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Edexcel AEA 2008 June Q2
12 marks Challenging +1.2
2.The points \(( x , y )\) on the curve \(C\) satisfy $$( x + 1 ) ( x + 2 ) \frac { d y } { d x } = x y$$ The line with equation \(y = 2 x + 5\) is the tangent to \(C\) at a point \(P\) .
  1. Find the coordinates of \(P\) .
  2. Find the equation of \(C\) ,giving your answer in the form \(y = \mathrm { f } ( x )\) .
Edexcel AEA 2008 June Q3
12 marks Challenging +1.8
3.(a)Prove that \(\tan 15 ^ { \circ } = 2 - \sqrt { 3 }\) (b)Solve,for \(0 \leqslant \theta < 360 ^ { \circ }\) , $$\sin \left( \theta + 60 ^ { \circ } \right) \sin \left( \theta - 60 ^ { \circ } \right) = ( 1 - \sqrt { } 3 ) \cos ^ { 2 } \theta$$
\includegraphics[max width=\textwidth, alt={}]{280c36e3-6bb5-44b7-8222-17ce25c1bdbe-3_1008_1343_369_465}
Figure 1 shows a sketch of the curve \(C\) with equation $$y = \cos x \ln ( \sec x ) , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } .$$ The points \(A\) and \(B\) are maximum points on \(C\) .
(a)Find the coordinates of \(B\) in terms of e . The finite region \(R\) lies between \(C\) and the line \(A B\) .
(b)Show that the area of \(R\) is $$\frac { 2 } { \mathrm { e } } \arccos \left( \frac { 1 } { \mathrm { e } } \right) + 2 \ln \left( \mathrm { e } + \sqrt { } \left( \mathrm { e } ^ { 2 } - 1 \right) \right) - \frac { 4 } { \mathrm { e } } \sqrt { } \left( \mathrm { e } ^ { 2 } - 1 \right) .$$ \(\left[ \arccos x \right.\) is an alternative notation for \(\left. \cos ^ { - 1 } x \right]\)
Edexcel AEA 2008 June Q5
14 marks Challenging +1.8
5. (i) Anna, who is confused about the rules for logarithms, states that $$\left( \log _ { 3 } p \right) ^ { 2 } = \log _ { 3 } \left( p ^ { 2 } \right)$$ and \(\quad \log _ { 3 } ( p + q ) = \log _ { 3 } p + \log _ { 3 } q\).
However, there is a value for \(p\) and a value for \(q\) for which both statements are correct.
Find the value of \(p\) and the value of \(q\).
(ii) Solve $$\frac { \log _ { 3 } \left( 3 x ^ { 3 } - 23 x ^ { 2 } + 40 x \right) } { \log _ { 3 } 9 } = 0.5 + \log _ { 3 } ( 3 x - 8 ) .$$
Edexcel AEA 2008 June Q6
15 marks Challenging +1.8
6. $$\mathrm { f } ( x ) = \frac { a x + b } { x + 2 } ; \quad x \in \mathbb { R } , x \neq - 2$$ where \(a\) and \(b\) are constants and \(b > 0\) .
  1. Find \(f ^ { - 1 } ( x )\) .
  2. Hence,or otherwise,find the value of \(a\) so that \(\operatorname { ff } ( x ) = x\) . The curve \(C\) has equation \(y = \mathrm { f } ( x )\) and \(\mathrm { f } ( x )\) satisfies \(\mathrm { ff } ( x ) = x\) .
  3. On separate axes sketch
    1. \(y = \mathrm { f } ( x )\) ,
    2. \(y = \mathrm { f } ( x - 2 ) + 2\) . On each sketch you should indicate the equations of any asymptotes and the coordinates,in terms of \(b\) ,of any intersections with the axes. The normal to \(C\) at the point \(P\) has equation \(y = 4 x - 39\) .The normal to \(C\) at the point \(Q\) has equation \(y = 4 x + k\) ,where \(k\) is a constant.
  4. By considering the images of the normals to \(C\) on the curve with equation \(y = \mathrm { f } ( x - 2 ) + 2\) ,or otherwise,find the value of \(k\) .
