Questions (33218 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
OCR MEI FP2 2007 January Q3
18 marks Challenging +1.2
3 Let \(\mathbf { P } = \left( \begin{array} { r r r } 4 & 2 & k \\ 1 & 1 & 3 \\ 1 & 0 & - 1 \end{array} \right) (\) where \(k \neq 4 )\) and \(\mathbf { M } = \left( \begin{array} { r r r } 2 & - 2 & - 6 \\ - 1 & 3 & 1 \\ 1 & - 2 & - 2 \end{array} \right)\).
  1. Find \(\mathbf { P } ^ { - 1 }\) in terms of \(k\), and show that, when \(k = 2 , \mathbf { P } ^ { - 1 } = \frac { 1 } { 2 } \left( \begin{array} { r r r } - 1 & 2 & 4 \\ 4 & - 6 & - 10 \\ - 1 & 2 & 2 \end{array} \right)\).
  2. Verify that \(\left( \begin{array} { l } 4 \\ 1 \\ 1 \end{array} \right) , \left( \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right)\) are eigenvectors of \(\mathbf { M }\), and find the corresponding eigenvalues.
  3. Show that \(\mathbf { M } ^ { n } = \left( \begin{array} { r r r } 4 & - 6 & - 10 \\ 2 & - 3 & - 5 \\ 0 & 0 & 0 \end{array} \right) + 2 ^ { n - 1 } \left( \begin{array} { r r r } - 2 & 4 & 4 \\ - 3 & 6 & 6 \\ 1 & - 2 & - 2 \end{array} \right)\). Section B (18 marks)
OCR MEI FP2 2007 January Q4
18 marks Challenging +1.2
4
  1. Show that \(\operatorname { arcosh } x = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\).
  2. Find \(\int _ { 2.5 } ^ { 3.9 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 9 } } \mathrm {~d} x\), giving your answer in the form \(a \ln b\), where \(a\) and \(b\) are rational numbers.
  3. There are two points on the curve \(y = \frac { \cosh x } { 2 + \sinh x }\) at which the gradient is \(\frac { 1 } { 9 }\). Show that one of these points is \(\left( \ln ( 1 + \sqrt { 2 } ) , \frac { 1 } { 3 } \sqrt { 2 } \right)\), and find the coordinates of the other point, in a similar form.
OCR MEI FP2 2007 January Q5
18 marks Challenging +1.2
5 Cartesian coordinates \(( x , y )\) and polar coordinates \(( r , \theta )\) are set up in the usual way, with the pole at the origin and the initial line along the positive \(x\)-axis, so that \(x = r \cos \theta\) and \(y = r \sin \theta\). A curve has polar equation \(r = k + \cos \theta\), where \(k\) is a constant with \(k \geqslant 1\).
  1. Use your graphical calculator to obtain sketches of the curve in the three cases $$k = 1 , k = 1.5 \text { and } k = 4$$
  2. Name the special feature which the curve has when \(k = 1\).
  3. For each of the three cases, state the number of points on the curve at which the tangent is parallel to the \(y\)-axis.
  4. Express \(x\) in terms of \(k\) and \(\theta\), and find \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\). Hence find the range of values of \(k\) for which there are just two points on the curve where the tangent is parallel to the \(y\)-axis. The distance between the point ( \(r , \theta\) ) on the curve and the point ( 1,0 ) on the \(x\)-axis is \(d\).
  5. Use the cosine rule to express \(d ^ { 2 }\) in terms of \(k\) and \(\theta\), and deduce that \(k ^ { 2 } \leqslant d ^ { 2 } \leqslant k ^ { 2 } + 1\).
  6. Hence show that, when \(k\) is large, the shape of the curve is very nearly circular.
OCR MEI FP2 2008 January Q1
18 marks Standard +0.8
1
  1. Fig. 1 shows the curve with polar equation \(r = a ( 1 - \cos 2 \theta )\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{43b4c7ed-3556-4d87-8aef-0111fe747982-2_529_620_577_799} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Find the area of the region enclosed by the curve.
