Questions — Edexcel (9671 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 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 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
Sort by: Default | Easiest first | Hardest first
Edexcel C2 Q9
11 marks Standard +0.3
  1. A pencil holder is in the shape of an open circular cylinder of radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The surface area of the cylinder (including the base) is \(250 \mathrm {~cm} ^ { 2 }\).
    1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by \(V = 125 r - \frac { \pi r ^ { 3 } } { 2 }\).
    2. Use calculus to find the value of \(r\) for which \(V\) has a stationary value.
    3. Prove that the value of \(r\) you found in part (b) gives a maximum value for \(V\).
    4. Calculate, to the nearest \(\mathrm { cm } ^ { 3 }\), the maximum volume of the pencil holder.
    9. continuedLeave blank
Edexcel C2 Q10
12 marks Standard +0.3
10. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{9e4e1626-238b-4afd-b81c-68c5ab1624c2-16_525_928_312_621}
\end{figure} Figure 2 shows part of the curve \(C\) with equation $$y = 9 - 2 x - \frac { 2 } { \sqrt { x } } , \quad x > 0$$ The point \(A ( 1,5 )\) lies on \(C\) and the curve crosses the \(x\)-axis at \(B ( b , 0 )\), where \(b\) is a constant and \(b > 0\).
  1. Verify that \(b = 4\). The tangent to \(C\) at the point \(A\) cuts the \(x\)-axis at the point \(D\), as shown in Fig. 2 .
  2. Show that an equation of the tangent to \(C\) at \(A\) is \(y + x = 6\).
  3. Find the coordinates of the point \(D\). The shaded region \(R\), shown in Fig. 2, is bounded by \(C\), the line \(A D\) and the \(x\)-axis.
  4. Use integration to find the area of \(R\).
    1. continued
    2. continued
Edexcel C2 Specimen Q1
4 marks Moderate -0.8
Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 + 3 x ) ^ { 6 }\).
(4)
Edexcel C2 Specimen Q4
7 marks Standard +0.3
4. Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation \(3 \sin ^ { 2 } x = 1 + \cos x\), giving your answers to the nearest degree.
Edexcel C2 Specimen Q5
7 marks Moderate -0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{afaf76d8-2a1f-4239-8275-70fad4f418c1-1_351_663_1529_693}
\end{figure} The shaded area in Fig. 1 shows a badge \(A B C\), where \(A B\) and \(A C\) are straight lines, with \(A B = A C = 8 \mathrm {~mm}\). The curve \(B C\) is an arc of a circle, centre \(O\), where \(O B = O C =\) 8 mm and \(O\) is in the same plane as \(A B C\). The angle \(B A C\) is 0.9 radians.
  1. Find the perimeter of the badge.
  2. Find the area of the badge.
Edexcel C2 Specimen Q6
10 marks Moderate -0.3
6. At the beginning of the year 2000 a company bought a new machine for \(\pounds 15000\). Each year the value of the machine decreases by \(20 \%\) of its value at the start of the year.
  1. Show that at the start of the year 2002, the value of the machine was \(\pounds 9600\). When the value of the machine falls below \(\pounds 500\), the company will replace it.
  2. Find the year in which the machine will be replaced. To plan for a replacement machine, the company pays \(\pounds 1000\) at the start of each year into a savings account. The account pays interest at a fixed rate of \(5 \%\) per annum. The first payment was made when the machine was first bought and the last payment will be made at the start of the year in which the machine is replaced.
  3. Using your answer to part (b), find how much the savings account will be worth immediately after the payment at the start of the year in which the machine is replaced.
Edexcel C2 Specimen Q7
12 marks Standard +0.3
7. (a) Use the factor theorem to show that \(( x + 1 )\) is a factor of \(x ^ { 3 } - x ^ { 2 } - 10 x - 8\).
(b) Find all the solutions of the equation \(x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0\).
(c) Prove that the value of \(x\) that satisfies $$2 \log _ { 2 } x + \log _ { 2 } ( x - 1 ) = 1 + \log _ { 2 } ( 5 x + 4 )$$ is a solution of the equation $$x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0$$ (d) State, with a reason, the value of \(x\) that satisfies equation (I).
