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Edexcel P2 2023 January Q6
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } + 8 x - 4 y = 0$$
  1. Find
    1. the coordinates of the centre of \(C\),
    2. the exact radius of \(C\). The point \(P\) lies on \(C\).
      Given that the tangent to \(C\) at \(P\) has equation \(x + 2 y + 10 = 0\)
  2. find the coordinates of \(P\)
  3. Find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.
Edexcel P2 2023 January Q7
  1. A geometric sequence has first term \(a\) and common ratio \(r\), where \(r > 0\)
Given that
  • the 3rd term is 20
  • the 5th term is 12.8
    1. show that \(r = 0.8\)
    2. Hence find the value of \(a\).
Given that the sum of the first \(n\) terms of this sequence is greater than 156
  • find the smallest possible value of \(n\).
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
    •
    Edexcel P2 2023 January Q8
    1. In this question you must show all stages of your working.
    Solutions based entirely on calculator technology are not acceptable.
    1. Solve, for \(- \frac { \pi } { 2 } < x < \pi\), the equation $$5 \sin ( 3 x + 0.1 ) + 2 = 0$$ giving your answers, in radians, to 2 decimal places.
    2. Solve, for \(0 < \theta < 360 ^ { \circ }\), the equation $$2 \tan \theta \sin \theta = 5 + \cos \theta$$ giving your answers, in degrees, to one decimal place.
    Edexcel P2 2023 January Q9
    1. In this question you must show all stages of your working.
    \section*{Solutions based entirely on calculator technology are not acceptable.} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f6af51c1-5f85-4952-b3c4-9dca42b2a309-26_761_940_411_566} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows
    • the curve \(C\) with equation \(y = x ^ { 2 } - 4 x + 5\)
    • the line \(l\) with equation \(y = 2\)
    The curve \(C\) intersects the \(y\)-axis at the point \(D\).
    1. Write down the coordinates of \(D\). The curve \(C\) intersects the line \(l\) at the points \(E\) and \(F\), as shown in Figure 3.
    2. Find the \(x\) coordinate of \(E\) and the \(x\) coordinate of \(F\). Shown shaded in Figure 3 is
      • the region \(R _ { 1 }\) which is bounded by \(C , l\) and the \(y\)-axis
      • the region \(R _ { 2 }\) which is bounded by \(C\) and the line segments \(E F\) and \(D F\)
      Given that \(\frac { \text { area of } R _ { 1 } } { \text { area of } R _ { 2 } } = k\), where \(k\) is a constant,
    3. use algebraic integration to find the exact value of \(k\), giving your answer as a simplified fraction.
    Edexcel P2 2023 January Q10
    1. A student was asked to prove by exhaustion that
      if \(n\) is an integer then \(2 n ^ { 2 } + n + 1\) is not divisible by 3
    The start of the student's proof is shown in the box below. Consider the case when \(n = 3 k\) $$2 n ^ { 2 } + n + 1 = 18 k ^ { 2 } + 3 k + 1 = 3 \left( 6 k ^ { 2 } + k \right) + 1$$ which is not divisible by 3 Complete this proof.
    Edexcel P2 2024 January Q1
    1. $$f ( x ) = a x ^ { 3 } + 3 x ^ { 2 } - 8 x + 2 \quad \text { where } a \text { is a constant }$$ Given that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is 3 , find the value of \(a\).
    Edexcel P2 2024 January Q2
    1. Find the coefficient of the term in \(x ^ { 7 }\) of the binomial expansion of
    $$\left( \frac { 3 } { 8 } + 4 x \right) ^ { 12 }$$ giving your answer in simplest form.
    Edexcel P2 2024 January Q3
    1. The circle \(C\)
    • has centre \(A ( 3,5 )\)
    • passes through the point \(B ( 8 , - 7 )\)
      1. Find an equation for \(C\).
    The points \(M\) and \(N\) lie on \(C\) such that \(M N\) is a chord of \(C\).
    Given that \(M N\)
    • lies above the \(x\)-axis
    • is parallel to the \(x\)-axis
    • has length \(4 \sqrt { 22 }\)
    • find an equation for the line passing through points \(M\) and \(N\).
    Edexcel P2 2024 January Q4
    1. (a) Sketch the curve with equation
    $$y = a ^ { - x } + 4$$ where \(a\) is a constant and \(a > 1\)
    On your sketch show
    • the coordinates of the point of intersection of the curve with the \(y\)-axis
    • the equation of the asymptote to the curve.
    \(x\)- 4- 1.513.568.5
    \(y\)136.2804.5774.1464.0374.009
    The table above shows corresponding values of \(x\) and \(y\) for \(y = 3 ^ { - \frac { 1 } { 2 } x } + 4\)
    The values of \(y\) are given to four significant figures, as appropriate.
