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 2023 June Q17
  1. In this question \(p\) and \(q\) are positive integers with \(q > p\)
Statement 1: \(q ^ { 3 } - p ^ { 3 }\) is never a multiple of 5
  1. Show, by means of a counter example, that Statement 1 is not true. Statement 2: When \(p\) and \(q\) are consecutive even integers \(q ^ { 3 } - p ^ { 3 }\) is a multiple of 8
  2. Prove, using algebra, that Statement 2 is true.
Edexcel AS Paper 1 2024 June Q1
  1. Find
$$\int \frac { 2 \sqrt { x } - 3 } { x ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form.
Edexcel AS Paper 1 2024 June Q2
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = 2 x ^ { 3 } - 3 a x ^ { 2 } + b x + 8 a$$ where \(a\) and \(b\) are constants.
Given that ( \(x - 4\) ) is a factor of \(\mathrm { f } ( x )\),
  1. use the factor theorem to show that $$10 a = 32 + b$$ Given also that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\),
  2. express \(\mathrm { f } ( x )\) in the form $$f ( x ) = ( 2 x + k ) ( x - 4 ) ( x - 2 )$$ where \(k\) is a constant to be found.
  3. Hence,
    1. state the number of real roots of the equation \(\mathrm { f } ( x ) = 0\)
    2. write down the largest root of the equation \(\mathrm { f } \left( \frac { 1 } { 3 } x \right) = 0\)
Edexcel AS Paper 1 2024 June Q3
  1. Relative to a fixed origin \(O\),
  • point \(P\) has position vector \(9 \mathbf { i } - 8 \mathbf { j }\)
  • point \(Q\) has position vector \(3 \mathbf { i } - 5 \mathbf { j }\)
    1. Find \(\overrightarrow { P Q }\)
Given that \(R\) is the point such that \(\overrightarrow { Q R } = 9 \mathbf { i } + 18 \mathbf { j }\)
  • show that angle \(P Q R = 90 ^ { \circ }\) Given also that \(S\) is the point such that \(\overrightarrow { P S } = 3 \overrightarrow { Q R }\)
  • find the exact area of \(P Q R S\)
    •
    Edexcel AS Paper 1 2024 June Q4
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-10_547_1475_306_294} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of triangle \(A B D\) and triangle \(B C D\)
    Given that
    • \(A D C\) is a straight line
    • \(B D = ( x + 3 ) \mathrm { cm }\)
    • \(B C = x \mathrm {~cm}\)
    • angle \(B D C = 30 ^ { \circ }\)
    • angle \(B C D = 140 ^ { \circ }\)
      1. show that \(x = 10.5\) correct to 3 significant figures.
    Given also that \(A D = ( x - 2 ) \mathrm { cm }\)
  • find the length of \(A B\), giving your answer to 3 significant figures.
    •
    Edexcel AS Paper 1 2024 June Q5
    1. The curve \(C _ { 1 }\) has equation
    $$y = \frac { 6 } { x } + 3$$
      1. Sketch \(C _ { 1 }\) stating the coordinates of any points where the curve cuts the coordinate axes.
      2. State the equations of any asymptotes to the curve \(C _ { 1 }\) The curve \(C _ { 2 }\) has equation $$y = 3 x ^ { 2 } - 4 x - 10$$
    2. Show that \(C _ { 1 }\) and \(C _ { 2 }\) intersect when $$3 x ^ { 3 } - 4 x ^ { 2 } - 13 x - 6 = 0$$ Given that the \(x\) coordinate of one of the points of intersection is \(- \frac { 2 } { 3 }\)
    3. use algebra to find the \(x\) coordinates of the other points of intersection between \(C _ { 1 }\) and \(C _ { 2 }\)
      (Solutions relying on calculator technology are not acceptable.)
    Edexcel AS Paper 1 2024 June Q6
    1. The binomial expansion of
    $$( 1 + a x ) ^ { 12 }$$ up to and including the term in \(x ^ { 2 }\) is $$1 - \frac { 15 } { 2 } x + k x ^ { 2 }$$ where \(a\) and \(k\) are constants.
    1. Show that \(a = - \frac { 5 } { 8 }\)
    2. Hence find the value of \(k\) Using the expansion and making your method clear,
    3. find an estimate for the value of \(\left( \frac { 17 } { 16 } \right) ^ { 12 }\), giving your answer to 4 decimal places.
    Edexcel AS Paper 1 2024 June Q7
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-18_614_878_296_555} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A chimney emits smoke particles.
