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Edexcel AEA 2020 June Q2
13 marks Challenging +1.8
2.The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 2 \sqrt { 1 - \mathrm { e } ^ { - x } } & x \in \mathbb { R } , x \geqslant 0 \\ \mathrm {~g} ( x ) = \ln \left( 4 - x ^ { 2 } \right) & x \in \mathbb { R } , - 2 < x < 2 \end{array}$$
    1. Explain why fg cannot be formed as a composite function.
    2. Explain why gf can be formed as a composite function.
    1. Find \(\mathrm { gf } ( x )\) ,giving the answer in the form \(\mathrm { gf } ( x ) = a + b x\) ,where \(a\) and \(b\) are constants.
    2. State the domain and range of gf.
  3. Sketch the graph of the function gf.
    On your sketch,you should show the coordinates of any points where the graph meets or crosses the coordinate axes. The circle \(C\) with centre \(( 0 , - \ln 9 )\) touches the line with equation \(y = \operatorname { gf } ( x )\) at precisely one point.
  4. Find an equation of the circle \(C\) .
Edexcel AEA 2020 June Q3
13 marks Challenging +1.8
3.(a)(i)Write down the binomial series expansion of $$\left( 1 + \frac { 2 } { n } \right) ^ { n } \quad n \in \mathbb { N } , n > 2$$ in powers of \(\left( \frac { 2 } { n } \right)\) up to and including the term in \(\left( \frac { 2 } { n } \right) ^ { 3 }\) (ii)Hence prove that,for \(n \in \mathbb { N } , n \geqslant 3\) $$\left( 1 + \frac { 2 } { n } \right) ^ { n } \geqslant \frac { 19 } { 3 } - \frac { 6 } { n }$$ (b)Use the binomial series expansion of \(\left( 1 - \frac { x } { 4 } \right) ^ { \frac { 1 } { 2 } }\) to show that \(\sqrt { 3 } < \frac { 7 } { 4 }\) $$\mathrm { f } ( x ) = \left( 1 + \frac { 2 } { x } \right) ^ { x } - 3 ^ { \frac { x } { 6 } } \quad x \in \mathbb { R } , x > 0$$ Given that the function \(\mathrm { f } ( x )\) is continuous and that \(\sqrt [ 6 ] { 3 } > \frac { 6 } { 5 }\) (c)prove that \(\mathrm { f } ( x ) = 0\) has a root in the interval[9,10]
Edexcel AEA 2020 June Q4
17 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d5b914c-28b2-4485-a42e-627c95fa16e2-16_581_961_251_552} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the parabola with equation \(y = \frac { 1 } { 2 } x ( 10 - x ) , 0 \leqslant x \leqslant 10\) This question concerns rectangles that lie under the parabola in the first quadrant.The bottom edge of each rectangle lies along the \(x\)-axis and the top left vertex lies on the parabola.Some examples are shown in Figure 2. Let the \(x\) coordinate of the top left vertex be \(a\) .
  1. Explain why the width,\(w\) ,of such a rectangle must satisfy \(w \leqslant 10 - 2 a\)
  2. Find the value of \(a\) that gives the maximum area for such a rectangle. Given that the rectangle must be a square,
  3. find the value of \(a\) that gives the maximum area for such a square. Given that the area of the rectangles is fixed as 36
  4. find the range of possible values for \(a\) . \includegraphics[max width=\textwidth, alt={}, center]{4d5b914c-28b2-4485-a42e-627c95fa16e2-16_2255_50_311_1980}
Edexcel AEA 2020 June Q5
22 marks Challenging +1.8
5.(a)The box below shows a student's attempt to prove the following identity for \(a > b > 0\) $$\arctan a - \arctan b \equiv \arctan \frac { a - b } { 1 + a b }$$ Let \(x = \arctan a\) and \(y = \arctan b\) ,so that \(a = \tan x\) and \(b = \tan y\) $$\begin{aligned} \text { So } \tan ( \arctan a - \arctan b ) & \equiv \tan ( x - y ) \\ & \equiv \frac { \tan x - \tan y } { 1 - \tan ^ { 2 } ( x y ) } \\ & \equiv \frac { a - b } { 1 - ( a b ) ^ { 2 } } \\ & \equiv \frac { a - a b + a b - b } { ( 1 - a b ) ( 1 + a b ) } \\ & \equiv \frac { a ( 1 - a b ) - b ( 1 - a b ) } { ( 1 - a b ) ( 1 + a b ) } \\ & \equiv \frac { a - b } { 1 + a b } \end{aligned}$$ Taking arctan of both sides gives \(\arctan a - \arctan b \equiv \arctan \frac { a - b } { 1 + a b }\) as required. There are three errors in the proof where the working does not follow from the previous line.
