Questions — Edexcel (9670 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 January Q4
6 marks Moderate -0.3
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
6 marks Moderate -0.3
  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
11 marks Moderate -0.3
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
12 marks Standard +0.3
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
13 marks Standard +0.3
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
10 marks Standard +0.3
  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
4 marks Easy -1.2
  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
6 marks Moderate -0.8
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
9 marks Moderate -0.8
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.
Edexcel C2 2013 January Q4
7 marks Moderate -0.3
4. Solve, for \(0 \leqslant x < 180 ^ { \circ }\), $$\cos \left( 3 x - 10 ^ { \circ } \right) = - 0.4$$ giving your answers to 1 decimal place. You should show each step in your working.
Edexcel C2 2013 January Q5
9 marks Moderate -0.3
5. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 24 y + 195 = 0$$ The centre of \(C\) is at the point \(M\).
  1. Find
    1. the coordinates of the point \(M\),
    2. the radius of the circle \(C\).
      \(N\) is the point with coordinates \(( 25,32 )\).
  2. Find the length of the line \(M N\). The tangent to \(C\) at a point \(P\) on the circle passes through point \(N\).
  3. Find the length of the line \(N P\).
Edexcel C2 2013 January Q6
7 marks Moderate -0.3
6. Given that $$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$
  1. Show that $$x ^ { 2 } - 34 x + 225 = 0$$
  2. Hence, or otherwise, solve the equation $$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$
Edexcel C2 2013 January Q7
12 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f31b6f1-33b5-4bca-9030-cf93760b454d-09_432_656_210_644} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The triangle \(X Y Z\) in Figure 1 has \(X Y = 6 \mathrm {~cm} , Y Z = 9 \mathrm {~cm} , Z X = 4 \mathrm {~cm}\) and angle \(Z X Y = \alpha\). The point \(W\) lies on the line \(X Y\). The circular arc \(Z W\), in Figure 1 is a major arc of the circle with centre \(X\) and radius 4 cm .
  1. Show that, to 3 significant figures, \(\alpha = 2.22\) radians.
  2. Find the area, in \(\mathrm { cm } ^ { 2 }\), of the major sector \(X Z W X\). The region enclosed by the major arc \(Z W\) of the circle and the lines \(W Y\) and \(Y Z\) is shown shaded in Figure 1. Calculate
  3. the area of this shaded region,
  4. the perimeter \(Z W Y Z\) of this shaded region.
Edexcel C2 2013 January Q8
9 marks Moderate -0.3
8. The curve \(C\) has equation \(y = 6 - 3 x - \frac { 4 } { x ^ { 3 } } , x \neq 0\)
  1. Use calculus to show that the curve has a turning point \(P\) when \(x = \sqrt { } 2\)
  2. Find the \(x\)-coordinate of the other turning point \(Q\) on the curve.
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  4. Hence or otherwise, state with justification, the nature of each of these turning points \(P\) and \(Q\).
Edexcel C2 2013 January Q9
12 marks Standard +0.3
9. \(y\) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f31b6f1-33b5-4bca-9030-cf93760b454d-13_895_1308_207_294} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The finite region \(R\), as shown in Figure 2, is bounded by the \(x\)-axis and the curve with equation $$y = 27 - 2 x - 9 \sqrt { } x - \frac { 16 } { x ^ { 2 } } , \quad x > 0$$ The curve crosses the \(x\)-axis at the points \(( 1,0 )\) and \(( 4,0 )\).
  1. Complete the table below, by giving your values of \(y\) to 3 decimal places.
    \(x\)11.522.533.54
    \(y\)05.8665.2101.8560
  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 2014 January Q1
5 marks Moderate -0.8
  1. The first three terms in ascending powers of \(x\) in the binomial expansion of \(( 1 + p x ) ^ { 12 }\) are given by
$$1 + 18 x + q x ^ { 2 }$$ where \(p\) and \(q\) are constants.
Find the value of \(p\) and the value of \(q\).
