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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
Edexcel C12 2016 June Q9
  1. The first term of a geometric series is 6 and the common ratio is 0.92
For this series, find
    1. the \(25 ^ { \text {th } }\) term, giving your answer to 2 significant figures,
    2. the sum to infinity. The sum to \(n\) terms of this series is greater than 72
  2. Calculate the smallest possible value of \(n\).
    VJYV SIHI NITIIIUMION, OC
    VILV SIHI NI JAHM ION OO
    VILV SIHI NI JIIIM ION OO
Edexcel C12 2016 June Q10
10. The curve \(C\) has equation \(y = \sin \left( x + \frac { \pi } { 4 } \right) , \quad 0 \leqslant x \leqslant 2 \pi\)
  1. On the axes below, sketch the curve \(C\).
  2. Write down the exact coordinates of all the points at which the curve \(C\) meets or intersects the \(x\)-axis and the \(y\)-axis.
  3. Solve, for \(0 \leqslant x \leqslant 2 \pi\), the equation $$\sin \left( x + \frac { \pi } { 4 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answers in the form \(k \pi\), where \(k\) is a rational number.
    \includegraphics[max width=\textwidth, alt={}, center]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-14_677_1031_1446_445}
Edexcel C12 2016 June Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-16_892_825_228_548} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale Figure 2 shows the design for a sail \(A P B C A\). The curved edge \(A P B\) of the sail is an arc of a circle centre \(O\) and radius \(r \mathrm {~m}\). The straight edge \(A C B\) is a chord of the circle. The height \(A B\) of the sail is 2.4 m . The maximum width \(C P\) of the sail is 0.4 m .
  1. Show that \(r = 2\)
  2. Show, to 4 decimal places, that angle \(A O B = 1.2870\) radians.
  3. Hence calculate the area of the sail, giving your answer, in \(\mathrm { m } ^ { 2 }\), to 3 decimal places.
Edexcel C12 2016 June Q12
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-18_636_887_274_534} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a circle \(C\)
\(C\) touches the \(y\)-axis and has centre at the point ( \(a , 0\) ) where \(a\) is a positive constant.
  1. Write down an equation for \(C\) in terms of \(a\) Given that the point \(P ( 4 , - 3 )\) lies on \(C\),
  2. find the value of \(a\)
Edexcel C12 2016 June Q13
  1. (a) Show that the equation
$$2 \log _ { 2 } y = 5 - \log _ { 2 } x \quad x > 0 , y > 0$$ may be written in the form \(y ^ { 2 } = \frac { k } { x }\) where \(k\) is a constant to be found.
(b) Hence, or otherwise, solve the simultaneous equations $$\begin{gathered} 2 \log _ { 2 } y = 5 - \log _ { 2 } x
\log _ { x } y = - 3 \end{gathered}$$ for \(x > 0 , y > 0\)
Edexcel C12 2016 June Q14
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-21_831_919_127_509} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the graph of \(y = g ( x ) , - 3 \leqslant x \leqslant 4\) and part of the line \(l\) with equation \(y = \frac { 1 } { 2 } x\) The graph of \(y = \mathrm { g } ( x )\) consists of three line segments, from \(P ( - 3,4 )\) to \(Q ( 0,4 )\), from \(Q ( 0,4 )\) to \(R ( 2,0 )\) and from \(R ( 2,0 )\) to \(S ( 4,10 )\). The line \(l\) intersects \(y = \mathrm { g } ( x )\) at the points \(A\) and \(B\) as shown in Figure 4.
  1. Use algebra to find the \(x\) coordinate of the point \(A\) and the \(x\) coordinate of the point \(B\). Show each step of your working and give your answers as exact fractions.
  2. Sketch the graph with equation $$y = \frac { 3 } { 2 } g ( x ) , \quad - 3 \leqslant x \leqslant 4$$ On your sketch show the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.
Edexcel C12 2016 June Q15
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-23_609_493_223_762} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a design for a water barrel.
It is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and radius \(r \mathrm {~cm}\). The barrel has a base but has no lid, is open at the top and is made of material of negligible thickness. The barrel is designed to hold \(60000 \mathrm {~cm} ^ { 3 }\) of water when full.
