Questions — Edexcel P2 (157 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
Edexcel P2 2019 October Q2
2. The adult population of a town at the start of 2019 is 25000 A model predicts that the adult population will increase by \(2 \%\) each year, so that the number of adults in the population at the start of each year following 2019 will form a geometric sequence.
  1. Find, according to the model, the adult population of the town at the start of 2032 It is also modelled that every member of the adult population gives \(\pounds 5\) to local charity at the start of each year.
  2. Find, according to these models, the total amount of money that would be given to local charity by the adult population of the town from 2019 to 2032 inclusive. Give your answer to the nearest \(\pounds 1000\)
Edexcel P2 2019 October Q3
3. (a) Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$\left( 1 + \frac { x } { 4 } \right) ^ { 12 }$$ giving each coefficient in its simplest form.
(b) Find the term independent of \(x\) in the expansion of $$\left( \frac { x ^ { 2 } + 8 } { x ^ { 5 } } \right) \left( 1 + \frac { x } { 4 } \right) ^ { 12 }$$
Edexcel P2 2019 October Q4
4. \(\mathrm { f } ( x ) = ( x - 3 ) \left( 3 x ^ { 2 } + x + a \right) - 35\) where \(a\) is a constant
  1. State the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\). Given \(( 3 x - 2 )\) is a factor of \(\mathrm { f } ( x )\),
  2. show that \(a = - 17\)
  3. Using algebra and showing each step of your working, fully factorise \(\mathrm { f } ( x )\).
Edexcel P2 2019 October Q5
5. (a) Given \(0 < a < 1\), sketch the curve with equation $$y = a ^ { x }$$ showing the coordinates of the point at which the curve crosses the \(y\)-axis.
\(x\)22.533.54
\(y\)4.256.4279.12512.3416.06
The table above shows corresponding values of \(x\) and \(y\) for \(y = x ^ { 2 } + \left( \frac { 1 } { 2 } \right) ^ { x }\) The values of \(y\) are given to 4 significant figures as appropriate.
Using the trapezium rule with all the values of \(y\) in the given table,
(b) obtain an estimate for \(\int _ { 2 } ^ { 4 } \left( x ^ { 2 } + \left( \frac { 1 } { 2 } \right) ^ { x } \right) \mathrm { d } x\) Using your answer to part (b) and making your method clear, estimate
(c) \(\quad \int _ { 2 } ^ { 4 } \left( x ( x - 3 ) + \left( \frac { 1 } { 2 } \right) ^ { x } \right) \mathrm { d } x\)
Edexcel P2 2019 October Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bfeb1724-9a00-4a36-9606-520395792b2b-16_677_826_258_559} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a circle \(C\) with centre \(N ( 4 , - 1 )\). The line \(l\) with equation \(y = \frac { 1 } { 2 } x\) is a tangent to \(C\) at the point \(P\). Find
  1. the equation of line \(P N\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
  2. the equation of \(C\).
    \includegraphics[max width=\textwidth, alt={}, center]{bfeb1724-9a00-4a36-9606-520395792b2b-16_2256_52_311_1978}
Edexcel P2 2019 October Q7
  1. Given \(\log _ { a } b = k\), find, in simplest form in terms of \(k\),
    1. \(\log _ { a } \left( \frac { \sqrt { a } } { b } \right)\)
    2. \(\frac { \log _ { a } a ^ { 2 } b } { \log _ { a } b ^ { 3 } }\)
    3. \(\sum _ { n = 1 } ^ { 50 } \left( k + \log _ { a } b ^ { n } \right)\)
Edexcel P2 2019 October Q8
8. Solutions relying on calculator technology are not acceptable in this question.
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bfeb1724-9a00-4a36-9606-520395792b2b-22_556_822_351_561} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of a curve with equation $$y = \frac { 8 \sqrt { x } - 5 } { 2 x ^ { 2 } } \quad x > 0$$ The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 4\) Find the exact area of \(R\).
  2. Find the value of the constant \(k\) such that $$\int _ { - 3 } ^ { 6 } \left( \frac { 1 } { 2 } x ^ { 2 } + k \right) \mathrm { d } x = 55$$
Edexcel P2 2019 October Q9
9. Solutions based entirely on graphical or numerical methods are not acceptable in this question.
  1. Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$3 \sin \left( 2 \theta - 10 ^ { \circ } \right) = 1$$ giving your answers to one decimal place.
  2. The first three terms of an arithmetic sequence are $$\sin \alpha , \frac { 1 } { \tan \alpha } \text { and } 2 \sin \alpha$$ where \(\alpha\) is a constant.
    (a) Show that \(2 \cos \alpha = 3 \sin ^ { 2 } \alpha\) Given that \(\pi < \alpha < 2 \pi\),
    (b) find, showing all working, the value of \(\alpha\) to 3 decimal places.
Edexcel P2 2019 October Q10
10. The curve \(C\) has equation $$y = a x ^ { 3 } - 3 x ^ { 2 } + 3 x + b$$ where \(a\) and \(b\) are constants. Given that
  • the point \(( 2,5 )\) lies on \(C\)
  • the gradient of the curve at \(( 2,5 )\) is 7
    1. find the value of \(a\) and the value of \(b\).
