Questions AS Paper 1 (363 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 2018 June Q10
  1. Prove, from first principles, that the derivative of \(x ^ { 3 }\) is \(3 x ^ { 2 }\)
Edexcel AS Paper 1 2018 June Q11
  1. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 2 - \frac { x } { 16 } \right) ^ { 9 }$$ giving each term in its simplest form. $$f ( x ) = ( a + b x ) \left( 2 - \frac { x } { 16 } \right) ^ { 9 } , \text { where } a \text { and } b \text { are constants }$$ Given that the first two terms, in ascending powers of \(x\), in the series expansion of \(\mathrm { f } ( x )\) are 128 and \(36 x\),
(b) find the value of \(a\),
(c) find the value of \(b\).
Edexcel AS Paper 1 2018 June Q12
  1. (a) Show that the equation
$$4 \cos \theta - 1 = 2 \sin \theta \tan \theta$$ can be written in the form $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ (b) Hence solve, for \(0 \leqslant x < 90 ^ { \circ }\) $$4 \cos 3 x - 1 = 2 \sin 3 x \tan 3 x$$ giving your answers, where appropriate, to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel AS Paper 1 2018 June Q13
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-36_563_1019_244_523} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The value of a rare painting, \(\pounds V\), is modelled by the equation \(V = p q ^ { t }\), where \(p\) and \(q\) are constants and \(t\) is the number of years since the value of the painting was first recorded on 1st January 1980. The line \(l\) shown in Figure 3 illustrates the linear relationship between \(t\) and \(\log _ { 10 } V\) since 1st January 1980. The equation of line \(l\) is \(\log _ { 10 } V = 0.05 t + 4.8\)
  1. Find, to 4 significant figures, the value of \(p\) and the value of \(q\).
  2. With reference to the model interpret
    1. the value of the constant \(p\),
    2. the value of the constant \(q\).
  3. Find the value of the painting, as predicted by the model, on 1st January 2010, giving your answer to the nearest hundred thousand pounds.
Edexcel AS Paper 1 2018 June Q14
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0$$
  1. Find
    1. the coordinates of the centre of \(C\)
    2. the radius of \(C\) The line with equation \(y = k x\), where \(k\) is a constant, cuts \(C\) at two distinct points.
  2. Find the range of values for \(k\).
Edexcel AS Paper 1 2018 June Q15
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-44_595_977_242_536} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 32 } { x ^ { 2 } } + 3 x - 8 , \quad x > 0$$ The point \(P ( 4,6 )\) lies on \(C\).
The line \(l\) is the normal to \(C\) at the point \(P\).
The region \(R\), shown shaded in Figure 4, is bounded by the line \(l\), the curve \(C\), the line with equation \(x = 2\) and the \(x\)-axis. Show that the area of \(R\) is 46
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel AS Paper 1 2019 June Q1
  1. The line \(l _ { 1 }\) has equation \(2 x + 4 y - 3 = 0\)
The line \(l _ { 2 }\) has equation \(y = m x + 7\), where \(m\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular,
  1. find the value of \(m\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\).
  2. Find the \(x\) coordinate of \(P\).
    \includegraphics[max width=\textwidth, alt={}, center]{deba6a2b-1821-4110-bde8-bde18a5f9be9-02_2258_48_313_1980}
Edexcel AS Paper 1 2019 June Q2
  1. Find, using algebra, all real solutions to the equation
    1. \(16 a ^ { 2 } = 2 \sqrt { a }\)
    2. \(b ^ { 4 } + 7 b ^ { 2 } - 18 = 0\)
Edexcel AS Paper 1 2019 June Q3
  1. (a) Given that \(k\) is a constant, find
$$\int \left( \frac { 4 } { x ^ { 3 } } + k x \right) \mathrm { d } x$$ simplifying your answer.
(b) Hence find the value of \(k\) such that $$\int _ { 0.5 } ^ { 2 } \left( \frac { 4 } { x ^ { 3 } } + k x \right) \mathrm { d } x = 8$$
Edexcel AS Paper 1 2019 June Q4
  1. A tree was planted in the ground.
Its height, \(H\) metres, was measured \(t\) years after planting.
Exactly 3 years after planting, the height of the tree was 2.35 metres.
Exactly 6 years after planting, the height of the tree was 3.28 metres.
Using a linear model,
  1. find an equation linking \(H\) with \(t\). The height of the tree was approximately 140 cm when it was planted.
  2. Explain whether or not this fact supports the use of the linear model in part (a).
Edexcel AS Paper 1 2019 June Q5
  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. 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
  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
  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
  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
  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
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 $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$