Questions — Edexcel (10514 questions)

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Edexcel P2 2023 October Q3
7 marks Standard +0.3
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\) the equation $$2 \tan \theta + 3 \sin \theta = 0$$ giving your answers, as appropriate, to one decimal place.
  2. Hence, or otherwise, find the smallest positive solution of $$2 \tan \left( 2 x + 40 ^ { \circ } \right) + 3 \sin \left( 2 x + 40 ^ { \circ } \right) = 0$$ giving your answer to one decimal place.
Edexcel P2 2023 October Q4
9 marks Moderate -0.3
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } + a x ^ { 2 } - 29 x + b$$ where \(a\) and \(b\) are constants.
Given that \(( 2 x + 1 )\) is a factor of \(\mathrm { f } ( x )\),
  1. show that $$a + 4 b = - 56$$ Given also that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is - 25
  2. find a second simplified equation linking \(a\) and \(b\).
  3. Hence, using algebra and showing your working,
    1. find the value of \(a\) and the value of \(b\),
    2. fully factorise \(\mathrm { f } ( x )\).
Edexcel P2 2023 October Q5
6 marks Moderate -0.3
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Solve $$3 ^ { a } = 70$$ giving the answer to 3 decimal places.
  2. Find the exact value of \(b\) such that $$4 + 3 \log _ { 3 } b = \log _ { 3 } 5 b$$
Edexcel P2 2023 October Q6
6 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{66abdef1-072e-41eb-a933-dd51a96330ff-14_488_1511_246_278} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A river is being studied.
At one particular place, the river is 15 m wide.
The depth, \(y\) metres, of the river is measured at a point \(x\) metres from one side of the river. Figure 1 shows a plot of the cross-section of the river and the coordinate values \(( x , y )\)
  1. Use the trapezium rule with all the \(y\) values given in Figure 1 to estimate the cross-sectional area of the river. The water in the river is modelled as flowing at a constant speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) across the whole of the cross-section.
  2. Use the model and the answer to part (a) to estimate the volume of water flowing through this section of the river each minute, giving your answer in \(\mathrm { m } ^ { 3 }\) to 2 significant figures. Assuming the model,
  3. state, giving a reason for your answer, whether your answer for part (b) is an overestimate or an underestimate of the true volume of water flowing through this section of the river each minute.
Edexcel P2 2023 October Q7
8 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{66abdef1-072e-41eb-a933-dd51a96330ff-16_949_940_246_566} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of
  • the circle \(C\) with centre \(X ( 4 , - 3 )\)
  • the line \(l\) with equation \(y = \frac { 5 } { 2 } x - \frac { 55 } { 2 }\)
Given that \(l\) is the tangent to \(C\) at the point \(N\),
  1. show that an equation for the straight line passing through \(X\) and \(N\) is $$2 x + 5 y + 7 = 0$$
  2. Hence find
    1. the coordinates of \(N\),
    2. an equation for \(C\).
Edexcel P2 2023 October Q8
7 marks Moderate -0.8
  1. In a large theatre there are \(n\) rows of seats, where \(n\) is a constant.
The number of seats in the first row is \(a\), where \(a\) is a constant.
In each subsequent row there are 4 more seats than in the previous row so that
  • in the 2 nd row there are \(( a + 4 )\) seats
  • in the 3rd row there are ( \(a + 8\) ) seats
  • the number of seats in each row form an arithmetic sequence
Given that the total number of seats in the first 10 rows is 360
  1. find the value of \(a\). Given also that the total number of seats in the \(n\) rows is 2146
  2. show that $$n ^ { 2 } + 8 n - 1073 = 0$$
  3. Hence
    1. state the number of rows of seats in the theatre,
    2. find the maximum number of seats in any one row.
Edexcel P2 2023 October Q9
12 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{66abdef1-072e-41eb-a933-dd51a96330ff-24_803_1050_251_511} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 2 } { 3 } x ^ { 2 } - 9 \sqrt { x } + 13 \quad x \geqslant 0$$
  1. Find, using calculus, the range of values of \(x\) for which \(y\) is increasing. The point \(P\) lies on \(C\) and has coordinates (9, 40).
    The line \(l\) is the tangent to \(C\) at the point \(P\).
    The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the line \(l\), the \(x\)-axis and the \(y\)-axis.
