Questions P2 (856 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
CAIE P2 2019 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{9c26457d-4b65-4cd4-a9b9-128aba92dbf4-04_586_734_260_701} The variables \(x\) and \(y\) satisfy the equation \(y = k x ^ { a }\), where \(k\) and \(a\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points ( \(0.22,3.96\) ) and ( \(1.32,2.43\) ), as shown in the diagram. Find the values of \(k\) and \(a\) correct to 3 significant figures.
CAIE P2 2019 November Q4
4 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) defined by $$x _ { 1 } = 1 , \quad x _ { n + 1 } = \frac { x _ { n } } { \ln \left( 2 x _ { n } \right) }$$ converges to the value \(\alpha\).
  1. Use the iterative formula to find the value of \(\alpha\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
  2. State an equation satisfied by \(\alpha\) and hence determine the exact value of \(\alpha\).
CAIE P2 2019 November Q5
5 Find the exact coordinates of the stationary point of the curve with equation \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } ( 2 x + 5 )\).
CAIE P2 2019 November Q6
6
  1. Show that \(\int _ { 2 } ^ { 18 } \frac { 3 } { 2 x } \mathrm {~d} x = \ln 27\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 4 \sin ^ { 2 } \left( \frac { 3 } { 2 } x \right) \mathrm { d } x\). Show all necessary working.
CAIE P2 2019 November Q7
7 The parametric equations of a curve are $$x = 3 \sin 2 \theta , \quad y = 1 + 2 \tan 2 \theta$$ for \(0 \leqslant \theta < \frac { 1 } { 4 } \pi\).
  1. Find the exact gradient of the curve at the point for which \(\theta = \frac { 1 } { 6 } \pi\).
  2. Find the value of \(\theta\) at the point where the gradient of the curve is 2 , giving the value correct to 3 significant figures.
CAIE P2 2019 November Q8
8
  1. Express \(0.5 \cos \theta - 1.2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(0.5 \cos \theta - 1.2 \sin \theta = 0.8\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
  3. Determine the greatest and least possible values of \(( 3 - \cos \theta + 2.4 \sin \theta ) ^ { 2 }\) as \(\theta\) varies.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2019 November Q1
1 Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9) \section*{READ THESE INSTRUCTIONS FIRST} Write your centre number, candidate number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
Answer all the questions in the space provided. If additional space is required, you should use the lined page at the end of this booklet. The question number(s) must be clearly shown.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50. 1
  1. Solve the inequality \(| 2 x - 7 | < | 2 x - 9 |\).
  2. Hence find the largest integer \(n\) satisfying the inequality \(| 2 \ln n - 7 | < | 2 \ln n - 9 |\).
CAIE P2 Specimen Q1
1 Use logarithms to solve the equation $$5 ^ { x + 3 } = 7 ^ { x - 1 }$$ giving the answer correct to 3 significant figures.
CAIE P2 Specimen Q2
2 A curve has equation $$y = \frac { 3 x + 1 } { x - 5 }$$ Find the coordinates of the points on the curve at which the gradient is - 4 .
CAIE P2 Specimen Q3
3
  1. Express \(8 \sin \theta + 15 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$8 \sin \theta + 15 \cos \theta = 6$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 Specimen Q4
4
  1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 4 - \frac { 1 } { 2 } x$$ has exactly one real root, \(\alpha\).
  2. Verify by calculation that \(4.5 < \alpha < 5.0\).
  3. Use the iterative formula \(x _ { n + 1 } = 8 - 2 \ln x _ { n }\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 Specimen Q5
5
  1. Find \(\int \left( \tan ^ { 2 } x + \sin 2 x \right) \mathrm { d } x\).
  2. Find the exact value of \(\int _ { 0 } ^ { 1 } 3 \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\).
CAIE P2 Specimen Q6
6
  1. Find the quotient and remainder when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + 12 x + 6$$ is divided by ( \(x ^ { 2 } - x + 4\) ).
  2. It is given that, when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q$$ is divided by \(\left( x ^ { 2 } - x + 4 \right)\), the remainder is zero. Find the values of the constants \(p\) and \(q\).
  3. When \(p\) and \(q\) have these values, show that there is exactly one real value of \(x\) satisfying the equation $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q = 0$$ and state what that value is.
CAIE P2 Specimen Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{77672e56-a268-47b8-ab8b-cd84b4b3de4f-10_551_689_258_726} The parametric equations of a curve are $$x = 6 \sin ^ { 2 } t , \quad y = 2 \sin 2 t + 3 \cos 2 t$$ for \(0 \leqslant t < \pi\). The curve crosses the \(x\)-axis at points \(B\) and \(D\) and the stationary points are \(A\) and \(C\), as shown in the diagram.
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 3 } \cot 2 t - 1\).
  2. Find the values of \(t\) at \(A\) and \(C\), giving each answer correct to 3 decimal places.
  3. Find the value of the gradient of the curve at \(B\).
Edexcel P2 2022 June Q7
7
6. In a geometric sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\)
  • the common ratio is \(r\)
  • \(u _ { 2 } + u _ { 3 } = 6\)
  • \(u _ { 4 } = 8\)
    1. Show that \(r\) satisfies
$$3 r ^ { 2 } - 4 r - 4 = 0$$ Given that the geometric sequence has a sum to infinity,
  • find \(u _ { 1 }\)
  • find \(S _ { \infty }\) 7. $$f ( x ) = A x ^ { 3 } + 6 x ^ { 2 } - 4 x + B$$ where \(A\) and \(B\) are constants. Given that
    • ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\)
    • \(\int _ { 3 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x = 176\)
      find the value of \(A\) and the value of \(B\).