    1. Given that \(\mathrm { f } ( x ) = \arctan ( \sqrt { 3 } + x )\), find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
    2. Hence find the Maclaurin series for \(\arctan ( \sqrt { 3 } + x )\), as far as the term in \(x ^ { 2 }\).
    3. Hence show that, if \(h\) is small, \(\int _ { - h } ^ { h } x \arctan ( \sqrt { 3 } + x ) \mathrm { d } x \approx \frac { 1 } { 6 } h ^ { 3 }\).
OCR MEI FP2 2008 January Q2
18 marks Challenging +1.2
2
  1. Find the 4th roots of 16j, in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Illustrate the 4th roots on an Argand diagram.
    1. Show that \(\left( 1 - 2 \mathrm { e } ^ { \mathrm { j } \theta } \right) \left( 1 - 2 \mathrm { e } ^ { - \mathrm { j } \theta } \right) = 5 - 4 \cos \theta\). Series \(C\) and \(S\) are defined by $$\begin{aligned} & C = 2 \cos \theta + 4 \cos 2 \theta + 8 \cos 3 \theta + \ldots + 2 ^ { n } \cos n \theta \\ & S = 2 \sin \theta + 4 \sin 2 \theta + 8 \sin 3 \theta + \ldots + 2 ^ { n } \sin n \theta \end{aligned}$$
    2. Show that \(C = \frac { 2 \cos \theta - 4 - 2 ^ { n + 1 } \cos ( n + 1 ) \theta + 2 ^ { n + 2 } \cos n \theta } { 5 - 4 \cos \theta }\), and find a similar expression for \(S\).
OCR MEI FP2 2008 January Q3
18 marks Standard +0.3
3 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 7 & 3 \\ - 4 & - 1 \end{array} \right)\).
  1. Find the eigenvalues, and corresponding eigenvectors, of the matrix \(\mathbf { M }\).
  2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }\).
  3. Given that \(\mathbf { M } ^ { n } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\), show that \(a = - \frac { 1 } { 2 } + \frac { 3 } { 2 } \times 5 ^ { n }\), and find similar expressions for \(b , c\) and \(d\). Section B (18 marks)
OCR MEI FP2 2008 January Q4
18 marks Standard +0.8
4
  1. Given that \(k \geqslant 1\) and \(\cosh x = k\), show that \(x = \pm \ln \left( k + \sqrt { k ^ { 2 } - 1 } \right)\).
  2. Find \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 1 } } \mathrm {~d} x\), giving the answer in an exact logarithmic form.
  3. Solve the equation \(6 \sinh x - \sinh 2 x = 0\), giving the answers in an exact form, using logarithms where appropriate.
  4. Show that there is no point on the curve \(y = 6 \sinh x - \sinh 2 x\) at which the gradient is 5 .
OCR MEI FP2 2008 January Q5
18 marks Challenging +1.2
5 A curve has parametric equations \(x = \frac { t ^ { 2 } } { 1 + t ^ { 2 } } , y = t ^ { 3 } - \lambda t\), where \(\lambda\) is a constant.
  1. Use your calculator to obtain a sketch of the curve in each of the cases $$\lambda = - 1 , \quad \lambda = 0 \quad \text { and } \quad \lambda = 1 .$$ Name any special features of these curves.
  2. By considering the value of \(x\) when \(t\) is large, write down the equation of the asymptote. For the remainder of this question, assume that \(\lambda\) is positive.
  3. Find, in terms of \(\lambda\), the coordinates of the point where the curve intersects itself.
  4. Show that the two points on the curve where the tangent is parallel to the \(x\)-axis have coordinates $$\left( \frac { \lambda } { 3 + \lambda } , \pm \sqrt { \frac { 4 \lambda ^ { 3 } } { 27 } } \right)$$ Fig. 5 shows a curve which intersects itself at the point ( 2,0 ) and has asymptote \(x = 8\). The stationary points A and B have \(y\)-coordinates 2 and - 2 . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{43b4c7ed-3556-4d87-8aef-0111fe747982-4_791_609_1482_769} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure}
  5. For the curve sketched in Fig. 5, find parametric equations of the form \(x = \frac { a t ^ { 2 } } { 1 + t ^ { 2 } } , y = b \left( t ^ { 3 } - \lambda t \right)\), where \(a , \lambda\) and \(b\) are to be determined.