  2. Find, using calculus, the exact area of \(R\).
Edexcel P2 2023 October Q10
12 marks Standard +0.3
  1. (a) Find, in ascending powers of \(x\), the 2nd, 3rd and 5th terms of the binomial expansion of $$( 3 + 2 x ) ^ { 6 }$$ For a particular value of \(x\), these three terms form consecutive terms in a geometric series.
    (b) Find this value of \(x\).
  2. In a different geometric series,
    • the first term is \(\sin ^ { 2 } \theta\)
    • the common ratio is \(2 \cos \theta\)
    • the sum to infinity is \(\frac { 8 } { 5 }\) (a) Show that
    $$5 \cos ^ { 2 } \theta - 16 \cos \theta + 3 = 0$$ (b) Hence find the exact value of the 2nd term in the series.
Edexcel P2 2018 Specimen Q1
7 marks Moderate -0.8
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 \(b\)
    VIIIV SIHI NI JIIIM ION OCVIIV SIHI NI JINAM ION OCVEYV SIHI NI JULIM ION OO
Edexcel P2 2018 Specimen Q2
8 marks Moderate -0.3
2. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\). The sum to infinity of the series is \(S _ { \infty }\)
  1. Find the value of \(S _ { \infty }\) The sum to \(N\) terms of the series is \(S _ { N }\)
  2. Find, to 1 decimal place, the value of \(S _ { 12 }\)
  3. Find the smallest value of \(N\), for which \(S _ { \infty } - S _ { N } < 0.5\) 2. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\). The sum to infinity
    of the series is \(S _ { \infty }\)
Edexcel P2 2018 Specimen Q3
7 marks Moderate -0.8
3. $$y = \sqrt { \left( 3 ^ { x } + x \right) }$$
  1. Complete the table below, giving the values of \(y\) to 3 decimal places.
    \(x\)00.250.50.751
    \(y\)11.2512
  2. Use the trapezium rule with all the values of \(y\) from your table to find an approximation for the value of $$\int _ { 0 } ^ { 1 } \sqrt { \left( 3 ^ { x } + x \right) } \mathrm { d } x$$ You must show clearly how you obtained your answer.
  3. Explain how the trapezium rule could be used to obtain a more accurate estimate for the value of $$\int _ { 0 } ^ { 1 } \sqrt { \left( 3 ^ { x } + x \right) } d x$$
    \includegraphics[max width=\textwidth, alt={}]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-10_2673_1948_107_118}
Edexcel P2 2018 Specimen Q4
4 marks Moderate -0.8
Given \(n \in \mathbb { N }\), prove, by exhaustion, that \(n ^ { 2 } + 2\) is not divisible by 4 . \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-12_2658_1943_111_118}
Edexcel P2 2018 Specimen Q6
7 marks Moderate -0.8
6.
  1. Find the exact value of \(x\) for which $$\log _ { 2 } ( 2 x ) = \log _ { 2 } ( 5 x + 4 ) - 3$$
  2. Given that $$\log _ { a } y + 3 \log _ { a } 2 = 5$$ express \(y\) in terms of \(a\). Give your answer in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-18_2674_1948_107_118}
Edexcel P2 2018 Specimen Q7
10 marks Moderate -0.5
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-19_739_871_260_532} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The circle with equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 16 y + 139 = 0$$ had centre \(C\) and radius \(r\).
  1. Find the coordinates of \(C\).
  2. Show that \(r = 5\) The line with equation \(x = 13\) crosses the circle at the points \(P\) and \(Q\) as shown in Figure 1 .
  3. Find the \(y\) coordinate of \(P\) and the \(y\) coordinate of \(Q\). A tangent to the circle from \(O\) touches the circle at point \(X\).
  4. Find, in surd form, the length \(O X\). \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-22_2673_1948_107_118}
Edexcel P2 2018 Specimen Q8
12 marks Moderate -0.3
8. Figure 2 Figure 2 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations $$\begin{array} { l l } C _ { 1 } : y = 10 x - x ^ { 2 } - 8 & x > 0 \\ C _ { 2 } : y = x ^ { 3 } & x > 0 \end{array}$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(A\) and \(B\).
  1. Verify that the point \(A\) has coordinates (1, 1)
  2. Use algebra to find the coordinates of the point \(B\) The finite region \(R\) is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
  3. Use calculus to find the exact area of \(R\) \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-23_936_759_118_582} \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-26_2674_1948_107_118}
Edexcel P2 2018 Specimen Q9
9 marks Moderate -0.3
9.
