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Edexcel AEA 2014 June Q5
15 marks Challenging +1.8
The square-based pyramid \(P\) has vertices \(A, B, C, D\) and \(E\). The position vectors of \(A, B, C\) and \(D\) are \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) and \(\mathbf{d}\) respectively where $$\mathbf{a} = \begin{pmatrix} -2 \\ 3 \\ -1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 8 \\ -6 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 2 \\ 5 \\ 3 \end{pmatrix}, \quad \mathbf{d} = \begin{pmatrix} 6 \\ 1 \\ 1 \end{pmatrix}$$
  1. Find the vectors \(\overrightarrow{AB}\), \(\overrightarrow{AC}\), \(\overrightarrow{AD}\), \(\overrightarrow{BC}\), \(\overrightarrow{BD}\) and \(\overrightarrow{CD}\). [3]
  2. Find
    1. the length of a side of the square base of \(P\),
    2. the cosine of the angle between one of the slanting edges of \(P\) and its base,
    3. the height of \(P\),
    4. the position vector of \(E\).
    [9] A second pyramid, identical to \(P\), is attached by its square base to the base of \(P\) to form an octahedron.
  3. Find the position vector of the other vertex of this octahedron. [3]
Edexcel AEA 2014 June Q6
20 marks Hard +2.3
  1. A curve with equation \(y = f(x)\) has \(f(x) \geq 0\) for \(x \geq a\) and $$A = \int_a^b f(x) \, dx \quad \text{and} \quad V = \pi \int_a^b [f(x)]^2 \, dx$$ where \(a\) and \(b\) are constants with \(b > a\). Use integration by substitution to show that for the positive constants \(r\) and \(h\) $$\pi \int_{a+h}^{b+h} [r + f(x - h)]^2 \, dx = \pi r^2 (b - a) + 2\pi rA + V$$ [3]
  2. % \includegraphics{figure_1} - Shows a curve with vertical asymptotes at x=m and x=n, crossing y-axis at point p Figure 1 shows part of the curve \(C\) with equation \(y = 4 + \frac{2}{\sqrt{3}\cos x + \sin x}\) This curve has asymptotes \(x = m\) and \(x = n\) and crosses the \(y\)-axis at \((0, p)\). (a) Find the value of \(p\), the value of \(m\) and the value of \(n\). [4] (b) Show that the equation of \(C\) can be written in the form \(y = r + f(x - h)\) and specify the function \(f\) and the constants \(r\) and \(h\). [4] The region bounded by \(C\), the \(x\)-axis and the lines \(x = \frac{\pi}{6}\) and \(x = \frac{\pi}{3}\) is rotated through \(2\pi\) radians about the \(x\)-axis. (c) Find the volume of the solid formed. [9]
Edexcel AEA 2014 June Q7
23 marks Hard +2.3
% \includegraphics{figure_2} - Shows a circular tower with center T at (0,1), a goat at point G attached to the base at O, with string along arc OA then tangent AG A circular tower stands in a large horizontal field of grass. A goat is attached to one end of a string and the other end of the string is attached to the fixed point \(O\) at the base of the tower. Taking the point \(O\) as the origin \((0, 0)\), the centre of the base of the tower is at the point \(T(0, 1)\). The radius of the base of the tower is 1. The string has length \(\pi\) and you may ignore the size of the goat. The curve \(C\) represents the edge of the region that the goat can reach as shown in Figure 2.
  1. Write down the equation of \(C\) for \(y < 0\). [1] When the goat is at the point \(G(x, y)\), with \(x > 0\) and \(y > 0\), as shown in Figure 2, the string lies along \(OAG\) where \(OA\) is an arc of the circle with angle \(OTA = \theta\) radians and \(AG\) is a tangent to the circle at \(A\).