  1. Describe these three errors.
  2. Write out a correct proof of the identity.
    (b)[In this question take \(g\) to be \(9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) ] \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4d5b914c-28b2-4485-a42e-627c95fa16e2-22_244_1267_1870_504} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Balls are projected,one after another,from a point,\(A\) ,one metre above horizontal ground. Each ball travels in a vertical plane towards a 6 metre high vertical wall of negligible thickness,which is a horizontal distance of \(10 \sqrt { 2 }\) metres from \(A\) . The balls are modelled as particles and it is assumed that there is no air resistance.
    Each ball is projected with an initial speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at a random angle \(\theta\) to the horizontal,where \(0 < \theta < 90 ^ { \circ }\) Given that a ball will pass over the wall precisely when \(\alpha \leqslant \theta \leqslant \beta\)
  3. find, in degrees, the angle \(\beta - \alpha\)
  4. Deduce that the probability that a particular ball will pass over the wall is \(\frac { 2 } { 3 }\)
  5. Hence find the probability that exactly 2 of the first 10 balls projected pass over the wall. You should give your answer in the form \(\frac { P } { Q ^ { k } }\) where \(P , Q\) and \(k\) are integers and \(P\) is not a multiple of \(Q\).
  6. Explain whether taking air resistance into account would increase or decrease the probability in (b)(iii).
  7. find, in degrees, the angle \(\beta - \alpha\)
Edexcel AEA 2020 June Q6
23 marks Hard +2.3
  1. (a) Given that f is a function such that the integrals exist,
    1. use the substitution \(u = a - x\) to show that
    $$\int _ { 0 } ^ { a } \mathrm { f } ( x ) \mathrm { d } x = \int _ { 0 } ^ { a } \mathrm { f } ( a - x ) \mathrm { d } x$$
  2. Hence use symmetry of \(\mathrm { f } ( \sin x )\) on the interval \([ 0 , \pi ]\) to show that $$\int _ { 0 } ^ { \pi } x \mathrm { f } ( \sin x ) \mathrm { d } x = \pi \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { f } ( \sin x ) \mathrm { d } x$$ (b) Use the result of (a)(i) to show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin ^ { n } x } { \sin ^ { n } x + \cos ^ { n } x } \mathrm {~d} x$$ is independent of \(n\), and find the value of this integral.
    (c) (i) Prove that $$\frac { \cos x } { 1 + \cos x } \equiv 1 - \frac { 1 } { 2 } \sec ^ { 2 } \left( \frac { x } { 2 } \right)$$
  3. Hence use the results from (a) to find $$\int _ { 0 } ^ { \pi } \frac { x \sin x } { 1 + \sin x } \mathrm {~d} x$$ (d) Find $$\int _ { 0 } ^ { \pi } \frac { x \sin ^ { 4 } x } { \sin ^ { 4 } x + \cos ^ { 4 } x } \mathrm {~d} x$$
Edexcel AEA 2022 June Q1
5 marks Challenging +1.2
1. $$\mathrm { f } ( x ) = x ^ { \left( x ^ { 2 } \right) } \quad x > 0$$ Use logarithms to find the \(x\) coordinate of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\) .
Edexcel AEA 2022 June Q2
10 marks Challenging +1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{71cd126f-1c7d-4e37-a26d-7ff98a74fd79-04_456_508_255_781} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a regular hexagon \(O P Q R S T\).
