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 2009 June Q4
4. (a) Complete the table below, giving values of \(\sqrt { } \left( 2 ^ { x } + 1 \right)\) to 3 decimal places.
\(x\)00.511.522.53
\(\sqrt { } \left( 2 ^ { x } + 1 \right)\)1.4141.5541.7321.9573
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{78a994ba-50c5-434f-a060-9596edb505cd-05_653_595_616_676} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the region \(R\) which is bounded by the curve with equation \(y = \sqrt { } \left( 2 ^ { x } + 1 \right)\), the \(x\)-axis and the lines \(x = 0\) and \(x = 3\)
(b) Use the trapezium rule, with all the values from your table, to find an approximation for the area of \(R\).
(c) By reference to the curve in Figure 1 state, giving a reason, whether your approximation in part (b) is an overestimate or an underestimate for the area of \(R\).
Edexcel C2 2009 June Q5
  1. The third term of a geometric sequence is 324 and the sixth term is 96
    1. Show that the common ratio of the sequence is \(\frac { 2 } { 3 }\)
    2. Find the first term of the sequence.
    3. Find the sum of the first 15 terms of the sequence.
    4. Find the sum to infinity of the sequence.
Edexcel C2 2009 June Q6
6. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 4 y = 12$$
  1. Find the centre and the radius of \(C\). The point \(P ( - 1,1 )\) and the point \(Q ( 7 , - 5 )\) both lie on \(C\).
  2. Show that \(P Q\) is a diameter of \(C\). The point \(R\) lies on the positive \(y\)-axis and the angle \(P R Q = 90 ^ { \circ }\).
  3. Find the coordinates of \(R\).
Edexcel C2 2009 June Q7
7. (i) Solve, for \(- 180 ^ { \circ } \leqslant \theta < 180 ^ { \circ }\), $$( 1 + \tan \theta ) ( 5 \sin \theta - 2 ) = 0$$ (ii) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \sin x = 3 \tan x .$$
Edexcel C2 2009 June Q8
8. (a) Find the value of \(y\) such that $$\log _ { 2 } y = - 3$$ (b) Find the values of \(x\) such that $$\frac { \log _ { 2 } 32 + \log _ { 2 } 16 } { \log _ { 2 } x } = \log _ { 2 } x$$
Edexcel C2 2009 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{78a994ba-50c5-434f-a060-9596edb505cd-14_554_454_212_744} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height \(h \mathrm {~cm}\). The cross section is a sector of a circle. The sector has radius \(r \mathrm {~cm}\) and angle 1 radian. The volume of the box is \(300 \mathrm {~cm} ^ { 3 }\).
  1. Show that the surface area of the box, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = r ^ { 2 } + \frac { 1800 } { r }$$
  2. Use calculus to find the value of \(r\) for which \(S\) is stationary.
  3. Prove that this value of \(r\) gives a minimum value of \(S\).
  4. Find, to the nearest \(\mathrm { cm } ^ { 2 }\), this minimum value of \(S\).
Edexcel C2 2010 June Q1
1. $$y = 3 ^ { x } + 2 x$$
  1. Complete the table below, giving the values of \(y\) to 2 decimal places.
    \(x\)00.20.40.60.81
    \(y\)11.655
  2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximate value for \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 2 x \right) d x\).
Edexcel C2 2010 June Q2
2. $$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 58 x + 40$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ). Given that \(( x - 5 )\) is a factor of \(\mathrm { f } ( x )\),
  2. find all the solutions of \(\mathrm { f } ( x ) = 0\).
Edexcel C2 2010 June Q3
3. $$y = x ^ { 2 } - k \sqrt { } x , \text { where } k \text { is a constant. }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Given that \(y\) is decreasing at \(x = 4\), find the set of possible values of \(k\).
Edexcel C2 2010 June Q4
4. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + a x ) ^ { 7 }\), where \(a\) is a constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 2 }\) in this expansion is 525 ,
(b) find the possible values of \(a\).
Edexcel C2 2010 June Q5
5. (a) Given that \(5 \sin \theta = 2 \cos \theta\), find the value of \(\tan \theta\).
(b) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$5 \sin 2 x = 2 \cos 2 x$$ giving your answers to 1 decimal place.
Edexcel C2 2010 June Q6
6.
\includegraphics[max width=\textwidth, alt={}, center]{571780c2-945b-4636-b7c3-0bd558d28710-07_458_809_258_569} \section*{Figure 1} Figure 1 shows the sector \(O A B\) of a circle with centre \(O\), radius 9 cm and angle 0.7 radians.
  1. Find the length of the arc \(A B\).
  2. Find the area of the sector \(O A B\). The line \(A C\) shown in Figure 1 is perpendicular to \(O A\), and \(O B C\) is a straight line.
  3. Find the length of \(A C\), giving your answer to 2 decimal places. The region \(H\) is bounded by the arc \(A B\) and the lines \(A C\) and \(C B\).
  4. Find the area of \(H\), giving your answer to 2 decimal places.
    \section*{LU}
Edexcel C2 2010 June Q7
7. (a) Given that $$2 \log _ { 3 } ( x - 5 ) - \log _ { 3 } ( 2 x - 13 ) = 1$$ show that \(x ^ { 2 } - 16 x + 64 = 0\).
(b) Hence, or otherwise, solve \(2 \log _ { 3 } ( x - 5 ) - \log _ { 3 } ( 2 x - 13 ) = 1\).
