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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 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 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 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 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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
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Edexcel C2 2014 June Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6b490c0-80c4-4e15-b587-ac052ee27db7-07_531_1127_264_411} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The shape \(A B C D E A\), as shown in Figure 2, consists of a right-angled triangle \(E A B\) and a triangle \(D B C\) joined to a sector \(B D E\) of a circle with radius 5 cm and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(B C = 7.5 \mathrm {~cm}\).
Angle \(E A B = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.4\) radians and \(C D = 6.1 \mathrm {~cm}\).
  1. Find, in \(\mathrm { cm } ^ { 2 }\), the area of the sector \(B D E\).
  2. Find the size of the angle \(D B C\), giving your answer in radians to 3 decimal places.
  3. Find, in \(\mathrm { cm } ^ { 2 }\), the area of the shape \(A B C D E A\), giving your answer to 3 significant figures.
Edexcel C2 2014 June Q6
8 marks Standard +0.3
6. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\) The sum to infinity of the series is \(S _ { \infty }\)
  1. Find the value of \(S _ { \infty }\) The sum to \(N\) terms of the series is \(S _ { N }\)
  2. Find, to 1 decimal place, the value of \(S _ { 12 }\)
  3. Find the smallest value of \(N\), for which $$S _ { \infty } - S _ { N } < 0.5$$
Edexcel C2 2014 June Q7
9 marks Moderate -0.3
7. (i) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$9 \sin \left( \theta + 60 ^ { \circ } \right) = 4$$ giving your answers to 1 decimal place.
You must show each step of your working.
(ii) Solve, for \(- \pi \leqslant x < \pi\), the equation $$2 \tan x - 3 \sin x = 0$$ giving your answers to 2 decimal places where appropriate. [Solutions based entirely on graphical or numerical methods are not acceptable.]
Edexcel C2 2014 June Q8
7 marks Moderate -0.8
8. (a) Sketch the graph of $$y = 3 ^ { x } , \quad x \in \mathbb { R }$$ showing the coordinates of any points at which the graph crosses the axes.
(b) Use algebra to solve the equation $$3 ^ { 2 x } - 9 \left( 3 ^ { x } \right) + 18 = 0$$ giving your answers to 2 decimal places where appropriate.
Edexcel C2 2014 June Q9
5 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6b490c0-80c4-4e15-b587-ac052ee27db7-15_761_1082_210_424} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a circle \(C\) with centre \(Q\) and radius 4 and the point \(T\) which lies on \(C\). The tangent to \(C\) at the point \(T\) passes through the origin \(O\) and \(O T = 6 \sqrt { } 5\) Given that the coordinates of \(Q\) are \(( 11 , k )\), where \(k\) is a positive constant, (a) find the exact value of \(k\),
(b) find an equation for \(C\).
Edexcel C2 2014 June Q10
14 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6b490c0-80c4-4e15-b587-ac052ee27db7-17_929_584_237_287} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6b490c0-80c4-4e15-b587-ac052ee27db7-17_716_544_452_1069} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 4 shows a closed letter box \(A B F E H G C D\), which is made to be attached to a wall of a house. The letter box is a right prism of length \(y \mathrm {~cm}\) as shown in Figure 4. The base \(A B F E\) of the prism is a rectangle. The total surface area of the six faces of the prism is \(S \mathrm {~cm} ^ { 2 }\). The cross section \(A B C D\) of the letter box is a trapezium with edges of lengths \(D A = 9 x \mathrm {~cm}\), \(A B = 4 x \mathrm {~cm} , B C = 6 x \mathrm {~cm}\) and \(C D = 5 x \mathrm {~cm}\) as shown in Figure 5.
The angle \(D A B = 90 ^ { \circ }\) and the angle \(A B C = 90 ^ { \circ }\). The volume of the letter box is \(9600 \mathrm {~cm} ^ { 3 }\).
  1. Show that $$y = \frac { 320 } { x ^ { 2 } }$$
  2. Hence show that the surface area of the letter box, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 60 x ^ { 2 } + \frac { 7680 } { x }$$
  3. Use calculus to find the minimum value of \(S\).
  4. Justify, by further differentiation, that the value of \(S\) you have found is a minimum.
Edexcel C2 2015 June Q1
4 marks Moderate -0.8
  1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$ giving each term in its simplest form.
Edexcel C2 2015 June Q2
7 marks Moderate -0.8
2. A circle \(C\) with centre at the point \(( 2 , - 1 )\) passes through the point \(A\) at \(( 4 , - 5 )\).
  1. Find an equation for the circle \(C\).
  2. Find an equation of the tangent to the circle \(C\) at the point \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C2 2015 June Q3
9 marks Moderate -0.3
3. \(\mathrm { f } ( x ) = 6 x ^ { 3 } + 3 x ^ { 2 } + A x + B\), where \(A\) and \(B\) are constants. Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is 45 ,
  1. show that \(B - A = 48\) Given also that ( \(2 x + 1\) ) is a factor of \(\mathrm { f } ( x )\),
  2. find the value of \(A\) and the value of \(B\).
