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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 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 \(\tan \frac { 3 \pi } { 8 } = 1 + \sqrt { } 2\) .
Edexcel AEA 2003 June Q2
8 marks Challenging +1.2
2.Find the values of \(\tan \theta\) such that $$2 \sin ^ { 2 } \theta - \sin \theta \sec \theta = 2 \sin 2 \theta - 2 .$$
Edexcel AEA 2003 June Q3
11 marks Challenging +1.8
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{25f0c7cc-0701-4836-931e-0eff5145e029-2_441_1111_1598_551}
\end{figure} Figure 2 shows a sketch of a part of the curve \(C\) with parametric equations $$x = t ^ { 3 } , y = t ^ { 2 } .$$ The tangent at the point \(P ( 8,4 )\) cuts \(C\) at the point \(Q\) .
Find the area of the shaded region between \(P Q\) and \(C\) .
Edexcel AEA 2003 June Q4
11 marks Challenging +1.2
4. $$f ( x ) = \frac { 1 - 3 x } { \left( 1 + 3 x ^ { 2 } \right) ( 1 - x ) ^ { 2 } } , x \neq 1$$
  1. Find the constants \(A , B , C\) and \(D\) such that $$\mathrm { f } ( x ) \equiv \frac { A x + B } { 1 + 3 x ^ { 2 } } + \frac { C } { 1 - x } + \frac { D } { ( 1 - x ) ^ { 2 } }$$
  2. Find a series expansion for \(\mathrm { f } ( x )\) in ascending powers of \(x\) ,up to and including the term in \(x ^ { 4 }\) .
  3. Find an equation of the tangent to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 0\) .
Edexcel AEA 2003 June Q5
17 marks Hard +2.3
5.The function \(f\) is given by $$f ( x ) = \frac { 1 } { \lambda } \left( x ^ { 2 } - 4 \right) \left( x ^ { 2 } - 25 \right)$$ where \(x\) is real and \(\lambda\) is a positive integer.
  1. Sketch the graph of \(y = \mathrm { f } ( x )\) showing clearly where the graph crosses the coordinate axes.
  2. Find,in terms of \(\lambda\) ,the range of f .
  3. Find the sets of positive integers \(k\) and \(\lambda\) such that the equation $$k = | \mathrm { f } ( x ) |$$ has exactly \(k\) distinct real roots.
Edexcel AEA 2003 June Q6
19 marks Challenging +1.8
6.(a)Show that $$\sqrt { 2 + \sqrt { 3 } } - \sqrt { 2 - \sqrt { 3 } } = \sqrt { 2 }$$ (b)Hence prove that $$\log _ { \frac { 1 } { 8 } } ( \sqrt { 2 + \sqrt { 3 } } - \sqrt { 2 - \sqrt { 3 } } ) = - \frac { 1 } { 6 } .$$ (c)Find all possible pairs of integers \(a\) and \(n\) such that $$\log _ { \frac { 1 } { n } } ( \sqrt { a + \sqrt { 15 } } - \sqrt { a - \sqrt { 15 } } ) = - \frac { 1 } { 2 } .$$
Edexcel AEA 2003 June Q7
22 marks Challenging +1.8
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{25f0c7cc-0701-4836-931e-0eff5145e029-4_446_1131_1093_567}
\end{figure} Figure 3 shows a sketch of part of the curve \(C\) with question $$y = \mathrm { e } ^ { - x } \sin x , \quad x \geq 0 .$$
  1. Find the coordinates of the points \(P , Q\) and \(R\) where \(C\) cuts the positive axis.
  2. Use integration by parts to show that $$\int \mathrm { e } ^ { - x } \sin x \mathrm {~d} x = - \frac { 1 } { 2 } \mathrm { e } ^ { - x } ( \sin x + \cos x ) + \text { constant }$$ The terms of the sequence \(A _ { 1 } , A _ { 2 } , \ldots , A _ { n } , \ldots\) represent areas between \(C\) and the \(x\)-axis for successive portions of \(C\) where \(y\) is positive.The area represented by \(A _ { 1 }\) and \(A _ { 2 }\) are shown in Figure 3.
  3. Find an expression for \(A _ { n }\) in terms of \(n\) and \(\pi\) .
    (6)
  4. Show that \(A _ { 1 } + A _ { 2 } + \ldots + A _ { n } + \ldots\) is a geometric series with sum to infinity $$\frac { \mathrm { e } ^ { \pi } } { 2 \left( \mathrm { e } ^ { \pi } - 1 \right) } .$$
  5. Given that $$\int _ { 0 } ^ { \infty } \mathrm { e } ^ { - x } \sin x \mathrm {~d} x = \frac { 1 } { 2 }$$ find the exact value of $$\int _ { 0 } ^ { \infty } \left| e ^ { - x } \sin x \right| d x$$ and simplify your answer. END
Edexcel AEA 2005 June Q1
6 marks Standard +0.8
1.A point \(P\) lies on the curve with equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 8 y = 24$$ Find the greatest and least possible values of the length \(O P\) ,where \(O\) is the origin.
Edexcel AEA 2005 June Q2
8 marks Challenging +1.2
2.Solve,for \(0 < \theta < 2 \pi\) , $$\sin 2 \theta + \cos 2 \theta + 1 = \sqrt { 6 } \cos \theta$$ giving your answers in terms of \(\pi\) .