    On a particular day, the concentration of smoke particles in the air emitted by this chimney, \(P\) parts per million, is measured at various distances, \(x \mathrm {~km}\), from the chimney. Figure 2 shows a sketch of the linear relationship between \(\log _ { 10 } P\) and \(x\) that is used to model this situation. The line passes through the point ( \(0,3.3\) ) and the point ( \(6,2.1\) )
    1. Find a complete equation for the model in the form $$P = a b ^ { x }$$ where \(a\) and \(b\) are constants. Give the value of \(a\) and the value of \(b\) each to 4 significant figures.
    2. With reference to the model, interpret the value of \(a b\)
    Edexcel AS Paper 1 2024 June Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-20_915_924_303_580} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of the curve \(C\) with equation $$y = x ^ { 3 } - 14 x + 23$$ The line \(l\) is the tangent to \(C\) at the point \(A\), also shown in Figure 3.
    Given that \(l\) has equation \(y = - 2 x + 7\)
    1. show, using calculus, that the \(x\) coordinate of \(A\) is 2 The line \(l\) cuts \(C\) again at the point \(B\).
    2. Verify that the \(x\) coordinate of \(B\) is - 4 The finite region, \(R\), shown shaded in Figure 3, is bounded by \(C\) and \(l\).
      Using algebraic integration,
    3. show that the area of \(R\) is 108
    Edexcel AS Paper 1 2024 June Q9
    9. $$\begin{aligned} p & = \log _ { a } 16
    q & = \log _ { a } 25 \end{aligned}$$ where \(a\) is a constant.
    Find in terms of \(p\) and/or \(q\),
    1. \(\log _ { a } 256\)
    2. \(\log _ { a } 100\)
    3. \(\log _ { a } 80 \times \log _ { a } 3.2\)
    Edexcel AS Paper 1 2024 June Q10
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-26_748_764_296_646} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of the circle \(C\)
    • the point \(P ( - 1 , k + 8 )\) is the centre of \(C\)
    • the point \(Q \left( 3 , k ^ { 2 } - 2 k \right)\) lies on \(C\)
    • \(k\) is a positive constant
    • the line \(l\) is the tangent to \(C\) at \(Q\)
    Given that the gradient of \(l\) is - 2
    1. show that $$k ^ { 2 } - 3 k - 10 = 0$$
    2. Hence find an equation for \(C\)
    Edexcel AS Paper 1 2024 June Q11
    1. The prices of two precious metals are being monitored.
    The price per gram of metal \(A , \pounds V _ { A }\), is modelled by the equation $$V _ { A } = 100 + 20 \mathrm { e } ^ { 0.04 t }$$ where \(t\) is the number of months after monitoring began.
    The price per gram of metal \(B , \pounds V _ { B }\), is modelled by the equation $$V _ { B } = p \mathrm { e } ^ { - 0.02 t }$$ where \(p\) is a positive constant and \(t\) is the number of months after monitoring began.
    Given that \(V _ { B } = 2 V _ { A }\) when \(t = 0\)
    1. find the value of \(p\) When \(t = T\), the rate of increase in the price per gram of metal \(A\) was equal to the rate of decrease in the price per gram of metal \(B\)
    2. Find the value of \(T\), giving your answer to one decimal place.
      (Solutions based entirely on calculator technology are not acceptable.)
    Edexcel AS Paper 1 2024 June Q12
    12. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-34_494_499_306_778} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} Figure 5 shows the plan view of the design for a swimming pool.
    The pool is modelled as a quarter of a circle joined to two equal sized rectangles as shown. Given that
    • the quarter circle has radius \(x\) metres
    • the rectangles each have length \(x\) metres and width \(y\) metres
    • the total surface area of the swimming pool is \(100 \mathrm {~m} ^ { 2 }\)
      1. show that, according to the model, the perimeter \(P\) metres of the swimming pool is given by
    $$P = 2 x + \frac { 200 } { x }$$
  • Use calculus to find the value of \(x\) for which \(P\) has a stationary value.
  • Prove, by further calculus, that this value of \(x\) gives a minimum value for \(P\) Access to the pool is by side \(A B\) shown in Figure 5.
    Given that \(A B\) must be at least one metre,
  • determine, according to the model, whether the swimming pool with the minimum perimeter would be suitable.
    •
    Edexcel AS Paper 1 2024 June Q13
    1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
      1. Show that the equation
      $$\sin \theta ( 7 \sin \theta - 4 \cos \theta ) = 4$$ can be written as $$3 \tan ^ { 2 } \theta - 4 \tan \theta - 4 = 0$$
    2. Hence solve, for \(0 < x < 360 ^ { \circ }\) $$\sin x ( 7 \sin x - 4 \cos x ) = 4$$ giving your answers to one decimal place.
