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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 $$\begin{gathered} \log _ { x } y = \log _ { y } x \\ \log _ { x } ( x - y ) = \log _ { y } ( x + y ) \end{gathered}$$
Edexcel AEA 2006 June Q4
14 marks Challenging +1.2
4.The line with equation \(y = m x\) is a tangent to the circle \(C _ { 1 }\) with equation $$( x + 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 13$$
  1. Show that \(m\) satisfies the equation $$3 m ^ { 2 } + 56 m + 36 = 0$$ The tangents from the origin \(O\) to \(C _ { 1 }\) touch \(C _ { 1 }\) at the points \(A\) and \(B\) .
  2. Find the coordinates of the points \(A\) and \(B\) .
    (8)
    Another circle \(C _ { 2 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 13\) .The tangents from the point \(( 4 , - 7 )\) to \(C _ { 2 }\) touch it at the points \(P\) and \(Q\) .
  3. Find the coordinates of either the point \(P\) or the point \(Q\) .
    (2)
Edexcel AEA 2006 June Q5
15 marks Challenging +1.8
5.The lines \(L _ { 1 }\) and \(L _ { 2 }\) have vector equations \(L _ { 1 } : \quad \mathbf { r } = - 2 \mathbf { i } + 11.5 \mathbf { j } + \lambda ( 3 \mathbf { i } - 4 \mathbf { j } - \mathbf { k } )\), \(L _ { 2 } : \quad \mathbf { r } = 11.5 \mathbf { i } + 3 \mathbf { j } + 8.5 \mathbf { k } + \mu ( 7 \mathbf { i } + 8 \mathbf { j } - 11 \mathbf { k } )\),
where \(\lambda\) and \(\mu\) are parameters.
  1. Show that \(L _ { 1 }\) and \(L _ { 2 }\) do not intersect.
  2. Show that the vector \(( 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\) is perpendicular to both \(L _ { 1 }\) and \(L _ { 2 }\) . The point \(A\) lies on \(L _ { 1 }\) ,the point \(B\) lies on \(L _ { 2 }\) and \(A B\) is perpendicular to both \(L _ { 1 }\) and \(L _ { 2 }\) .
  3. Find the position vector of the point \(A\) and the position vector of the point \(B\) .
    (8) \includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-4_554_1017_404_571} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \sin ( \ln x ) , \quad x \geq 1 .$$ The point \(Q\) ,on \(C\) ,is a maximum.
Edexcel AEA 2006 June Q7
20 marks Hard +2.3
7. \includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-5_648_1590_296_275} The circle \(C _ { 1 }\) has centre \(O\) and radius \(R\). The tangents \(A P\) and \(B P\) to \(C _ { 1 }\) meet at the point \(P\) and angle \(A P B = 2 \alpha , 0 < \alpha < \frac { \pi } { 2 }\). A sequence of circles \(C _ { 1 } , C _ { 2 } , \ldots , C _ { n } , \ldots\) is drawn so that each new circle \(C _ { n + 1 }\) touches each of \(C _ { n } , A P\) and \(B P\) for \(n = 1,2,3 , \ldots\) as shown in Figure 2. The centre of each circle lies on the line \(O P\).
  1. Show that the radii of the circles form a geometric sequence with common ratio $$\frac { 1 - \sin \alpha } { 1 + \sin \alpha }$$
  2. Find, in terms of \(R\) and \(\alpha\), the total area enclosed by all the circles, simplifying your answer. The area inside the quadrilateral \(P A O B\), not enclosed by part of \(C _ { 1 }\) or any of the other circles, is \(S\).
  3. Show that $$S = R ^ { 2 } \left( \alpha + \cot \alpha - \frac { \pi } { 4 } \operatorname { cosec } \alpha - \frac { \pi } { 4 } \sin \alpha \right) .$$
  4. Show that, as \(\alpha\) varies, $$\frac { \mathrm { d } S } { \mathrm {~d} \alpha } = R ^ { 2 } \cot ^ { 2 } \alpha \left( \frac { \pi } { 4 } \cos \alpha - 1 \right)$$
  5. Find, in terms of \(R\), the least value of \(S\) for \(\frac { \pi } { 6 } \leq \alpha \leq \frac { \pi } { 4 }\).
Edexcel AEA 2007 June Q1
9 marks Challenging +1.2
1.(a)Write down the binomial expansion of \(\frac { 1 } { ( 1 - y ) ^ { 2 } } , | y | < 1\) ,in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\) .
(b)Hence,or otherwise,show that $$\frac { 1 } { 4 } \operatorname { cosec } ^ { 4 } \left( \frac { \theta } { 2 } \right) = 1 + 2 \cos \theta + 3 \cos ^ { 2 } \theta + 4 \cos ^ { 3 } \theta + \ldots + ( r + 1 ) \cos ^ { r } \theta + \ldots$$ and state the values of \(\theta\) for which this result is not valid.
(4)
Find
(c) $$\begin{aligned} & 1 + \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } + \frac { 4 } { 2 ^ { 3 } } + \ldots + \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots \\ & 1 - \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } - \frac { 4 } { 2 ^ { 3 } } + \ldots + ( - 1 ) ^ { r } \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots \end{aligned}$$ (d)
Edexcel AEA 2007 June Q2
10 marks Challenging +1.8
2.(a)On the same diagram,sketch \(y = x\) and \(y = \sqrt { } x\) ,for \(x \geq 0\) ,and mark clearly the coordinates of the points of intersection of the two graphs.
