Questions — Edexcel (10514 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
Edexcel C3 2014 June Q7
15 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be00fdaa-2fe3-4f06-a710-08ec67fb911e-13_456_881_214_534} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(C\), with equation \(y = 6 \cos x + 2.5 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\)
  1. Express \(6 \cos x + 2.5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\) to 3 decimal places.
  2. Find the coordinates of the points on the graph where the curve \(C\) crosses the coordinate axes. A student records the number of hours of daylight each Sunday throughout the year. She starts on the last Sunday in May with a recording of 18 hours, and continues until her final recording 52 weeks later. She models her results with the continuous function given by $$H = 12 + 6 \cos \left( \frac { 2 \pi t } { 52 } \right) + 2.5 \sin \left( \frac { 2 \pi t } { 52 } \right) , \quad 0 \leqslant t \leqslant 52$$ where \(H\) is the number of hours of daylight and \(t\) is the number of weeks since her first recording. Use this function to find
  3. the maximum and minimum values of \(H\) predicted by the model,
  4. the values for \(t\) when \(H = 16\), giving your answers to the nearest whole number.
    [0pt] [You must show your working. Answers based entirely on graphical or numerical methods are not acceptable.] \includegraphics[max width=\textwidth, alt={}, center]{be00fdaa-2fe3-4f06-a710-08ec67fb911e-14_40_58_2460_1893}
Edexcel C3 2014 June Q1
6 marks Moderate -0.8
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where
$$f ( x ) = \frac { 4 x + 1 } { x - 2 } , \quad x > 2$$
  1. Show that $$f ^ { \prime } ( x ) = \frac { - 9 } { ( x - 2 ) ^ { 2 } }$$ Given that \(P\) is a point on \(C\) such that \(\mathrm { f } ^ { \prime } ( x ) = - 1\),
  2. find the coordinates of \(P\).
Edexcel C3 2014 June Q2
6 marks Moderate -0.8
2. Find the exact solutions, in their simplest form, to the equations
  1. \(2 \ln ( 2 x + 1 ) - 10 = 0\)
  2. \(3 ^ { x } \mathrm { e } ^ { 4 x } = \mathrm { e } ^ { 7 }\)
Edexcel C3 2014 June Q3
8 marks Standard +0.3
3. The curve \(C\) has equation \(x = 8 y \tan 2 y\) The point \(P\) has coordinates \(\left( \pi , \frac { \pi } { 8 } \right)\)
  1. Verify that \(P\) lies on \(C\).
  2. Find the equation of the tangent to \(C\) at \(P\) in the form \(a y = x + b\), where the constants \(a\) and \(b\) are to be found in terms of \(\pi\).
Edexcel C3 2014 June Q4
7 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{16c69ee4-255e-4d77-955a-92e1eb2f7d3e-05_665_776_233_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the graph with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(Q ( 6 , - 1 )\). The graph crosses the \(y\)-axis at the point \(P ( 0,11 )\). Sketch, on separate diagrams, the graphs of
  1. \(y = | f ( x ) |\)
  2. \(y = 2 f ( - x ) + 3\) On each diagram, show the coordinates of the points corresponding to \(P\) and \(Q\).
    Given that \(\mathrm { f } ( x ) = a | x - b | - 1\), where \(a\) and \(b\) are constants,
  3. state the value of \(a\) and the value of \(b\).
Edexcel C3 2014 June Q5
10 marks Standard +0.8
5. $$\mathrm { g } ( x ) = \frac { x } { x + 3 } + \frac { 3 ( 2 x + 1 ) } { x ^ { 2 } + x - 6 } , \quad x > 3$$
  1. Show that \(\mathrm { g } ( x ) = \frac { x + 1 } { x - 2 } , \quad x > 3\)
  2. Find the range of g.
  3. Find the exact value of \(a\) for which \(\mathrm { g } ( a ) = \mathrm { g } ^ { - 1 } ( a )\).
Edexcel C3 2014 June Q6
8 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{16c69ee4-255e-4d77-955a-92e1eb2f7d3e-09_458_1164_239_383} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = 2 \cos \left( \frac { 1 } { 2 } x ^ { 2 } \right) + x ^ { 3 } - 3 x - 2$$ The curve crosses the \(x\)-axis at the point \(Q\) and has a minimum turning point at \(R\).