    •
    Edexcel P2 2020 January Q1
    1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \log _ { 2 } ( 2 x )\)
    The values of \(y\) are given to 2 decimal places as appropriate. Using the trapezium rule with all the values of \(y\) in the given table,
    1. obtain an estimate for \(\int _ { 2 } ^ { 14 } \log _ { 2 } ( 2 x ) \mathrm { d } x\), giving your answer to one decimal place. Using your answer to part (a) and making your method clear, estimate
      1. \(\quad \int _ { 2 } ^ { 14 } \frac { \log _ { 2 } \left( 4 x ^ { 2 } \right) } { 5 } \mathrm {~d} x\)
      2. \(\int _ { 2 } ^ { 14 } \log _ { 2 } \left( \frac { 2 } { x } \right) \mathrm { d } x\)
        \(x\)2581114
        \(y\)23.3244.464.81
    Edexcel P2 2020 January Q2
    2. One of the terms in the binomial expansion of \(( 3 + a x ) ^ { 6 }\), where \(a\) is a constant, is \(540 x ^ { 4 }\)
    1. Find the possible values of \(a\).
    2. Hence find the term independent of \(x\) in the expansion of $$\left( \frac { 1 } { 81 } + \frac { 1 } { x ^ { 6 } } \right) ( 3 + a x ) ^ { 6 }$$
    Edexcel P2 2020 January Q3
    3. $$f ( x ) = 6 x ^ { 3 } + 17 x ^ { 2 } + 4 x - 12$$
    1. Use the factor theorem to show that ( \(2 x + 3\) ) is a factor of \(\mathrm { f } ( x )\).
    2. Hence, using algebra, write \(\mathrm { f } ( x )\) as a product of three linear factors.
    3. Solve, for \(\frac { \pi } { 2 } < \theta < \pi\), the equation $$6 \tan ^ { 3 } \theta + 17 \tan ^ { 2 } \theta + 4 \tan \theta - 12 = 0$$ giving your answers to 3 significant figures.
    Edexcel P2 2020 January Q4
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{08aac50c-7317-4510-927a-7f5f2e00f485-08_858_654_118_671} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve with equation $$y = 2 x ^ { 2 } + 7 \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(y\)-axis and the line with equation \(y = 17\) Find the exact area of \(R\).
    Edexcel P2 2020 January Q5
    5. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A colony of bees is being studied. The number of bees in the colony at the start of the study was 30000 Three years after the start of the study, the number of bees in the colony is 34000 A model predicts that the number of bees in the colony will increase by \(p \%\) each year, so that the number of bees in the colony at the end of each year of study forms a geometric sequence. Assuming the model,
    1. find the value of \(p\), giving your answer to 2 decimal places. According to the model, at the end of \(N\) years of study the number of bees in the colony exceeds 75000
    2. Find, showing all steps in your working, the smallest integer value of \(N\).
    Edexcel P2 2020 January Q6
    6. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 6 x - 4 y - 14 = 0$$
    1. Find
      1. the coordinates of the centre of \(C\),
      2. the exact radius of \(C\). The line with equation \(y = k\), where \(k\) is a constant, is a tangent to \(C\).
    2. Find the possible values of \(k\). The line with equation \(y = p\), where \(p\) is a negative constant, is a chord of \(C\).
      Given that the length of this chord is 4 units,
    3. find the value of \(p\).
      VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
    Edexcel P2 2020 January Q7
    7. (a) Show that the equation $$8 \tan \theta = 3 \cos \theta$$ may be rewritten in the form $$3 \sin ^ { 2 } \theta + 8 \sin \theta - 3 = 0$$ (b) Hence solve, for \(0 \leqslant x \leqslant 90 ^ { \circ }\), the equation $$8 \tan 2 x = 3 \cos 2 x$$ giving your answers to 2 decimal places.
    Edexcel P2 2020 January Q8
    8. (i) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum to \(n\) terms of this series is $$\frac { n } { 2 } \{ 2 a + ( n - 1 ) d \}$$ (ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is given by $$u _ { n } = 5 n + 3 ( - 1 ) ^ { n }$$ Find the value of
    1. \(u _ { 5 }\)
    2. \(\sum _ { n = 1 } ^ { 59 } u _ { n }\)
    Edexcel P2 2020 January Q9
    9. (a) Sketch the curve with equation $$y = 3 \times 4 ^ { x }$$ showing the coordinates of any points of intersection with the coordinate axes. The curve with equation \(y = 6 ^ { 1 - x }\) meets the curve with equation \(y = 3 \times 4 ^ { x }\) at the point \(P\).
    (b) Show that the \(x\) coordinate of \(P\) is \(\frac { \log _ { 10 } 2 } { \log _ { 10 } 24 }\)
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
    Edexcel P2 2020 January Q10
    10. A curve \(C\) has equation $$y = 4 x ^ { 3 } - 9 x + \frac { k } { x } \quad x > 0$$ where \(k\) is a constant.
    The point \(P\) with \(x\) coordinate \(\frac { 1 } { 2 }\) lies on \(C\).
    Given that \(P\) is a stationary point of \(C\),
    1. show that \(k = - \frac { 3 } { 2 }\)
    2. Determine the nature of the stationary point at \(P\), justifying your answer. The curve \(C\) has a second stationary point.
    3. Using algebra, find the \(x\) coordinate of this second stationary point.
      \includegraphics[max width=\textwidth, alt={}, center]{08aac50c-7317-4510-927a-7f5f2e00f485-26_2255_50_312_1980}