Questions — Edexcel (9671 questions)

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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 2013 June Q9
12 marks Standard +0.3
  1. (i) Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\)
$$\sin \left( 2 \theta - 30 ^ { \circ } \right) + 1 = 0.4$$ giving your answers to 1 decimal place.
(ii) Find all the values of \(x\), in the interval \(0 \leqslant x < 360 ^ { \circ }\), for which $$9 \cos ^ { 2 } x - 11 \cos x + 3 \sin ^ { 2 } x = 0$$ giving your answers to 1 decimal place. You must show clearly how you obtained your answers.
Edexcel C2 2013 June Q1
4 marks Moderate -0.8
  1. The first three terms of a geometric series are
$$18,12 \text { and } p$$ respectively, where \(p\) is a constant. Find
  1. the value of the common ratio of the series,
  2. the value of \(p\),
  3. the sum of the first 15 terms of the series, giving your answer to 3 decimal places.
Edexcel C2 2013 June Q2
5 marks Easy -1.2
2. (a) Use the binomial theorem to find all the terms of the expansion of $$( 2 + 3 x ) ^ { 4 }$$ Give each term in its simplest form.
(b) Write down the expansion of $$( 2 - 3 x ) ^ { 4 }$$ in ascending powers of \(x\), giving each term in its simplest form.
Edexcel C2 2013 June Q3
9 marks Standard +0.3
3. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + a x + 18$$ where \(a\) is a constant. Given that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\),
  1. show that \(a = - 9\)
  2. factorise \(\mathrm { f } ( x )\) completely. Given that $$\mathrm { g } ( y ) = 2 \left( 3 ^ { 3 y } \right) - 5 \left( 3 ^ { 2 y } \right) - 9 \left( 3 ^ { y } \right) + 18$$
  3. find the values of \(y\) that satisfy \(\mathrm { g } ( y ) = 0\), giving your answers to 2 decimal places where appropriate.
Edexcel C2 2013 June Q4
7 marks Moderate -0.8
4. $$y = \frac { 5 } { \left( x ^ { 2 } + 1 \right) }$$
  1. Complete the table below, giving the missing value of \(y\) to 3 decimal places.
    \(x\)00.511.522.53
    \(y\)542.510.6900.5
    (1) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1c51b071-5cb1-4841-b031-80bde9027433-06_732_1118_826_411} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the region \(R\) which is bounded by the curve with equation \(y = \frac { 5 } { \left( x ^ { 2 } + 1 \right) }\),
    the \(x\)-axis and the lines \(x = 0\) and \(x = 3\) the \(x\)-axis and the lines \(x = 0\) and \(x = 3\)
  2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximate value for the area of \(R\).
  3. Use your answer to part (b) to find an approximate value for $$\int _ { 0 } ^ { 3 } \left( 4 + \frac { 5 } { \left( x ^ { 2 } + 1 \right) } \right) d x$$ giving your answer to 2 decimal places.
Edexcel C2 2013 June Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c51b071-5cb1-4841-b031-80bde9027433-08_598_1297_118_319} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a plan view of a garden.
The plan of the garden \(A B C D E A\) consists of a triangle \(A B E\) joined to a sector \(B C D E\) of a circle with radius 12 m and centre \(B\).
The points \(A , B\) and \(C\) lie on a straight line with \(A B = 23 \mathrm {~m}\) and \(B C = 12 \mathrm {~m}\).
Given that the size of angle \(A B E\) is exactly 0.64 radians, find
  1. the area of the garden, giving your answer in \(\mathrm { m } ^ { 2 }\), to 1 decimal place,
  2. the perimeter of the garden, giving your answer in metres, to 1 decimal place.
Edexcel C2 2013 June Q6
8 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c51b071-5cb1-4841-b031-80bde9027433-10_697_1182_210_386} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = x ( x + 4 ) ( x - 2 )$$ The curve \(C\) crosses the \(x\)-axis at the origin \(O\) and at the points \(A\) and \(B\).
  1. Write down the \(x\)-coordinates of the points \(A\) and \(B\). The finite region, shown shaded in Figure 3, is bounded by the curve \(C\) and the \(x\)-axis.
