Questions — Edexcel (10514 questions)

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Edexcel P2 2020 October Q7
7 marks Standard +0.3
7.
  1. Show that $$\tan \theta + \frac { 1 } { \tan \theta } \equiv \frac { 1 } { \sin \theta \cos \theta } \quad \theta \neq \frac { \mathrm { n } \pi } { 2 } \quad n \in \mathbb { Z }$$
  2. Solve, for \(0 \leqslant x < 90 ^ { \circ }\), the equation $$3 \cos ^ { 2 } \left( 2 x + 10 ^ { \circ } \right) = 1$$ giving your answers in degrees to one decimal place.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel P2 2020 October Q8
8 marks Moderate -0.8
8. A geometric series has first term \(a\) and common ratio \(r\).
  1. Prove that the sum of the first \(n\) terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ The second term of a geometric series is - 320 and the fifth term is \(\frac { 512 } { 25 }\)
  2. Find the value of the common ratio.
  3. Hence find the sum of the first 13 terms of the series, giving your answer to 2 decimal places.
Edexcel P2 2020 October Q9
10 marks Moderate -0.3
9.
  1. Find the exact value of \(x\) for which $$\log _ { 3 } ( x + 5 ) - 4 = \log _ { 3 } ( 2 x - 1 )$$
  2. Given that $$3 ^ { y + 3 } \times 2 ^ { 1 - 2 y } = 108$$
    1. show that $$0.75 ^ { y } = 2$$
    2. Hence find the value of \(y\), giving your answer to 3 decimal places.
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Edexcel P2 2021 October Q1
6 marks Moderate -0.8
  1. The first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + k x ) ^ { 16 }\) are
$$1 , - 4 x \text { and } p x ^ { 2 }$$ where \(k\) and \(p\) are constants.
  1. Find, in simplest form,
    1. the value of \(k\)
    2. the value of \(p\) $$g ( x ) = \left( 2 + \frac { 16 } { x } \right) ( 1 + k x ) ^ { 16 }$$ Using the value of \(k\) found in part (a),
  2. find the term in \(x ^ { 2 }\) in the expansion of \(\mathrm { g } ( x )\). $$\begin{aligned} u _ { 1 } & = 6 \\ u _ { n + 1 } & = k u _ { n } + 3 \end{aligned}$$ where \(k\) is a positive constant.
  1. Find, in terms of \(k\), an expression for \(u _ { 3 }\) Given that \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 117\)
  2. find the value of \(k\).
Edexcel P2 2021 October Q2
5 marks Moderate -0.5
2. A sequence is defined by
Edexcel P2 2021 October Q3
8 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-06_725_668_118_639} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \log _ { 10 } x\) The region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 14\) Using the trapezium rule with four strips of equal width,
  1. show that the area of \(R\) is approximately 10.10
  2. Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(R\).
  3. Using the answer to part (a) and making your method clear, estimate the value of
    1. \(\quad \int _ { 2 } ^ { 14 } \log _ { 10 } \sqrt { x } \mathrm {~d} x\)
    2. \(\int _ { 2 } ^ { 14 } \log _ { 10 } 100 x ^ { 3 } \mathrm {~d} x\)
Edexcel P2 2021 October Q4
8 marks Moderate -0.3
4. $$f ( x ) = \left( x ^ { 2 } - 2 \right) ( 2 x - 3 ) - 21$$
  1. State the value of the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 3\) )
  2. Use the factor theorem to show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\)
  3. Hence,
    1. factorise \(\mathrm { f } ( x )\)
    2. show that the equation \(\mathrm { f } ( x ) = 0\) has only one real root.
Edexcel P2 2021 October Q5
6 marks Standard +0.3
5. A company that owned a silver mine
  • extracted 480 tonnes of silver from the mine in year 1
  • extracted 465 tonnes of silver from the mine in year 2
  • extracted 450 tonnes of silver from the mine in year 3
    and so on, forming an arithmetic sequence.
    1. Find the mass of silver extracted in year 14
After a total of 7770 tonnes of silver was extracted, the company stopped mining. Given that this occurred at the end of year \(N\),
  • show that $$N ^ { 2 } - 65 N + 1036 = 0$$
  • Hence, state the value of \(N\).
    •
    Edexcel P2 2021 October Q6
    8 marks Moderate -0.3
    6. (i) The circle \(C _ { 1 }\) has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 12 y = k \quad \text { where } k \text { is a constant }$$
    1. Find the coordinates of the centre of \(C _ { 1 }\)
    2. State the possible range in values for \(k\).
      (ii) The point \(P ( p , 0 )\), the point \(Q ( - 2,10 )\) and the point \(R ( 8 , - 14 )\) lie on a different circle, \(C _ { 2 }\) Given that
    Edexcel P2 2021 October Q7
    10 marks Moderate -0.3
    7.
    1. A geometric sequence has first term 4 and common ratio 6 Given that the \(n ^ { \text {th } }\) term is greater than \(10 ^ { 100 }\), find the minimum possible value of \(n\).