  1. Show that the \(x\) coordinate of \(Q\) lies between 2.1 and 2.2
  2. Show that the \(x\) coordinate of \(R\) is a solution of the equation $$x = \sqrt { 1 + \frac { 2 } { 3 } x \sin \left( \frac { 1 } { 2 } x ^ { 2 } \right) }$$ Using the iterative formula $$x _ { n + 1 } = \sqrt { 1 + \frac { 2 } { 3 } x _ { n } \sin \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } \right) } , \quad x _ { 0 } = 1.3$$
  3. find the values of \(x _ { 1 }\) and \(x _ { 2 }\) to 3 decimal places.
Edexcel C3 2014 June Q7
10 marks Standard +0.8
7.
  1. Show that $$\operatorname { cosec } 2 x + \cot 2 x = \cot x , \quad x \neq 90 n ^ { \circ } , \quad n \in \mathbb { Z }$$
  2. Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), $$\operatorname { cosec } \left( 4 \theta + 10 ^ { \circ } \right) + \cot \left( 4 \theta + 10 ^ { \circ } \right) = \sqrt { 3 }$$ You must show your working.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C3 2014 June Q8
11 marks Standard +0.8
8. A rare species of primrose is being studied. The population, \(P\), of primroses at time \(t\) years after the study started is modelled by the equation $$P = \frac { 800 \mathrm { e } ^ { 0.1 t } } { 1 + 3 \mathrm { e } ^ { 0.1 t } } , \quad t \geqslant 0 , \quad t \in \mathbb { R }$$
  1. Calculate the number of primroses at the start of the study.
  2. Find the exact value of \(t\) when \(P = 250\), giving your answer in the form \(a \ln ( b )\) where \(a\) and \(b\) are integers.
  3. Find the exact value of \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) when \(t = 10\). Give your answer in its simplest form.
  4. Explain why the population of primroses can never be 270
Edexcel C3 2014 June Q9
9 marks Standard +0.3
9.
  1. Express \(2 \sin \theta - 4 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the value of \(\alpha\) to 3 decimal places. $$H ( \theta ) = 4 + 5 ( 2 \sin 3 \theta - 4 \cos 3 \theta ) ^ { 2 }$$ Find
    1. the maximum value of \(\mathrm { H } ( \theta )\),
    2. the smallest value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this maximum value occurs. Find
    1. the minimum value of \(\mathrm { H } ( \theta )\),
    2. the largest value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this minimum value occurs.
Edexcel C3 2015 June Q1
6 marks Moderate -0.3
  1. Given that
$$\tan \theta ^ { \circ } = p , \text { where } p \text { is a constant, } p \neq \pm 1$$ use standard trigonometric identities, to find in terms of \(p\),
  1. \(\tan 2 \theta ^ { \circ }\)
  2. \(\cos \theta ^ { \circ }\)
  3. \(\cot ( \theta - 45 ) ^ { \circ }\) Write each answer in its simplest form.
Edexcel C3 2015 June Q2
10 marks Moderate -0.8
2. Given that $$\mathrm { f } ( x ) = 2 \mathrm { e } ^ { x } - 5 , \quad x \in \mathbb { R }$$
  1. sketch, on separate diagrams, the curve with equation
    1. \(y = \mathrm { f } ( x )\)
    2. \(y = | \mathrm { f } ( x ) |\) On each diagram, show the coordinates of each point at which the curve meets or cuts the axes. On each diagram state the equation of the asymptote.
  2. Deduce the set of values of \(x\) for which \(\mathrm { f } ( x ) = | \mathrm { f } ( x ) |\)
  3. Find the exact solutions of the equation \(| \mathrm { f } ( x ) | = 2\)
Edexcel C3 2015 June Q3
10 marks Standard +0.3
3. $$g ( \theta ) = 4 \cos 2 \theta + 2 \sin 2 \theta$$ Given that \(\mathrm { g } ( \theta ) = R \cos ( 2 \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),
  1. find the exact value of \(R\) and the value of \(\alpha\) to 2 decimal places.
