Questions — Edexcel (9685 questions)

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Edexcel AS Paper 1 2019 June Q5
5 marks Moderate -0.3
  1. A curve has equation
$$y = 3 x ^ { 2 } + \frac { 24 } { x } + 2 \quad x > 0$$
  1. Find, in simplest form, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Hence find the exact range of values of \(x\) for which the curve is increasing.
Edexcel AS Paper 1 2019 June Q6
6 marks Moderate -0.3
6. Figure 1 Figure 1 shows a sketch of a triangle \(A B C\) with \(A B = 3 x \mathrm {~cm} , A C = 2 x \mathrm {~cm}\) and angle \(C A B = 60 ^ { \circ }\) Given that the area of triangle \(A B C\) is \(18 \sqrt { 3 } \mathrm {~cm} ^ { 2 }\)
  1. show that \(x = 2 \sqrt { 3 }\)
  2. Hence find the exact length of BC, giving your answer as a simplified surd.
Edexcel AS Paper 1 2019 June Q7
8 marks Standard +0.3
  1. The curve \(C\) has equation
$$y = \frac { k ^ { 2 } } { x } + 1 \quad x \in \mathbb { R } , x \neq 0$$ where \(k\) is a constant.
  1. Sketch \(C\) stating the equation of the horizontal asymptote. The line \(l\) has equation \(y = - 2 x + 5\)
  2. Show that the \(x\) coordinate of any point of intersection of \(l\) with \(C\) is given by a solution of the equation $$2 x ^ { 2 } - 4 x + k ^ { 2 } = 0$$
  3. Hence find the exact values of \(k\) for which \(l\) is a tangent to \(C\).
Edexcel AS Paper 1 2019 June Q8
5 marks Moderate -0.8
  1. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 2 + \frac { 3 x } { 4 } \right) ^ { 6 }$$ giving each term in its simplest form.
(b) Explain how you could use your expansion to estimate the value of \(1.925 ^ { 6 }\) You do not need to perform the calculation.
Edexcel AS Paper 1 2019 June Q9
6 marks Moderate -0.8
  1. A company started mining tin in Riverdale on 1st January 2019.
A model to find the total mass of tin that will be mined by the company in Riverdale is given by the equation $$T = 1200 - 3 ( n - 20 ) ^ { 2 }$$ where \(T\) tonnes is the total mass of tin mined in the \(n\) years after the start of mining.
Using this model,
  1. calculate the mass of tin that will be mined up to 1st January 2020,
  2. deduce the maximum total mass of tin that could be mined,
  3. calculate the mass of tin that will be mined in 2023.
  4. State, giving reasons, the limitation on the values of \(n\).
Edexcel AS Paper 1 2019 June Q10
5 marks Moderate -0.3
  1. A circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } - 4 x + 8 y - 8 = 0$$
  1. Find
    1. the coordinates of the centre of \(C\),
    2. the exact radius of \(C\). The straight line with equation \(x = k\), where \(k\) is a constant, is a tangent to \(C\).
  2. Find the possible values for \(k\).
Edexcel AS Paper 1 2019 June Q11
10 marks Standard +0.3
11. $$f ( x ) = 2 x ^ { 3 } - 13 x ^ { 2 } + 8 x + 48$$
  1. Prove that \(( x - 4 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Hence, using algebra, show that the equation \(\mathrm { f } ( x ) = 0\) has only two distinct roots. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{deba6a2b-1821-4110-bde8-bde18a5f9be9-24_727_1059_566_504} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
  3. Deduce, giving reasons for your answer, the number of real roots of the equation $$2 x ^ { 3 } - 13 x ^ { 2 } + 8 x + 46 = 0$$ Given that \(k\) is a constant and the curve with equation \(y = \mathrm { f } ( x + k )\) passes through the origin, (d) find the two possible values of \(k\).
Edexcel AS Paper 1 2019 June Q12
7 marks Standard +0.8
  1. (a) Show that
$$\frac { 10 \sin ^ { 2 } \theta - 7 \cos \theta + 2 } { 3 + 2 \cos \theta } \equiv 4 - 5 \cos \theta$$ (b) Hence, or otherwise, solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$\frac { 10 \sin ^ { 2 } x - 7 \cos x + 2 } { 3 + 2 \cos x } = 4 + 3 \sin x$$
Edexcel AS Paper 1 2019 June Q13
7 marks Standard +0.3
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{deba6a2b-1821-4110-bde8-bde18a5f9be9-32_800_787_244_644} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 2 x ^ { 3 } - 17 x ^ { 2 } + 40 x$$ The curve has a minimum turning point at \(x = k\).
