Questions AEA (167 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
Edexcel AEA 2012 June Q3
10 marks Hard +2.3
3.The angle \(\theta , 0 < \theta < \frac { \pi } { 2 }\) ,satisfies $$\tan \theta \tan 2 \theta = \sum _ { r = 0 } ^ { \infty } 2 \cos ^ { r } 2 \theta$$
  1. Show that \(\tan \theta = 3 ^ { p }\) ,where \(p\) is a rational number to be found.
  2. Hence show that \(\frac { \pi } { 6 } < \theta < \frac { \pi } { 4 }\)
Edexcel AEA 2012 June Q6
16 marks Challenging +1.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fc5d0d07-b750-4646-bdcb-419a290200c9-4_433_1011_221_529} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = ( x + a ) ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are positive constants. The curve cuts the \(x\)-axis at \(P\) and has a maximum point at \(S\) and a minimum point at \(Q\).
  1. Write down the coordinates of \(P\) and \(Q\) in terms of \(a\) and \(b\).
  2. Show that \(G\), the area of the shaded region between the curve \(P S Q\) and the \(x\)-axis, is given by \(G = \frac { ( a + b ) ^ { 4 } } { 12 }\). The rectangle \(P Q R S T\) has \(R S T\) parallel to \(Q P\) and both \(P T\) and \(Q R\) are parallel to the \(y\)-axis.
  3. Show that \(\frac { G } { \text { Area of } P Q R S T } = k\), where \(k\) is a constant independent of \(a\) and \(b\) and find the value of \(k\).
Edexcel AEA 2013 June Q1
6 marks Challenging +1.2
1.In the binomial expansion of $$\left( 1 + \frac { 12 n } { 5 } x \right) ^ { n }$$ the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) are equal and non-zero.
  1. Find the possible values of \(n\) .
    (4)
  2. State,giving a reason,which value of \(n\) gives a valid expansion when \(x = \frac { 1 } { 2 }\) (2)
Edexcel AEA 2013 June Q2
8 marks Standard +0.8
2.(a)Use the formula for \(\sin ( A - B )\) to show that \(\sin \left( 90 ^ { \circ } - x \right) = \cos x\) (b)Solve for \(0 < \theta < 360 ^ { \circ }\) $$2 \sin \left( \theta + 17 ^ { \circ } \right) = \frac { \cos \left( \theta + 8 ^ { \circ } \right) } { \cos \left( \theta + 17 ^ { \circ } \right) }$$
Edexcel AEA 2013 June Q3
13 marks Challenging +1.2
3.The lines \(L _ { 1 }\) and \(L _ { 2 }\) have equations given by \(L _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { c } - 7 \\ 7 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ 0 \\ - 3 \end{array} \right)\) and \(L _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { c } 7 \\ p \\ - 6 \end{array} \right) + \mu \left( \begin{array} { c } 10 \\ - 4 \\ - 1 \end{array} \right)\) where \(\lambda\) and \(\mu\) are parameters and \(p\) is a constant.
The two lines intersect at the point \(C\) .
  1. Find
    1. the value of \(p\) ,
    2. the position vector of \(C\) .
  2. Show that the point \(B\) with position vector \(\left( \begin{array} { c } - 13 \\ 11 \\ - 4 \end{array} \right)\) lies on \(L _ { 2 }\) . The point \(A\) with position vector \(\left( \begin{array} { c } - 7 \\ 7 \\ 1 \end{array} \right)\) lies on \(L _ { 1 }\) .
  3. Find \(\cos ( \angle A C B )\) ,giving your answer as an exact fraction. The line \(L _ { 3 }\) bisects the angle \(A C B\) .
  4. Find a vector equation of \(L _ { 3 }\) .
Edexcel AEA 2013 June Q4
13 marks Challenging +1.2
4.A sequence of positive integers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) has \(r\) th term given by $$a _ { r } = 2 ^ { r } - 1$$
