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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}
    •
    Edexcel P2 2021 October Q8
    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
    9. (a) Prove that for all positive values of \(x\) and \(y\), $$\frac { x + y } { 2 } \geqslant \sqrt { x y }$$ (b) 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. 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
    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.
    \(а\)\(b\)\(c\)
    1
    2
    Edexcel P2 2022 October Q2
    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
    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
    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
    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
    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 curve \(C _ { 1 }\)
      • the curve \(C _ { 2 }\) with equation \(y = 2 - \frac { 1 } { 4 } x ^ { 2 }\)
      • the line with equation \(x = 2\)
      • the \(y\)-axis
      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
    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
    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.