    2. Prove that \(C\) has no turning points.
Edexcel P2 2020 October Q1
  1. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$ giving each term in its simplest form.
(b) Hence find the constant term in the series expansion of $$\left( 3 - \frac { 1 } { x } \right) ^ { 2 } \left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$
Edexcel P2 2020 October Q2
2. $$y = \frac { 2 ^ { x } } { \sqrt { \left( 5 x ^ { 2 } + 3 \right) } }$$
  1. Complete the table below,giving the values of \(y\) to 3 decimal places.
    \(x\)- 0.2500.250.50.75
    \(y\)0.4620.6530.698
  2. Use the trapezium rule,with all the values of \(y\) from the completed table,to find an approximate value for
    . $$\int _ { - 0.25 } ^ { 0.75 } \frac { 2 ^ { x } } { \sqrt { \left( 5 x ^ { 2 } + 3 \right) } } \mathrm { d } x$$
Edexcel P2 2020 October Q3
3. $$f ( x ) = a x ^ { 3 } - x ^ { 2 } + b x + 4$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x + 4\) ), the remainder is - 108
  1. Use the remainder theorem to show that $$16 a + b = 24$$ Given also that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\),
  2. find the value of \(a\) and the value of \(b\).
  3. Find \(\mathrm { f } ^ { \prime } ( x )\).
  4. Hence find the exact coordinates of the stationary points of the curve with equation \(y = \mathrm { f } ( x )\).
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel P2 2020 October Q4
4. The points \(P\) and \(Q\) have coordinates \(( - 11,6 )\) and \(( - 3,12 )\) respectively. Given that \(P Q\) is a diameter of the circle \(C\),
    1. find the coordinates of the centre of \(C\),
    2. find the radius of \(C\).
  2. Hence find an equation of \(C\).
  3. Find an equation of the tangent to \(C\) at the point \(Q\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.
    \includegraphics[max width=\textwidth, alt={}, center]{0e107b51-2fb3-4ad7-8542-5aa0da13b127-13_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel P2 2020 October Q5
5. Ben is saving for the deposit for a house over a period of 60 months. Ben saves \(\pounds 100\) in the first month and in each subsequent month, he saves \(\pounds 5\) more than the previous month, so that he saves \(\pounds 105\) in the second month, \(\pounds 110\) in the third month, and so on, forming an arithmetic sequence.
  1. Find the amount Ben saves in the 40th month.
  2. Find the total amount Ben saves over the 60 -month period. Lina is also saving for a deposit for a house.
    Lina saves \(\pounds 600\) in the first month and in each subsequent month, she saves \(\pounds 10\) less than the previous month, so that she saves \(\pounds 590\) in the second month, \(\pounds 580\) in the third month, and so on, forming an arithmetic sequence. Given that, after \(n\) months, Lina will have saved exactly \(\pounds 18200\) for her deposit,
  3. form an equation in \(n\) and show that it can be written as $$n ^ { 2 } - 121 n + 3640 = 0$$
  4. Solve the equation in part (c).
  5. State, with a reason, which of the solutions to the equation in part (c) is not a sensible value for \(n\).
Edexcel P2 2020 October Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e107b51-2fb3-4ad7-8542-5aa0da13b127-20_978_1292_267_328} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations $$\begin{array} { l l } C _ { 1 } : y = x ^ { 3 } - 6 x + 9 & x \geqslant 0
C _ { 2 } : y = - 2 x ^ { 2 } + 7 x - 1 & x \geqslant 0 \end{array}$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(A\) and \(B\) as shown in Figure 1 .
The point \(A\) has coordinates (1,4). Using algebra and showing all steps of your working,
  1. find the coordinates of the point \(B\). The finite region \(R\), shown shaded in Figure 1, is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
  2. Use algebraic integration to find the exact area of \(R\).
Edexcel P2 2020 October Q7
7. (i) Show that $$\tan \theta + \frac { 1 } { \tan \theta } \equiv \frac { 1 } { \sin \theta \cos \theta } \quad \theta \neq \frac { \mathrm { n } \pi } { 2 } \quad n \in \mathbb { Z }$$ (ii) 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. 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
9. (i) Find the exact value of \(x\) for which $$\log _ { 3 } ( x + 5 ) - 4 = \log _ { 3 } ( 2 x - 1 )$$ (ii) 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.
    VIHV SIHII NI I IIIM I ON OCVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel P2 2021 October Q1
  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.
  3. Find, in terms of \(k\), an expression for \(u _ { 3 }\) Given that \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 117\)
  4. find the value of \(k\).
Edexcel P2 2021 October Q2
2. A sequence is defined by
Edexcel P2 2021 October Q3
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
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
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
    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
      • \(p\) is a positive constant
      • \(Q R\) is a diameter of \(C _ { 2 }\)
        find the exact value of \(p\).
    Edexcel P2 2021 October Q7
    7. (i) 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 \(n\).
    (ii) 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
    $$25 r ^ { 2 } - 25 r - 6 = 0$$
  • Write down the solutions of $$25 r ^ { 2 } - 25 r - 6 = 0$$ Hence,
  • state the value of \(r\), giving a reason for your answer,
  • 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}
    •