  1. Write down the first 6 terms of this sequence.
  2. Verify that \(a _ { r + 1 } = 2 a _ { r } + 1\)
  3. Find \(\sum _ { r = 1 } ^ { n } a _ { r }\)
  4. Show that \(\frac { 1 } { a _ { r + 1 } } < \frac { 1 } { 2 } \times \frac { 1 } { a _ { r } }\)
  5. Hence show that \(1 + \frac { 1 } { 3 } + \frac { 1 } { 7 } + \frac { 1 } { 15 } + \frac { 1 } { 31 } + \ldots < 1 + \frac { 1 } { 3 } + \left( \frac { 1 } { 7 } + \frac { \frac { 1 } { 2 } } { 7 } + \frac { \frac { 1 } { 4 } } { 7 } + \ldots \right)\)
  6. Show that \(\frac { 31 } { 21 } < \sum _ { r = 1 } ^ { \infty } \frac { 1 } { a _ { r } } < \frac { 34 } { 21 }\)
Edexcel AEA 2013 June Q5
15 marks Hard +2.3
5.In this question u and v are functions of \(x\) .Given that \(\int \mathrm { u } \mathrm { d } x , \int \mathrm { v } \mathrm { d } x\) and \(\int \mathrm { uv } \mathrm { d } x\) satisfy $$\int \text { uv } \mathrm { d } x = \left( \int \mathrm { u } \mathrm {~d} x \right) \times \left( \int \mathrm { v } \mathrm {~d} x \right) \quad \text { uv } \neq 0$$
  1. show that \(1 = \frac { \int \mathrm { u } \mathrm { d } x } { \mathrm { u } } + \frac { \int \mathrm { v } \mathrm { d } x } { \mathrm { v } }\) Given also that \(\frac { \int \mathrm { u } \mathrm { d } x } { \mathrm { u } } = \mathrm { sin } ^ { 2 } x\),
  2. use part(a)to write down an expression,in terms of \(x\) ,for \(\frac { \int \mathrm { v } \mathrm { d } x } { \mathrm { v } }\) ,
  3. show that $$\frac { 1 } { \mathrm { u } } \frac { \mathrm { du } } { \mathrm {~d} x } = \frac { 1 - 2 \sin x \cos x } { \sin ^ { 2 } x }$$
  4. hence use integration to show that \(\mathrm { u } = A \mathrm { e } ^ { - \cot x } \operatorname { cosec } ^ { 2 } x\) ,where \(A\) is an arbitrary constant.
  5. By differentiating \(\mathrm { e } ^ { \tan x }\) find a similar expression for v .
Edexcel AEA 2013 June Q6
16 marks Hard +2.3
6.(a)Starting from \([ \mathrm { f } ( x ) - \lambda \mathrm { g } ( x ) ] ^ { 2 } \geqslant 0\) show that \(\lambda\) satisfies the quadratic inequality $$\left( \int _ { a } ^ { b } [ \operatorname { g } ( x ) ] ^ { 2 } \mathrm {~d} x \right) \lambda ^ { 2 } - 2 \left( \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { g } ( x ) \mathrm { d } x \right) \lambda + \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x \geqslant 0$$ where \(a\) and \(b\) are constants and \(\lambda\) can take any real value.
(2)
(b)Hence prove that $$\left[ \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { g } ( x ) \mathrm { d } x \right] ^ { 2 } \leqslant \left[ \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x \right] \times \left[ \int _ { a } ^ { b } [ \mathrm {~g} ( x ) ] ^ { 2 } \mathrm {~d} x \right]$$ (c)By letting \(\mathrm { f } ( x ) = 1\) and \(\mathrm { g } ( x ) = \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } }\) show that $$\int _ { - 1 } ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x \leqslant \frac { 9 } { 2 }$$ (d)Show that \(\int _ { - 1 } ^ { 2 } x ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 4 } } \mathrm {~d} x = \frac { 12 \sqrt { } 3 } { 5 }\) (e)Hence show that $$\frac { 144 } { 55 } \leqslant \int _ { - 1 } ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x$$
Edexcel AEA 2013 June Q7
22 marks Challenging +1.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8bd0bc33-e69e-4e51-aae7-288810c5db07-6_643_1374_173_351} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { x } { 3 } + \frac { 12 } { x } \quad x \neq 0$$ The lines \(x = 0\) and \(y = \frac { x } { 3 }\) are asymptotes to \(C _ { 1 }\). The point \(A\) on \(C _ { 1 }\) is a minimum and the point \(B\) on \(C _ { 1 }\) is a maximum.
  1. Find the coordinates of \(A\) and \(B\). There is a normal to \(C _ { 1 }\), which does not intersect \(C _ { 1 }\) a second time, that has equation \(x = k\), where \(k > 0\).
  2. Write down the value of \(k\). The point \(P ( \alpha , \beta ) , \alpha > 0\) and \(\alpha \neq k\), lies on \(C _ { 1 }\). The normal to \(C _ { 1 }\) at \(P\) does not intersect \(C _ { 1 }\) a second time.
  3. Find the value of \(\alpha\), leaving your answer in simplified surd form.
  4. Find the equation of this normal. The curve \(C _ { 2 }\) has equation \(y = | \mathrm { f } ( x ) |\)
  5. Sketch \(C _ { 2 }\) stating the coordinates of any turning points and the equations of any asymptotes. The line with equation \(y = m x + 1\) does not touch or intersect \(C _ { 2 }\).