Edexcel AEA 2008 June Q7
22 marks Challenging +1.8
7. Relative to a fixed origin \(O\), the position vectors of the points \(A , B\) and \(C\) are $$\overrightarrow { O A } = - 3 \mathbf { i } + \mathbf { j } - 9 \mathbf { k } , \quad \overrightarrow { O B } = \mathbf { i } - \mathbf { k } , \quad \overrightarrow { O C } = 5 \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k } \text { respectively. }$$
  1. Find the cosine of angle \(A B C\). The line \(L\) is the angle bisector of angle \(A B C\).
  2. Show that an equation of \(L\) is \(\mathbf { r } = \mathbf { i } - \mathbf { k } + t ( \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k } )\).
  3. Show that \(| \overrightarrow { A B } | = | \overrightarrow { A C } |\). The circle \(S\) lies inside triangle \(A B C\) and each side of the triangle is a tangent to \(S\).
  4. Find the position vector of the centre of \(S\).
  5. Find the radius of \(S\).
Edexcel AEA 2009 June Q1
8 marks Challenging +1.2
  1. (a) On the same diagram, sketch
$$y = ( x + 1 ) ( 2 - x ) \quad \text { and } \quad y = x ^ { 2 } - 2 | x |$$ Mark clearly the coordinates of the points where these curves cross the coordinate axes.
(b) Find the \(x\)-coordinates of the points of intersection of these two curves.
Edexcel AEA 2009 June Q2
9 marks Hard +2.3
2. The curve \(C\) has equation \(y = x ^ { \sin x } , \quad x > 0\).
  1. Find the equation of the tangent to \(C\) at the point where \(x = \frac { \pi } { 2 }\).
  2. Prove that this tangent touches \(C\) at infinitely many points.
Edexcel AEA 2009 June Q3
12 marks Challenging +1.2
3. (a) Solve, for \(0 \leqslant \theta < 2 \pi\), $$\sin \left( \frac { \pi } { 3 } - \theta \right) = \frac { 1 } { \sqrt { } 3 } \cos \theta$$ (b) Find the value of \(x\) for which $$\begin{aligned} & \arcsin ( 1 - 2 x ) = \frac { \pi } { 3 } - \arcsin x , \quad 0 < x < 0.5 \\ & { \left[ \arcsin x \text { is an alternative notation for } \sin ^ { - 1 } x \right] } \end{aligned}$$
Edexcel AEA 2009 June Q4
14 marks Challenging +1.8
  1. (a) The function \(\mathrm { f } ( x )\) has \(\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { u } ( x ) } { \mathrm { v } ( x ) }\). Given that \(\mathrm { f } ^ { \prime } ( k ) = 0\), show that \(\mathrm { f } ^ { \prime \prime } ( k ) = \frac { \mathrm { u } ^ { \prime } ( k ) } { \mathrm { v } ( k ) }\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dfb57dc0-5831-4bbb-b1e5-58c4798215cb-3_874_879_486_593} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} (b) The curve \(C\) with equation $$y = \frac { 2 x ^ { 2 } + 3 } { x ^ { 2 } - 1 }$$ crosses the \(y\)-axis at the point \(A\). Figure 1 shows a sketch of \(C\) together with its 3 asymptotes.
  1. Find the coordinates of the point \(A\).
  2. Find the equations of the asymptotes of \(C\). The point \(P ( a , b ) , a > 0\) and \(b > 0\), lies on \(C\). The point \(Q\) also lies on \(C\) with \(P Q\) parallel to the \(x\)-axis and \(A P = A Q\).
  3. Show that the area of triangle \(P A Q\) is given by \(\frac { 5 a ^ { 3 } } { a ^ { 2 } - 1 }\).
  4. Find, as \(a\) varies, the minimum area of triangle \(P A Q\), giving your answer in its simplest form.
Edexcel AEA 2009 June Q5
15 marks Challenging +1.8
5.(a)The sides of the triangle \(A B C\) have lengths \(B C = a , A C = b\) and \(A B = c\) ,where \(a < b < c\) .The sizes of the angles \(A , B\) and \(C\) form an arithmetic sequence.
