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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 \((0, p)\).
    1. Find the value of \(p\), the value of \(m\) and the value of \(n\). [4]
    2. 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.
    3. 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]
CAIE M2 2014 June Q3
Standard +0.3
3 A light elastic string has natural length 0.8 m and modulus of elasticity 16 N . One end of the string is attached to a fixed point \(O\), and a particle \(P\) of mass 0.4 kg is attached to the other end of the string. The particle \(P\) hangs in equilibrium vertically below \(O\).
  1. Show that the extension of the string is 0.2 m . \(P\) is projected vertically downwards from the equilibrium position. \(P\) first comes to instantaneous rest at the point where \(O P = 1.4 \mathrm {~m}\).
  2. Calculate the speed at which \(P\) is projected.
  3. Find the speed of \(P\) at the first instant when the string subsequently becomes slack.
CAIE M2 2014 June Q4
Standard +0.8
4 A particle \(P\) is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground.
  1. Find the height of \(P\) above the ground when \(P\) has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Calculate the length of time for which the speed of \(P\) is less than \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and find the horizontal distance travelled by \(P\) during this time.
CAIE M2 2013 June Q1
Easy -1.8
1 hour 15 minutes \section*{
\includegraphics[max width=\textwidth, alt={}]{10abedc3-c814-47c0-8ed4-849ef325feca-1_403_143_792_68}
} Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9) \section*{READ THESE INSTRUCTIONS FIRST} If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
Where a numerical value for the acceleration due to gravity is needed, use \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.
CAIE M2 2013 June Q2
Standard +0.8
2 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 45 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at \(O\) and falls vertically. Find the extension of the string when \(P\) is at its lowest position.
CAIE M2 2013 June Q3
Moderate -0.8
3 A ball is projected horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top of a tower which is 30 m high. The tower stands on horizontal ground.
  1. Find the speed and direction of motion of the ball when it reaches the ground.
  2. Calculate the distance from the foot of the tower to the point where the ball reaches the ground.
CAIE M2 2013 June Q7
Challenging +1.2
7 A small ball \(B\) of mass 0.2 kg moves in a narrow fixed smooth cylindrical tube \(O A\) of length 1 m , closed at the end \(A\). When the ball has displacement \(x \mathrm {~m}\) from \(O\), it has velocity \(v \mathrm {~ms} ^ { - 1 }\) in the direction \(O A\) and experiences a resisting force of magnitude \(\frac { k } { 1 - x } \mathrm {~N}\).
  1. \includegraphics[max width=\textwidth, alt={}, center]{10abedc3-c814-47c0-8ed4-849ef325feca-4_186_805_488_715} The tube is fixed in a horizontal position and \(B\) is projected from \(O\) towards \(A\) with velocity \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Given that \(B\) comes to instantaneous rest after travelling 0.55 m , show that \(k = 0.1803\), correct to 4 significant figures.
  2. The tube is now fixed in a vertical position with \(O\) above \(A\). The ball \(B\) is released from rest at \(O\). Calculate the speed of \(B\) after it has descended 0.1 m . \end{document}
CAIE FP2 2013 November Q3
Standard +0.8
3 hours
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF10) \section*{READ THESE INSTRUCTIONS FIRST} If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
Where a numerical value is necessary, take the acceleration due to gravity to be \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
The use of a calculator is expected, where appropriate.
Results obtained solely from a graphic calculator, without supporting working or reasoning, will not receive credit.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
CAIE FP2 2013 November Q1
Challenging +1.2
1 \includegraphics[max width=\textwidth, alt={}, center]{c1aae41e-530c-4db4-8959-8afe223c4dbc-2_547_423_260_861} Three identical uniform rods, \(A B , B C\) and \(C D\), each of mass \(M\) and length \(2 a\), are rigidly joined to form three sides of a square. A uniform circular disc, of mass \(\frac { 2 } { 3 } M\) and radius \(a\), has the opposite ends of one of its diameters attached to \(A\) and \(D\) respectively. The disc and the rods all lie in the same plane (see diagram). Find the moment of inertia of the system about the axis \(A D\).
CAIE FP2 2013 November Q4
Challenging +1.8
4 \includegraphics[max width=\textwidth, alt={}, center]{c1aae41e-530c-4db4-8959-8afe223c4dbc-3_563_572_258_785} A uniform circular disc, with centre \(O\) and weight \(W\), rests in equilibrium on a horizontal floor and against a vertical wall. The plane of the disc is vertical and perpendicular to the wall. The disc is in contact with the floor at \(A\) and with the wall at \(B\). A force of magnitude \(P\) acts tangentially on the disc at the point \(C\) on the edge of the disc, where the radius \(O C\) makes an angle \(\theta\) with the upward vertical, and \(\tan \theta = \frac { 4 } { 3 }\) (see diagram). The coefficient of friction between the disc and the floor and between the disc and the wall is \(\frac { 1 } { 2 }\). Show that the sum of the magnitudes of the frictional forces at \(A\) and \(B\) is equal to \(P\). Given that the equilibrium is limiting at both \(A\) and \(B\),
  1. show that \(P = \frac { 15 } { 34 } \mathrm {~W}\),
  2. find the ratio of the magnitude of the normal reaction at \(A\) to the magnitude of the normal reaction at \(B\).