Edexcel M2 Q2
Standard +0.3
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{173a2029-a0b8-437f-9339-5a1b6f30a8e3-004_378_652_294_630}
\end{figure} A thin uniform wire, of total length 20 cm , is bent to form a frame. The frame is in the shape of a trapezium \(A B C D\), where \(A B = A D = 4 \mathrm {~cm} , C D = 5 \mathrm {~cm}\), and \(A B\) is perpendicular to \(B C\) and \(A D\), as shown in Figure 1.
  1. Find the distance of the centre of mass of the frame from \(A B\). The frame has mass \(M\). A particle of mass \(k M\) is attached to the frame at \(C\). When the frame is freely suspended from the mid-point of \(B C\), the frame hangs in equilibrium with \(B C\) horizontal.
  2. Find the value of \(k\).
Edexcel M2 Q3
Standard +0.3
3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$ where \(c\) is a positive constant.When \(t = 1.5\) ,the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) .Find
  1. the value of \(c\) ,
  2. the acceleration of \(P\) when \(t = 1.5\) . \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\begin{aligned} \mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } , \\ \text { where } c \text { is a positive constant.When } t = 1.5 \text { ,the speed of } P \text { is } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 } \text { .Find } \end{aligned}$$ (a)the value of \(c\) , 3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is D墐
    (b)the acceleration of \(P\) when \(t = 1.5\) .
Edexcel M2 Q6
Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{173a2029-a0b8-437f-9339-5a1b6f30a8e3-010_442_689_292_632}
\end{figure} A uniform pole \(A B\), of mass 30 kg and length 3 m , is smoothly hinged to a vertical wall at one end \(A\). The pole is held in equilibrium in a horizontal position by a light rod CD. One end \(C\) of the rod is fixed to the wall vertically below \(A\). The other end \(D\) is freely jointed to the pole so that \(\angle A C D = 30 ^ { \circ }\) and \(A D = 0.5 \mathrm {~m}\), as shown in Figure 2. Find
  1. the thrust in the rod \(C D\),
  2. the magnitude of the force exerted by the wall on the pole at \(A\). The rod \(C D\) is removed and replaced by a longer light rod \(C M\), where \(M\) is the mid-point of \(A B\). The rod is freely jointed to the pole at \(M\). The pole \(A B\) remains in equilibrium in a horizontal position.
  3. Show that the force exerted by the wall on the pole at \(A\) now acts horizontally.
Edexcel AEA 2002 Specimen Q1
7 marks Standard +0.8
1.(a)By considering the series $$1 + t + t ^ { 2 } + t ^ { 3 } + \ldots + t ^ { n }$$ or otherwise,sum the series $$1 + 2 t + 3 t ^ { 2 } + 4 t ^ { 3 } + \ldots + n t ^ { n - 1 }$$ for \(t \neq 1\) .
(b)Hence find and simplify an expression for $$1 + 2 \times 3 + 3 \times 3 ^ { 2 } + 4 \times 3 ^ { 3 } + \ldots + 2001 \times 3 ^ { 2000 }$$ (c)Write down an expression for both the sums of the series in part(a)for the case where \(t = 1\) .
Edexcel AEA 2002 Specimen Q2
9 marks Challenging +1.2
2.Given that \(S = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { 2 x } \sin x \mathrm {~d} x\) and \(C = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { 2 x } \cos x \mathrm {~d} x\) ,
  1. show that \(S = 1 + 2 C\) ,
  2. find the exact value of \(S\) .
