Questions — Edexcel C2 (476 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
Edexcel C2 2012 June Q1
Find expansion of
Edexcel C2 2012 June Q2
2. Find the values of \(x\) such that $$2 \log _ { 3 } x - \log _ { 3 } ( x - 2 ) = 2$$
Edexcel C2 2012 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1ef99f0-4ad4-49d8-bee7-d5bb9cc84660-04_734_1262_237_315} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The circle \(C\) with centre \(T\) and radius \(r\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 16 y + 139 = 0$$
  1. Find the coordinates of the centre of \(C\).
  2. Show that \(r = 5\) The line \(L\) has equation \(x = 13\) and crosses \(C\) at the points \(P\) and \(Q\) as shown in Figure 1.
  3. Find the \(y\) coordinate of \(P\) and the \(y\) coordinate of \(Q\). Given that, to 3 decimal places, the angle \(P T Q\) is 1.855 radians,
  4. find the perimeter of the sector \(P T Q\).
Edexcel C2 2012 June Q4
4. $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 10 x + 24$$
  1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Factorise f(x) completely.
Edexcel C2 2012 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1ef99f0-4ad4-49d8-bee7-d5bb9cc84660-07_823_1081_267_404} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the line with equation \(y = 10 - x\) and the curve with equation \(y = 10 x - x ^ { 2 } - 8\) The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.
  1. Calculate the coordinates of \(A\) and the coordinates of \(B\). The shaded area \(R\) is bounded by the line and the curve, as shown in Figure 2.
  2. Calculate the exact area of \(R\).
Edexcel C2 2012 June Q6
  1. (a) Show that the equation
$$\tan 2 x = 5 \sin 2 x$$ can be written in the form $$( 1 - 5 \cos 2 x ) \sin 2 x = 0$$ (b) Hence solve, for \(0 \leqslant x \leqslant 180 ^ { \circ }\), $$\tan 2 x = 5 \sin 2 x$$ giving your answers to 1 decimal place where appropriate.
You must show clearly how you obtained your answers.
Edexcel C2 2012 June Q7
7. $$y = \sqrt { } \left( 3 ^ { x } + x \right)$$
  1. Complete the table below, giving the values of \(y\) to 3 decimal places.
    \(x\)00.250.50.751
    \(y\)11.2512
  2. Use the trapezium rule with all the values of \(y\) from your table to find an approximation for the value of \(\int _ { 0 } ^ { 1 } \sqrt { } \left( 3 ^ { x } + x \right) \mathrm { d } x\) You must show clearly how you obtained your answer.
Edexcel C2 2012 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1ef99f0-4ad4-49d8-bee7-d5bb9cc84660-11_305_446_223_749} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A manufacturer produces pain relieving tablets. Each tablet is in the shape of a solid circular cylinder with base radius \(x \mathrm {~mm}\) and height \(h \mathrm {~mm}\), as shown in Figure 3. Given that the volume of each tablet has to be \(60 \mathrm {~mm} ^ { 3 }\),
  1. express \(h\) in terms of \(x\),
  2. show that the surface area, \(A \mathrm {~mm} ^ { 2 }\), of a tablet is given by \(A = 2 \pi x ^ { 2 } + \frac { 120 } { x }\) The manufacturer needs to minimise the surface area \(A \mathrm {~mm} ^ { 2 }\), of a tablet.
  3. Use calculus to find the value of \(x\) for which \(A\) is a minimum.
  4. Calculate the minimum value of \(A\), giving your answer to the nearest integer.
  5. Show that this value of \(A\) is a minimum.
Edexcel C2 2012 June Q9
  1. A geometric series is \(a + a r + a r ^ { 2 } + \ldots\)
    1. Prove that the sum of the first \(n\) terms of this series is given by
    $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ The third and fifth terms of a geometric series are 5.4 and 1.944 respectively and all the terms in the series are positive. For this series find,
  2. the common ratio,
  3. the first term,
  4. the sum to infinity.
Edexcel C2 2013 June Q1
  1. Using calculus, find the coordinates of the stationary point on the curve with equation
$$y = 2 x + 3 + \frac { 8 } { x ^ { 2 } } , \quad x > 0$$
Edexcel C2 2013 June Q2
2. $$y = \frac { x } { \sqrt { ( 1 + x ) } }$$
  1. Complete the table below with the value of \(y\) corresponding to \(x = 1.3\), giving your answer to 4 decimal places.
    \(x\)11.11.21.31.41.5
    \(y\)0.70710.75910.80900.90370.9487
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an approximate value for $$\int _ { 1 } ^ { 1.5 } \frac { x } { \sqrt { } ( 1 + x ) } \mathrm { d } x$$ giving your answer to 3 decimal places.
    You must show clearly each stage of your working.
Edexcel C2 2013 June Q3
3. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 1 } { 2 } x \right) ^ { 8 }$$ giving each term in its simplest form.
Edexcel C2 2013 June Q4
4. \(\mathrm { f } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } + b x + 4\), where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ) the remainder is 55
When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 9
  1. Find the value of \(a\) and the value of \(b\). Given that \(( 3 x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
  2. factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2013 June Q5
5. The first three terms of a geometric series are \(4 p , ( 3 p + 15 )\) and ( \(5 p + 20\) ) respectively, where \(p\) is a positive constant.
  1. Show that \(11 p ^ { 2 } - 10 p - 225 = 0\)
  2. Hence show that \(p = 5\)
  3. Find the common ratio of this series.
  4. Find the sum of the first ten terms of the series, giving your answer to the nearest integer.
Edexcel C2 2013 June Q6
6. Given that \(\log _ { 3 } x = a\), find in terms of \(a\),
  1. \(\log _ { 3 } ( 9 x )\)
  2. \(\log _ { 3 } \left( \frac { x ^ { 5 } } { 81 } \right)\)
    giving each answer in its simplest form.
  3. Solve, for \(x\), $$\log _ { 3 } ( 9 x ) + \log _ { 3 } \left( \frac { x ^ { 5 } } { 81 } \right) = 3$$ giving your answer to 4 significant figures. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4f4eac7b-8908-480f-bb39-049944203fff-10_775_1605_221_159} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The line with equation \(y = 10\) cuts the curve with equation \(y = x ^ { 2 } + 2 x + 2\) at the points \(A\) and \(B\) as shown in Figure 1. The figure is not drawn to scale.
Edexcel C2 2013 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4f4eac7b-8908-480f-bb39-049944203fff-12_556_1392_210_283} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the design for a triangular garden \(A B C\) where \(A B = 7 \mathrm {~m} , A C = 13 \mathrm {~m}\) and \(B C = 10 \mathrm {~m}\). Given that angle \(B A C = \theta\) radians,
  1. show that, to 3 decimal places, \(\theta = 0.865\) The point \(D\) lies on \(A C\) such that \(B D\) is an arc of the circle centre \(A\), radius 7 m .
    The shaded region \(S\) is bounded by the arc \(B D\) and the lines \(B C\) and \(D C\). The shaded region \(S\) will be sown with grass seed, to make a lawned area. Given that 50 g of grass seed are needed for each square metre of lawn,
  2. find the amount of grass seed needed, giving your answer to the nearest 10 g .
Edexcel C2 2013 June Q9
  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
  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
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
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
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
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
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. (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
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.