  1. Show that the area of triangle \(A B C\) is \(a c \frac { \sqrt { 3 } } { 4 }\) . Given that \(a = 2\) and \(\sin A = \frac { \sqrt { } 15 } { 5 }\) ,find
  2. the value of \(b\) ,
  3. the value of \(c\) .
    (b)The internal angles of an \(n\)-sided polygon form an arithmetic sequence with first term \(143 ^ { \circ }\) and common difference \(2 ^ { \circ }\) . Given that all of the internal angles are less than \(180 ^ { \circ }\) ,find the value of \(n\) .
Edexcel AEA 2009 June Q6
17 marks Challenging +1.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dfb57dc0-5831-4bbb-b1e5-58c4798215cb-5_700_684_246_694} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 2 \sin t , \quad y = \ln ( \sec t ) , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The tangent to \(C\) at the point \(P\) ,where \(t = \frac { \pi } { 3 }\) ,cuts the \(x\)-axis at \(A\) .
  1. Show that the \(x\)-coordinate of \(A\) is \(\frac { \sqrt { } 3 } { 3 } ( 3 - \ln 2 )\) . The shaded region \(R\) lies between \(C\) ,the positive \(x\)-axis and the tangent \(A P\) as shown in Figure 2 .
  2. Show that the area of \(R\) is \(\sqrt { 3 } ( 1 + \ln 2 ) - 2 \ln ( 2 + \sqrt { 3 } ) - \frac { \sqrt { 3 } } { 6 } ( \ln 2 ) ^ { 2 }\) .
Edexcel AEA 2009 June Q7
18 marks Challenging +1.8
7.Relative to a fixed origin \(O\) the points \(A , B\) and \(C\) have position vectors $$\mathbf { a } = - \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { and } \mathbf { c } = 6 \mathbf { i } + \frac { 16 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { respectively. }$$
  1. Find the cosine of angle \(A B C\) . The quadrilateral \(A B C D\) is a kite \(K\) .
  2. Find the area of \(K\) . A circle is drawn inside \(K\) so that it touches each of the 4 sides of \(K\) .
  3. Find the radius of the circle,giving your answer in the form \(p \sqrt { } ( q ) - q \sqrt { } ( p )\) ,where \(p\) and \(q\) are positive integers.
  4. Find the position vector of the point \(D\) .
    (Total 18 marks)
Edexcel AEA 2010 June Q1
12 marks Standard +0.8
1.(a)Solve the equation $$\sqrt { } ( 3 x + 16 ) = 3 + \sqrt { } ( x + 1 )$$ (b)Solve the equation $$\log _ { 3 } ( x - 7 ) - \frac { 1 } { 2 } \log _ { 3 } x = 1 - \log _ { 3 } 2$$
Edexcel AEA 2010 June Q2
11 marks Challenging +1.2
2.The sum of the first \(p\) terms of an arithmetic series is \(q\) and the sum of the first \(q\) terms of the same arithmetic series is \(p\) ,where \(p\) and \(q\) are positive integers and \(p \neq q\) . Giving simplified answers in terms of \(p\) and \(q\) ,find
  1. the common difference of the terms in this series,
  2. the first term of the series,
  3. the sum of the first \(( p + q )\) terms of the series.
Edexcel AEA 2010 June Q3
11 marks Challenging +1.2
3.The curve \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + f x y = g ^ { 2 }$$ where \(f\) and \(g\) are constants and \(g \neq 0\) .
  1. Find an expression in terms of \(\alpha , \beta\) and \(f\) for the gradient of \(C\) at the point \(( \alpha , \beta )\) . Given that \(f < 2\) and \(f \neq - 2\) and that the gradient of \(C\) at the point \(( \alpha , \beta )\) is 1 ,
  2. show that \(\alpha = - \beta = \frac { \pm g } { \sqrt { } ( 2 - f ) }\) . Given that \(f = - 2\) ,
  3. sketch \(C\) .
Edexcel AEA 2010 June Q4
16 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-3_643_332_246_870} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a cuboid \(O A B C D E F G\), where \(O\) is the origin, \(A\) has position vector \(5 \mathbf { i } , C\) has position vector \(10 \mathbf { j }\) and \(D\) has position vector \(20 \mathbf { k }\).