    2. A different geometric sequence has first term \(a\) and common ratio \(r\). Given that
      • the second term of the sequence is - 6
      • the sum to infinity of the series is 25
        1. show that
      $$25 r ^ { 2 } - 25 r - 6 = 0$$
    3. Write down the solutions of $$25 r ^ { 2 } - 25 r - 6 = 0$$ Hence,
    4. state the value of \(r\), giving a reason for your answer,
    5. find the sum of the first 4 terms of the series. \includegraphics[max width=\textwidth, alt={}, center]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-23_70_37_2617_1914}
    Edexcel P2 2021 October Q8
    10 marks Standard +0.2
    8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-24_739_736_411_605} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 4 } { 3 } x ^ { 3 } - 11 x ^ { 2 } + k x \quad \text { where } k \text { is a constant }$$ The point \(M\) is the maximum turning point of \(C\) and is shown in Figure 2.
    Given that the \(x\) coordinate of \(M\) is 2
    1. show that \(k = 28\)
    2. Determine the range of values of \(x\) for which \(y\) is increasing. The line \(l\) passes through \(M\) and is parallel to the \(x\)-axis.
      The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(y\)-axis.
    3. Find, by algebraic integration, the exact area of \(R\).
    Edexcel P2 2021 October Q9
    4 marks Moderate -0.5
    9.
    1. Prove that for all positive values of \(x\) and \(y\), $$\frac { x + y } { 2 } \geqslant \sqrt { x y }$$
    2. Prove by counter-example that this inequality does not hold when \(x\) and \(y\) are both negative.
      (1)
      \includegraphics[max width=\textwidth, alt={}, center]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-29_61_54_2608_1852}
    Edexcel P2 2021 October Q10
    10 marks Standard +0.3
    10. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    1. Solve, for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\) $$\tan ^ { 2 } \left( 2 x + \frac { \pi } { 4 } \right) = 3$$
    2. Solve, for \(0 < \theta < 360 ^ { \circ }\) $$( 2 \sin \theta - \cos \theta ) ^ { 2 } = 1$$ giving your answers, as appropriate, to one decimal place.
    Edexcel P2 2022 October Q1
    3 marks Easy -1.2
    1. Given that \(a , b\) and \(c\) are integers greater than 0 such that
    • \(c = b + 2\)
    • \(a + b + c = 10\)
    Prove, by exhaustion, that the product of \(a , b\) and \(c\) is always even.
    You may use the table below to illustrate your answer. You may not need to use all rows of this table.
    \(a\)\(b\)\(c\)
    1
    2
    Edexcel P2 2022 October Q2
    7 marks Moderate -0.3
    1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where
    $$f ( x ) = ( 2 - k x ) ^ { 5 }$$ and \(k\) is a constant.
    Given that when \(\mathrm { f } ( x )\) is divided by \(( 4 x - 5 )\) the remainder is \(\frac { 243 } { 32 }\)
    1. show that \(k = \frac { 2 } { 5 }\)
    2. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 2 } { 5 } x \right) ^ { 5 }$$ giving each term in simplest form. Using the solution to part (b) and making your method clear,
    3. find the gradient of \(C\) at the point where \(x = 0\)
    Edexcel P2 2022 October Q3
    7 marks Standard +0.3
    1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
    $$a _ { n } = \cos ^ { 2 } \left( \frac { \mathrm { n } \pi } { 3 } \right)$$ Find the exact values of
      1. \(a _ { 1 }\)
      2. \(a _ { 2 }\)
      3. \(a _ { 3 }\)
    2. Hence find the exact value of 50 $$n + \cos ^ { 2 } \frac { n \pi } { 3 }$$ You must make your method clear.
    Edexcel P2 2022 October Q4
    8 marks Moderate -0.3
    1. The weight of a baby mammal is monitored over a 16 -month period.
    The weight of the mammal, \(w \mathrm {~kg}\), is given by $$w = \log _ { a } ( t + 5 ) - \log _ { a } 4 \quad 2 \leqslant t \leqslant 18$$ where \(t\) is the age of the mammal in months and \(a\) is a constant.
    Given that the weight of the mammal was 10 kg when \(t = 3\)
    1. show that \(a = 1.072\) correct to 3 decimal places. Using \(a = 1.072\)
    2. find an equation for \(t\) in terms of \(w\)
    3. find the value of \(t\) when \(w = 15\), giving your answer to 3 significant figures.
    Edexcel P2 2022 October Q5
    8 marks Standard +0.3
    1. In this question you must show detailed reasoning.
    Solutions relying entirely on calculator technology are not acceptable.
    1. Show that the equation $$( 3 \cos \theta - \tan \theta ) \cos \theta = 2$$ can be written as $$3 \sin ^ { 2 } \theta + \sin \theta - 1 = 0$$
    2. Hence solve for \(- \frac { \pi } { 2 } \leqslant x \leqslant \frac { \pi } { 2 }\) $$( 3 \cos 2 x - \tan 2 x ) \cos 2 x = 2$$
    Edexcel P2 2022 October Q6
    7 marks Moderate -0.8
    1. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\).
