left( \begin{array} { c } 2
0
- 3 \end{array} \right)\) and \(L _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { c } 7
p
- 6 \end{array} \right) + \mu \left( \begin{array} { c } 10
- 4
- 1 \end{array} \right)\)
where \(\lambda\) and \(\mu\) are parameters and \(p\) is a constant．
The two lines intersect at the point \(C\) ．
（a）Find
（i）the value of \(p\) ，
（ii）the position vector of \(C\) ．
（b）Show that the point \(B\) with position vector \(\left( \begin{array} { c } - 13
11
- 4 \end{array} \right)\) lies on \(L _ { 2 }\) ． The point \(A\) with position vector \(\left( \begin{array} { c } - 7
7
1 \end{array} \right)\) lies on \(L _ { 1 }\) ．
（c）Find \(\cos ( \angle A C B )\) ，giving your answer as an exact fraction． The line \(L _ { 3 }\) bisects the angle \(A C B\) ．
（d）Find a vector equation of \(L _ { 3 }\) ．

4．A sequence of positive integers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) has \(r\) th term given by $$a _ { r } = 2 ^ { r } - 1$$ （a）Write down the first 6 terms of this sequence．
（b）Verify that \(a _ { r + 1 } = 2 a _ { r } + 1\)
（c）Find \(\sum _ { r = 1 } ^ { n } a _ { r }\)
（d）Show that \(\frac { 1 } { a _ { r + 1 } } < \frac { 1 } { 2 } \times \frac { 1 } { a _ { r } }\)
（e）Hence show that \(1 + \frac { 1 } { 3 } + \frac { 1 } { 7 } + \frac { 1 } { 15 } + \frac { 1 } { 31 } + \ldots < 1 + \frac { 1 } { 3 } + \left( \frac { 1 } { 7 } + \frac { \frac { 1 } { 2 } } { 7 } + \frac { \frac { 1 } { 4 } } { 7 } + \ldots \right)\)
（f）Show that \(\frac { 31 } { 21 } < \sum _ { r = 1 } ^ { \infty } \frac { 1 } { a _ { r } } < \frac { 34 } { 21 }\)

5．In this question u and v are functions of \(x\) ．Given that \(\int \mathrm { u } \mathrm { d } x , \int \mathrm { v } \mathrm { d } x\) and \(\int \mathrm { uv } \mathrm { d } x\) satisfy $$\int \text { uv } \mathrm { d } x = \left( \int \mathrm { u } \mathrm {~d} x \right) \times \left( \int \mathrm { v } \mathrm {~d} x \right) \quad \text { uv } \neq 0$$ （a）show that \(1 = \frac { \int \mathrm { u } \mathrm { d } x } { \mathrm { u } } + \frac { \int \mathrm { v } \mathrm { d } x } { \mathrm { v } }\) Given also that \(\frac { \int \mathrm { u } \mathrm { d } x } { \mathrm { u } } = \mathrm { sin } ^ { 2 } x\),
（b）use part（a）to write down an expression，in terms of \(x\) ，for \(\frac { \int \mathrm { v } \mathrm { d } x } { \mathrm { v } }\) ，
（c）show that $$\frac { 1 } { \mathrm { u } } \frac { \mathrm { du } } { \mathrm {~d} x } = \frac { 1 - 2 \sin x \cos x } { \sin ^ { 2 } x }$$ （d）hence use integration to show that \(\mathrm { u } = A \mathrm { e } ^ { - \cot x } \operatorname { cosec } ^ { 2 } x\) ，where \(A\) is an arbitrary constant．
（e）By differentiating \(\mathrm { e } ^ { \tan x }\) find a similar expression for v ．

6．（a）Starting from \([ \mathrm { f } ( x ) - \lambda \mathrm { g } ( x ) ] ^ { 2 } \geqslant 0\) show that \(\lambda\) satisfies the quadratic inequality $$\left( \int _ { a } ^ { b } [ \operatorname { g } ( x ) ] ^ { 2 } \mathrm {~d} x \right) \lambda ^ { 2 } - 2 \left( \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { g } ( x ) \mathrm { d } x \right) \lambda + \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x \geqslant 0$$ where \(a\) and \(b\) are constants and \(\lambda\) can take any real value．
（2）
（b）Hence prove that $$\left[ \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { g } ( x ) \mathrm { d } x \right] ^ { 2 } \leqslant \left[ \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x \right] \times \left[ \int _ { a } ^ { b } [ \mathrm {~g} ( x ) ] ^ { 2 } \mathrm {~d} x \right]$$ （c）By letting \(\mathrm { f } ( x ) = 1\) and \(\mathrm { g } ( x ) = \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } }\) show that $$\int _ { - 1 } ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x \leqslant \frac { 9 } { 2 }$$ （d）Show that \(\int _ { - 1 } ^ { 2 } x ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 4 } } \mathrm {~d} x = \frac { 12 \sqrt { } 3 } { 5 }\)
（e）Hence show that $$\frac { 144 } { 55 } \leqslant \int _ { - 1 } ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x$$

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { x } { 3 } + \frac { 12 } { x } \quad x \neq 0$$ The lines \(x = 0\) and \(y = \frac { x } { 3 }\) are asymptotes to \(C _ { 1 }\). The point \(A\) on \(C _ { 1 }\) is a minimum and the point \(B\) on \(C _ { 1 }\) is a maximum.

1. Find the coordinates of \(A\) and \(B\). There is a normal to \(C _ { 1 }\), which does not intersect \(C _ { 1 }\) a second time, that has equation \(x = k\), where \(k > 0\).
2. Write down the value of \(k\). The point \(P ( \alpha , \beta ) , \alpha > 0\) and \(\alpha \neq k\), lies on \(C _ { 1 }\). The normal to \(C _ { 1 }\) at \(P\) does not intersect \(C _ { 1 }\) a second time.
3. Find the value of \(\alpha\), leaving your answer in simplified surd form.
4. Find the equation of this normal. The curve \(C _ { 2 }\) has equation \(y = | \mathrm { f } ( x ) |\)
5. Sketch \(C _ { 2 }\) stating the coordinates of any turning points and the equations of any asymptotes. The line with equation \(y = m x + 1\) does not touch or intersect \(C _ { 2 }\).
6. Find the set of possible values for \(m\).