Edexcel C2 2010 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{571780c2-945b-4636-b7c3-0bd558d28710-10_611_831_210_575} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 10 x ^ { 2 } + k x$$ where \(k\) is a constant. The point \(P\) on \(C\) is the maximum turning point.
Given that the \(x\)-coordinate of \(P\) is 2 ,
  1. show that \(k = 28\). The line through \(P\) parallel to the \(x\)-axis cuts the \(y\)-axis at the point \(N\). The region \(R\) is bounded by \(C\), the \(y\)-axis and \(P N\), as shown shaded in Figure 2.
  2. Use calculus to find the exact area of \(R\).
Edexcel C2 2010 June Q9
9. The adult population of a town is 25000 at the end of Year 1. A model predicts that the adult population of the town will increase by \(3 \%\) each year, forming a geometric sequence.
  1. Show that the predicted adult population at the end of Year 2 is 25750.
  2. Write down the common ratio of the geometric sequence. The model predicts that Year \(N\) will be the first year in which the adult population of the town exceeds 40000.
  3. Show that $$( N - 1 ) \log 1.03 > \log 1.6$$
  4. Find the value of \(N\). At the end of each year, each member of the adult population of the town will give \(\pounds 1\) to a charity fund. Assuming the population model,
  5. find the total amount that will be given to the charity fund for the 10 years from the end of Year 1 to the end of Year 10, giving your answer to the nearest \(\pounds 1000\).
Edexcel C2 2010 June Q10
10. The circle \(C\) has centre \(A ( 2,1 )\) and passes through the point \(B ( 10,7 )\).
  1. Find an equation for \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(B\).
  2. Find an equation for \(l _ { 1 }\). The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the mid-point of \(A B\).
    Given that \(l _ { 2 }\) intersects \(C\) at the points \(P\) and \(Q\),
  3. find the length of \(P Q\), giving your answer in its simplest surd form.
    \includegraphics[max width=\textwidth, alt={}, center]{571780c2-945b-4636-b7c3-0bd558d28710-15_115_127_2461_1814}
Edexcel C2 2011 June Q1
1. $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 5 x + 4$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\).
  2. Use the factor theorem to show that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\).
  3. Factorise f(x) completely.
Edexcel C2 2011 June Q2
2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 3 + b x ) ^ { 5 }$$ where \(b\) is a non-zero constant. Give each term in its simplest form. Given that, in this expansion, the coefficient of \(x ^ { 2 }\) is twice the coefficient of \(x\),
(b) find the value of \(b\).
Edexcel C2 2011 June Q3
3. Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which
  1. \(5 ^ { x } = 10\),
  2. \(\log _ { 3 } ( x - 2 ) = - 1\).
Edexcel C2 2011 June Q4
4. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - 11 = 0$$ Find
  1. the coordinates of the centre of \(C\),
  2. the radius of \(C\),
  3. the coordinates of the points where \(C\) crosses the \(y\)-axis, giving your answers as simplified surds.
Edexcel C2 2011 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c9758792-ca4c-4837-bd7c-e695fe0c0cdf-06_426_417_260_760} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The shape shown in Figure 1 is a pattern for a pendant. It consists of a sector \(O A B\) of a circle centre \(O\), of radius 6 cm , and angle \(A O B = \frac { \pi } { 3 }\). The circle \(C\), inside the sector, touches the two straight edges, \(O A\) and \(O B\), and the \(\operatorname { arc } A B\) as shown. Find
  1. the area of the sector \(O A B\),
  2. the radius of the circle \(C\). The region outside the circle \(C\) and inside the sector \(O A B\) is shown shaded in Figure 1.
  3. Find the area of the shaded region.
Edexcel C2 2011 June Q6
  1. The second and third terms of a geometric series are 192 and 144 respectively.
For this series, find
  1. the common ratio,
  2. the first term,
  3. the sum to infinity,
  4. the smallest value of \(n\) for which the sum of the first \(n\) terms of the series exceeds 1000.
Edexcel C2 2011 June Q7
  1. (a) Solve for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers in degrees to 1 decimal place,
$$3 \sin \left( x + 45 ^ { \circ } \right) = 2$$ (b) Find, for \(0 \leqslant x < 2 \pi\), all the solutions of $$2 \sin ^ { 2 } x + 2 = 7 \cos x$$ giving your answers in radians.
You must show clearly how you obtained your answers.
Edexcel C2 2011 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c9758792-ca4c-4837-bd7c-e695fe0c0cdf-12_662_719_127_609} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A cuboid has a rectangular cross-section where the length of the rectangle is equal to twice its width, \(x \mathrm {~cm}\), as shown in Figure 2.
The volume of the cuboid is 81 cubic centimetres.
  1. Show that the total length, \(L \mathrm {~cm}\), of the twelve edges of the cuboid is given by $$L = 12 x + \frac { 162 } { x ^ { 2 } }$$
  2. Use calculus to find the minimum value of \(L\).
  3. Justify, by further differentiation, that the value of \(L\) that you have found is a minimum.
Edexcel C2 2011 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c9758792-ca4c-4837-bd7c-e695fe0c0cdf-14_360_956_278_504} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The straight line with equation \(y = x + 4\) cuts the curve with equation \(y = - x ^ { 2 } + 2 x + 24\) at the points \(A\) and \(B\), as shown in Figure 3.
  1. Use algebra to find the coordinates of the points \(A\) and \(B\). The finite region \(R\) is bounded by the straight line and the curve and is shown shaded in Figure 3.
  2. Use calculus to find the exact area of \(R\).