  1. Find the cosine of angle \(C A F\). Given that the point \(X\) lies on \(A C\) and that \(F X\) is perpendicular to \(A C\),
  2. find the position vector of point \(X\) and the distance \(F X\). The line \(l _ { 1 }\) passes through \(O\) and through the midpoint of the face \(A B F E\). The line \(l _ { 2 }\) passes through \(A\) and through the midpoint of the edge \(F G\).
  3. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of the point of intersection.
Edexcel AEA 2010 June Q5
12 marks Challenging +1.8
5. $$I = \int \frac { 1 } { ( x - 1 ) \sqrt { } \left( x ^ { 2 } - 1 \right) } \mathrm { d } x , \quad x > 1$$
  1. Use the substitution \(x = 1 + u ^ { - 1 }\) to show that $$I = - \left( \frac { x + 1 } { x - 1 } \right) ^ { \frac { 1 } { 2 } } + c$$
  2. Hence show that $$\int _ { \sec \alpha } ^ { \sec \beta } \frac { 1 } { ( x - 1 ) \sqrt { } \left( x ^ { 2 } - 1 \right) } \mathrm { d } x = \cot \left( \frac { \alpha } { 2 } \right) - \cot \left( \frac { \beta } { 2 } \right) , \quad 0 < \alpha < \beta < \frac { \pi } { 2 }$$
Edexcel AEA 2010 June Q6
10 marks Challenging +1.8
6.(a)Given that \(x ^ { 4 } + y ^ { 4 } = 1\) ,prove that \(x ^ { 2 } + y ^ { 2 }\) is a maximum when \(x = \pm y\) ,and find the maximum and minimum values of \(x ^ { 2 } + y ^ { 2 }\) .
(b)On the same diagram,sketch the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations \(x ^ { 4 } + y ^ { 4 } = 1\) and \(x ^ { 2 } + y ^ { 2 } = 1\) respectively.
(c)Write down the equation of the circle \(C _ { 3 }\) ,centre the origin,which touches the curve \(C _ { 1 }\) at the points where \(x = \pm y\) .
Edexcel AEA 2010 June Q7
21 marks Challenging +1.2
7. $$\mathrm { f } ( x ) = \left[ 1 + \cos \left( x + \frac { \pi } { 4 } \right) \right] \left[ 1 + \sin \left( x + \frac { \pi } { 4 } \right) \right] , \quad 0 \leqslant x \leqslant 2 \pi$$
  1. Show that \(\mathrm { f } ( x )\) may be written in the form $$f ( x ) = \left( \frac { 1 } { \sqrt { 2 } } + \cos x \right) ^ { 2 } , \quad 0 \leqslant x \leqslant 2 \pi$$
  2. Find the range of the function \(\mathrm { f } ( x )\). The graph of \(y = \mathrm { f } ( x )\) is shown in Figure 2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_426_938_849_591} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure}
  3. Find the coordinates of all the maximum and minimum points on this curve. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_432_942_1535_589} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The region \(R\), bounded by \(y = 2\) and \(y = \mathrm { f } ( x )\), is shown shaded in Figure 3.
  4. Find the area of \(R\).
Edexcel AEA 2011 June Q1
4 marks Standard +0.3
1.Solve for \(0 \leqslant \theta \leqslant 180 ^ { \circ }\) $$\tan \left( \theta + 35 ^ { \circ } \right) = \cot \left( \theta - 53 ^ { \circ } \right)$$
Edexcel AEA 2011 June Q2
7 marks Challenging +1.2
2.Given that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \left( 1 + \tan \left[ \frac { 1 } { 2 } x \right] \right) ^ { 2 } \mathrm {~d} x = a + \ln b$$ find the value of \(a\) and the value of \(b\) .
