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 2010 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e3faf018-37a8-48ef-b100-81402a8ec87f-09_696_821_205_516} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The curve \(C\) has equation \(y = x ^ { 2 } - 5 x + 4\). It cuts the \(x\)-axis at the points \(L\) and \(M\) as shown in Figure 2.
  1. Find the coordinates of the point \(L\) and the point \(M\).
  2. Show that the point \(N ( 5,4 )\) lies on \(C\).
  3. Find \(\int \left( x ^ { 2 } - 5 x + 4 \right) \mathrm { d } x\). The finite region \(R\) is bounded by \(L N , L M\) and the curve \(C\) as shown in Figure 2.
  4. Use your answer to part (c) to find the exact value of the area of \(R\).
    \section*{LU}
Edexcel C2 2010 January Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e3faf018-37a8-48ef-b100-81402a8ec87f-11_1262_1178_203_386} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the circle \(C\) with centre \(N\) and equation $$( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = \frac { 169 } { 4 }$$
  1. Write down the coordinates of \(N\).
  2. Find the radius of \(C\). The chord \(A B\) of \(C\) is parallel to the \(x\)-axis, lies below the \(x\)-axis and is of length 12 units as shown in Figure 3.
  3. Find the coordinates of \(A\) and the coordinates of \(B\).
  4. Show that angle \(A N B = 134.8 ^ { \circ }\), to the nearest 0.1 of a degree. The tangents to \(C\) at the points \(A\) and \(B\) meet at the point \(P\).
  5. Find the length \(A P\), giving your answer to 3 significant figures.
Edexcel C2 2010 January Q9
9. The curve \(C\) has equation \(y = 12 \sqrt { } ( x ) - x ^ { \frac { 3 } { 2 } } - 10 , \quad x > 0\)
  1. Use calculus to find the coordinates of the turning point on \(C\).
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  3. State the nature of the turning point.
Edexcel C2 2011 January Q1
1. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is 7 .
  1. Show that \(a + b = 3\). When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\), the remainder is - 8 .
  2. Find the value of \(a\) and the value of \(b\).
Edexcel C2 2011 January Q2
2. In the triangle \(A B C , A B = 11 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and \(C A = 8 \mathrm {~cm}\).
  1. Find the size of angle \(C\), giving your answer in radians to 3 significant figures.
  2. Find the area of triangle \(A B C\), giving your answer in \(\mathrm { cm } ^ { 2 }\) to 3 significant figures.
Edexcel C2 2011 January Q3
3. The second and fifth terms of a geometric series are 750 and - 6 respectively. Find
  1. the common ratio of the series,
  2. the first term of the series,
  3. the sum to infinity of the series.
Edexcel C2 2011 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be8f9187-055a-476f-974d-22e8e16e9996-05_547_798_251_575} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = ( x + 1 ) ( x - 5 )$$ The curve crosses the \(x\)-axis at the points \(A\) and \(B\).
  1. Write down the \(x\)-coordinates of \(A\) and \(B\). The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis.
  2. Use integration to find the area of \(R\).
Edexcel C2 2011 January Q5
  1. Given that \(\binom { 40 } { 4 } = \frac { 40 ! } { 4 ! b ! }\),
    1. write down the value of \(b\).
    In the binomial expansion of \(( 1 + x ) ^ { 40 }\), the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) are \(p\) and \(q\) respectively.
  2. Find the value of \(\frac { q } { p }\).
Edexcel C2 2011 January Q6
6. $$y = \frac { 5 } { 3 x ^ { 2 } - 2 }$$
  1. Complete the table below, giving the values of \(y\) to 2 decimal places.
    \(x\)22.252.52.753
    \(y\)0.50.380.2
  2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximate value for \(\int _ { 2 } ^ { 3 } \frac { 5 } { 3 x ^ { 2 } - 2 } \mathrm {~d} x\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{be8f9187-055a-476f-974d-22e8e16e9996-08_537_743_941_603} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \frac { 5 } { 3 x ^ { 2 } - 2 } , x > 1\).