Edexcel C2 Specimen Q8
12 marks Moderate -0.3
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{afaf76d8-2a1f-4239-8275-70fad4f418c1-2_616_712_1658_713}
\end{figure} The line with equation \(y = x + 5\) cuts the curve with equation \(y = x ^ { 2 } - 3 x + 8\) at the points \(A\) and \(B\), as shown in Fig. 2.
  1. Find the coordinates of the points \(A\) and \(B\).
  2. Find the area of the shaded region between the curve and the line, as shown in Fig. 2.
Edexcel C2 Specimen Q9
13 marks Standard +0.3
9. Figure 3 $$( x + 1 ) ^ { 2 }$$ Figure 3 shows a triangle \(P Q R\). The size of angle \(Q P R\) is \(30 ^ { \circ }\), the length of \(P Q\) is \(( x + 1 )\) and the length of \(P R\) is \(( 4 - x ) ^ { 2 }\), where \(X \in \Re\).
  1. Show that the area \(A\) of the triangle is given by \(A = \frac { 1 } { 4 } \left( x ^ { 3 } - 7 x ^ { 2 } + 8 x + 16 \right)\)
  2. Use calculus to prove that the area of \(\triangle P Q R\) is a maximum when \(x = \frac { 2 } { 3 }\). Explain clearly how you know that this value of \(x\) gives the maximum area.
  3. Find the maximum area of \(\triangle P Q R\).
  4. Find the length of \(Q R\) when the area of \(\triangle P Q R\) is a maximum. END
Edexcel P3 2020 January Q1
6 marks Standard +0.3
  1. A population of a rare species of toad is being studied.
The number of toads, \(N\), in the population, \(t\) years after the start of the study, is modelled by the equation $$N = \frac { 900 \mathrm { e } ^ { 0.12 t } } { 2 \mathrm { e } ^ { 0.12 t } + 1 } \quad t \geqslant 0 , t \in \mathbb { R }$$ According to this model,
  1. calculate the number of toads in the population at the start of the study,
  2. find the value of \(t\) when there are 420 toads in the population, giving your answer to 2 decimal places.
  3. Explain why, according to this model, the number of toads in the population can never reach 500
Edexcel P3 2020 January Q2
8 marks Moderate -0.3
2. The function \(f\) and the function \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 12 } { x + 1 } & x > 0 , x \in \mathbb { R } \\ \mathrm {~g} ( x ) = \frac { 5 } { 2 } \ln x & x > 0 , x \in \mathbb { R } \end{array}$$
  1. Find, in simplest form, the value of \(\mathrm { fg } \left( \mathrm { e } ^ { 2 } \right)\)
  2. Find f-1
  3. Hence, or otherwise, find all real solutions of the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )$$
Edexcel P3 2020 January Q3
5 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c700103-ecab-4a08-b411-3f445ed88885-08_599_883_299_536} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a linear relationship between \(\log _ { 10 } y\) and \(\log _ { 10 } x\)
The line passes through the points \(( 0,4 )\) and \(( 6,0 )\) as shown.
  1. Find an equation linking \(\log _ { 10 } y\) with \(\log _ { 10 } x\)
  2. Hence, or otherwise, express \(y\) in the form \(p x ^ { q }\), where \(p\) and \(q\) are constants to be found.
Edexcel P3 2020 January Q4
11 marks Standard +0.3
4. (i) $$f ( x ) = \frac { ( 2 x + 5 ) ^ { 2 } } { x - 3 } \quad x \neq 3$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\) in the form \(\frac { P ( x ) } { Q ( x ) }\) where \(P ( x )\) and \(Q ( x )\) are fully factorised quadratic expressions.
  2. Hence find the range of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
    (ii) $$g ( x ) = x \sqrt { \sin 4 x } \quad 0 \leqslant x < \frac { \pi } { 4 }$$ The curve with equation \(y = g ( x )\) has a maximum at the point \(M\). Show that the \(x\) coordinate of \(M\) satisfies the equation $$\tan 4 x + k x = 0$$ where \(k\) is a constant to be found.
Edexcel P3 2020 January Q5
8 marks Standard +0.8
5. (a) Use the substitution \(t = \tan x\) to show that the equation $$12 \tan 2 x + 5 \cot x \sec ^ { 2 } x = 0$$ can be written in the form $$5 t ^ { 4 } - 24 t ^ { 2 } - 5 = 0$$ (b) Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$12 \tan 2 x + 5 \cot x \sec ^ { 2 } x = 0$$ Show each stage of your working and give your answers to one decimal place.