The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = k\). Show that the area of \(R\) is \(\frac { 256 } { 3 }\) (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel AS Paper 1 2019 June Q14
9 marks Standard +0.3
  1. The value of a car, \(\pounds V\), can be modelled by the equation
$$V = 15700 \mathrm { e } ^ { - 0.25 t } + 2300 \quad t \in \mathbb { R } , t \geqslant 0$$ where the age of the car is \(t\) years.
Using the model,
  1. find the initial value of the car. Given the model predicts that the value of the car is decreasing at a rate of \(\pounds 500\) per year at the instant when \(t = T\),
    1. show that $$3925 \mathrm { e } ^ { - 0.25 T } = 500$$
    2. Hence find the age of the car at this instant, giving your answer in years and months to the nearest month.
      (Solutions based entirely on graphical or numerical methods are not acceptable.) The model predicts that the value of the car approaches, but does not fall below, \(\pounds A\).
  3. State the value of \(A\).
  4. State a limitation of this model.
Edexcel AS Paper 1 2019 June Q15
4 marks Standard +0.8
  1. Given \(n \in \mathbb { N }\), prove that \(n ^ { 3 } + 2\) is not divisible by 8
Edexcel AS Paper 1 2019 June Q16
5 marks Standard +0.3
  1. (i) Two non-zero vectors, \(\mathbf { a }\) and \(\mathbf { b }\), are such that
$$| \mathbf { a } + \mathbf { b } | = | \mathbf { a } | + | \mathbf { b } |$$ Explain, geometrically, the significance of this statement.
(ii) Two different vectors, \(\mathbf { m }\) and \(\mathbf { n }\), are such that \(| \mathbf { m } | = 3\) and \(| \mathbf { m } - \mathbf { n } | = 6\) The angle between vector \(\mathbf { m }\) and vector \(\mathbf { n }\) is \(30 ^ { \circ }\) Find the angle between vector \(\mathbf { m }\) and vector \(\mathbf { m } - \mathbf { n }\), giving your answer, in degrees, to one decimal place.
Edexcel AS Paper 1 2020 June Q1
5 marks Moderate -0.8
  1. A curve has equation
$$y = 2 x ^ { 3 } - 4 x + 5$$ Find the equation of the tangent to the curve at the point \(P ( 2,13 )\).
Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found.
Solutions relying on calculator technology are not acceptable.
(5)
Edexcel AS Paper 1 2020 June Q2
6 marks Moderate -0.8
  1. \hspace{0pt} [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.]
A coastguard station \(O\) monitors the movements of a small boat.
At 10:00 the boat is at the point \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km }\) relative to \(O\).
At 12:45 the boat is at the point \(( - 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km }\) relative to \(O\).
The motion of the boat is modelled as that of a particle moving in a straight line at constant speed.
  1. Calculate the bearing on which the boat is moving, giving your answer in degrees to one decimal place.
  2. Calculate the speed of the boat, giving your answer in \(\mathrm { kmh } ^ { - 1 }\)
Edexcel AS Paper 1 2020 June Q3
6 marks Moderate -0.8
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. Solve the equation $$x \sqrt { 2 } - \sqrt { 18 } = x$$ writing the answer as a surd in simplest form.
  2. Solve the equation $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$
Edexcel AS Paper 1 2020 June Q4
6 marks Easy -1.2
  1. In 1997 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK was \(190 \mathrm {~g} / \mathrm { km }\).
In 2005 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK had fallen to \(169 \mathrm {~g} / \mathrm { km }\).
Given \(\mathrm { Ag } / \mathrm { km }\) is the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK \(n\) years after 1997 and using a linear model,
  1. form an equation linking \(A\) with \(n\). In 2016 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK was \(120 \mathrm {~g} / \mathrm { km }\).
  2. Comment on the suitability of your model in light of this information.
Edexcel AS Paper 1 2020 June Q5
6 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bcbd842f-b2e2-4587-ab4c-15a57a449e5d-10_360_1164_260_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the design for a structure used to support a roof.
The structure consists of four steel beams, \(A B , B D , B C\) and \(A D\).
Given \(A B = 12 \mathrm {~m} , B C = B D = 7 \mathrm {~m}\) and angle \(B A C = 27 ^ { \circ }\)
  1. find, to one decimal place, the size of angle \(A C B\). The steel beams can only be bought in whole metre lengths.
  2. Find the minimum length of steel that needs to be bought to make the complete structure.
Edexcel AS Paper 1 2020 June Q6
6 marks Moderate -0.3
  1. (a) Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of
$$( 1 + k x ) ^ { 10 }$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. Given that in the expansion of \(( 1 + k x ) ^ { 10 }\) the coefficient \(x ^ { 3 }\) is 3 times the coefficient of \(x\), (b) find the possible values of \(k\).