  2. Hence solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), $$4 \cos 2 \theta + 2 \sin 2 \theta = 1$$ giving your answers to one decimal place. Given that \(k\) is a constant and the equation \(\mathrm { g } ( \theta ) = k\) has no solutions,
  3. state the range of possible values of \(k\).
Edexcel C3 2015 June Q4
7 marks Moderate -0.3
  1. Water is being heated in an electric kettle. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of the water \(t\) seconds after the kettle is switched on, is modelled by the equation
$$\theta = 120 - 100 \mathrm { e } ^ { - \lambda t } , \quad 0 \leqslant t \leqslant T$$
  1. State the value of \(\theta\) when \(t = 0\) Given that the temperature of the water in the kettle is \(70 ^ { \circ } \mathrm { C }\) when \(t = 40\),
  2. find the exact value of \(\lambda\), giving your answer in the form \(\frac { \ln a } { b }\), where \(a\) and \(b\) are integers. When \(t = T\), the temperature of the water reaches \(100 ^ { \circ } \mathrm { C }\) and the kettle switches off.
  3. Calculate the value of \(T\) to the nearest whole number.
Edexcel C3 2015 June Q5
7 marks Standard +0.3
5. The point \(P\) lies on the curve with equation $$x = ( 4 y - \sin 2 y ) ^ { 2 }$$ Given that \(P\) has \(( x , y )\) coordinates \(\left( p , \frac { \pi } { 2 } \right)\), where \(p\) is a constant,
  1. find the exact value of \(p\). The tangent to the curve at \(P\) cuts the \(y\)-axis at the point \(A\).
  2. Use calculus to find the coordinates of \(A\).
Edexcel C3 2015 June Q6
8 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{57ea7a94-6939-4c12-a6fd-01cd6af73310-10_1004_1120_260_420} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 is a sketch showing part of the curve with equation \(y = 2 ^ { x + 1 } - 3\) and part of the line with equation \(y = 17 - x\). The curve and the line intersect at the point \(A\).
  1. Show that the \(x\) coordinate of \(A\) satisfies the equation $$x = \frac { \ln ( 20 - x ) } { \ln 2 } - 1$$
  2. Use the iterative formula $$x _ { n + 1 } = \frac { \ln \left( 20 - x _ { n } \right) } { \ln 2 } - 1 , \quad x _ { 0 } = 3$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
  3. Use your answer to part (b) to deduce the coordinates of the point \(A\), giving your answers to one decimal place.
Edexcel C3 2015 June Q7
10 marks Standard +0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{57ea7a94-6939-4c12-a6fd-01cd6af73310-12_632_873_294_532} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation $$\mathrm { g } ( x ) = x ^ { 2 } ( 1 - x ) \mathrm { e } ^ { - 2 x } , \quad x \geqslant 0$$
  1. Show that \(\mathrm { g } ^ { \prime } ( x ) = \mathrm { f } ( x ) \mathrm { e } ^ { - 2 x }\), where \(\mathrm { f } ( x )\) is a cubic function to be found.
  2. Hence find the range of g .
  3. State a reason why the function \(\mathrm { g } ^ { - 1 } ( x )\) does not exist.
Edexcel C3 2015 June Q8
9 marks Standard +0.8
  1. Prove that $$\sec 2 A + \tan 2 A \equiv \frac { \cos A + \sin A } { \cos A - \sin A } , \quad A \neq \frac { ( 2 n + 1 ) \pi } { 4 } , n \in \mathbb { Z }$$
  2. Hence solve, for \(0 \leqslant \theta < 2 \pi\), $$\sec 2 \theta + \tan 2 \theta = \frac { 1 } { 2 }$$ Give your answers to 3 decimal places.