  6. Find the set of possible values for \(m\).
Edexcel AEA 2016 June Q1
7 marks Standard +0.8
1.The function f is given by $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 9 \quad x \in \mathbb { R } , x \geqslant 3$$
  1. Find the range of f . The function g is given by $$\operatorname { g } ( x ) = \frac { 10 } { x + 1 } \quad x \in \mathbb { R } , x \geqslant 4$$
  2. Find an expression for \(\operatorname { gf } ( x )\) .
  3. Find the domain and range of gf.
Edexcel AEA 2016 June Q2
7 marks Challenging +1.8
2.Find the value of $$\arccos \left( \frac { 1 } { \sqrt { 2 } } \right) + \arcsin \left( \frac { 1 } { 3 } \right) + 2 \arctan \left( \frac { 1 } { \sqrt { 2 } } \right)$$ Give your answer as a multiple of \(\pi\) . $$\text { (arccos } x \text { is an alternative notion for } \cos ^ { - 1 } x \text { etc.) }$$
Edexcel AEA 2016 June Q3
9 marks Challenging +1.2
3.The points \(A , B , C , D\) and \(E\) are five of the vertices of a rectangular cuboid and \(A E\) is a diagonal of the cuboid.With respect to a fixed origin \(O\) ,the position vectors of \(A , B , C\) and \(D\) are \(\mathbf { a , b , c }\) and \(\mathbf{d}\) respectively,where $$\mathbf { a } = \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { c } 0 \\ - 3 \\ - 8 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { c } 4 \\ - 1 \\ - 10 \end{array} \right)$$
Edexcel AEA 2016 June Q4
11 marks Challenging +1.8
\text { and } \mathbf { d } = \left( \begin{array} { c } - 4
2
- 11 \end{array} \right)$$
  1. Find the position vector of \(E\) . The volume of a tetrahedron is given by the formula $$\text { volume } = \frac { 1 } { 3 } ( \text { area of base } ) \times ( \text { height } )$$
  2. Find the volume of the tetrahedron \(A B C D\) . 4.(a)Given that \(x > 0 , y > 0 , x \neq 1\) and \(n > 0\) ,show that $$\log _ { x } y = \log _ { x ^ { n } } y ^ { n }$$
  3. Solve the following,leaving your answers in the form \(2 ^ { p }\) ,where \(p\) is a rational number.
    1. \(\log _ { 2 } u + \log _ { 4 } u ^ { 2 } + \log _ { 8 } u ^ { 3 } + \log _ { 16 } u ^ { 4 } = 5\)
    2. \(\log _ { 2 } v + \log _ { 4 } v + \log _ { 8 } v + \log _ { 16 } v = 5\)
    3. \(\log _ { 4 } w ^ { 2 } + \frac { 3 \log _ { 8 } 64 } { \log _ { 2 } w } = 5\)
Edexcel AEA 2016 June Q5
13 marks Challenging +1.8
5.(a)Show that $$\sum _ { r = 0 } ^ { n } x ^ { - r } = \frac { x } { x - 1 } - \frac { x ^ { - n } } { x - 1 } \quad \text { where } x \neq 0 \text { and } x \neq 1$$ (b)Hence find an expression in terms of \(x\) and \(n\) for \(\sum _ { r = 0 } ^ { n } r x ^ { - ( r + 1 ) }\) for \(x \neq 0\) and \(x \neq 1\) Simplify your answer.
(c)Find \(\sum _ { r = 0 } ^ { n } \left( \frac { 3 + 5 r } { 2 ^ { r } } \right)\) Give your answer in the form \(a - \frac { b + c n } { 2 ^ { n } }\) ,where \(a , b\) and \(c\) are integers.
Edexcel AEA 2016 June Q6
22 marks Challenging +1.2
6. \includegraphics[max width=\textwidth, alt={}, center]{0214eebf-93f2-4338-9222-443000115225-4_346_1040_303_548} \section*{Figure 1} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation $$y = \cos ( \cos x ) \sin x \quad \text { for } \quad 0 \leqslant x \leqslant \pi$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Hence verify that the turning point is at \(x = \frac { \pi } { 2 }\) and find the \(y\) coordinate of this point.