Edexcel P3 2020 January Q6
9 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c700103-ecab-4a08-b411-3f445ed88885-18_736_1102_258_427} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 2 x - 5 | + 3 \quad x \geqslant 0$$ The vertex of the graph is at point \(P\) as shown.
  1. State the coordinates of \(P\).
  2. Solve the equation \(\mathrm { f } ( x ) = 3 x - 2\) Given that the equation $$f ( x ) = k x + 2$$ where \(k\) is a constant, has exactly two roots,
  3. find the range of values of \(k\).
Edexcel P3 2020 January Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c700103-ecab-4a08-b411-3f445ed88885-22_707_1047_264_463} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 2 \cos 3 x - 3 x + 4 \quad x > 0$$ where \(x\) is measured in radians. The curve crosses the \(x\)-axis at the point \(P\), as shown in Figure 3.
Given that the \(x\) coordinate of \(P\) is \(\alpha\),
  1. show that \(\alpha\) lies between 0.8 and 0.9 The iteration formula $$x _ { n + 1 } = \frac { 1 } { 3 } \arccos \left( 1.5 x _ { n } - 2 \right)$$ can be used to find an approximate value for \(\alpha\).
  2. Using this iteration formula with \(x _ { 1 } = 0.8\) find, to 4 decimal places, the value of
    1. \(X _ { 2 }\)
    2. \(X _ { 5 }\) The point \(Q\) and the point \(R\) are local minimum points on the curve, as shown in Figure 3.
      Given that the \(x\) coordinates of \(Q\) and \(R\) are \(\beta\) and \(\lambda\) respectively, and that they are the two smallest values of \(x\) at which local minima occur,
  3. find, using calculus, the exact value of \(\beta\) and the exact value of \(\lambda\).
Edexcel P3 2020 January Q8
10 marks Moderate -0.3
8. (i) Find, using algebraic integration, the exact value of $$\int _ { 3 } ^ { 42 } \frac { 2 } { 3 x - 1 } \mathrm {~d} x$$ giving your answer in simplest form.
(ii) $$\mathrm { h } ( x ) = \frac { 2 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 1 } { ( x - 1 ) ^ { 2 } } \quad x > 1$$ Given \(\mathrm { h } ( x ) = A x + B + \frac { C } { ( x - 1 ) ^ { 2 } }\) where \(A , B\) and \(C\) are constants to be found, find $$\int \mathrm { h } ( x ) \mathrm { d } x$$ \includegraphics[max width=\textwidth, alt={}, center]{1c700103-ecab-4a08-b411-3f445ed88885-26_2258_47_312_1985}
Edexcel P3 2020 January Q9
7 marks Standard +0.3
9. $$\mathrm { f } ( \theta ) = 5 \cos \theta - 4 \sin \theta \quad \theta \in \mathbb { R }$$
  1. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 decimal places. The curve with equation \(y = \cos \theta\) is transformed onto the curve with equation \(y = \mathrm { f } ( \theta )\) by a sequence of two transformations. Given that the first transformation is a stretch and the second a translation,
    1. describe fully the transformation that is a stretch,
    2. describe fully the transformation that is a translation. Given $$g ( \theta ) = \frac { 90 } { 4 + ( f ( \theta ) ) ^ { 2 } } \quad \theta \in \mathbb { R }$$
  3. find the range of g.
    Leave blank
    Q9

    \hline END &
    \hline \end{tabular}
Edexcel P3 2021 January Q1
3 marks Moderate -0.8
  1. Find
$$\int \frac { x ^ { 2 } - 5 } { 2 x ^ { 3 } } \mathrm {~d} x \quad x > 0$$ giving your answer in simplest form.