Edexcel AS Paper 1 2020 June Q7
8 marks Moderate -0.3
  1. Given that \(k\) is a positive constant and \(\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4\)
    1. show that \(3 k + 5 \sqrt { k } - 12 = 0\)
    2. Hence, using algebra, find any values of \(k\) such that
    $$\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4$$
Edexcel AS Paper 1 2020 June Q8
9 marks Moderate -0.8
  1. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of a cup of tea \(t\) minutes after it was placed on a table in a room, is modelled by the equation
$$\theta = 18 + 65 \mathrm { e } ^ { - \frac { t } { 8 } } \quad t \geqslant 0$$ Find, according to the model,
  1. the temperature of the cup of tea when it was placed on the table,
  2. the value of \(t\), to one decimal place, when the temperature of the cup of tea was \(35 ^ { \circ } \mathrm { C }\).
  3. Explain why, according to this model, the temperature of the cup of tea could not fall to \(15 ^ { \circ } \mathrm { C }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bcbd842f-b2e2-4587-ab4c-15a57a449e5d-16_675_951_973_573} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The temperature, \(\mu ^ { \circ } \mathrm { C }\), of a second cup of tea \(t\) minutes after it was placed on a table in a different room, is modelled by the equation $$\mu = A + B \mathrm { e } ^ { - \frac { t } { 8 } } \quad t \geqslant 0$$ where \(A\) and \(B\) are constants.
    Figure 2 shows a sketch of \(\mu\) against \(t\) with two data points that lie on the curve.
    The line \(l\), also shown on Figure 2, is the asymptote to the curve.
    Using the equation of this model and the information given in Figure 2
  4. find an equation for the asymptote \(l\).
Edexcel AS Paper 1 2020 June Q9
8 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bcbd842f-b2e2-4587-ab4c-15a57a449e5d-20_810_1214_255_427} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of the curve with equation \(y = 3 \cos x ^ { \circ }\).
The point \(P ( c , d )\) is a minimum point on the curve with \(c\) being the smallest negative value of \(x\) at which a minimum occurs.
  1. State the value of \(c\) and the value of \(d\).
  2. State the coordinates of the point to which \(P\) is mapped by the transformation which transforms the curve with equation \(y = 3 \cos x ^ { \circ }\) to the curve with equation
    1. \(y = 3 \cos \left( \frac { x ^ { \circ } } { 4 } \right)\)
    2. \(y = 3 \cos ( x - 36 ) ^ { \circ }\)
  3. Solve, for \(450 ^ { \circ } \leqslant \theta < 720 ^ { \circ }\), $$3 \cos \theta = 8 \tan \theta$$ giving your solution to one decimal place.
    In part (c) you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable.
Edexcel AS Paper 1 2020 June Q10
10 marks Standard +0.3
10. $$g ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 41 x - 70$$
  1. Use the factor theorem to show that \(\mathrm { g } ( x )\) is divisible by \(( x - 5 )\).
  2. Hence, showing all your working, write \(\mathrm { g } ( x )\) as a product of three linear factors. The finite region \(R\) is bounded by the curve with equation \(y = \mathrm { g } ( x )\) and the \(x\)-axis, and lies below the \(x\)-axis.
  3. Find, using algebraic integration, the exact value of the area of \(R\).
Edexcel AS Paper 1 2020 June Q11
9 marks Moderate -0.3
  1. (i) A circle \(C _ { 1 }\) has equation
$$x ^ { 2 } + y ^ { 2 } + 18 x - 2 y + 30 = 0$$ The line \(l\) is the tangent to \(C _ { 1 }\) at the point \(P ( - 5,7 )\).
Find an equation of \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.
(ii) A different circle \(C _ { 2 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 8 x + 12 y + k = 0$$ where \(k\) is a constant.
Given that \(C _ { 2 }\) lies entirely in the 4th quadrant, find the range of possible values for \(k\).
Edexcel AS Paper 1 2020 June Q12
7 marks Moderate -0.3
  1. An advertising agency is monitoring the number of views of an online advert.
The equation $$\log _ { 10 } V = 0.072 t + 2.379 \quad 1 \leqslant t \leqslant 30 , t \in \mathbb { N }$$ is used to model the total number of views of the advert, \(V\), in the first \(t\) days after the advert went live.
  1. Show that \(V = a b ^ { t }\) where \(a\) and \(b\) are constants to be found. Give the value of \(a\) to the nearest whole number and give the value of \(b\) to 3 significant figures.
  2. Interpret, with reference to the model, the value of \(a b\). Using this model, calculate
  3. the total number of views of the advert in the first 20 days after the advert went live. Give your answer to 2 significant figures.
Edexcel AS Paper 1 2020 June Q13
5 marks Moderate -0.3
  1. (a) Prove that for all positive values of \(a\) and \(b\)
$$\frac { 4 a } { b } + \frac { b } { a } \geqslant 4$$ (b) Prove, by counter example, that this is not true for all values of \(a\) and \(b\).