    At the points \(A\) and \(B\) on the curve, \(x = 2\) and \(x = 3\) respectively.
    The region \(S\) is bounded by the curve, the straight line through \(B\) and ( 2,0 ), and the line through \(A\) parallel to the \(y\)-axis. The region \(S\) is shown shaded in Figure 2.
  3. Use your answer to part (b) to find an approximate value for the area of \(S\).
Edexcel C2 2011 January Q7
  1. (a) Show that the equation
$$3 \sin ^ { 2 } x + 7 \sin x = \cos ^ { 2 } x - 4$$ can be written in the form $$4 \sin ^ { 2 } x + 7 \sin x + 3 = 0$$ (b) Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$3 \sin ^ { 2 } x + 7 \sin x = \cos ^ { 2 } x - 4$$ giving your answers to 1 decimal place where appropriate.
Edexcel C2 2011 January Q8
  1. (a) Sketch the graph of \(y = 7 ^ { x } , x \in \mathbb { R }\), showing the coordinates of any points at which the graph crosses the axes.
    (b) Solve the equation
$$7 ^ { 2 x } - 4 \left( 7 ^ { x } \right) + 3 = 0$$ giving your answers to 2 decimal places where appropriate.
Edexcel C2 2011 January Q9
9. The points \(A\) and \(B\) have coordinates \(( - 2,11 )\) and \(( 8,1 )\) respectively. Given that \(A B\) is a diameter of the circle \(C\),
  1. show that the centre of \(C\) has coordinates \(( 3,6 )\),
  2. find an equation for \(C\).
  3. Verify that the point \(( 10,7 )\) lies on \(C\).
  4. Find an equation of the tangent to \(C\) at the point (10, 7), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C2 2011 January Q10
  1. The volume \(V \mathrm {~cm} ^ { 3 }\) of a box, of height \(x \mathrm {~cm}\), is given by
$$V = 4 x ( 5 - x ) ^ { 2 } , \quad 0 < x < 5$$
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
  2. Hence find the maximum volume of the box.
  3. Use calculus to justify that the volume that you found in part (b) is a maximum.
Edexcel C2 2012 January Q1
  1. A geometric series has first term \(a = 360\) and common ratio \(r = \frac { 7 } { 8 }\)
Giving your answers to 3 significant figures where appropriate, find
  1. the 20 th term of the series,
  2. the sum of the first 20 terms of the series,
  3. the sum to infinity of the series.
Edexcel C2 2012 January Q2
2. A circle \(C\) has centre \(( - 1,7 )\) and passes through the point \(( 0,0 )\). Find an equation for \(C\).
(4)
Edexcel C2 2012 January Q3
3. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$\left( 1 + \frac { x } { 4 } \right) ^ { 8 }$$ giving each term in its simplest form.
(b) Use your expansion to estimate the value of \(( 1.025 ) ^ { 8 }\), giving your answer to 4 decimal places.
Edexcel C2 2012 January Q4
4. Given that \(y = 3 x ^ { 2 }\),
  1. show that \(\log _ { 3 } y = 1 + 2 \log _ { 3 } x\)
  2. Hence, or otherwise, solve the equation $$1 + 2 \log _ { 3 } x = \log _ { 3 } ( 28 x - 9 )$$
Edexcel C2 2012 January Q5
  1. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 3\), where \(a\) and \(b\) are constants.
Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is 7 ,
  1. show that \(2 a - b = 6\) Given also that when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is 4 ,
  2. find the value of \(a\) and the value of \(b\).
Edexcel C2 2012 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{42116a65-60ec-4dff-a05e-bab529939e1e-07_611_1326_280_310} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of the curve with equation $$y = \frac { 16 } { x ^ { 2 } } - \frac { x } { 2 } + 1 , \quad x > 0$$ The finite region \(R\), bounded by the lines \(x = 1\), the \(x\)-axis and the curve, is shown shaded in Figure 1. The curve crosses the \(x\)-axis at the point \(( 4,0 )\).