    Using the trapezium rule with all the values of \(y\) in the table,
    (b) find an approximate value for $$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) d x$$ giving your answer to two significant figures.
    (c) Using the answer to part (b), find an approximate value for
    1. \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x\)
    2. \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x + \int _ { - 8.5 } ^ { 4 } \left( 3 ^ { \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x\)
    Edexcel P2 2024 January Q5
      1. Find the value of
    $$\sum _ { r = 1 } ^ { \infty } 6 \times ( 0.25 ) ^ { r }$$ (3)
    (ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$\begin{aligned} u _ { 1 } & = 3
    u _ { n + 1 } & = \frac { u _ { n } - 3 } { u _ { n } - 2 } \quad n \in \mathbb { N } \end{aligned}$$
    1. Show that this sequence is periodic.
    2. State the order of this sequence.
    3. Hence find $$\sum _ { n = 1 } ^ { 70 } u _ { n }$$
    Edexcel P2 2024 January Q6
    1. (a) Given that
    $$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$ show that $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$ (b) Given also that - 1 is a root of the equation $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$
    1. use algebra to find the other two roots of the equation.
    2. Hence solve $$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$
    Edexcel P2 2024 January Q7
    1. Wheat is grown on a farm.
    • In year 1 , the farm produced 300 tonnes of wheat.
    • In year 12 , the farm is predicted to produce 4000 tonnes of wheat.
    Model \(A\) assumes that the amount of wheat produced on the farm will increase by the same amount each year.
    1. Using model \(A\), find the amount of wheat produced on the farm in year 4. Give your answer to the nearest 10 tonnes. Model \(B\) assumes that the amount of wheat produced on the farm will increase by the same percentage each year.
    2. Using model \(B\), find the amount of wheat produced on the farm in year 2. Give your answer to the nearest 10 tonnes.
    3. Calculate, according to the two models, the difference between the total amounts of wheat predicted to be produced on the farm from year 1 to year 12 inclusive. Give your answer to the nearest 10 tonnes.
    Edexcel P2 2024 January Q8
    1. (i) Use a counter example to show that the following statement is false
    $$\text { " } n ^ { 2 } + 3 n + 1 \text { is prime for all } n \in \mathbb { N } \text { " }$$ (ii) Use algebra to prove by exhaustion that for all \(n \in \mathbb { N }\) $$\text { " } n ^ { 2 } - 2 \text { is not a multiple of } 4 \text { " }$$
    Edexcel P2 2024 January Q9
    1. In this question you must show detailed reasoning.
    Solutions relying entirely on calculator technology are not acceptable.
    1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$\sin x \tan x = 5$$ giving your answers to one decimal place.
    2. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{0e08d931-aa1c-48a8-8b39-47096f981950-26_643_736_721_660} \captionsetup{labelformat=empty} \caption{Figure 1}
      \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = A \sin \left( 2 \theta - \frac { 3 \pi } { 8 } \right) + 2$$ where \(A\) is a constant and \(\theta\) is measured in radians.
      The points \(P , Q\) and \(R\) lie on the curve and are shown in Figure 1.
      Given that the \(y\) coordinate of \(P\) is 7
      (a) state the value of \(A\),
      (b) find the exact coordinates of \(Q\),
      (c) find the value of \(\theta\) at \(R\), giving your answer to 3 significant figures.
    Edexcel P2 2024 January Q10
    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]{0e08d931-aa1c-48a8-8b39-47096f981950-30_646_741_376_662} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the curve with equation $$y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 1458 } { \sqrt { x ^ { 3 } } } - 74 \quad x > 0$$ The point \(P\) is the only stationary point on the curve.
    1. Use calculus to show that the \(x\) coordinate of \(P\) is 9 The line \(l\) passes through the point \(P\) and is parallel to the \(x\)-axis.
      The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line \(l\) and the line with equation \(x = 4\)
    2. Use algebraic integration to find the exact area of \(R\).
    Edexcel P2 2019 June Q1
    1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
    $$\begin{aligned} a _ { n + 1 } & = 4 - a _ { n }
    a _ { 1 } & = 3 \end{aligned}$$ Find the value of
      1. \(a _ { 2 }\)
      2. \(a _ { 107 }\)
    2. \(\sum _ { n = 1 } ^ { 200 } \left( 2 a _ { n } - 1 \right)\)
    Edexcel P2 2019 June Q2
    2. A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 4 x - 10 y - 21 = 0$$ Find
      1. the coordinates of the centre of \(C\),
      2. the exact value of the radius of \(C\). The point \(P ( 5,4 )\) lies on \(C\).