  2. Use integration to find the total area of the finite region shown shaded in Figure 3.
Edexcel C2 2013 June Q7
7 marks Moderate -0.3
7. (i) Find the exact value of \(x\) for which $$\log _ { 2 } ( 2 x ) = \log _ { 2 } ( 5 x + 4 ) - 3$$ (ii) Given that $$\log _ { a } y + 3 \log _ { a } 2 = 5$$ express \(y\) in terms of \(a\).
Give your answer in its simplest form.
Edexcel C2 2013 June Q8
11 marks Standard +0.3
8. (i) Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), $$\tan \left( x - 40 ^ { \circ } \right) = 1.5$$ giving your answers to 1 decimal place.
(ii) (a) Show that the equation $$\sin \theta \tan \theta = 3 \cos \theta + 2$$ can be written in the form $$4 \cos ^ { 2 } \theta + 2 \cos \theta - 1 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$\sin \theta \tan \theta = 3 \cos \theta + 2$$ showing each stage of your working.
Edexcel C2 2013 June Q9
9 marks Moderate -0.3
  1. The curve with equation
$$y = x ^ { 2 } - 32 \sqrt { } ( x ) + 20 , \quad x > 0$$ has a stationary point \(P\). Use calculus
  1. to find the coordinates of \(P\),
  2. to determine the nature of the stationary point \(P\).
Edexcel C2 2013 June Q10
6 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c51b071-5cb1-4841-b031-80bde9027433-16_723_979_207_495} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The circle \(C\) has radius 5 and touches the \(y\)-axis at the point \(( 0,9 )\), as shown in Figure 4.
  1. Write down an equation for the circle \(C\), that is shown in Figure 4. A line through the point \(P ( 8 , - 7 )\) is a tangent to the circle \(C\) at the point \(T\).
  2. Find the length of \(P T\).
Edexcel C2 2014 June Q1
4 marks Moderate -0.8
  1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 1 + \frac { 3 x } { 2 } \right) ^ { 8 }$$ giving each term in its simplest form.
Edexcel C2 2014 June Q2
8 marks Standard +0.3
2. A geometric series has first term \(a\), where \(a \neq 0\), and common ratio \(r\). The sum to infinity of this series is 6 times the first term of the series.
  1. Show that \(r = \frac { 5 } { 6 }\) Given that the fourth term of this series is 62.5
  2. find the value of \(a\),
  3. find the difference between the sum to infinity and the sum of the first 30 terms, giving your answer to 3 significant figures.
Edexcel C2 2014 June Q3
5 marks Easy -1.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07cc9ed-a820-46c8-a3a3-3c780cf20fa7-05_821_1273_118_338} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt { } ( 2 x - 1 ) , x \geqslant 0.5\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines with equations \(x = 2\) and \(x = 10\). The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { } ( 2 x - 1 )\).
\(x\)246810
\(y\)\(\sqrt { } 3\)\(\sqrt { } 11\)\(\sqrt { } 19\)
  1. Complete the table with the values of \(y\) corresponding to \(x = 4\) and \(x = 8\).
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to find an approximate value for the area of \(R\), giving your answer to 2 decimal places.
  3. State whether your approximate value in part (b) is an overestimate or an underestimate for the area of \(R\).
Edexcel C2 2014 June Q4
7 marks Moderate -0.3
4.
\(\mathrm { f } ( x ) = - 4 x ^ { 3 } + a x ^ { 2 } + 9 x - 18\), where \(a\) is a constant. Given that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\),
  1. find the value of \(a\),
  2. factorise \(\mathrm { f } ( x )\) completely,
  3. find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ).
Edexcel C2 2014 June Q5
6 marks Moderate -0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07cc9ed-a820-46c8-a3a3-3c780cf20fa7-08_566_725_127_614} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the shape \(A B C D E A\) which consists of a right-angled triangle \(B C D\) joined to a sector \(A B D E A\) of a circle with radius 7 cm and centre \(B\).
\(A , B\) and \(C\) lie on a straight line with \(A B = 7 \mathrm {~cm}\).
Given that the size of angle \(A B D\) is exactly 2.1 radians,
  1. find, in cm, the length of the arc \(D E A\),
  2. find, in cm, the perimeter of the shape \(A B C D E A\), giving your answer to 1 decimal place.