Edexcel AEA 2011 June Q3
13 marks Challenging +1.8
3.A sequence \(\left\{ u _ { n } \right\}\) is given by $$\begin{aligned} u _ { 1 } & = k & & \\ u _ { 2 n } & = u _ { 2 n - 1 } \times p & & n \geqslant 1 \\ u _ { 2 n + 1 } & = u _ { 2 n } \times q & & n \geqslant 1 \end{aligned}$$ where \(k , p\) and \(q\) are positive constants with \(p q \neq 1\)
  1. Write down the first 6 terms of this sequence.
  2. Show that \(\sum _ { r = 1 } ^ { 2 n } u _ { r } = \frac { k ( 1 + p ) \left( 1 - ( p q ) ^ { n } \right) } { 1 - p q }\) In part(c) \([ x ]\) means the integer part of \(x\) ,so for example \([ 2.73 ] = 2 , [ 4 ] = 4\) and \([ 0 ] = 0\)
  3. Find \(\sum _ { r = 1 } ^ { \infty } 6 \times \left( \frac { 4 } { 3 } \right) ^ { \left[ \frac { r } { 2 } \right] } \times \left( \frac { 3 } { 5 } \right) ^ { \left[ \frac { r - 1 } { 2 } \right] }\)
Edexcel AEA 2011 June Q4
13 marks Challenging +1.2
4. The curve \(C\) has parametric equations $$\begin{gathered} x = \cos ^ { 2 } t \\ y = \cos t \sin t \end{gathered}$$ where \(0 \leqslant t < \pi\)
  1. Show that \(C\) is a circle and find its centre and its radius. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-3_668_750_726_660} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of \(C\). The point \(P\), with coordinates \(\left( \cos ^ { 2 } \alpha , \cos \alpha \sin \alpha \right) , \quad 0 < \alpha < \frac { \pi } { 2 }\), lies on \(C\). The rectangle \(R\) has one side on the \(x\)-axis, one side on the \(y\)-axis and \(O P\) as a diagonal, where \(O\) is the origin.
  2. Show that the area of \(R\) is \(\sin \alpha \cos ^ { 3 } \alpha\)
  3. Find the maximum area of \(R\), as \(\alpha\) varies.
Edexcel AEA 2011 June Q5
17 marks Challenging +1.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-4_739_1397_187_335} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} 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 \(U\). The point \(P\) with \(x\)-coordinate \(a ( a \neq 0 )\) lies on \(C\).
  2. Show that the normal to \(C\) at \(P\) cuts the \(y\)-axis at the point $$\left( 0 , \left[ \frac { a ^ { 2 } - 2 } { a ^ { 2 } - 4 } - \frac { \left( a ^ { 2 } - 4 \right) ^ { 2 } } { 4 } \right] \right)$$ The circle \(E\), with centre on the \(y\)-axis, touches all three branches of \(C\).
    1. Show that $$\left[ \frac { a ^ { 2 } } { 2 \left( a ^ { 2 } - 4 \right) } - \frac { \left( a ^ { 2 } - 4 \right) ^ { 2 } } { 4 } \right] ^ { 2 } = a ^ { 2 } + \frac { \left( a ^ { 2 } - 4 \right) ^ { 4 } } { 16 }$$
    2. Hence, show that $$\left( a ^ { 2 } - 4 \right) ^ { 2 } = 1$$
    3. Find the centre and radius of \(E\).
Edexcel AEA 2011 June Q6
19 marks Challenging +1.8
6.The line \(L\) has equation $$\mathbf { r } = \left( \begin{array} { r } 13 \\ - 3 \\ - 8 \end{array} \right) + t \left( \begin{array} { r } - 5 \\ 3 \\ 4 \end{array} \right)$$ The point \(P\) has position vector \(\left( \begin{array} { r } - 7 \\ 2 \\ 7 \end{array} \right)\) .
The point \(P ^ { \prime }\) is the reflection of \(P\) in \(L\) .
  1. Find the position vector of \(P ^ { \prime }\) .
  2. Show that the point \(A\) with position vector \(\left( \begin{array} { r } - 7 \\ 9 \\ 8 \end{array} \right)\) lies on \(L\) .
  3. Show that angle \(P A P ^ { \prime } = 120 ^ { \circ }\) . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-5_483_1367_1263_347} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The point \(B\) lies on \(L\) and \(A P B P ^ { \prime }\) forms a kite as shown in Figure 3.
    The area of the kite is \(50 \sqrt { } 3\)
  4. Find the position vector of the point \(B\) .
  5. Show that angle \(B P A = 90 ^ { \circ }\) . The circle \(C\) passes through the points \(A , P , P ^ { \prime }\) and \(B\) .
  6. Find the position vector of the centre of \(C\) .