    2. Find the equation of the tangent to \(C\) at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
    Edexcel P2 2019 June Q3
    3. (i) Use algebra to prove that for all real values of \(x\) $$( x - 4 ) ^ { 2 } \geqslant 2 x - 9$$ (ii) Show that the following statement is untrue. $$2 ^ { n } + 1 \text { is a prime number for all values of } n , n \in \mathbb { N }$$
    Edexcel P2 2019 June Q4
    4. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 1 } { 4 } x \right) ^ { 6 }$$ (b) Given that \(x\) is small, so terms in \(x ^ { 4 }\) and higher powers of \(x\) may be ignored, show $$\left( 2 - \frac { 1 } { 4 } x \right) ^ { 6 } + \left( 2 + \frac { 1 } { 4 } x \right) ^ { 6 } = a + b x ^ { 2 }$$ where \(a\) and \(b\) are constants to be found.
    Edexcel P2 2019 June Q5
    5. A company makes a particular type of watch. The annual profit made by the company from sales of these watches is modelled by the equation $$P = 12 x - x ^ { \frac { 3 } { 2 } } - 120$$ where \(P\) is the annual profit measured in thousands of pounds and \(\pounds x\) is the selling price of the watch. According to this model,
    1. find, using calculus, the maximum possible annual profit.
    2. Justify, also using calculus, that the profit you have found is a maximum.
    Edexcel P2 2019 June Q6
    6. \(\mathrm { f } ( x ) = k x ^ { 3 } - 15 x ^ { 2 } - 32 x - 12\) where \(k\) is a constant Given ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\),
    1. show that \(k = 9\)
    2. Using algebra and showing each step of your working, fully factorise \(\mathrm { f } ( x )\).
    3. Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$9 \cos ^ { 3 } \theta - 15 \cos ^ { 2 } \theta - 32 \cos \theta - 12 = 0$$ giving your answers to one decimal place.
    Edexcel P2 2019 June Q7
    7. Kim starts working for a company.
    • In year 1 her annual salary will be \(\pounds 16200\)
    • In year 10 her annual salary is predicted to be \(\pounds 31500\)
    Model \(A\) assumes that her annual salary will increase by the same amount each year.
    1. According to model \(A\), determine Kim's annual salary in year 2 . Model \(B\) assumes that her annual salary will increase by the same percentage each year.
    2. According to model \(B\), determine Kim's annual salary in year 2 . Give your answer to the nearest \(\pounds 10\)
    3. Calculate, according to the two models, the difference between the total amounts that Kim is predicted to earn from year 1 to year 10 inclusive. Give your answer to the nearest £10
    Edexcel P2 2019 June Q8
    8. (i) Find the exact solution of the equation $$8 ^ { 2 x + 1 } = 6$$ giving your answer in the form \(a + b \log _ { 2 } 3\), where \(a\) and \(b\) are constants to be found.
    (ii) Using the laws of logarithms, solve $$\log _ { 5 } ( 7 - 2 y ) = 2 \log _ { 5 } ( y + 1 ) - 1$$
    Edexcel P2 2019 June Q9
    9. (a) Show that the equation $$\cos \theta - 1 = 4 \sin \theta \tan \theta$$ can be written in the form $$5 \cos ^ { 2 } \theta - \cos \theta - 4 = 0$$ (b) Hence solve, for \(0 \leqslant x < \frac { \pi } { 2 }\) $$\cos 2 x - 1 = 4 \sin 2 x \tan 2 x$$ giving your answers, where appropriate, to 2 decimal places.
    Edexcel P2 2019 June Q10
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fc9cd828-f9bc-4cad-8a70-4214697b1c6a-11_707_855_255_539} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 36 } { x ^ { 2 } } + 2 x - 13 \quad x > 0$$ Using calculus,
    1. find the range of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing,
    2. show that \(\int _ { 2 } ^ { 9 } \left( \frac { 36 } { x ^ { 2 } } + 2 x - 13 \right) \mathrm { d } x = 0\) The point \(P ( 2,0 )\) and the point \(Q ( 6,0 )\) lie on \(C\).
      Given \(\int _ { 2 } ^ { 6 } \left( \frac { 36 } { x ^ { 2 } } + 2 x - 13 \right) \mathrm { d } x = - 8\)
      1. state the value of \(\int _ { 6 } ^ { 9 } \left( \frac { 36 } { x ^ { 2 } } + 2 x - 13 \right) \mathrm { d } x\)
      2. find the value of the constant \(k\) such that \(\int _ { 2 } ^ { 6 } \left( \frac { 36 } { x ^ { 2 } } + 2 x + k \right) \mathrm { d } x = 0\)