Edexcel AEA (Advanced Extension Award) 2011 June

Solve for \(0 \leq \theta \leq 180°\) $$\tan(\theta + 35°) = \cot(\theta - 53°)$$ [Total 4 marks]

Given that $$\int_0^{\frac{\pi}{2}} (1 + \tan\left[\frac{1}{2}x\right])^2 \, dx = a + \ln b$$ find the value of \(a\) and the value of \(b\). [Total 7 marks]

A sequence \(\{u_n\}\) is given by $$u_1 = k$$ $$u_{2n} = u_{2n-1} \times p \qquad n \geq 1$$ $$u_{2n+1} = u_{2n} \times q \qquad n \geq 1$$ where \(k\), \(p\) and \(q\) are positive constants with \(pq \neq 1\)

1. Write down the first 6 terms of this sequence. [3]
2. Show that \(\sum_{r=1}^{2n} u_r = \frac{k(1+p)(1-(pq)^n)}{1-pq}\) [6]

In part (c) \([x]\) means the integer part of \(x\), so for example \([2.73] = 2\), \([4] = 4\) and \([0] = 0\)

1. Find \(\sum_{r=1}^{\infty} 6 \times \left(\frac{4}{3}\right)^{\left[\frac{r}{2}\right]} \times \left(\frac{3}{5}\right)^{\left[\frac{r-1}{2}\right]}\) [4]

[Total 13 marks]

The curve \(C\) has parametric equations $$x = \cos^2 t$$ $$y = \cos t \sin t$$ where \(0 \leq t < \pi\)

1. Show that \(C\) is a circle and find its centre and its radius. [5]

% Figure 1 shows a sketch of C with point P, rectangle R with diagonal OP \includegraphics{figure_1} Figure 1 Figure 1 shows a sketch of \(C\). The point \(P\), with coordinates \((\cos^2 \alpha, \cos\alpha \sin \alpha)\), \(0 < \alpha < \frac{\pi}{2}\), lies on \(C\). The rectangle \(R\) has one side on the \(x\)-axis, one side on the \(y\)-axis and \(OP\) as a diagonal, where \(O\) is the origin.

1. Show that the area of \(R\) is \(\sin\alpha \cos^3 \alpha\) [1]
2. Find the maximum area of \(R\), as \(\alpha\) varies. [7]

[Total 13 marks]

% Figure 2 shows curve with vertical asymptotes at x = -2 and x = 2, horizontal asymptote at y = 1, with U-shaped region between asymptotes \includegraphics{figure_2} Figure 2 Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac{x^2 - 2}{x^2 - 4}\) and \(x \neq \pm 2\). The curve cuts the \(y\)-axis at \(U\).

1. Write down the coordinates of the point \(U\). [1]

The point \(P\) with \(x\)-coordinate \(a\) (\(a \neq 0\)) lies on \(C\).

1. Show that the normal to \(C\) at \(P\) cuts the \(y\)-axis at the point $$\left(0, \frac{a^2 - 2}{a^2 - 4} - \frac{(a^2 - 4)^2}{4}\right)$$ [6]

The circle \(E\), with centre on the \(y\)-axis, touches all three branches of \(C\).

1. 1. Show that $$\frac{a^2}{2(a^2-4)} - \frac{(a^2-4)^2}{4} = a^2 + \frac{(a^2-4)^4}{16}$$
   2. Hence, show that $$(a^2 - 4)^2 = 1$$
   3. Find the centre and radius of \(E\).
   [10]

[Total 17 marks]

The line \(L\) has equation $$\mathbf{r} = \begin{pmatrix} 13 \\ -3 \\ -8 \end{pmatrix} + t \begin{pmatrix} -5 \\ 3 \\ 4 \end{pmatrix}$$ The point \(P\) has position vector \(\begin{pmatrix} -7 \\ 2 \\ 7 \end{pmatrix}\). The point \(P'\) is the reflection of \(P\) in \(L\).

1. Find the position vector of \(P'\). [6]
2. Show that the point \(A\) with position vector \(\begin{pmatrix} -7 \\ 9 \\ 8 \end{pmatrix}\) lies on \(L\). [1]
3. Show that angle \(PAP' = 120°\). [3]

% Figure 3 shows kite APBP' with angle at A = 120° \includegraphics{figure_3} Figure 3 The point \(B\) lies on \(L\) and \(APBP'\) forms a kite as shown in Figure 3. The area of the kite is \(50\sqrt{3}\)

1. Find the position vector of the point \(B\). [5]
2. Show that angle \(BPA = 90°\). [2]

The circle \(C\) passes through the points \(A\), \(P\), \(P'\) and \(B\).

1. Find the position vector of the centre of \(C\). [2]

[Total 19 marks]

% Figure 4 shows curves with asymptotic behavior at x = 3 \includegraphics{figure_4} Figure 4

1. Figure 4 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = \frac{x^2 - 5}{3-x}, \quad x \in \mathbb{R}, x \neq 3$$ The curve has a minimum at the point \(A\), with \(x\)-coordinate \(\alpha\), and a maximum at the point \(B\), with \(x\)-coordinate \(\beta\). Find the value of \(\alpha\), the value of \(\beta\) and the \(y\)-coordinates of the points \(A\) and \(B\). [5]
2. The functions \(g\) and \(h\) are defined as follows $$g: x \to x + p \quad x \in \mathbb{R}$$ $$h: x \to |x| \quad x \in \mathbb{R}$$ where \(p\) is a constant. % Figure 5 shows curve with minimum points at C and D symmetric about y-axis \includegraphics{figure_5} Figure 5 Figure 5 shows a sketch of the curve with equation \(y = h(fg(x) + q)\), \(x \in \mathbb{R}\), \(x \neq 0\), where \(q\) is a constant. The curve is symmetric about the \(y\)-axis and has minimum points at \(C\) and \(D\).
   1. Find the value of \(p\) and the value of \(q\).
   2. Write down the coordinates of \(D\).
   [5]
3. The function \(\mathrm{m}\) is given by $$\mathrm{m}(x) = \frac{x^2 - 5}{3-x} \quad x \in \mathbb{R}, x < \alpha$$ where \(\alpha\) is the \(x\)-coordinate of \(A\) as found in part (a).
   1. Find \(\mathrm{m}^{-1}\)
   2. Write down the domain of \(\mathrm{m}^{-1}\)
   3. Find the value of \(t\) such that \(\mathrm{m}(t) = \mathrm{m}^{-1}(t)\)
   [10]

[Total 20 marks]