Edexcel P3 2021 January Q2
6 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{624e9e2f-b6b8-47ce-accc-31dcd5f0554e-04_903_1148_123_399} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where \(x \in \mathbb { R }\) and \(\mathrm { f } ( x )\) is a polynomial. The curve passes through the origin and touches the \(x\)-axis at the point \(( 3,0 )\) There is a maximum turning point at \(( 1,2 )\) and a minimum turning point at \(( 3,0 )\) On separate diagrams, sketch the curve with equation
  1. \(y = 3 f ( 2 x )\)
  2. \(y = \mathrm { f } ( - x ) - 1\) On each sketch, show clearly the coordinates of
    • the point where the curve crosses the \(y\)-axis
    • any maximum or minimum turning points
Edexcel P3 2021 January Q3
8 marks Standard +0.3
3. $$f ( x ) = 3 - \frac { x - 2 } { x + 1 } + \frac { 5 x + 26 } { 2 x ^ { 2 } - 3 x - 5 } \quad x > 4$$
  1. Show that $$\mathrm { f } ( x ) = \frac { a x + b } { c x + d } \quad x > 4$$ where \(a , b , c\) and \(d\) are integers to be found.
  2. Hence find \(\mathrm { f } ^ { - 1 } ( x )\)
  3. Find the domain of \(\mathrm { f } ^ { - 1 }\)
Edexcel P3 2021 January Q4
9 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{624e9e2f-b6b8-47ce-accc-31dcd5f0554e-10_646_762_264_593} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = | 3 x + a | + a$$ and where \(a\) is a positive constant. The graph has a vertex at the point \(P\), as shown in Figure 2 .
  1. Find, in terms of \(a\), the coordinates of \(P\).
  2. Sketch the graph with equation \(y = g ( x )\), where $$g ( x ) = | x + 5 a |$$ On your sketch, show the coordinates, in terms of \(a\), of each point where the graph cuts or meets the coordinate axes. The graph with equation \(y = \mathrm { g } ( x )\) intersects the graph with equation \(y = \mathrm { f } ( x )\) at two points.
  3. Find, in terms of \(a\), the coordinates of the two points. \includegraphics[max width=\textwidth, alt={}, center]{624e9e2f-b6b8-47ce-accc-31dcd5f0554e-11_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel P3 2021 January Q5
11 marks Standard +0.3
5. The temperature, \(\theta ^ { \circ } \mathrm { C }\), inside an oven, \(t\) minutes after the oven is switched on, is given by $$\theta = A - 180 \mathrm { e } ^ { - k t }$$ where \(A\) and \(k\) are positive constants. Given that the temperature inside the oven is initially \(18 ^ { \circ } \mathrm { C }\),
  1. find the value of \(A\). The temperature inside the oven, 5 minutes after the oven is switched on, is \(90 ^ { \circ } \mathrm { C }\).
  2. Show that \(k = p \ln q\) where \(p\) and \(q\) are rational numbers to be found. Hence find
  3. the temperature inside the oven 9 minutes after the oven is switched on, giving your answer to 3 significant figures,
  4. the rate of increase of the temperature inside the oven 9 minutes after the oven is switched on. Give your answer in \({ } ^ { \circ } \mathrm { C } \min ^ { - 1 }\) to 3 significant figures.
Edexcel P3 2021 January Q6
8 marks Standard +0.8
6. $$\mathrm { f } ( x ) = x \cos \left( \frac { x } { 3 } \right) \quad x > 0$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\)
  2. Show that the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\) can be written as $$x = k \arctan \left( \frac { k } { x } \right)$$ where \(k\) is an integer to be found.
  3. Starting with \(x _ { 1 } = 2.5\) use the iteration formula $$x _ { n + 1 } = k \arctan \left( \frac { k } { x _ { n } } \right)$$ with the value of \(k\) found in part (b), to calculate the values of \(x _ { 2 }\) and \(x _ { 6 }\) giving your answers to 3 decimal places.
  4. Using a suitable interval and a suitable function that should be stated, show that a root of \(\mathrm { f } ^ { \prime } ( x ) = 0\) is 2.581 correct to 3 decimal places.
    In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Edexcel P3 2021 January Q7
9 marks Standard +0.8
7. (a) Prove that $$\frac { \sin 2 x } { \cos x } + \frac { \cos 2 x } { \sin x } \equiv \operatorname { cosec } x \quad x \neq \frac { n \pi } { 2 } n \in \mathbb { Z }$$ (b) Hence solve, for \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\) $$7 + \frac { \sin 4 \theta } { \cos 2 \theta } + \frac { \cos 4 \theta } { \sin 2 \theta } = 3 \cot ^ { 2 } 2 \theta$$ giving your answers in radians to 3 significant figures where appropriate.