    3. Hence find the smallest solution of the equation $$\sin 4 \alpha ( 7 \sin 4 \alpha - 4 \cos 4 \alpha ) = 4$$ in the range \(720 ^ { \circ } < \alpha < 1080 ^ { \circ }\), giving your answer to one decimal place.
    Edexcel AS Paper 1 2024 June Q14
    1. Prove, using algebra, that
    $$n ^ { 2 } + 5 n$$ is even for all \(n \in \mathbb { N }\)
    Edexcel AS Paper 1 2021 November Q1
    1. In this question you should show all stages of your working.
    Solutions relying on calculator technology are not acceptable.
    Using algebra, solve the inequality $$x ^ { 2 } - x > 20$$ writing your answer in set notation.
    Edexcel AS Paper 1 2021 November Q2
    1. In this question you should show all stages of your working.
    Solutions relying on calculator technology are not acceptable.
    Given $$\frac { 9 ^ { x - 1 } } { 3 ^ { y + 2 } } = 81$$ express \(y\) in terms of \(x\), writing your answer in simplest form.
    Edexcel AS Paper 1 2021 November Q3
    1. Find
    $$\int \frac { 3 x ^ { 4 } - 4 } { 2 x ^ { 3 } } d x$$ writing your answer in simplest form.
    Edexcel AS Paper 1 2021 November Q4
    1. \hspace{0pt} [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.]
    A stone slides horizontally across ice.
    Initially the stone is at the point \(A ( - 24 \mathbf { i } - 10 \mathbf { j } ) \mathrm { m }\) relative to a fixed point \(O\).
    After 4 seconds the stone is at the point \(B ( 12 \mathbf { i } + 5 \mathbf { j } )\) m relative to the fixed point \(O\).
    The motion of the stone is modelled as that of a particle moving in a straight line at constant speed. Using the model,
    1. prove that the stone passes through \(O\),
    2. calculate the speed of the stone.
    Edexcel AS Paper 1 2021 November Q5
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{235cd1dc-a3ab-473a-bf77-3e41b274dfd8-10_680_684_255_694} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows part of the curve with equation \(y = 3 x ^ { 2 } - 2\)
    The point \(P ( 2,10 )\) lies on the curve.
    1. Find the gradient of the tangent to the curve at \(P\). The point \(Q\) with \(x\) coordinate \(2 + h\) also lies on the curve.
    2. Find the gradient of the line \(P Q\), giving your answer in terms of \(h\) in simplest form.
    3. Explain briefly the relationship between part (b) and the answer to part (a).
    Edexcel AS Paper 1 2021 November Q6
    1. In this question you should show all stages of your working.
    Solutions relying on calculator technology are not acceptable.
    1. Using algebra, find all solutions of the equation $$3 x ^ { 3 } - 17 x ^ { 2 } - 6 x = 0$$
    2. Hence find all real solutions of $$3 ( y - 2 ) ^ { 6 } - 17 ( y - 2 ) ^ { 4 } - 6 ( y - 2 ) ^ { 2 } = 0$$
    Edexcel AS Paper 1 2021 November Q7
    1. A parallelogram \(P Q R S\) has area \(50 \mathrm {~cm} ^ { 2 }\)
    Given
    • \(P Q\) has length 14 cm
    • \(Q R\) has length 7 cm
    • angle \(S P Q\) is obtuse
      find
      1. the size of angle \(S P Q\), in degrees, to 2 decimal places,
      2. the length of the diagonal \(S Q\), in cm , to one decimal place.
    Edexcel AS Paper 1 2021 November Q8
    8. $$g ( x ) = ( 2 + a x ) ^ { 8 } \quad \text { where } a \text { is a constant }$$ Given that one of the terms in the binomial expansion of \(\mathrm { g } ( x )\) is \(3402 x ^ { 5 }\)
    1. find the value of \(a\). Using this value of \(a\),
    2. find the constant term in the expansion of $$\left( 1 + \frac { 1 } { x ^ { 4 } } \right) ( 2 + a x ) ^ { 8 }$$
    Edexcel AS Paper 1 2021 November Q9
    1. Find the value of the constant \(k , 0 < k < 9\), such that
    $$\int _ { k } ^ { 9 } \frac { 6 } { \sqrt { x } } \mathrm {~d} x = 20$$
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    Edexcel AS Paper 1 2021 November Q10
    1. A student is investigating the following statement about natural numbers.
    \begin{displayquote} " \(n ^ { 3 } - n\) is a multiple of 4 "
    1. Prove, using algebra, that the statement is true for all odd numbers.
    2. Use a counterexample to show that the statement is not always true. \end{displayquote}