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Edexcel P2 2023 June Q9
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Show that $$3 \cos \theta ( \tan \theta \sin \theta + 3 ) = 11 - 5 \cos \theta$$ may be written as $$3 \cos ^ { 2 } \theta - 14 \cos \theta + 8 = 0$$
  2. Hence solve, for \(0 < x < 360 ^ { \circ }\) $$3 \cos 2 x ( \tan 2 x \sin 2 x + 3 ) = 11 - 5 \cos 2 x$$ giving your answers to one decimal place.
Edexcel P2 2023 June Q10
  1. The curve \(C\) has equation
$$y = \frac { ( x - k ) ^ { 2 } } { \sqrt { x } } \quad x > 0$$ where \(k\) is a positive constant.
  1. Show that $$\int _ { 1 } ^ { 16 } \frac { ( x - k ) ^ { 2 } } { \sqrt { x } } \mathrm {~d} x = a k ^ { 2 } + b k + \frac { 2046 } { 5 }$$ where \(a\) and \(b\) are integers to be found. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0e3b364c-151b-471d-acb6-01afb018fb75-26_645_670_904_699} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve \(C\) and the line \(l\).
    Given that \(l\) intersects \(C\) at the point \(A ( 1,9 )\) and at the point \(B ( 16 , q )\) where \(q\) is a constant,
  2. show that \(k = 4\) The region \(R\), shown shaded in Figure 1, is bounded by \(C\) and \(l\)
    Using the answers to parts (a) and (b),
  3. find the area of region \(R\)
Edexcel P2 2023 June Q11
  1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
$$\begin{aligned} u _ { n + 1 } & = b - a u _ { n }
u _ { 1 } & = 3 \end{aligned}$$ where \(a\) and \(b\) are constants.
  1. Find, in terms of \(a\) and \(b\),
    1. \(u _ { 2 }\)
    2. \(u _ { 3 }\) Given
      • \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 153\)
  2. \(b = a + 9\)
  3. show that
    4. $$a ^ { 2 } - 5 a - 66 = 0$$
  4. Hence find the larger possible value of \(u _ { 2 }\)
Edexcel P2 2024 June Q1
  1. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving each term in simplest form.
(b) Hence find the coefficient of \(x ^ { 3 }\) in the expansion of $$( 10 x + 3 ) \left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving the answer in simplest form.
Edexcel P2 2024 June Q2
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
In an arithmetic series,
  • the sixth term is 2
  • the sum of the first ten terms is - 80
For this series,
  1. find the value of the first term and the value of the common difference.
  2. Hence find the smallest value of \(n\) for which $$S _ { n } > 8000$$
Edexcel P2 2024 June Q3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    1. Using the laws of logarithms, solve
    $$2 \log _ { 2 } ( 2 - x ) = 4 + \log _ { 2 } ( x + 10 )$$
  2. Find the value of $$\log _ { \sqrt { a } } a ^ { 6 }$$ where \(a\) is a positive constant greater than 1
Edexcel P2 2024 June Q4
4. $$f ( x ) = ( x - 2 ) \left( 2 x ^ { 2 } + 5 x + k \right) + 21$$ where \(k\) is a constant.
  1. State the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) Given that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\)
  2. show that \(k = 11\)
  3. Hence
    1. fully factorise \(\mathrm { f } ( x )\),
    2. find the number of real solutions of the equation $$\mathrm { f } ( x ) = 0$$ giving a reason for your answer.
Edexcel P2 2024 June Q5
  1. In this question you must show detailed reasoning.
    1. Given that \(x\) and \(y\) are positive numbers such that
    $$( x - y ) ^ { 3 } > x ^ { 3 } - y ^ { 3 }$$ prove that $$y > x$$
  2. Using a counter example, show that the result in part (a) is not true for all real numbers.
Edexcel P2 2024 June Q6
  1. (a) Sketch the curve with equation
$$y = a ^ { x } + 4$$ where \(a\) is a positive constant greater than 1
On your sketch, show
  • the coordinates of the point of intersection of the curve with the \(y\)-axis
  • the equation of the asymptote of the curve
\(x\)22.32.62.93.23.5
\(y\)00.32460.86291.66432.78964.3137
The table shows corresponding values of \(x\) and \(y\) for $$y = 2 ^ { x } - 2 x$$ with the values of \(y\) given to 4 decimal places as appropriate.