  1. Complete the table with the values of \(y\) corresponding to \(x = 2\) and 2.5
    \(x\)11.522.533.54
    \(y\)16.57.3611.2780.5560
  2. Use the trapezium rule with all the values in the completed table to find an approximate value for the area of \(R\), giving your answer to 2 decimal places.
  3. Use integration to find the exact value for the area of \(R\).
Edexcel C2 2012 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{42116a65-60ec-4dff-a05e-bab529939e1e-09_408_435_262_756} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows \(A B C\), a sector of a circle of radius 6 cm with centre \(A\). Given that the size of angle \(B A C\) is 0.95 radians, find
  1. the length of the \(\operatorname { arc } B C\),
  2. the area of the sector \(A B C\). The point \(D\) lies on the line \(A C\) and is such that \(A D = B D\). The region \(R\), shown shaded in Figure 2, is bounded by the lines \(C D , D B\) and the \(\operatorname { arc } B C\).
  3. Show that the length of \(A D\) is 5.16 cm to 3 significant figures. Find
  4. the perimeter of \(R\),
  5. the area of \(R\), giving your answer to 2 significant figures.
Edexcel C2 2012 January Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{42116a65-60ec-4dff-a05e-bab529939e1e-11_403_440_262_744} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius \(x\) metres with two equal rectangles attached to it along its radii. Each rectangle has length equal to \(x\) metres and width equal to \(y\) metres. Given that the area of the flowerbed is \(4 \mathrm {~m} ^ { 2 }\),
  1. show that $$y = \frac { 16 - \pi x ^ { 2 } } { 8 x }$$
  2. Hence show that the perimeter \(P\) metres of the flowerbed is given by the equation $$P = \frac { 8 } { x } + 2 x$$
  3. Use calculus to find the minimum value of \(P\).
  4. Find the width of each rectangle when the perimeter is a minimum. Give your answer to the nearest centimetre.
Edexcel C2 2012 January Q9
  1. (i) Find the solutions of the equation \(\sin \left( 3 x - 15 ^ { \circ } \right) = \frac { 1 } { 2 }\), for which \(0 \leqslant x \leqslant 180 ^ { \circ }\)
    (6)
    (ii)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{42116a65-60ec-4dff-a05e-bab529939e1e-13_476_1141_495_406} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows part of the curve with equation $$y = \sin ( a x - b ) , \text { where } a > 0,0 < b < \pi$$ The curve cuts the \(x\)-axis at the points \(P , Q\) and \(R\) as shown.
Given that the coordinates of \(P , Q\) and \(R\) are \(\left( \frac { \pi } { 10 } , 0 \right) , \left( \frac { 3 \pi } { 5 } , 0 \right)\) and \(\left( \frac { 11 \pi } { 10 } , 0 \right)\) respectively, find the values of \(a\) and \(b\).
Edexcel C2 2013 January Q1
  1. Find the first 3 terms, in ascending powers of \(x\), in the binomial expansion of
$$( 2 - 5 x ) ^ { 6 }$$ Give each term in its simplest form.
Edexcel C2 2013 January Q2
2. \(\mathrm { f } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 4 x - 3\), where \(a\) and \(b\) are constants. Given that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\),
  1. show that $$a + b = 7$$ Given also that, when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\), the remainder is 9 ,
  2. find the value of \(a\) and the value of \(b\), showing each step in your working.
Edexcel C2 2013 January Q3
3. A company predicts a yearly profit of \(\pounds 120000\) in the year 2013 . The company predicts that the yearly profit will rise each year by \(5 \%\). The predicted yearly profit forms a geometric sequence with common ratio 1.05
  1. Show that the predicted profit in the year 2016 is \(\pounds 138915\)
  2. Find the first year in which the yearly predicted profit exceeds \(\pounds 200000\)
  3. Find the total predicted profit for the years 2013 to 2023 inclusive, giving your answer to the nearest pound.