Questions — Edexcel AS Paper 1 (150 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel AS Paper 1 2019 June Q11
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
  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
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
  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
  1. Given \(n \in \mathbb { N }\), prove that \(n ^ { 3 } + 2\) is not divisible by 8
Edexcel AS Paper 1 2019 June Q16
  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
  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
  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
  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
  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
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
  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
  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
  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
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. $$g ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 41 x - 70$$
  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
  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
  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
  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 \(b\).
Edexcel AS Paper 1 2020 June Q14
  1. A curve has equation \(y = \mathrm { g } ( x )\).
Given that
  • \(\mathrm { g } ( x )\) is a cubic expression in which the coefficient of \(x ^ { 3 }\) is equal to the coefficient of \(x\)
  • the curve with equation \(y = \mathrm { g } ( x )\) passes through the origin
  • the curve with equation \(y = \mathrm { g } ( x )\) has a stationary point at \(( 2,9 )\)
    1. find \(\mathrm { g } ( x )\),
    2. prove that the stationary point at \(( 2,9 )\) is a maximum.
Edexcel AS Paper 1 2022 June Q1
  1. Find
$$\int \left( 8 x ^ { 3 } - \frac { 3 } { 2 \sqrt { x } } + 5 \right) \mathrm { d } x$$ giving your answer in simplest form.
Edexcel AS Paper 1 2022 June Q2
2. $$f ( x ) = 2 x ^ { 3 } + 5 x ^ { 2 } + 2 x + 15$$
  1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Find the constants \(a\), \(b\) and \(c\) such that $$f ( x ) = ( x + 3 ) \left( a x ^ { 2 } + b x + c \right)$$
  3. Hence show that \(\mathrm { f } ( x ) = 0\) has only one real root.
  4. Write down the real root of the equation \(\mathrm { f } ( x - 5 ) = 0\)
Edexcel AS Paper 1 2022 June Q3
  1. The triangle \(P Q R\) is such that \(\overrightarrow { P Q } = 3 \mathbf { i } + 5 \mathbf { j }\) and \(\overrightarrow { P R } = 13 \mathbf { i } - 15 \mathbf { j }\)
    1. Find \(\overrightarrow { Q R }\)
    2. Hence find \(| \overrightarrow { Q R } |\) giving your answer as a simplified surd.
    The point \(S\) lies on the line segment \(Q R\) so that \(Q S : S R = 3 : 2\)
  2. Find \(\overrightarrow { P S }\)
Edexcel AS Paper 1 2022 June Q4
4. Figure 1 Figure 1 shows a sketch of triangle \(A B C\) with \(A B = ( x + 2 ) \mathrm { cm } , B C = ( 3 x + 10 ) \mathrm { cm }\), \(A C = 7 x \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A C B = \theta ^ { \circ }\)
    1. Show that \(17 x ^ { 2 } - 35 x - 48 = 0\)
    2. Hence find the value of \(x\).
  2. Hence find the value of \(\theta\) giving your answer to one decimal place.
Edexcel AS Paper 1 2022 June Q5
  1. The mass, \(A\) kg, of algae in a small pond, is modelled by the equation
$$A = p q ^ { t }$$ where \(p\) and \(q\) are constants and \(t\) is the number of weeks after the mass of algae was first recorded. Data recorded indicates that there is a linear relationship between \(t\) and \(\log _ { 10 } A\) given by the equation $$\log _ { 10 } A = 0.03 t + 0.5$$
  1. Use this relationship to find a complete equation for the model in the form $$A = p q ^ { t }$$ giving the value of \(p\) and the value of \(q\) each to 4 significant figures.
  2. With reference to the model, interpret
    1. the value of the constant \(p\),
    2. the value of the constant \(q\).
  3. Find, according to the model,
    1. the mass of algae in the pond when \(t = 8\), giving your answer to the nearest 0.5 kg ,
    2. the number of weeks it takes for the mass of algae in the pond to reach 4 kg .
  4. State one reason why this may not be a realistic model in the long term.