The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are defined by \(\mathbf { p } = \overrightarrow { O P }\) and \(\mathbf { q } = \overrightarrow { O Q }\) Forces, in Newtons, \(\mathbf { F } _ { P } = ( \overrightarrow { O P } ) , \mathbf { F } _ { Q } = 2 \times ( \overrightarrow { O Q } ) , \mathbf { F } _ { R } = 3 \times ( \overrightarrow { O R } ) , \mathbf { F } _ { S } = 4 \times ( \overrightarrow { O S } )\) and \(\mathbf { F } _ { T } = 5 \times ( \overrightarrow { O T } )\) are applied to a particle.
  1. Find, in terms of \(\mathbf { p }\) and \(\mathbf { q }\), the resultant force on the particle. The magnitude of the acceleration of the particle due to these forces is \(13 \mathrm {~ms} ^ { - 2 }\) Given that the mass of the particle is 3 kg ,
  2. find \(| \mathbf { p } |\) \includegraphics[max width=\textwidth, alt={}, center]{71cd126f-1c7d-4e37-a26d-7ff98a74fd79-04_2255_56_310_1980}
Edexcel AEA 2022 June Q3
12 marks Challenging +1.8
3.(a)Use the formulae for \(\sin ( A \pm B )\) and \(\cos ( A \pm B )\) to prove that \(\tan \left( 90 ^ { \circ } - \theta \right) \equiv \cot \theta\) (b)Solve for \(0 < \theta < 360 ^ { \circ }\) $$2 - \sec ^ { 2 } \left( \theta + 11 ^ { \circ } \right) = 2 \tan \left( \theta + 11 ^ { \circ } \right) \tan \left( \theta - 34 ^ { \circ } \right)$$ Give each answer as an integer in degrees.
Edexcel AEA 2022 June Q4
14 marks
4.Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { x ^ { 3 } - 2 x }\)
  1. find \(\mathrm { f } ^ { \prime } ( x )\) The curves \(C _ { 1 }\) and \(C _ { 2 }\) are defined by the functions g and h respectively,where $$\begin{array} { l l } \mathrm { g } ( x ) = 8 x ^ { 3 } \mathrm { e } ^ { x ^ { 3 } - 2 x } & x \in \mathbb { R } , x > 0 \\ \mathrm {~h} ( x ) = \left( 3 x ^ { 5 } + 4 x \right) \mathrm { e } ^ { x ^ { 3 } - 2 x } & x \in \mathbb { R } , x > 0 \end{array}$$
  2. Find the \(x\) coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) Given that \(C _ { 1 }\) lies above \(C _ { 2 }\) between these points of intersection,
  3. find the area of the region bounded by the curves between these two points.
    Give your answer in the form \(A + B \mathrm { e } ^ { C }\) where \(A , B\) ,and \(C\) are exact real numbers to be found.
Edexcel AEA 2022 June Q5
11 marks Challenging +1.8
  1. An aeroplane leaves a runway and moves with a constant speed of \(V \mathrm {~km} / \mathrm { h }\) due north along a straight path inclined at an angle \(\arctan \left( \frac { 3 } { 4 } \right)\) to the horizontal.
A light aircraft is moving due north in a straight horizontal line in the same vertical plane as the aeroplane, at a height of 3 km above the runway. The light aircraft is travelling with a constant speed of \(2 V \mathrm {~km} / \mathrm { h }\).
At the moment the aeroplane leaves the runway, the light aircraft is at a horizontal distance \(d \mathrm {~km}\) behind the aeroplane. Both aircraft continue to move with the same trajectories due north.
  1. Show that the distance, \(D \mathrm {~km}\), between the two aircraft \(t\) hours after the aeroplane leaves the runway satisfies $$D ^ { 2 } = \left( \frac { 6 } { 5 } V t - d \right) ^ { 2 } + \left( \frac { 3 } { 5 } V t - 3 \right) ^ { 2 }$$ Given that the distance between the two aircraft is never less than 2 km ,
  2. find the range of possible values for \(d\).
Edexcel AEA 2022 June Q6
24 marks Hard +2.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{71cd126f-1c7d-4e37-a26d-7ff98a74fd79-22_481_1139_189_463} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the first few iterations in the construction of a curve, \(L\).