  1. Show that the total external surface area, \(S \mathrm {~cm} ^ { 2 }\), of the barrel is given by the formula $$S = \pi r ^ { 2 } + \frac { 120000 } { r }$$
  2. Use calculus to find the minimum value of \(S\), giving your answer to 3 significant figures.
  3. Justify that the value of \(S\) you found in part (b) is a minimum.
Edexcel C12 2016 June Q16
16. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-25_739_1308_278_328} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} Figure 6 shows a sketch of part of the curve \(C\) with equation $$y = x ( x - 1 ) ( x - 2 )$$ The point \(P\) lies on \(C\) and has \(x\) coordinate \(\frac { 1 } { 2 }\)
The line \(l\), as shown on Figure 6, is the tangent to \(C\) at \(P\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Use part (a) to find an equation for \(l\) in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. The finite region \(R\), shown shaded in Figure 6, is bounded by the line \(l\), the curve \(C\) and the \(x\)-axis. The line \(l\) meets the curve again at the point \(( 2,0 )\)
  3. Use integration to find the exact area of the shaded region \(R\).
Edexcel C12 2017 June Q1
  1. An arithmetic sequence has first term 6 and common difference 10 Find
    1. the 15th term of the sequence,
    2. the sum of the first 20 terms of the sequence.
Edexcel C12 2017 June Q2
  1. Simplify the following expressions fully.
    1. \(\left( \frac { 1 } { 9 } x ^ { 4 } \right) ^ { 0.5 }\)
    2. \(\left( \frac { x } { \sqrt { 2 } } \right) ^ { - 2 }\)
    3. \(x \sqrt { 3 } \div \sqrt { \frac { 48 } { x ^ { 4 } } }\)
Edexcel C12 2017 June Q3
  1. The line \(l _ { 1 }\) has equation \(2 x + 3 y = 6\)
The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(( 3 , - 5 )\).
Find the equation for the line \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C12 2017 June Q4
4. The curve \(C\) has equation \(y = 4 x \sqrt { x } + \frac { 48 } { \sqrt { x } } - \sqrt { 8 } , x > 0\)
  1. Find, simplifying each term,
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
  2. Use part (a) to find the exact coordinates of the stationary point of \(C\).
  3. Determine whether the stationary point of \(C\) is a maximum or minimum, giving a reason for your answer.
Edexcel C12 2017 June Q5
5. $$f ( x ) = - 4 x ^ { 3 } + 16 x ^ { 2 } - 13 x + 3$$
  1. Use the remainder theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ).
  2. Use the factor theorem to show that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\).
  3. Hence fully factorise \(\mathrm { f } ( x )\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-12_581_636_731_657} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
  4. Use your answer to part (c) and the sketch to deduce the set of values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 0\)
Edexcel C12 2017 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-16_364_689_214_630} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of a design for a triangular garden \(A B C\). The garden has sides \(B A\) with length \(10 \mathrm {~m} , B C\) with length 6 m and \(C A\) with length 12 m . The point \(D\) lies on \(A C\) such that \(B D\) is an arc of the circle centre \(A\), radius 10 m . A flowerbed \(B C D\) is shown shaded in Figure 2.
  1. Find the size of angle \(B A C\), in radians, to 4 decimal places.
  2. Find the perimeter of the flowerbed \(B C D\), in m , to 2 decimal places.
  3. Find the area of the flowerbed \(B C D\), in \(\mathrm { m } ^ { 2 }\), to 2 decimal places.
Edexcel C12 2017 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-20_588_839_219_550} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(y\)-axis at the point \(( 0,8 )\). The line with equation \(y = 10\) is the only asymptote to the curve.
The curve has a single turning point, a minimum point at \(( 2,5 )\), as shown in Figure 3.
  1. State the coordinates of the minimum point of the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 4 } x \right)\)
  2. State the equation of the asymptote to the curve with equation \(y = \mathrm { f } ( x ) - 3\) The curve with equation \(y = \mathrm { f } ( x )\) meets the line with equation \(y = k\), where \(k\) is a constant, at two distinct points.
  3. State the set of possible values for \(k\).
  4. Sketch the curve with equation \(y = - \mathrm { f } ( x )\). On your sketch, show clearly the coordinates of the turning point, the coordinates of the intersection with the \(y\)-axis and the equation of the asymptote. \section*{\textbackslash section*\{D\}}
Edexcel C12 2017 June Q8
8. (a) Find \(\int \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x\), simplifying each term. Given that \(b\) is a constant and $$\int _ { b } ^ { 4 } \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x = 36$$ (b) show that \(b ^ { 3 } + 2 b ^ { 2 } - 15 b = 0\)
(c) Hence find the possible values of \(b\).
