Questions P2 (867 questions)

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Edexcel P2 2023 January Q10
4 marks Moderate -0.8
  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
3 marks Easy -1.2
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
3 marks Moderate -0.8
  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
6 marks Standard +0.3
  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
9 marks Standard +0.3
  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
8 marks Standard +0.8
    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
8 marks Standard +0.3
  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
9 marks Moderate -0.3
  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
6 marks Moderate -0.8
  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
14 marks Standard +0.3
  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
9 marks Standard +0.3
  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
4 marks Easy -1.3
  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
7 marks Moderate -0.3
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
4 marks Moderate -0.8
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
7 marks Moderate -0.8
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
8 marks Moderate -0.3
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
8 marks Standard +0.3
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
9 marks Moderate -0.3
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
9 marks Moderate -0.3
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
8 marks Standard +0.3
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
11 marks Moderate -0.3
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 \(\int _ { 2 } ^ { 6 } \left( \frac { 36 } { x ^ { 2 } } + 2 x + k \right) \mathrm { d } x = 0\)
Edexcel P2 2021 June Q1
7 marks Easy -1.2
  1. Adina is saving money to buy a new computer. She saves \(\pounds 5\) in week \(1 , \pounds 5.25\) in week 2 , \(\pounds 5.50\) in week 3 and so on until she has enough money, in total, to buy the computer.
She decides to model her savings using either an arithmetic series or a geometric series.
Using the information given,
    1. state with a reason whether an arithmetic series or a geometric series should be used,
    2. write down an expression, in terms of \(n\), for the amount, in pounds ( \(\pounds\) ), saved in week \(n\). Given that the computer Adina wants to buy costs \(\pounds 350\)
  2. find the number of weeks it will take for Adina to save enough money to buy the computer.
    VIAV SIHI NI III IM ION OCVIIN SIHI NI III M M O N OOVIAV SIHI NI IIIIM I ION OC
Edexcel P2 2021 June Q2
8 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-04_1001_1481_267_221} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = 4 ^ { x }\) A copy of Figure 1, labelled Diagram 1, is shown on the next page.
  1. On Diagram 1, sketch the curve with equation
    1. \(y = 2 ^ { x }\)
    2. \(y = 4 ^ { x } - 6\) Label clearly the coordinates of any points of intersection with the coordinate axes. The curve with equation \(y = 2 ^ { x }\) meets the curve with equation \(y = 4 ^ { x } - 6\) at the point \(P\).
  2. Using algebra, find the exact coordinates of \(P\).
    \includegraphics[max width=\textwidth, alt={}]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-05_1009_1490_264_219}
    \section*{Diagram 1}
Edexcel P2 2021 June Q3
5 marks Moderate -0.3
3. (i) Prove that for all single digit prime numbers, \(p\), $$p ^ { 3 } + p \text { is a multiple of } 10$$ (ii) Show, using algebra, that for \(n \in \mathbb { N }\) $$( n + 1 ) ^ { 3 } - n ^ { 3 } \text { is not a multiple of } 3$$
Edexcel P2 2021 June Q4
8 marks Moderate -0.3
  1. (a) Find, in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), the binomial expansion of
$$\left( 2 + \frac { x } { 8 } \right) ^ { 13 }$$ fully simplifying each coefficient.
(b) Use the answer to part (a) to find an approximation for \(2.0125 ^ { 13 }\) Give your answer to 3 decimal places. Without calculating \(2.0125 { } ^ { 13 }\) (c) state, with a reason, whether the answer to part (b) is an overestimate or an underestimate.