  2. With the aid of a suitable diagram show that $$x = \sin \theta + (\pi - \theta) \cos \theta$$ $$y = 1 - \cos \theta + (\pi - \theta) \sin \theta$$ [5]
  3. By considering \(\int y \frac{dx}{d\theta} d\theta\), show that the area between \(C\), the positive \(x\)-axis and the positive \(y\)-axis can be expressed in the form $$\int_0^{\pi} u \sin u \, du + \int_0^{\pi} u^2 \sin^2 u \, du + \int_0^{\pi} u \sin u \cos u \, du$$ [5]
  4. Show that \(\int_0^{\pi} u^2 \sin^2 u \, du = \frac{\pi^3}{6} + \int_0^{\pi} u \sin u \cos u \, du\) [4]
  5. Hence find the area of grass that can be reached by the goat. [8]
Edexcel AEA 2011 June Q1
Standard +0.3
Solve for \(0 \leq \theta \leq 180°\) $$\tan(\theta + 35°) = \cot(\theta - 53°)$$ [Total 4 marks]
Edexcel AEA 2011 June Q2
Challenging +1.8
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]
Edexcel AEA 2011 June Q3
17 marks Challenging +1.8
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]
Edexcel AEA 2011 June Q4
13 marks Challenging +1.2
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]
Edexcel AEA 2011 June Q5
17 marks Challenging +1.8
% 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. 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]
Edexcel AEA 2011 June Q6
19 marks Hard +2.3
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]
Edexcel AEA 2011 June Q7
20 marks Challenging +1.8
% 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]
Edexcel AEA 2015 June Q1
6 marks Moderate -0.5
  1. Sketch the graph of the curve with equation $$y = \ln(2x + 5), \quad x > -\frac{5}{2}$$ On your sketch you should clearly state the equations of any asymptotes and mark the coordinates of points where the curve meets the coordinate axes. [3]
  2. Solve the equation \(\ln(2x + 5) = \ln 9\) [3]
Edexcel AEA 2015 June Q2
9 marks Challenging +1.8
  1. Show that \((x + 1)\) is a factor of \(2x^3 + 3x^2 - 1\) [1]
  2. Solve the equation $$\sqrt{x^2 + 2x + 5} = x + \sqrt{2x + 3}$$ [8]
Edexcel AEA 2015 June Q3
9 marks Challenging +1.8
Solve for \(0 < x < 360°\) $$\cot 2x - \tan 78° = \frac{(\sec x)(\sec 78°)}{2}$$ where \(x\) is not an integer multiple of \(90°\) [9]
Edexcel AEA 2015 June Q4
15 marks Challenging +1.8
  1. Find the binomial series expansion for \((4 + y)^{\frac{1}{2}}\) in ascending powers of \(y\) up to and including the term in \(y^3\). Simplify the coefficient of each term. [3]
  2. Hence show that the binomial series expansion for \((4 + 5x + x^2)^{\frac{1}{2}}\) in ascending powers of \(x\) up to and including the term in \(x^3\) is $$2 + \frac{5x}{4} - \frac{9x^2}{64} + \frac{45x^3}{512}$$ [3]
  3. Show that the binomial series expansion of \((4 + 5x + x^2)^{\frac{1}{2}}\) will converge for \(-\frac{1}{2} < x \leq \frac{1}{2}\) [6]
  4. Use the result in part (b) to estimate $$\int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{4 + 5x + x^2} \, dx$$ Give your answer as a single fraction. [3]
Edexcel AEA 2015 June Q5
16 marks Challenging +1.2
% Figure shows a curve with maximum at point A, passing through origin O, with horizontal asymptote \includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\) where $$f(x) = \frac{x^2 + 16}{3x} \quad x \neq 0$$ The curve has a maximum at the point \(A\) with coordinates \((a, b)\).
  1. Find the value of \(a\) and the value of \(b\). [4] The function g is defined as $$g : x \mapsto \frac{x^2 + 16}{3x} \quad a \leq x < 0$$ where \(a\) is the value found in part (a).
  2. Write down the range of g. [1]
  3. On the same axes sketch \(y = g(x)\) and \(y = g^{-1}(x)\). [3]
  4. Find an expression for \(g^{-1}(x)\) and state the domain of \(g^{-1}\) [5]
  5. Solve the equation \(g(x) = g^{-1}(x)\). [3]
Edexcel AEA 2015 June Q6
19 marks Challenging +1.8
The lines \(L_1\) and \(L_2\) have vector equations $$L_1 : \mathbf{r} = \begin{pmatrix} 1 \\ 10 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -5 \\ 4 \end{pmatrix}$$ $$L_2 : \mathbf{r} = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$$
  1. Show that \(L_1\) and \(L_2\) are perpendicular. [2]
  2. Show that \(L_1\) and \(L_2\) are skew lines. [3] The point \(A\) with position vector \(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) lies on \(L_2\) and the point \(X\) lies on \(L_1\) such that \(\overrightarrow{AX}\) is perpendicular to \(L_1\)
  3. Find the position vector of \(X\). [5]
  4. Find \(|\overrightarrow{AX}|\) [2] The point \(B\) (distinct from \(A\)) also lies on \(L_2\) and \(|\overrightarrow{BX}| = |\overrightarrow{AX}|\)
  5. Find the position vector of \(B\). [5]
  6. Find the cosine of angle \(AXB\). [2]
Edexcel AEA 2015 June Q7
19 marks Hard +2.3
  1. Use the substitution \(x = \sec\theta\) to show that $$\int_{\sqrt{2}}^{2} \frac{1}{(x^2 - 1)^{\frac{3}{2}}} \, dx = \frac{\sqrt{6} - 2}{\sqrt{3}}$$ [5]
  2. Use integration by parts to show that $$\int \cos\theta \cot^2\theta \, d\theta = \frac{1}{2}[\ln|\cos\theta + \cot\theta| - \cos\theta \cot\theta] + c$$ [6] % Figure shows a curve y = 1/(x^2-1)^(1/2) for x > 1, with shaded region R between x = sqrt(2) and x = 2 \includegraphics{figure_2} Figure 2 shows a sketch of part of the curve with equation \(y = \frac{1}{(x^2 - 1)^{\frac{1}{2}}}\) for \(x > 1\) The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the lines \(x = \sqrt{2}\) and \(x = 2\) The region \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.
  3. Show that the volume of the solid formed is $$\pi \left[\frac{3}{8}\ln\left(\frac{1 + \sqrt{2}}{\sqrt{3}}\right) + \frac{7}{36} - \frac{\sqrt{2}}{8}\right]$$ [8]