Edexcel AEA 2005 June Q3
9 marks Hard +2.3
3.Given that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( u \sqrt { } x ) = \frac { \mathrm { d } u } { \mathrm {~d} x } \times \frac { \mathrm { d } ( \sqrt { } x ) } { \mathrm { d } x } , \quad 0 < x < \frac { 1 } { 2 }$$ where \(u\) is a function of \(x\) ,and that \(u = 4\) when \(x = \frac { 3 } { 8 }\) ,find \(u\) in terms of \(x\) .
(9)
Edexcel AEA 2005 June Q4
13 marks Challenging +1.8
4.A rectangle \(A B C D\) is drawn so that \(A\) and \(B\) lie on the \(x\)-axis,and \(C\) and \(D\) lie on the curve with equation \(y = \cos x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\) .The point \(A\) has coordinates \(( p , 0 )\) ,where \(0 < p < \frac { \pi } { 2 }\) .
  1. Find an expression,in terms of \(p\) ,for the area of this rectangle. The maximum area of \(A B C D\) is \(S\) and occurs when \(p = \alpha\) .Show that
  2. \(\frac { \pi } { 4 } < \alpha < 1\) ,
  3. \(S = \frac { 2 \alpha ^ { 2 } } { \sqrt { } \left( 1 + \alpha ^ { 2 } \right) }\) ,
  4. \(\frac { \pi ^ { 2 } } { 2 \sqrt { } \left( 16 + \pi ^ { 2 } \right) } < S < \sqrt { } 2\) .
Edexcel AEA 2005 June Q5
19 marks Challenging +1.2
5.The point \(A\) has position vector \(7 \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k }\) and the point \(B\) has position vector \(12 \mathbf { i } + 3 \mathbf { j } - 15 \mathbf { k }\) .
  1. Find a vector for the line \(L _ { 1 }\) which passes through \(A\) and \(B\) . The line \(L _ { 2 }\) has vector equation $$\mathbf { r } = - 4 \mathbf { i } + 12 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { k } )$$
  2. Show that \(L _ { 2 }\) passes through the origin \(O\) .
  3. Show that \(L _ { 1 }\) and \(L _ { 2 }\) intersect at a point \(C\) and find the position vector of \(C\) .
  4. Find the cosine of \(\angle O C A\) .
  5. Hence,or otherwise,find the shortest distance from \(O\) to \(L _ { 1 }\) .
  6. Show that \(| \overrightarrow { C O } | = | \overrightarrow { A B } |\) .
  7. Find a vector equation for the line which bisects \(\angle O C A\) . \includegraphics[max width=\textwidth, alt={}, center]{f9d3e02c-cef2-435b-9cda-76c43fcac575-4_922_1054_279_586} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x \left( 12 - x ^ { 2 } \right) .$$ The curve cuts the \(x\)-axis at the points \(P , O\) and \(R\), and \(Q\) is the maximum point.
Edexcel AEA 2005 June Q7
19 marks Challenging +1.8
  1. (a) Use the substitution \(x = \sec \theta\) to show that
$$\int \sqrt { } \left( x ^ { 2 } - 1 \right) d x$$ can be written as $$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta$$ (3)
(b) Use integration by parts to show that $$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta = \frac { 1 } { 2 } [ \sec \theta \tan \theta - \ln | \sec \theta + \tan \theta | ] + \text { constant. }$$ (c) Evaluate \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin x \sqrt { } ( \cos 2 x ) \mathrm { d } x\).
Edexcel AEA 2006 June Q1
8 marks Challenging +1.8
1.(a)For \(| y | < 1\) ,write down the binomial series expansion of \(( 1 - y ) ^ { - 2 }\) in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\) .
(b)Hence,or otherwise,show that $$1 + \frac { 2 x } { 1 + x } + \frac { 3 x ^ { 2 } } { ( 1 + x ) ^ { 2 } } + \ldots + \frac { r x ^ { r - 1 } } { ( 1 + x ) ^ { r - 1 } } + \ldots$$ can be written in the form \(( a + x ) ^ { n }\) .Write down the values of the integers \(a\) and \(n\) .
(c)Find the set of values of \(x\) for which the series in part(b)is convergent.
Edexcel AEA 2006 June Q2
10 marks Challenging +1.2
2.Given that \(( \sin \theta + \cos \theta ) \neq 0\) ,find all the solutions of $$\frac { 2 \cos 2 \theta ( \sin 2 \theta - \sqrt { } 3 \cos 2 \theta ) } { \sin \theta + \cos \theta } = \sqrt { } 6 ( \sin 2 \theta - \sqrt { } 3 \cos 2 \theta )$$ for \(0 \leq \theta < 360 ^ { \circ }\) .
Edexcel AEA 2006 June Q3
11 marks Challenging +1.2
3.Given that \(x > y > 0\) ,
  1. by writing \(\log _ { y } x = z\) ,or otherwise,show that \(\log _ { y } x = \frac { 1 } { \log _ { x } y }\) .
  2. Given also that \(\log _ { x } y = \log _ { y } x\) ,show that \(y = \frac { 1 } { x }\) .
  3. Solve the simultaneous equations $$\begin{gathered} \log _ { x } y = \log _ { y } x \\ \log _ { x } ( x - y ) = \log _ { y } ( x + y ) \end{gathered}$$