Edexcel AEA 2002 Specimen Q3
12 marks Challenging +1.8
3.Solve for values of \(\theta\) ,in degrees,in the range \(0 \leq \theta \leq 360\) , $$\sqrt { } 2 \times ( \sin 2 \theta + \cos \theta ) + \cos 3 \theta = \sin 2 \theta + \cos \theta$$
Edexcel AEA 2002 Specimen Q4
12 marks Challenging +1.8
4.A curve \(C\) has equation \(y = \mathrm { f } ( x )\) with \(\mathrm { f } ^ { \prime } ( x ) > 0\) .The \(x\)-coordinate of the point \(P\) on the curve is \(a\) .The tangent and the normal to \(C\) are drawn at \(P\) .The tangent cuts the \(x\)-axis at the point \(A\) and the normal cuts the \(x\)-axis at the point \(B\) .
  1. Show that the area of \(\triangle A P B\) is $$\frac { 1 } { 2 } [ \mathrm { f } ( a ) ] ^ { 2 } \left( \frac { \left[ \mathrm { f } ^ { \prime } ( a ) \right] ^ { 2 } + 1 } { \mathrm { f } ^ { \prime } ( a ) } \right)$$
  2. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { 5 x }\) and the area of \(\triangle A P B\) is \(\mathrm { e } ^ { 5 a }\) ,find and simplify the exact value of \(a\) .
Edexcel AEA 2002 Specimen Q5
17 marks Challenging +1.8
5.The function f is defined on the domain \([ - 2,2 ]\) by: $$f ( x ) = \left\{ \begin{array} { r l r } - k x ( 2 + x ) & \text { if } & - 2 \leq x < 0 , \\ k x ( 2 - x ) & \text { if } & 0 \leq x \leq 2 , \end{array} \right.$$ where \(k\) is a positive constant.
The function g is defined on the domain \([ - 2,2 ]\) by \(\mathrm { g } ( x ) = ( 2.5 ) ^ { 2 } - x ^ { 2 }\) .
  1. Prove that there is a value of \(k\) such that the graph of f touches the graph of g .
  2. For this value of \(k\) sketch the graphs of the functions f and g on the same axes,stating clearly where the graphs touch.
  3. Find the exact area of the region bounded by the two graphs.
Edexcel AEA 2002 Specimen Q6
18 marks Hard +2.3
6.Given that the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 4 }\) in the expansion of \(( 1 + k x ) ^ { n }\) ,where \(n \geq 4\) and \(k\) is a positive constant,are the consecutive terms of a geometric series,
  1. show that \(k = \frac { 6 ( n - 1 ) } { ( n - 2 ) ( n - 3 ) }\) .
  2. Given further that both \(n\) and \(k\) are positive integers,find all possible pairs of values for \(n\) and \(k\) .You should show clearly how you know that you have found all possible pairs of values.
  3. For the case where \(k = 1.4\) ,find the value of the positive integer \(n\) .
  4. Given that \(k = 1.4 , n\) is a positive integer and that the first term of the geometric series is the coefficient of \(x\) ,estimate how many terms are required for the sum of the geometric series to exceed \(1.12 \times 10 ^ { 12 }\) .[You may assume that \(\log _ { 10 } 4 \approx 0.6\) and \(\log _ { 10 } 5 \approx 0.7\) .]
Edexcel AEA 2002 Specimen Q7
18 marks Hard +2.3
7.The variable \(y\) is defined by $$y = \ln \left( \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x \right) \text { for } 0 < x < \frac { \pi } { 2 } .$$ A student was asked to prove that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - 4 \cot 2 x .$$ The attempted proof was as follows: $$\begin{aligned} y & = \ln \left( \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x \right) \\ & = \ln \left( \sec ^ { 2 } x \right) + \ln \left( \operatorname { cosec } ^ { 2 } x \right) \\ & = 2 \ln \sec x + 2 \ln \operatorname { cosec } x \\ \frac { \mathrm {~d} y } { \mathrm {~d} x } & = 2 \tan x - 2 \cot x \\ & = \frac { 2 \left( \sin ^ { 2 } x - \cos ^ { 2 } x \right) } { \sin x \cos x } \\ & = \frac { - 2 \cos 2 x } { \frac { 1 } { 2 } \sin 2 x } \\ & = - 4 \cot 2 x \end{aligned}$$