Edexcel C3 2015 June Q9
8 marks Standard +0.3
9. Given that \(k\) is a negative constant and that the function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 2 - \frac { ( x - 5 k ) ( x - k ) } { x ^ { 2 } - 3 k x + 2 k ^ { 2 } } , \quad x \geqslant 0$$
  1. show that \(\mathrm { f } ( x ) = \frac { x + k } { x - 2 k }\)
  2. Hence find \(\mathrm { f } ^ { \prime } ( x )\), giving your answer in its simplest form.
  3. State, with a reason, whether \(\mathrm { f } ( x )\) is an increasing or a decreasing function. Justify your answer.
Edexcel C3 2016 June Q1
5 marks Standard +0.3
  1. The functions \(f\) and \(g\) are defined by
$$\begin{aligned} & \mathrm { f } : x \rightarrow 7 x - 1 , \quad x \in \mathbb { R } \\ & \mathrm {~g} : x \rightarrow \frac { 4 } { x - 2 } , \quad x \neq 2 , x \in \mathbb { R } \end{aligned}$$
  1. Solve the equation \(\operatorname { fg } ( x ) = x\)
  2. Hence, or otherwise, find the largest value of \(a\) such that \(\mathrm { g } ( a ) = \mathrm { f } ^ { - 1 } ( a )\)
Edexcel C3 2016 June Q2
7 marks Standard +0.3
2.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), writing your answer as a single fraction in its simplest form.
  2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } < 0\) 2. $$y = \frac { 4 x } { x ^ { 2 } + 5 }$$
Edexcel C3 2016 June Q3
10 marks Standard +0.8
  1. Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places.
  2. Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$\frac { 2 } { 2 \cos \theta - \sin \theta - 1 } = 15$$ Give your answers to one decimal place.
  3. Use your solutions to parts (a) and (b) to deduce the smallest positive value of \(\theta\) for which $$\frac { 2 } { 2 \cos \theta + \sin \theta - 1 } = 15$$ Give your answer to one decimal place.
Edexcel C3 2016 June Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d3ba2776-eedb-48f0-834f-41aa454afba3-06_675_1118_205_406} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = g ( x )\), where $$\mathrm { g } ( x ) = \left| 4 \mathrm { e } ^ { 2 x } - 25 \right| , \quad x \in \mathbb { R }$$ The curve cuts the \(y\)-axis at the point \(A\) and meets the \(x\)-axis at the point \(B\). The curve has an asymptote \(y = k\), where \(k\) is a constant, as shown in Figure 1
  1. Find, giving each answer in its simplest form,
    1. the \(y\) coordinate of the point \(A\),
    2. the exact \(x\) coordinate of the point \(B\),
    3. the value of the constant \(k\). The equation \(\mathrm { g } ( x ) = 2 x + 43\) has a positive root at \(x = \alpha\)
  2. Show that \(\alpha\) is a solution of \(x = \frac { 1 } { 2 } \ln \left( \frac { 1 } { 2 } x + 17 \right)\) The iteration formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 1 } { 2 } x _ { n } + 17 \right)$$ can be used to find an approximation for \(\alpha\)
  3. Taking \(x _ { 0 } = 1.4\) find the values of \(x _ { 1 }\) and \(x _ { 2 }\) Give each answer to 4 decimal places.
  4. By choosing a suitable interval, show that \(\alpha = 1.437\) to 3 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-07_2258_47_315_37}
Edexcel C3 2016 June Q5
10 marks Standard +0.3
5.
  1. Find, using calculus, the \(x\) coordinate of the turning point of the curve with equation $$y = \mathrm { e } ^ { 3 x } \cos 4 x , \quad \frac { \pi } { 4 } \leqslant x < \frac { \pi } { 2 }$$ Give your answer to 4 decimal places.
  2. Given \(x = \sin ^ { 2 } 2 y , \quad 0 < y < \frac { \pi } { 4 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(y\). Write your answer in the form $$\frac { \mathrm { d } y } { \mathrm {~d} x } = p \operatorname { cosec } ( q y ) , \quad 0 < y < \frac { \pi } { 4 }$$ where \(p\) and \(q\) are constants to be determined. \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-09_2258_47_315_37}
Edexcel C3 2016 June Q6
9 marks Standard +0.3
6. $$f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 3 x ^ { 2 } + 7 x - 6 } { x ^ { 2 } + x - 6 } , \quad x > 2 , x \in \mathbb { R }$$
  1. Given that $$\frac { x ^ { 4 } + x ^ { 3 } - 3 x ^ { 2 } + 7 x - 6 } { x ^ { 2 } + x - 6 } \equiv x ^ { 2 } + A + \frac { B } { x - 2 }$$ find the values of the constants \(A\) and \(B\).
  2. Hence or otherwise, using calculus, find an equation of the normal to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 3\)