Edexcel C12 2017 June Q9
9. (i) Find the exact value of \(x\) for which $$2 \log _ { 10 } ( x - 2 ) - \log _ { 10 } ( x + 5 ) = 0$$ (ii) Given $$\log _ { p } ( 4 y + 1 ) - \log _ { p } ( 2 y - 2 ) = 1 \quad p > 2 , y > 1$$ express \(y\) in terms of \(p\).
Edexcel C12 2017 June Q10
  1. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 2 - \frac { x } { 8 } \right) ^ { 10 }$$ giving each term in its simplest form. $$\mathrm { f } ( x ) = \left( 2 - \frac { x } { 8 } \right) ^ { 10 } ( a + b x ) , \text { where } a \text { and } b \text { are constants }$$ Given that the first two terms, in ascending powers of \(x\) in the series expansion of \(\mathrm { f } ( x )\), are 256 and \(352 x\),
(b) find the value of \(a\),
(c) find the value of \(b\).
Edexcel C12 2017 June Q11
11. Wheat is to be grown on a farm. A model predicts that the mass of wheat harvested on the farm will increase by \(1.5 \%\) per year, so that the mass of wheat harvested each year forms a geometric sequence. Given that the mass of wheat harvested during year one is 6000 tonnes,
  1. show that, according to the model, the mass of wheat harvested on the farm during year 4 will be approximately 6274 tonnes. During year \(N\), according to the model, there is predicted to be more than 8000 tonnes of wheat harvested on the farm.
  2. Find the smallest possible value of \(N\). It costs \(\pounds 5\) per tonne to harvest the wheat.
  3. Assuming the model, find the total amount that it would cost to harvest the wheat from year one to year 10 inclusive. Give your answer to the nearest \(\pounds 1000\).
Edexcel C12 2017 June Q12
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-40_814_713_219_612} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 9 x ^ { 2 } + 26 x - 18$$ The point \(P ( 4,6 )\) lies on \(C\).
  1. Use calculus to show that the normal to \(C\) at the point \(P\) has equation $$2 y + x = 16$$ The region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the normal to \(C\) at \(P\).
  2. Show that \(C\) cuts the \(x\)-axis at \(( 1,0 )\)
  3. Showing all your working, use calculus to find the exact area of \(R\).
Edexcel C12 2017 June Q13
13. (a) Show that the equation $$5 \cos x + 1 = \sin x \tan x$$ can be written in the form $$6 \cos ^ { 2 } x + \cos x - 1 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$5 \cos 2 \theta + 1 = \sin 2 \theta \tan 2 \theta$$ giving your answers, where appropriate, to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2017 June Q14
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-48_771_812_237_575} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of the circle \(C _ { 1 }\)
The points \(A ( 1,4 )\) and \(B ( 7,8 )\) lie on \(C _ { 1 }\)
Given that \(A B\) is a diameter of the circle \(C _ { 1 }\)
  1. find the coordinates for the centre of \(C _ { 1 }\)
  2. find the exact radius of \(C _ { 1 }\) simplifying your answer. Two distinct circles \(C _ { 2 }\) and \(C _ { 3 }\) each have centre \(( 0,0 )\).
    Given that each of these circles touch circle \(C _ { 1 }\)
  3. find the equation of circle \(C _ { 2 }\) and the equation of circle \(C _ { 3 }\)
Edexcel C12 2017 June Q15
15. The height of water, \(H\) metres, in a harbour on a particular day is given by the equation $$H = 4 + 1.5 \sin \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight, and \(\frac { \pi t } { 6 }\) is measured in radians.
  1. Show that the height of the water at 1 a.m. is 4.75 metres.
  2. Find the height of the water at 2 p.m.
  3. Find, to the nearest minute, the first two times when the height of the water is 3 metres.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2018 June Q1
  1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 1 } { \sqrt { ( x + 1 ) } }\), with the values
    for \(y\) rounded to 3 decimal places where necessary.
\(x\)03691215
\(y\)10.50.3780.3160.277
  1. Complete the table by giving the value of \(y\) corresponding to \(x = 15\)
  2. Use the trapezium rule with all the values of \(y\) from the completed table to find an approximate value for $$\int _ { 0 } ^ { 15 } \frac { 1 } { \sqrt { ( x + 1 ) } } \mathrm { d } x$$ giving your answer to 2 decimal places.
Edexcel C12 2018 June Q2
2. $$f ( x ) = a x ^ { 3 } + 2 x ^ { 2 } + b x - 3$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ) the remainder is 1
  1. Show that $$a + 4 b = 28$$ When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 17
  2. Find the value of \(a\) and the value of \(b\).