  1. Identify the error in this attempt at a proof.
  2. Give a correct version of the proof.
  3. Find and simplify a general relationship between \(p\) and \(q\) ,where \(p\) and \(q\) are variables that depend on \(x\) ,such that the student would obtain the correct result when differentiating \(\ln ( p + q )\) with respect to \(x\) by the above incorrect method.
  4. Given that \(p ( x ) = k \sec r x\) and \(q ( x ) = \operatorname { cosec } ^ { 2 } x\) ,where \(k\) and \(r\) are positive integers,find the values of \(k\) and \(r\) such that \(p\) and \(q\) satisfy the relationship found in part(c). \section*{END} Marks for presentation: 7
    TOTAL MARKS: 100
Edexcel AEA 2019 June Q1
7 marks Standard +0.8
1.(a)By writing \(u = \log _ { 4 } r\) ,where \(r > 0\) ,show that $$\log _ { 4 } r = \frac { 1 } { 2 } \log _ { 2 } r$$ (b)Solve the equation $$\log _ { 4 } \left( 5 x ^ { 2 } - 11 \right) = \log _ { 2 } ( 3 x - 5 )$$
Edexcel AEA 2019 June Q2
8 marks Challenging +1.8
2.The discrete random variable \(X\) follows the binomial distribution $$X \sim \mathrm {~B} ( n , p )$$ where \(0 < p < 1\) .The mode of \(X\) is \(m\) .
  1. Write down,in terms of \(m , n\) and \(p\) ,an expression for \(\mathrm { P } ( X = m )\)
  2. Determine,in terms of \(n\) and \(p\) ,an interval of width 1 ,in which \(m\) lies.
  3. Find a value of \(n\) where \(n > 100\) ,and a value of \(p\) where \(p < 0.2\) ,for which \(X\) has two modes. For your chosen values of \(n\) and \(p\) ,state these two modes.
Edexcel AEA 2019 June Q3
11 marks Challenging +1.8
3.Given that \(\phi = \frac { 1 } { 2 } ( \sqrt { 5 } + 1 )\) ,
  1. show that
    1. \(\phi ^ { 2 } = \phi + 1\)
    2. \(\frac { 1 } { \phi } = \phi - 1\)
  2. The equations of two curves are $$\begin{array} { r l r l } y & = \frac { 1 } { x } & x > 0 \\ \text { and } & y & = \ln x - x + k & x > 0 \end{array}$$ where \(k\) is a positive constant.
    The curves touch at the point \(P\) .
    Find in terms of \(\phi\)
    1. the coordinates of \(P\) ,
    2. the value of \(k\) .
Edexcel AEA 2019 June Q4
17 marks Challenging +1.8
4.(a)Prove the identity $$( \sin x + \cos y ) \cos ( x - y ) \equiv ( 1 + \sin ( x - y ) ) ( \cos x + \sin y )$$ (b)Hence,or otherwise,show that $$\frac { \sin 5 \theta + \cos 3 \theta } { \cos 5 \theta + \sin 3 \theta } \equiv \frac { 1 + \tan \theta } { 1 - \tan \theta }$$ (c)Given that \(k > 1\) ,show that the equation \(\frac { \sin 5 \theta + \cos 3 \theta } { \cos 5 \theta + \sin 3 \theta } = k\) has a unique solution in the interval \(0 < \theta < \frac { \pi } { 4 }\)
Edexcel AEA 2019 June Q5
16 marks Challenging +1.8
  1. Points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\), respectively, relative to an origin \(O\), and are such that \(O A B\) is a triangle with \(O A = a\) and \(O B = b\).
The point \(C\), with position vector \(\mathbf { c }\), lies on the line through \(O\) that bisects the angle \(A O B\).