  1. Solve, for \(0 \leqslant \theta < \pi\), the equation $$\sin 3 \theta - \sqrt { 3 } \cos 3 \theta = 0$$ giving your answers in terms of \(\pi\)
  2. Given that $$4 \sin ^ { 2 } x + \cos x = 4 - k , \quad 0 \leqslant k \leqslant 3$$
    1. find \(\cos x\) in terms of \(k\)
    2. When \(k = 3\), find the values of \(x\) in the range \(0 \leqslant x < 360 ^ { \circ }\)
      \includegraphics[max width=\textwidth, alt={}]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-30_2671_1942_107_121}
Edexcel C2 Q3
Moderate -0.8
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c1a3d21d-38fe-4619-9e99-5c4788cdb891-019_675_792_287_568}
\end{figure} In Figure \(1 , A ( 4,0 )\) and \(B ( 3,5 )\) are the end points of a diameter of the circle \(C\). Find
  1. the exact length of \(A B\),
  2. the coordinates of the midpoint \(P\) of \(A B\),
  3. an equation for the circle \(C\).
Edexcel C2 2005 January Q1
4 marks Easy -1.2
Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 3 + 2 x ) ^ { 5 }\), giving each term in its simplest form.
(4)
Edexcel C2 2005 January Q3
7 marks Moderate -0.8
3. Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which
  1. \(3 ^ { x } = 5\),
  2. \(\log _ { 2 } ( 2 x + 1 ) - \log _ { 2 } x = 2\).
Edexcel C2 2005 January Q4
7 marks Moderate -0.3
4.
  1. Show that the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ can be written as $$5 \sin ^ { 2 } x + 3 \sin x - 2 = 0 .$$
  2. Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ giving your answers to 1 decimal place where appropriate.
Edexcel C2 2005 January Q5
8 marks Moderate -0.8
  1. \(\quad \mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ), the remainder is 1 .
When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is 28 .
  1. Find the value of \(a\) and the value of \(b\).
  2. Show that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\).
Edexcel C2 2005 January Q6
8 marks Moderate -0.3
  1. The second and fourth terms of a geometric series are 7.2 and 5.832 respectively.
The common ratio of the series is positive.
For this series, find
  1. the common ratio,
  2. the first term,
  3. the sum of the first 50 terms, giving your answer to 3 decimal places,
  4. the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places.
Edexcel C2 2005 January Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{13bca882-27da-40f2-99d8-4fdeb6629c4e-12_707_1072_301_434}
\end{figure} Figure 1 shows the triangle \(A B C\), with \(A B = 8 \mathrm {~cm} , A C = 11 \mathrm {~cm}\) and \(\angle B A C = 0.7\) radians. The \(\operatorname { arc } B D\), where \(D\) lies on \(A C\), is an arc of a circle with centre \(A\) and radius 8 cm . The region \(R\), shown shaded in Figure 1, is bounded by the straight lines \(B C\) and \(C D\) and the \(\operatorname { arc } B D\). Find
  1. the length of the \(\operatorname { arc } B D\),
  2. the perimeter of \(R\), giving your answer to 3 significant figures,
  3. the area of \(R\), giving your answer to 3 significant figures.
Edexcel C2 2005 January Q8
12 marks Moderate -0.3
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{13bca882-27da-40f2-99d8-4fdeb6629c4e-14_1102_1317_308_340}
\end{figure} The line with equation \(y = 3 x + 20\) cuts the curve with equation \(y = x ^ { 2 } + 6 x + 10\) at the points \(A\) and \(B\), as shown in Figure 2.
  1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). The shaded region \(S\) is bounded by the line and the curve, as shown in Figure 2.
  2. Use calculus to find the exact area of \(S\).
Edexcel C2 2005 January Q9
12 marks Standard +0.3
9. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{13bca882-27da-40f2-99d8-4fdeb6629c4e-16_821_958_301_516}
\end{figure} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the rectangular part is \(2 x\) metres and the width is \(y\) metres. The diameter of the semicircular part is \(2 x\) metres. The perimeter of the stage is 80 m .
  1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the stage is given by $$A = 80 x - \left( 2 + \frac { \pi } { 2 } \right) x ^ { 2 } .$$
  2. Use calculus to find the value of \(x\) at which \(A\) has a stationary value.
  3. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\).
  4. Calculate, to the nearest \(\mathrm { m } ^ { 2 }\), the maximum area of the stage.