  3. Factorise f(x) fully.
Edexcel C2 2015 June Q4
8 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8a7593c3-4f0b-4351-afae-7bd98cfc351d-06_513_775_269_589} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a design for a scraper blade. The blade \(A O B C D A\) consists of an isosceles triangle \(C O D\) joined along its equal sides to sectors \(O B C\) and \(O D A\) of a circle with centre \(O\) and radius 8 cm . Angles \(A O D\) and \(B O C\) are equal. \(A O B\) is a straight line and is parallel to the line \(D C . D C\) has length 7 cm .
  1. Show that the angle \(C O D\) is 0.906 radians, correct to 3 significant figures.
  2. Find the perimeter of \(A O B C D A\), giving your answer to 3 significant figures.
  3. Find the area of \(A O B C D A\), giving your answer to 3 significant figures.
Edexcel C2 2015 June Q5
10 marks Standard +0.3
    1. All the terms of a geometric series are positive. The sum of the first two terms is 34 and the sum to infinity is 162
Find
  1. the common ratio,
  2. the first term.
    (ii) A different geometric series has a first term of 42 and a common ratio of \(\frac { 6 } { 7 }\). Find the smallest value of \(n\) for which the sum of the first \(n\) terms of the series exceeds 290
Edexcel C2 2015 June Q6
9 marks Moderate -0.3
6. (a) Find $$\int 10 x \left( x ^ { \frac { 1 } { 2 } } - 2 \right) \mathrm { d } x$$ giving each term in its simplest form. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8a7593c3-4f0b-4351-afae-7bd98cfc351d-10_401_1002_543_470} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = 10 x \left( x ^ { \frac { 1 } { 2 } } - 2 \right) , \quad x \geqslant 0$$ The curve \(C\) starts at the origin and crosses the \(x\)-axis at the point \(( 4,0 )\). The area, shown shaded in Figure 2, consists of two finite regions and is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 9\)
(b) Use your answer from part (a) to find the total area of the shaded regions.
Edexcel C2 2015 June Q7
9 marks Moderate -0.3
7. (i) Use logarithms to solve the equation \(8 ^ { 2 x + 1 } = 24\), giving your answer to 3 decimal places.
(ii) Find the values of \(y\) such that $$\log _ { 2 } ( 11 y - 3 ) - \log _ { 2 } 3 - 2 \log _ { 2 } y = 1 , \quad y > \frac { 3 } { 11 }$$
Edexcel C2 2015 June Q8
9 marks Moderate -0.3
8. (i) Solve, for \(0 \leqslant \theta < \pi\), the equation $$\sin 3 \theta - \sqrt { 3 } \cos 3 \theta = 0$$ giving your answers in terms of \(\pi\).
(ii) Given that $$4 \sin ^ { 2 } x + \cos x = 4 - k , \quad 0 \leqslant k \leqslant 3$$
  1. find \(\cos x\) in terms of \(k\).
  2. When \(k = 3\), find the values of \(x\) in the range \(0 \leqslant x < 360 ^ { \circ }\)
Edexcel C2 2015 June Q9
10 marks Standard +0.3
9. A solid glass cylinder, which is used in an expensive laser amplifier, has a volume of \(75 \pi \mathrm {~cm} ^ { 3 }\).
The cost of polishing the surface area of this glass cylinder is \(\pounds 2\) per \(\mathrm { cm } ^ { 2 }\) for the curved surface area and \(\pounds 3\) per \(\mathrm { cm } ^ { 2 }\) for the circular top and base areas. Given that the radius of the cylinder is \(r \mathrm {~cm}\),
  1. show that the cost of the polishing, \(\pounds C\), is given by $$C = 6 \pi r ^ { 2 } + \frac { 300 \pi } { r }$$
  2. Use calculus to find the minimum cost of the polishing, giving your answer to the nearest pound.
  3. Justify that the answer that you have obtained in part (b) is a minimum.
Edexcel C2 2016 June Q1
7 marks Moderate -0.8
  1. A geometric series has first term \(a\) and common ratio \(r = \frac { 3 } { 4 }\)
The sum of the first 4 terms of this series is 175
  1. Show that \(a = 64\)
  2. Find the sum to infinity of the series.
  3. Find the difference between the 9th and 10th terms of the series. Give your answer to 3 decimal places.
Edexcel C2 2016 June Q2
6 marks Moderate -0.8
2. The curve \(C\) has equation $$y = 8 - 2 ^ { x - 1 } , \quad 0 \leqslant x \leqslant 4$$
  1. Complete the table below with the value of \(y\) corresponding to \(x = 1\)
    \(x\)01234
    \(y\)7.5640
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to find an approximate value for \(\int _ { 0 } ^ { 4 } \left( 8 - 2 ^ { x - 1 } \right) \mathrm { d } x\) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{582cda45-80fc-43a8-90e6-1cae08cb1534-03_650_606_1016_671} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = 8 - 2 ^ { x - 1 } , \quad 0 \leqslant x \leqslant 4\)
    The curve \(C\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).