  3. Find the area of the region bounded by \(C _ { 1 }\) and the positive \(x\)-axis between \(x = 0\) and \(x = \pi\) Figure 2 shows a sketch of the curve \(C _ { 1 }\) and the curve \(C _ { 2 }\) with equation $$y = \sin ( \cos x ) \sin x \quad \text { for } \quad 0 \leqslant x \leqslant \pi$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0214eebf-93f2-4338-9222-443000115225-4_519_1065_1631_484} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the origin and the point \(A ( a , b )\) ,where \(a < \pi\)
  4. Find \(a\) and \(b\) ,giving \(b\) in a form not involving trigonometric functions.
  5. Find the area of the shaded region between \(C _ { 1 }\) and \(C _ { 2 }\)
Edexcel AEA 2016 June Q7
24 marks Challenging +1.8
7. (a) Find the set of values of \(k\) for which the equation $$\frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 } = k$$ has no real roots. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0214eebf-93f2-4338-9222-443000115225-5_718_869_511_603} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 }\) The curve has asymptotes \(x = a , x = b\) and \(y = c\), where \(a , b\) and \(c\) are integers.
(b) Find the value of \(a\), the value of \(b\) and the value of \(c\).
(c) Find the coordinates of the points of intersection of \(C _ { 1 }\) with the line \(y = 2\) (d) Find all the integer pairs \(( r , s )\) that satisfy \(s = \frac { r ^ { 2 } + 3 r + 8 } { r ^ { 2 } + r - 2 }\) The curve \(C _ { 2 }\) has equation \(y = \mathrm { g } ( x )\) where \(\mathrm { g } ( x ) = \frac { 2 x ^ { 2 } - 4 x + 6 } { x ^ { 2 } - 3 x }\) (e) Show that, for suitable integers \(m\) and \(n , \mathrm {~g} ( x )\) can be written in the form \(\mathrm { f } ( x + m ) + n\).
(f) Sketch \(C _ { 2 }\) showing any asymptotes and stating their equations.
Edexcel AEA 2017 June Q1
7 marks Standard +0.8
1.The function f is given by $$\mathrm { f } ( x ) = \sqrt { x + 2 } \quad \text { for } \quad x \in \mathbb { R } , x \geqslant 0$$
  1. Find \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\) The function g is given by $$\mathrm { g } ( x ) = x ^ { 2 } - 4 x + 5 \text { for } x \in \mathbb { R } , x \geqslant 0$$
  2. Find the range of g .
  3. Solve the equation \(\operatorname { fg } ( x ) = x\) .
Edexcel AEA 2017 June Q2
9 marks Challenging +1.8
2.(a)Show that the equation $$\tan x = \frac { \sqrt { 3 } } { 1 + 4 \cos x }$$ can be written in the form $$\sin 2 x = \sin \left( 60 ^ { \circ } - x \right)$$ (b)Solve,for \(0 < x < 180 ^ { \circ }\) $$\tan x = \frac { \sqrt { 3 } } { 1 + 4 \cos x }$$
Edexcel AEA 2017 June Q3
13 marks Challenging +1.3
  1. The line \(L _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { c } - 13 \\ 7 \\ - 1 \end{array} \right) + t \left( \begin{array} { c } 6 \\ - 2 \\ 3 \end{array} \right)\). The line \(L _ { 2 }\) passes through the point \(A\) with position vector \(\left( \begin{array} { c } 1 \\ p \\ 10 \end{array} \right)\) and is parallel to \(\left( \begin{array} { c } - 2 \\ 11 \\ - 5 \end{array} \right)\), where \(p\) is a constant. The lines \(L _ { 1 }\) and \(L _ { 2 }\) intersect at the point \(B\).
    1. Find
      1. the value of \(p\),
      2. the position vector of \(B\).
    The point \(C\) lies on \(L _ { 1 }\) and angle \(A C B\) is \(90 ^ { \circ }\)
  2. Find the position vector of \(C\). The point \(D\) also lies on \(L _ { 1 }\) and triangle \(A B D\) is isosceles with \(A B = A D\).
  3. Find the area of triangle \(A B D\).
Edexcel AEA 2017 June Q4
13 marks Challenging +1.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15e3f7f2-a77c-4ee4-8f0a-ac739e9fede5-4_332_454_201_810} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the equilateral triangle \(L M N\) of side 2 cm .The point \(P\) lies on \(L M\) such that \(L P = x \mathrm {~cm}\) and the point \(Q\) lies on \(L N\) such that \(L Q = y \mathrm {~cm}\) .The points \(P\) and \(Q\) are chosen so that the area of triangle \(L P Q\) is half the area of triangle \(L M N\) .
  1. Show that \(x y = 2\)
  2. Find the shortest possible length of \(P Q\) ,justifying your answer. Mathematicians know that for any closed curve or polygon enclosing a fixed area,the ratio \(\frac { \text { area enclosed } } { \text { perimeter } }\) is a maximum when the closed curve is a circle. By considering 6 copies of triangle \(L M N\) suitably arranged,
  3. find the length of the shortest line or curve that can be drawn from a point on \(L M\) to a point on \(L N\) to divide the area of triangle \(L M N\) in half.Justify your answer.