Starting with a straight line \(L _ { 0 }\) of length 4 , the middle half of this line is replaced by three sides of a trapezium above \(L _ { 0 }\) as shown, such that the length of each of these sides is \(\frac { 1 } { 4 }\) of the length of \(L _ { 0 }\) After the first iteration each line segment has length one.
In subsequent iterations, each line segment parallel to \(L _ { 0 }\) similarly has its middle half replaced by three sides of a trapezium above that line segment, with each side \(\frac { 1 } { 4 }\) the length of that line segment. Line segments in \(L _ { n }\) are either parallel to \(L _ { 0 }\) or are sloped.
  1. Show that the length of \(L _ { 2 }\) is \(\frac { 23 } { 4 }\)
  2. Write down the number of
    1. line segments in \(L _ { n }\) that are parallel to \(L _ { 0 }\)
    2. sloped line segments in \(L _ { 2 }\) that are not in \(L _ { 1 }\)
    3. new sloped line segments that are created by the ( \(n + 1\) )th iteration.
  3. Hence find the length of \(L _ { n }\) as \(n \rightarrow \infty\) The area enclosed between \(L _ { 0 }\) and \(L _ { n }\) is \(A _ { n }\)
  4. Find the value of \(A _ { 1 }\)
  5. Find, in terms of \(n\), an expression for \(A _ { n + 1 } - A _ { n }\)
  6. Hence find the value of \(A _ { n }\) as \(n \rightarrow \infty\) The same construction as described above is applied externally to the three sides of an equilateral triangle of side length \(a\).
    Given that the limit of the area of the resulting shape is \(26 \sqrt { 3 }\)
  7. find the value of \(a\).
Edexcel AEA 2022 June Q7
24 marks Challenging +1.8
7.A circle \(C\) has centre \(X ( a , b )\) and radius \(r\) .
A line \(l\) has equation \(y = m x + c\)
  1. Show that the \(x\) coordinates of the points where \(C\) and \(l\) intersect satisfy $$\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 2 ( a - m ( c - b ) ) x + a ^ { 2 } + ( c - b ) ^ { 2 } - r ^ { 2 } = 0$$ Given that \(l\) is a tangent to \(C\) ,
  2. show that $$c = b - m a \pm r \sqrt { m ^ { 2 } + 1 }$$ The circle \(C _ { 1 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 16 = 0$$ and the circle \(C _ { 2 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 10 y + 89 = 0$$
  3. Find the equations of any lines that are normal to both \(C _ { 1 }\) and \(C _ { 2 }\) ,justifying your answer.
  4. Find the equations of all lines that are a tangent to both \(C _ { 1 }\) and \(C _ { 2 }\) [You may find the following Pythagorean triple helpful in this part: \(7 ^ { 2 } + 24 ^ { 2 } = 25 ^ { 2 }\) ]
Edexcel AEA 2023 June Q1
6 marks Challenging +1.3
1.(a)Write down the exact value of \(\cos 405 ^ { \circ }\) (b)Hence,using a double angle identity for cosine,or otherwise,determine the exact value of \(\cos 101.25 ^ { \circ }\) ,giving your answer in the form $$a \sqrt { b + c \sqrt { 2 + \sqrt { 2 } } }$$ where \(a\) ,\(b\) and \(c\) are rational numbers.
Edexcel AEA 2023 June Q2
9 marks Challenging +1.8
2.A student is attempting to prove that there are infinitely many prime numbers.
The student's attempt to prove this is in the box below. Assume there are only finitely many prime numbers,then there is a biggest prime number,\(p\) . Let \(n = 2 p + 1\) .Then \(n\) is bigger than \(p\) and since \(2 p + 1\) is not divisible by \(p\) , \(n\) is a prime number. Hence \(n\) is a prime number bigger than \(p\) ,contradicting the initial assumption. So we conclude there are infinitely many prime numbers.