Edexcel C2 2014 June Q6
7 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07cc9ed-a820-46c8-a3a3-3c780cf20fa7-09_796_1132_121_397} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 1 } { 8 } x ^ { 3 } + \frac { 3 } { 4 } x ^ { 2 } , \quad x \in \mathbb { R }$$ The curve \(C\) has a maximum turning point at the point \(A\) and a minimum turning point at the origin \(O\). The line \(l\) touches the curve \(C\) at the point \(A\) and cuts the curve \(C\) at the point \(B\). The \(x\) coordinate of \(A\) is - 4 and the \(x\) coordinate of \(B\) is 2 . The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\) and the line \(l\).
Use integration to find the area of the finite region \(R\).
Edexcel C2 2014 June Q7
8 marks Standard +0.3
7. (i) Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$\frac { \sin 2 \theta } { ( 4 \sin 2 \theta - 1 ) } = 1$$ giving your answers to 1 decimal place.
(ii) Solve, for \(0 \leqslant x < 2 \pi\), the equation $$5 \sin ^ { 2 } x - 2 \cos x - 5 = 0$$ giving your answers to 2 decimal places. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C2 2014 June Q8
8 marks Moderate -0.3
8. (i) Solve $$5 ^ { y } = 8$$ giving your answer to 3 significant figures.
(ii) Use algebra to find the values of \(x\) for which $$\log _ { 2 } ( x + 15 ) - 4 = \frac { 1 } { 2 } \log _ { 2 } x$$
Edexcel C2 2014 June Q9
13 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07cc9ed-a820-46c8-a3a3-3c780cf20fa7-14_899_686_212_639} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows the plan of a pool. The shape of the pool \(A B C D E F A\) consists of a rectangle \(B C E F\) joined to an equilateral triangle \(B F A\) and a semi-circle \(C D E\), as shown in Figure 4. Given that \(A B = x\) metres, \(E F = y\) metres, and the area of the pool is \(50 \mathrm {~m} ^ { 2 }\),
  1. show that $$y = \frac { 50 } { x } - \frac { x } { 8 } ( \pi + 2 \sqrt { } 3 )$$
  2. Hence show that the perimeter, \(P\) metres, of the pool is given by $$P = \frac { 100 } { x } + \frac { x } { 4 } ( \pi + 8 - 2 \sqrt { } 3 )$$
  3. Use calculus to find the minimum value of \(P\), giving your answer to 3 significant figures.
  4. Justify, by further differentiation, that the value of \(P\) that you have found is a minimum.
Edexcel C2 2014 June Q10
9 marks Moderate -0.3
  1. The circle \(C\), with centre \(A\), passes through the point \(P\) with coordinates ( \(- 9,8\) ) and the point \(Q\) with coordinates \(( 15 , - 10 )\).
Given that \(P Q\) is a diameter of the circle \(C\),
  1. find the coordinates of \(A\),
  2. find an equation for \(C\). A point \(R\) also lies on the circle \(C\).
    Given that the length of the chord \(P R\) is 20 units,
  3. find the length of the shortest distance from \(A\) to the chord \(P R\). Give your answer as a surd in its simplest form.
  4. Find the size of the angle \(A R Q\), giving your answer to the nearest 0.1 of a degree.
Edexcel C2 2014 June Q1
5 marks Easy -1.2
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6b490c0-80c4-4e15-b587-ac052ee27db7-02_738_1257_274_340} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt { } \left( x ^ { 2 } + 1 \right) , x \geqslant 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) The table below shows corresponding values for \(x\) and \(y\) for \(y = \sqrt { } \left( x ^ { 2 } + 1 \right)\).
\(x\)11.251.51.752
\(y\)1.4141.8032.0162.236
  1. Complete the table above, giving the missing value of \(y\) to 3 decimal places.
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to find an approximate value for the area of \(R\), giving your answer to 2 decimal places.
Edexcel C2 2014 June Q2
6 marks Moderate -0.8
2. $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 4 x + 4$$
  1. Use the factor theorem to show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2014 June Q3
7 marks Moderate -0.8
3. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 - 3 x ) ^ { 6 }$$ giving each term in its simplest form.
(b) Hence, or otherwise, find the first 3 terms, in ascending powers of \(x\), of the expansion of $$\left( 1 + \frac { x } { 2 } \right) ( 2 - 3 x ) ^ { 6 }$$
Edexcel C2 2014 June Q4
5 marks Moderate -0.8
  1. Use integration to find
$$\int _ { 1 } ^ { \sqrt { 3 } } \left( \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 3 x ^ { 2 } } \right) \mathrm { d } x$$ giving your answer in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are constants to be determined.