    A table of values of \(x\) and \(y\) for \(y = \mathrm { f } ( x )\) is shown below, with the \(y\) values rounded to 4 decimal places where appropriate.
    \(x\)00.511.52
    \(y\)32.68332.42.14661.92
    1. Use the trapezium rule with all the values of \(y\) in the table to find an approximation for $$\int _ { 0 } ^ { 2 } f ( x ) d x$$ giving your answer to 3 decimal places. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{6f926d53-c6de-4eb7-9d18-596f61ec26e1-16_629_592_1105_402} \captionsetup{labelformat=empty} \caption{Figure 1}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{6f926d53-c6de-4eb7-9d18-596f61ec26e1-16_540_456_1194_1192} \captionsetup{labelformat=empty} \caption{Figure 2}
      \end{figure} The region \(R\), shown shaded in Figure 1, is bounded by
      The region \(R\) forms part of the design for a logo shown in Figure 2.
      The design consists of the shaded region \(R\) inside a rectangle of width 2 and height 3 Using calculus and the answer to part (a),
    2. calculate an estimate for the percentage of the logo which is shaded.
    Edexcel P2 2022 October Q7
    9 marks Standard +0.3
    1. The curve \(C\) has equation
    $$y = \frac { 12 x ^ { 3 } ( x - 7 ) + 14 x ( 13 x - 15 ) } { 21 \sqrt { x } } \quad x > 0$$
    1. Write the equation of \(C\) in the form $$y = a x ^ { \frac { 7 } { 2 } } + b x ^ { \frac { 5 } { 2 } } + c x ^ { \frac { 3 } { 2 } } + d x ^ { \frac { 1 } { 2 } }$$ where \(a , b , c\) and \(d\) are fully simplified constants. The curve \(C\) has three turning points.
      Using calculus,
    2. show that the \(x\) coordinates of the three turning points satisfy the equation $$2 x ^ { 3 } - 10 x ^ { 2 } + 13 x - 5 = 0$$ Given that the \(x\) coordinate of one of the turning points is 1
    3. find, using algebra, the exact \(x\) coordinates of the other two turning points.
      (Solutions based entirely on calculator technology are not acceptable.)
    Edexcel P2 2022 October Q8
    7 marks Moderate -0.3
    1. A geometric sequence has first term \(a\) and common ratio \(r\)
    Given that \(S _ { \infty } = 3 a\)
    1. show that \(r = \frac { 2 } { 3 }\) Given also that $$u _ { 2 } - u _ { 4 } = 16$$ where \(u _ { k }\) is the \(k ^ { \text {th } }\) term of this sequence,
    2. find the value of \(S _ { 10 }\) giving your answer to one decimal place.
    Edexcel P2 2022 October Q9
    12 marks Standard +0.3
    1. In this question you must show detailed reasoning.
    Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6f926d53-c6de-4eb7-9d18-596f61ec26e1-26_723_455_413_804} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows
    • the curve \(C _ { 1 }\) with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 3 x + 14\)
    • the circle \(C _ { 2 }\) with centre \(T\)
    The point \(T\) is the minimum turning point of \(C _ { 1 }\) Using Figure 3 and calculus,
    1. find the coordinates of \(T\) The curve \(C _ { 1 }\) intersects the circle \(C _ { 2 }\) at the point \(A\) with \(x\) coordinate 2
    2. Find an equation of the circle \(C _ { 2 }\) The line \(l\) shown in Figure 3, is the tangent to circle \(C _ { 2 }\) at \(A\)
    3. Show that an equation of \(l\) is $$y = \frac { 1 } { 3 } x + \frac { 22 } { 3 }$$ The region \(R\), shown shaded in Figure 3, is bounded by \(C _ { 1 } , l\) and the \(y\)-axis.
    4. Find the exact area of \(R\).
    Edexcel P2 2022 October Q10
    7 marks Moderate -0.3
    Given \(a = \log _ { 2 } 3\)
    1. write, in simplest form, in terms of \(a\),
      1. \(\log _ { 2 } 9\)
      2. \(\log _ { 2 } \left( \frac { \sqrt { 3 } } { 16 } \right)\)
      3. Solve $$3 ^ { x } \times 2 ^ { x + 4 } = 6$$ giving your answer, in simplest form, in terms of \(a\).
    Edexcel P2 2023 October Q1
    3 marks Easy -1.2
    1. Given that \(a , b\) and \(c\) are integers greater than 0 such that
    • \(c = 3 a + 1\)
    • \(a + b + c = 15\) prove, by exhaustion, that the product \(a b c\) is always a multiple of 4
      You may use the table below to illustrate your answer.
    You may not need to use all rows of this table.
    \(a\)\(b\)\(c\)\(a b c\)
    Edexcel P2 2023 October Q2
    5 marks Moderate -0.3
    1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
    $$\begin{gathered} u _ { 1 } = 3 \\ u _ { n + 1 } = 2 - \frac { 4 } { u _ { n } } \end{gathered}$$
    1. Find the value of \(u _ { 2 }\), the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\)
    2. Find the value of $$\sum _ { r = 1 } ^ { 100 } u _ { r }$$