  1. Use \(p = 7\) to show that the following claim made in the student's proof is not true: since \(2 p + 1\) is not divisible by \(p , n\) is a prime number. The student changes their proof to use \(n = 6 p + 1\) instead of \(n = 2 p + 1\)
  2. Show,by counter example,that this does not correct the student's proof.
  3. Write out a correct proof by contradiction to show that there are infinitely many prime numbers.
Edexcel AEA 2023 June Q3
10 marks Challenging +1.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{78ba3acc-4cca-4d15-8362-a27e425c5859-08_752_586_251_742} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(C\) given by the parametric equations $$x = \frac { 5 } { \sqrt { 3 } } \sin t \quad y = 5 ( 1 - \cos t ) \quad 0 \leqslant t \leqslant 2 \pi$$ The circle with centre at the origin \(O\) and with radius \(\frac { 5 \sqrt { 2 } } { 2 }\) meets the curve \(C\) at the points \(A\) and \(B\) as shown in Figure 1.
  1. Determine the value of \(t\) at the point \(B\) . The region \(R\) ,shown shaded in Figure 1,is bounded by the curve \(C\) and the circle.
  2. Determine the area of the region \(R\) .
Edexcel AEA 2023 June Q4
16 marks Challenging +1.2
4.(a)Use the trapezium rule with 4 strips to find an approximate value for $$\int _ { 0 } ^ { 1 } 16 ^ { x } d x$$ (b)Use the trapezium rule with \(n\) strips to write down an expression that would give an approximate value for $$\int _ { 0 } ^ { 1 } 16 ^ { x } d x$$ (c)Hence show that $$\int _ { 0 } ^ { 1 } 16 ^ { x } \mathrm {~d} x = \lim _ { n \rightarrow \infty } \left( \frac { 1 } { n } \left( 1 + 16 ^ { \frac { 1 } { n } } + \ldots + 16 ^ { \frac { n - 1 } { n } } \right) \right)$$ (d)Use integration to determine the exact value of $$\int _ { 0 } ^ { 1 } 16 ^ { x } d x$$ Given that the limit exists,
(e)use part(c)and the answer to part(d)to determine the exact value of $$\lim _ { x \rightarrow 0 } \frac { 16 ^ { x } - 1 } { x }$$
Edexcel AEA 2023 June Q5
21 marks Hard +2.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{78ba3acc-4cca-4d15-8362-a27e425c5859-16_517_881_210_593} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a partially completed Venn diagram of sports that a year group of students enjoy,where \(a , b , c , d\) and \(e\) are non-negative integers. The diagram shows how many students enjoy a combination of football( \(F\) ),golf( \(G\) ) and hockey \(( H )\) or none of these sports. There are \(n\) students in the year group.
It is known that
- \(\mathrm { P } ( F ) = \frac { 3 } { 7 }\) - \(\mathrm { P } ( H \mid G ) = \frac { 1 } { 3 }\) -\(F\) is independent of \(H \cap G\)
  1. Show that \(\mathrm { P } ( F \cap H \cap G ) = \frac { 1 } { 7 } \mathrm { P } ( G )\)
  2. Prove that if two events \(X\) and \(Y\) are independent,then \(X ^ { \prime }\) and \(Y\) are also independent.
  3. Hence find the value \(k\) such that \(\mathrm { P } \left( F ^ { \prime } \cap H \cap G \right) = k \mathrm { P } ( G )\)
  4. Show that \(c = \frac { 4 } { 3 } a\) Given further that \(\mathrm { P } ( F \mid H ) = \frac { 1 } { 5 }\)
  5. find an expression for \(d\) in terms of \(a\) ,and hence deduce the maximum possible value of \(a\) .
  6. Determine the possible values of \(n\) .
Edexcel AEA 2023 June Q6
23 marks Challenging +1.2
  1. \hspace{0pt} [In this question you may assume the following formulae for the volume and curved] surface area of a cone of base radius \(r\) and height \(h\) and of a sphere of radius \(r\).
Cone: volume \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) and curved surface area \(S = \pi r \sqrt { h ^ { 2 } + r ^ { 2 } }\) Sphere: volume \(V = \frac { 4 \pi } { 3 } r ^ { 3 }\) and curved surface area \(S = 4 \pi r ^ { 2 }\) \includegraphics[max width=\textwidth, alt={}, center]{78ba3acc-4cca-4d15-8362-a27e425c5859-22_782_755_637_657} Figure 3
Figure 3 shows the design for a garden ornament.
The ornament is made of a hemisphere on top of a truncated cone.