    The region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\) and the straight line through \(A\) and \(B\).
  3. Use your answer to part (b) to find an approximate value for the area of \(R\).
Edexcel C2 2016 June Q3
8 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{582cda45-80fc-43a8-90e6-1cae08cb1534-05_791_917_121_484} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale The circle \(C\) has centre \(P ( 7,8 )\) and passes through the point \(Q ( 10,13 )\), as shown in Figure 2.
  1. Find the length \(P Q\), giving your answer as an exact value.
  2. Hence write down an equation for \(C\). The line \(l\) is a tangent to \(C\) at the point \(Q\), as shown in Figure 2.
  3. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C2 2016 June Q4
8 marks Moderate -0.8
4. $$f ( x ) = 6 x ^ { 3 } + 13 x ^ { 2 } - 4$$
  1. Use the remainder theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 3\) ).
  2. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
  3. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2016 June Q5
9 marks Moderate -0.3
  1. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$( 2 - 9 x ) ^ { 4 }$$ giving each term in its simplest form. $$f ( x ) = ( 1 + k x ) ( 2 - 9 x ) ^ { 4 } , \text { where } k \text { is a constant }$$ The expansion, in ascending powers of \(x\), of \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 2 }\) is $$A - 232 x + B x ^ { 2 }$$ where \(A\) and \(B\) are constants.
(b) Write down the value of \(A\).
(c) Find the value of \(k\).
(d) Hence find the value of \(B\).
Edexcel C2 2016 June Q6
9 marks Moderate -0.3
6. (i) Solve, for \(- \pi < \theta \leqslant \pi\), $$1 - 2 \cos \left( \theta - \frac { \pi } { 5 } \right) = 0$$ giving your answers in terms of \(\pi\).
(ii) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \cos ^ { 2 } x + 7 \sin x - 2 = 0$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C2 2016 June Q7
6 marks Moderate -0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{582cda45-80fc-43a8-90e6-1cae08cb1534-12_563_812_244_630} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 3 x - x ^ { \frac { 3 } { 2 } } , \quad x \geqslant 0$$ The finite region \(S\), bounded by the \(x\)-axis and the curve, is shown shaded in Figure 3.
  1. Find $$\int \left( 3 x - x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x$$
  2. Hence find the area of \(S\).
Edexcel C2 2016 June Q8
7 marks Moderate -0.3
8. (i) Given that $$\log _ { 3 } ( 3 b + 1 ) - \log _ { 3 } ( a - 2 ) = - 1 , \quad a > 2$$ express \(b\) in terms of \(a\).
(ii) Solve the equation $$2 ^ { 2 x + 5 } - 7 \left( 2 ^ { x } \right) = 0$$ giving your answer to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C2 2016 June Q9
15 marks Standard +0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{582cda45-80fc-43a8-90e6-1cae08cb1534-15_831_1167_118_513} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a plan view of a sheep enclosure.
The enclosure \(A B C D E A\), as shown in Figure 4, consists of a rectangle \(B C D E\) joined to an equilateral triangle \(B F A\) and a sector \(F E A\) of a circle with radius \(x\) metres and centre \(F\). The points \(B , F\) and \(E\) lie on a straight line with \(F E = x\) metres and \(10 \leqslant x \leqslant 25\)
  1. Find, in \(\mathrm { m } ^ { 2 }\), the exact area of the sector \(F E A\), giving your answer in terms of \(x\), in its simplest form. Given that \(B C = y\) metres, where \(y > 0\), and the area of the enclosure is \(1000 \mathrm {~m} ^ { 2 }\),
  2. show that $$y = \frac { 500 } { x } - \frac { x } { 24 } ( 4 \pi + 3 \sqrt { 3 } )$$
  3. Hence show that the perimeter \(P\) metres of the enclosure is given by $$P = \frac { 1000 } { x } + \frac { x } { 12 } ( 4 \pi + 36 - 3 \sqrt { 3 } )$$
  4. Use calculus to find the minimum value of \(P\), giving your answer to the nearest metre.
  5. Justify, by further differentiation, that the value of \(P\) you have found is a minimum.
Edexcel C2 2017 June Q1
4 marks Easy -1.2
  1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 3 - \frac { 1 } { 3 } x \right) ^ { 5 }$$ giving each term in its simplest form.
\includegraphics[max width=\textwidth, alt={}, center]{752efc6c-8d0e-46a6-b75d-5125956969d8-03_104_107_2631_1774}