Edexcel C2 2014 January Q2
6 marks Moderate -0.3
2. \(\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants. Given that when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) the remainder is 25 ,
  1. show that \(2 a + b = 5\) Given also that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\),
  2. find the value of \(a\) and the value of \(b\).
    \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-05_90_97_2613_1784}
    \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-05_52_169_2709_1765}
Edexcel C2 2014 January Q3
11 marks Moderate -0.8
3. The curve \(C\) has equation $$y = 2 \sqrt { } x + \frac { 18 } { \sqrt { } x } - 1 , \quad x > 0$$
  1. Find
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
  2. Use calculus to find the coordinates of the stationary point of \(C\).
  3. Determine whether the stationary point is a maximum or minimum, giving a reason for your answer.
    \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-09_138_154_2597_1804}
Edexcel C2 2014 January Q4
8 marks Moderate -0.3
4. The first term of a geometric series is 5 and the common ratio is 1.2 For this series find, to 1 decimal place,
    1. the \(20 ^ { \text {th } }\) term,
    2. the sum of the first 20 terms. The sum of the first \(n\) terms of the series is greater than 3000
  2. Calculate the smallest possible value of \(n\).
Edexcel C2 2014 January Q5
7 marks Moderate -0.8
5. The height of water, \(H\) metres, in a harbour on a particular day is given by the equation $$H = 10 + 5 \sin \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight.
  1. Show that the height of the water 1 hour after midnight is 12.5 metres.
  2. Find, to the nearest minute, the times before midday when the height of the water is 9 metres.
Edexcel C2 2014 January Q6
5 marks Standard +0.3
6. Given that $$\log _ { x } ( 7 y + 1 ) - \log _ { x } ( 2 y ) = 1 , \quad x > 4 , \quad 0 < y < 1$$ express \(y\) in terms of \(x\).
Edexcel C2 2014 January Q7
13 marks Standard +0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e7043e7a-2c8f-425a-8471-f647828cc297-18_1109_958_214_502} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 5$$ The point \(P ( 4,9 )\) lies on \(C\).
  1. Show that the normal to \(C\) at the point \(P\) has equation $$x + 9 y = 85$$ The region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the \(y\)-axis and the normal to \(C\) at \(P\).
  2. Showing all your working, calculate the exact area of \(R\).
Edexcel C2 2014 January Q8
11 marks Moderate -0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e7043e7a-2c8f-425a-8471-f647828cc297-22_1015_1542_267_185} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a circle \(C\) with centre \(O\) and radius 5
  1. Write down the cartesian equation of \(C\). The points \(P ( - 3 , - 4 )\) and \(Q ( 3 , - 4 )\) lie on \(C\).
  2. Show that the tangent to \(C\) at the point \(Q\) has equation $$3 x - 4 y = 25$$
  3. Show that, to 3 decimal places, angle \(P O Q\) is 1.287 radians. The tangent to \(C\) at \(P\) and the tangent to \(C\) at \(Q\) intersect on the \(y\)-axis at the point \(R\).
  4. Find the area of the shaded region \(P Q R\) shown in Figure 2.
    \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-25_177_154_2576_1804}
Edexcel C2 2014 January Q9
9 marks Standard +0.3
9. (a) Show that the equation $$5 \sin x - \cos ^ { 2 } x + 2 \sin ^ { 2 } x = 1$$ can be written in the form $$3 \sin ^ { 2 } x + 5 \sin x - 2 = 0$$ (b) Hence solve, for \(- 180 ^ { \circ } \leqslant \theta < 180 ^ { \circ }\), the equation $$5 \sin 2 \theta - \cos ^ { 2 } 2 \theta + 2 \sin ^ { 2 } 2 \theta = 1$$ giving your answers to 2 decimal places.
Edexcel C2 2005 June Q1
4 marks Easy -1.2
Find the coordinates of the stationary point on the curve with equation \(y = 2 x ^ { 2 } - 12 x\).