The truncated cone has base radius \(2 r \mathrm {~cm}\), top radius \(r \mathrm {~cm}\) and height \(4 r \mathrm {~cm}\).
The hemisphere has radius \(R \mathrm {~cm}\).
Given that the volume of the ornament is \(2100 \pi \mathrm {~cm} ^ { 3 }\)
  1. show that $$R ^ { 3 } = 3150 - 14 r ^ { 3 }$$
  2. Find an expression involving \(\frac { \mathrm { d } R } { \mathrm {~d} r }\) in terms of \(r\) and/or \(R\). The base of the truncated cone of the ornament is fixed to the ground.
  3. Show that the visible surface area of the ornament, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = ( 3 \sqrt { 17 } - 1 ) \pi r ^ { 2 } + 3 \pi R ^ { 2 }$$
  4. Hence show that $$\frac { \mathrm { d } A } { \mathrm {~d} r } = \gamma \pi r - \frac { \delta \pi r ^ { 2 } } { R }$$ where \(\gamma\) and \(\delta\) are real numbers to be determined. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{78ba3acc-4cca-4d15-8362-a27e425c5859-23_705_803_625_630} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of \(A\) against \(r\), for \(r \geqslant 0\) There is a local minimum at \(r = 0\) and a local maximum at the point \(M\). The overall minimum point is at the point \(N\), where the gradient of the curve is undefined.
    1. Determine the \(r\) coordinate of the point \(N\).
    2. Explain why, for the ornament, \(r\) must be less than this value.
  6. Show that the \(r\) coordinate of the point \(M\) is $$\sqrt [ 3 ] { \frac { p ( 3 \sqrt { 17 } - 1 ) ^ { 3 } } { 3 q ^ { 2 } + ( 3 \sqrt { 17 } - 1 ) ^ { 3 } } }$$ where \(p\) and \(q\) are integers to be determined.
Edexcel AEA 2023 June Q7
15 marks Hard +2.3
  1. A sequence of non-zero real numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$a _ { n + 1 } = p + \frac { q } { a _ { n } } \quad n \in \mathbb { N }$$ where \(p\) and \(q\) are real numbers with \(q \neq 0\) It is known that
  • one of the terms of this sequence is a
  • the sequence is periodic
    1. Determine an equation for \(q\), in terms of \(p\) and \(a\), such that the sequence is constant (of period/order one).
    2. Determine the value of \(p\) that is necessary for the sequence to be of period/order 2.
    3. Give an example of a sequence that satisfies the condition in part (b), but is not of period/order 2.
    4. Determine an equation for \(q\), in terms of \(p\) only, such that the sequence has period/order 4.
Edexcel AEA 2002 June Q1
8 marks Challenging +1.2
1.Solve the following equation,for \(0 \leq x \leq \pi\) ,giving your answers in terms of \(\pi\) . $$\sin 5 x - \cos 5 x = \cos x - \sin x$$
Edexcel AEA 2002 June Q2
9 marks Challenging +1.8
2.In the binomial expansion of $$( 1 - 4 x ) ^ { p } , \quad | x | < \frac { 1 } { 4 }$$ the coefficient of \(x ^ { 2 }\) is equal to the coefficient of \(x ^ { 4 }\) and the coefficient of \(x ^ { 3 }\) is positive.
Find the value of \(p\) .
Edexcel AEA 2002 June Q3
11 marks Challenging +1.8
3.The curve \(C\) has parametric equations $$x = 15 t - t ^ { 3 } , \quad y = 3 - 2 t ^ { 2 }$$ Find the values of \(t\) at the points where the normal to \(C\) at \(( 14,1 )\) cuts \(C\) again.
Edexcel AEA 2002 June Q4
14 marks Challenging +1.2
4.Find the coordinates of the stationary points of the curve with equation $$x ^ { 3 } + y ^ { 3 } - 3 x y = 48$$ and determine their nature. \includegraphics[max width=\textwidth, alt={}, center]{7f1bc552-3850-43c5-b435-abc87b264f0a-3_553_749_401_618} Figure 1 shows a sketch of part of the curve with equation $$y = \sin ( \cos x ) .$$ The curve cuts the \(x\)-axis at the points \(A\) and \(C\) and the \(y\)-axis at the point \(B\).