(b)With reference to your sketch,explain why there exists a value \(a\) of \(x ( a > 1 )\) such that $$\int _ { 0 } ^ { a } x \mathrm {~d} x = \int _ { 0 } ^ { a } \sqrt { } x \mathrm {~d} x$$ (c)Find the exact value of \(a\) .
(d)Hence,or otherwise,find a non-constant function \(\mathrm { f } ( x )\) and a constant \(b ( b \neq 0 )\) such that $$\int _ { - b } ^ { b } \mathrm { f } ( x ) \mathrm { d } x = \int _ { - b } ^ { b } \sqrt { } [ \mathrm { f } ( x ) ] \mathrm { d } x$$
Edexcel AEA 2007 June Q3
11 marks Standard +0.8
3.(a)Solve,for \(0 \leq x < 2 \pi\) , $$\cos x + \cos 2 x = 0$$ (b)Find the exact value of \(x , x \geq 0\) ,for which $$\arccos x + \arccos 2 x = \frac { \pi } { 2 }$$ [ \(\arccos x\) is an alternative notation for \(\cos ^ { - 1 } x\) .]
Edexcel AEA 2007 June Q4
11 marks Hard +2.3
4.The function \(\mathrm { h } ( x )\) has domain \(\mathbb { R }\) and range \(\mathrm { h } ( x ) > 0\) ,and satisfies $$\sqrt { \int \mathrm { h } ( x ) \mathrm { d } x } = \int \sqrt { \mathrm { h } ( x ) } \mathrm { d } x$$
  1. By substituting \(\mathrm { h } ( x ) = \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 }\) ,show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( y + c ) ,$$ where \(c\) is constant.
  2. Hence find a general expression for \(y\) in terms of \(x\) .
  3. Given that \(\mathrm { h } ( 0 ) = 1\) ,find \(\mathrm { h } ( x )\) .
Edexcel AEA 2007 June Q5
15 marks Challenging +1.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2290882-b9a4-43ec-a38f-c44d46477242-4_493_1324_279_367} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of a sequence \(S _ { 1 } , S _ { 2 } , S _ { 3 } , \ldots\), of model snowflakes. The first term \(S _ { 1 }\) consists of a single square of side \(a\). To obtain \(S _ { 2 }\), the middle third of each edge is replaced with a new square, of side \(\frac { a } { 3 }\), as shown in Figure 1 . Subsequent terms are obtained by replacing the middle third of each external edge of a new square formed in the previous snowflake, by a square \(\frac { 1 } { 3 }\) of the size, as illustrated by \(S _ { 3 }\) in Figure 1.
  1. Deduce that to form \(S _ { 4 } , 36\) new squares of side \(\frac { a } { 27 }\) must be added to \(S _ { 3 }\).
  2. Show that the perimeters of \(S _ { 2 }\) and \(S _ { 3 }\) are \(\frac { 20 a } { 3 }\) and \(\frac { 28 a } { 3 }\) respectively.
  3. Find the perimeter of \(S _ { n }\).
  4. Describe what happens to the perimeter of \(S _ { n }\) as \(n\) increases.
  5. Find the areas of \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\).
  6. Find the smallest value of the constant \(S\) such that the area of \(S _ { n } < S\), for all values of \(n\). \includegraphics[max width=\textwidth, alt={}, center]{f2290882-b9a4-43ec-a38f-c44d46477242-5_590_1041_283_588} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \tan \frac { t } { 2 } , \quad 0 \leq t \leq \frac { \pi } { 2 }\).
    The point \(P\) on \(C\) has coordinates \(\left( x , \tan \frac { x } { 2 } \right)\).
    The vertices of rectangle \(R\) are at \(( x , 0 ) , \left( \frac { x } { 2 } , 0 \right) , \left( \frac { x } { 2 } , \tan \frac { x } { 2 } \right)\) and \(\left( x , \tan \frac { x } { 2 } \right)\) as shown in Figure 2.
Edexcel AEA 2007 June Q7
20 marks Challenging +1.8
7.The points \(O , P\) and \(Q\) lie on a circle \(C\) with diameter \(O Q\) .The position vectors of \(P\) and \(Q\) , relative to \(O\) ,are \(\mathbf { p }\) and \(\mathbf { q }\) respectively.
  1. Prove that \(\mathbf { p } . \mathbf { q } = | \mathbf { p } | ^ { 2 }\) . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f2290882-b9a4-43ec-a38f-c44d46477242-6_615_714_412_689} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The point \(R\) also lies on \(C\) and \(O P Q R\) is a kite \(K\) as shown in Figure 3.The point \(S\) has position vector,relative to \(O\) ,of \(\lambda \mathbf { q }\) ,where \(\lambda\) is a constant.Given that \(\mathbf { p } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \mathbf { q } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }\) and that \(O Q\) is perpendicular to \(P S\) ,find
  2. the value of \(\lambda\) ,
  3. the position vector of \(R\) ,
  4. the area of \(K\) . Another circle \(C _ { 1 }\) is drawn inside \(K\) so that the 4 sides of the kite are each tangents to \(C _ { 1 }\) .
  5. Find the radius of \(C _ { 1 }\) giving your answer in the form \(( \sqrt { } 2 - 1 ) \sqrt { } n\) ,where \(n\) is an integer. A second kite \(K _ { 1 }\) is similar to \(K\) and is drawn inside \(C _ { 1 }\) .
  6. Find that area of \(K _ { 1 }\) .
Edexcel AEA 2008 June Q1
5 marks Standard +0.8
1.The first and second terms of an arithmetic series are 200 and 197.5 respectively.
The sum to \(n\) terms of the series is \(S _ { n }\) . Find the largest positive value of \(S _ { n }\) .