    (6)
Edexcel AEA 2017 June Q5
14 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15e3f7f2-a77c-4ee4-8f0a-ac739e9fede5-5_946_1498_210_287} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 4 ( x - 1 ) } { x ( x - 3 ) }$$ The curve cuts the \(x\)-axis at \(( a , 0 )\). The lines \(y = 0 , x = 0\) and \(x = b\) are asymptotes to the curve.
  1. Write down the value of \(a\) and the value of \(b\).
    (2)
  2. On separate axes, sketch the curves with the following equations. On your sketches, you should mark the coordinates of any intersections with the coordinate axes and state the equations of any asymptotes.
    1. \(y = \mathrm { f } ( x + 2 ) - 4\)
    2. \(y = \mathrm { f } ( | x | ) - 3\)
Edexcel AEA 2017 June Q6
16 marks Challenging +1.8
6.(a)Show that $$\frac { \mathrm { d } } { \mathrm {~d} u } \ln \left( u + \sqrt { u ^ { 2 } - 1 } \right) = \frac { 1 } { \sqrt { u ^ { 2 } - 1 } }$$ (b)Use the result from part(a)and the substitution \(x + 3 = \frac { 1 } { t }\) to find $$\int \frac { 1 } { ( x + 3 ) \sqrt { 2 x + 7 } } \mathrm {~d} x$$ (6)
(c)Express \(\frac { 1 } { 2 x ^ { 2 } + 13 x + 21 }\) in partial fractions.
(d)Find $$\int _ { 1 } ^ { 9 } \frac { 1 } { \left( 2 x ^ { 2 } + 13 x + 21 \right) \sqrt { 2 x + 7 } } \mathrm {~d} x$$ giving your answer in the form \(\ln r - s\) where \(r\) and \(s\) are rational numbers.
Edexcel AEA 2017 June Q7
21 marks Challenging +1.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15e3f7f2-a77c-4ee4-8f0a-ac739e9fede5-7_583_1198_217_440} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of the curve \(C\) with equation \(y = x ^ { 4 } - 10 x ^ { 3 } + 33 x ^ { 2 } - 34 x\) and the line \(L\) with equation \(y = m x + c\) . The line \(L\) touches \(C\) at the points \(P\) and \(Q\) with \(x\) coordinates \(p\) and \(q\) respectively.
  1. Explain why $$x ^ { 4 } - 10 x ^ { 3 } + 33 x ^ { 2 } - ( 34 + m ) x - c = ( x - p ) ^ { 2 } ( x - q ) ^ { 2 }$$ The finite region \(R\) ,shown shaded in Figure 3,is bounded by \(C\) and \(L\) .
  2. Use integration by parts to show that the area of \(R\) is \(\frac { ( q - p ) ^ { 5 } } { 30 }\)
  3. Show that $$( x - p ) ^ { 2 } ( x - q ) ^ { 2 } = x ^ { 4 } - 2 ( p + q ) x ^ { 3 } + S x ^ { 2 } - T x + U$$ where \(S , T\) and \(U\) are expressions to be found in terms of \(p\) and \(q\) .
  4. Using part(a)and part(c)find the value of \(p\) ,the value of \(q\) and the equation of \(L\) .
Edexcel AEA 2017 Specimen Q1
8 marks Challenging +1.2
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)Show that when it is convergent,the series $$1 + \frac { 2 x } { x + 3 } + \frac { 3 x ^ { 2 } } { ( x + 3 ) ^ { 2 } } + \ldots + \frac { r x ^ { r - 1 } } { ( x + 3 ) ^ { r - 1 } } + \ldots$$ can be written in the form \(( 1 + a x ) ^ { n }\) ,where \(a\) and \(n\) are constants to be found.
(c)Find the set of values of \(x\) for which the series in part(b)is convergent.
Edexcel AEA 2017 Specimen Q2
11 marks Challenging +1.8
2.(a)On separate diagrams,sketch the curves with the following equations.On each sketch you should label the exact coordinates of the points where the curve meets the coordinate axes.
  1. \(y = 8 + 2 x - x ^ { 2 }\)
  2. \(y = 8 + 2 | x | - x ^ { 2 }\)
  3. \(y = 8 + x + | x | - x ^ { 2 }\) (b)Find the values of \(x\) for which $$\left| 8 + x + | x | - x ^ { 2 } \right| = 8 + 2 | x | - x ^ { 2 }$$