Using the trapezium rule with all the values of \(y\) in the given table,
(b) obtain an estimate for \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x } - 2 x \right) \mathrm { d } x\), giving your answer to 2 decimal places.
(c) Using your answer to part (b) and making your method clear, estimate
  1. \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x } + 2 x \right) \mathrm { d } x\)
  2. \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x + 1 } - 4 x \right) \mathrm { d } x\)
Edexcel P2 2024 June Q7
  1. The circle \(C _ { 1 }\) has equation
$$x ^ { 2 } + y ^ { 2 } + 8 x - 10 y = 29$$
    1. Find the coordinates of the centre of \(C _ { 1 }\)
    2. Find the exact value of the radius of \(C _ { 1 }\) In part (b) you must show detailed reasoning.
      The circle \(C _ { 2 }\) has equation $$( x - 5 ) ^ { 2 } + ( y + 8 ) ^ { 2 } = 52$$
  2. Prove that the circles \(C _ { 1 }\) and \(C _ { 2 }\) neither touch nor intersect.
Edexcel P2 2024 June Q8
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    1. Solve, for \(0 < x \leqslant \pi\), the equation
    $$5 \sin x \tan x + 13 = \cos x$$ giving your answer in radians to 3 significant figures.
  2. The temperature inside a greenhouse is monitored on one particular day. The temperature, \(H ^ { \circ } \mathrm { C }\), inside the greenhouse, \(t\) hours after midnight, is modelled by the equation $$H = 10 + 12 \sin ( k t + 18 ) ^ { \circ } \quad 0 \leqslant t < 24$$ where \(k\) is a constant.
    Use the equation of the model to answer parts (a) to (c).
    Given that
    • the temperature inside the greenhouse was \(20 ^ { \circ } \mathrm { C }\) at 6 am
    • \(0 < k < 20\)
      (a) find all possible values for \(k\), giving each answer to 2 decimal places.
    Given further that \(0 < k < 10\)
    (b) find the maximum temperature inside the greenhouse,
    (c) find the time of day at which this maximum temperature occurs. Give your answer to the nearest minute.
Edexcel P2 2024 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b822842d-ee62-40ce-a8de-967e556a80a8-26_915_912_255_580} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 is a sketch of the curve \(C\) with equation $$y = 2 x ^ { \frac { 3 } { 2 } } ( 4 - x ) \quad x \geqslant 0$$ The point \(P\) is the stationary point of \(C\).
  1. Find, using calculus, the \(x\) coordinate of \(P\). The region \(R _ { 1 }\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis.
    The region \(R _ { 2 }\), also shown shaded in Figure 1, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = k\), where \(k\) is a constant. Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
  2. find, using calculus, the exact value of \(k\).
Edexcel P2 2024 June Q10
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
The number of dormice and the number of voles on an island are being monitored.
Initially there are 2000 dormice on the island.
A model predicts that the number of dormice will increase by \(3 \%\) each year, so that the numbers of dormice on the island at the end of each year form a geometric sequence.
  1. Find, according to the model, the number of dormice on the island 6 years after monitoring began. Give your answer to 3 significant figures. The number of voles on the island is being monitored over the same period of time.
    Given that
    • 4 years after monitoring began there were 3690 voles on the island
    • 7 years after monitoring began there were 3470 voles on the island
    • the number of voles on the island at the end of each year is modelled as a geometric sequence
    • find the equation of this model in the form
    $$N = a b ^ { t }$$ where \(N\) is the number of voles, \(t\) years after monitoring began and \(a\) and \(b\) are constants. Give the value of \(a\) and the value of \(b\) to 2 significant figures. When \(t = T\), the number of dormice on the island is equal to the number of voles on the island.
  2. Find, according to the models, the value of \(T\), giving your answer to one decimal place.
Edexcel P2 2019 October Q1
  1. A curve \(C\) has equation \(y = 2 x ^ { 2 } ( x - 5 )\)
    1. Find, using calculus, the \(x\) coordinates of the stationary points of \(C\).
    2. Hence find the values of \(x\) for which \(y\) is increasing.
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
    . $$\int _ { - 0.25 } ^ { 0.75 } \frac { 2 ^ { x } } { \sqrt { \left( 5 x ^ { 2 } + 3 \right) } } \mathrm { d } x$$