  1. Prove that the vector \(b \mathbf { a } - a \mathbf { b }\) is perpendicular to \(\mathbf { c }\). The point \(D\), with position vector \(\mathbf { d }\), lies on the line \(A B\) between \(A\) and \(B\).
  2. Explain why \(\mathbf { d }\) can be expressed in the form \(\mathbf { d } = ( 1 - \lambda ) \mathbf { a } + \lambda \mathbf { b }\) for some scalar \(\lambda\) with \(0 < \lambda < 1\)
  3. Given that \(D\) is also on the line \(O C\), find an expression for \(\lambda\) in terms of \(a\) and \(b\) only and hence show that $$D A : D B = O A : O B$$
Edexcel AEA 2019 June Q6
19 marks Challenging +1.8
6.Figure 1 shows a sketch of part of the curve with equation \(y = x \sin ( \ln x ) , x \geqslant 1\) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{175528b0-6cd1-4d0d-a6b3-28ac980f74f3-18_451_1170_312_450} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} For \(x > 1\) ,the curve first crosses the \(x\)-axis at the point \(A\) .
  1. Find the \(x\) coordinate of \(A\) .
  2. Differentiate \(x \sin ( \ln x )\) and \(x \cos ( \ln x )\) with respect to \(x\) and hence find $$\int \sin ( \ln x ) \mathrm { d } x \text { and } \int \cos ( \ln x ) \mathrm { d } x$$
    1. Find \(\int x \sin ( \ln x ) \mathrm { d } x\) .
    2. Hence show that the area of the shaded region \(\boldsymbol { R }\) ,bounded by the curve and the \(x\)-axis between the points \(( 1,0 )\) and \(A\) ,is $$\frac { 1 } { 5 } \left( \mathrm { e } ^ { 2 \pi } + 1 \right)$$
Edexcel AEA 2019 June Q7
22 marks Challenging +1.8
7.Figure 2 shows a rectangular section of marshland,\(O A B C\) ,which is \(a\) metres long by \(b\) metres wide,where \(a > b\) . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{175528b0-6cd1-4d0d-a6b3-28ac980f74f3-22_360_847_340_609} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Edgar intends to get from \(O\) to \(B\) in the shortest possible time.In order to do this,he runs along edge \(O A\) for a distance \(x\) metres \(( 0 \leqslant x < a )\) to the point \(D\) before wading through the marsh directly from \(D\) to \(B\) . Edgar can wade through the marsh at a constant speed of \(1 \mathrm {~ms} ^ { - 1 }\) ,and he can run along the edge of the marsh at a constant speed of \(\lambda \mathrm { ms } ^ { - 1 }\) ,where \(\lambda > 1\)
  1. By finding an expression in terms of \(x\) for the time taken,\(t\) seconds,for Edgar to reach \(B\) from \(O\) ,show that $$\frac { \mathrm { d } t } { \mathrm {~d} x } = \frac { 1 } { \lambda } - \frac { a - x } { \sqrt { ( a - x ) ^ { 2 } + b ^ { 2 } } }$$
    1. Find,in terms of \(a , b\) and \(\lambda\) ,the value of \(x\) for which \(\frac { \mathrm { d } t } { \mathrm {~d} x } = 0\)
    2. Show that this value of \(x\) lies in the interval \(0 \leqslant x < a\) provided \(\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
    3. For \(\lambda\) in this range,show that the value of \(x\) found in(b)(i)gives a minimum value of \(t\) .
  3. Find the minimum time taken for Edgar to get from \(O\) to \(B\) if
    1. \(\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
    2. \(1 < \lambda < \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\) Edgar's friend,Frankie,also runs at a constant speed of \(\lambda \mathrm { m } \mathrm { s } ^ { - 1 }\) .Frankie runs along the edges \(O A\) and \(A B\) .Given that \(\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
  4. find the range of values of \(\lambda\) for which Frankie gets to \(B\) from \(O\) in a shorter time than Edgar's minimum time.