  1. Find the coordinates of the points \(A , B\) and \(C\).
  2. Prove that \(B\) is a stationary point. Given that the region \(O C B\) is convex,
  3. show that, for \(0 \leq x \leq \frac { \pi } { 2 }\), $$\sin ( \cos x ) \leq \cos x$$ and $$\left( 1 - \frac { 2 } { \pi } x \right) \sin 1 \leq \sin ( \cos x )$$ and state in each case the value or values of \(x\) for which equality is achieved.
  4. Hence show that $$\frac { \pi } { 4 } \sin 1 < \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ( \cos x ) d x < 1$$
    \includegraphics[max width=\textwidth, alt={}]{7f1bc552-3850-43c5-b435-abc87b264f0a-4_682_824_399_704}
    Figure 2 shows a sketch of part of two curves \(C _ { 1 }\) and \(C _ { 2 }\) for \(y \geq 0\).
    The equation of \(C _ { 1 }\) is \(y = m _ { 1 } - x ^ { n _ { 1 } }\) and the equation of \(C _ { 2 }\) is \(y = m _ { 2 } - x ^ { n _ { 2 } }\), where \(m _ { 1 }\), \(m _ { 2 } , n _ { 1 }\) and \(n _ { 2 }\) are positive integers with \(m _ { 2 } > m _ { 1 }\). Both \(C _ { 1 }\) and \(C _ { 2 }\) are symmetric about the line \(x = 0\) and they both pass through the points \(( 3,0 )\) and \(( - 3,0 )\). Given that \(n _ { 1 } + n _ { 2 } = 12\), find
Edexcel AEA 2002 June Q7
18 marks Challenging +1.8
7.A student was attempting to prove that \(x = \frac { 1 } { 2 }\) is the only real root of $$x ^ { 3 } + \frac { 3 } { 4 } x - \frac { 1 } { 2 } = 0$$ The attempted solution was as follows. $$\begin{array} { r l r } & x ^ { 3 } + \frac { 3 } { 4 } x & = \frac { 1 } { 2 } \\ \therefore & x \left( x ^ { 2 } + \frac { 3 } { 4 } \right) & = \frac { 1 } { 2 } \\ \therefore & x & = \frac { 1 } { 2 } \\ \text { or } & x ^ { 2 } + \frac { 3 } { 4 } & = \frac { 1 } { 2 } \\ \text { i.e. } & x ^ { 2 } & = - \frac { 1 } { 4 } \quad \text { no solution } \\ \therefore & \text { only real root is } x & = \frac { 1 } { 2 } \end{array}$$
  1. Explain clearly the error in the above attempt.
  2. Give a correct proof that \(x = \frac { 1 } { 2 }\) is the only real root of \(x ^ { 3 } + \frac { 3 } { 4 } x - \frac { 1 } { 2 } = 0\) . The equation $$x ^ { 3 } + \beta x - \alpha = 0$$ where \(\alpha , \beta\) are real,\(\alpha \neq 0\) ,has a real root at \(x = \alpha\) .
  3. Find and simplify an expression for \(\beta\) in terms of \(\alpha\) and prove that \(\alpha\) is the only real root provided \(| \alpha | < 2\) .
    (6)
    An examiner chooses a positive number \(\alpha\) so that \(\alpha\) is the only real root of equation(I) but the incorrect method used by the student produces 3 distinct real"roots".
  4. Find the range of possible values for \(\alpha\) . Marks for style,clarity and presentation: 7
Edexcel AEA 2003 June Q1
5 marks Challenging +1.2
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{25f0c7cc-0701-4836-931e-0eff5145e029-2_433_549_270_773}
\end{figure} The point \(A\) is a distance 1 unit from the fixed origin \(O\) .Its position vector is \(\mathbf { a } = \frac { 1 } { \sqrt { 2 } } ( \mathbf { i } + \mathbf { j } )\) . The point \(B\) has position vector \(\mathbf { a } + \mathbf { j }\) ,as shown in Figure 1. By considering \(\triangle O A B\) ,prove that \(\tan \frac { 3 \pi } { 8 } = 1 + \sqrt { } 2\) .