Questions AS Paper 1 (363 questions)

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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
    $$\int _ { k } ^ { 9 } \frac { 6 } { \sqrt { x } } \mathrm {~d} x = 20$$
    VI4V SIHIL NI III HM IONOOVIAV SIHI NI III M M O N OOVIIIV SIHI NI IIIIM I I ON OC
    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}
    Edexcel AS Paper 1 2021 November Q11
    1. The owners of a nature reserve decided to increase the area of the reserve covered by trees.
    Tree planting started on 1st January 2005.
    The area of the nature reserve covered by trees, \(A \mathrm {~km} ^ { 2 }\), is modelled by the equation $$A = 80 - 45 \mathrm { e } ^ { c t }$$ where \(c\) is a constant and \(t\) is the number of years after 1st January 2005.
    Using the model,
    1. find the area of the nature reserve that was covered by trees just before tree planting started. On 1st January 2019 an area of \(60 \mathrm {~km} ^ { 2 }\) of the nature reserve was covered by trees.
    2. Use this information to find a complete equation for the model, giving your value of \(c\) to 3 significant figures. On 1st January 2020, the owners of the nature reserve announced a long-term plan to have \(100 \mathrm {~km} ^ { 2 }\) of the nature reserve covered by trees.
    3. State a reason why the model is not appropriate for this plan.
    Edexcel AS Paper 1 2021 November Q12
    1. In this question you should show all stages of your working.
    \section*{Solutions relying entirely on calculator technology are not acceptable.}
    1. Solve, for \(0 < \theta \leqslant 450 ^ { \circ }\), the equation $$5 \cos ^ { 2 } \theta = 6 \sin \theta$$ giving your answers to one decimal place.
    2. (a) A student's attempt to solve the question
      "Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation \(3 \tan x - 5 \sin x = 0\) " is set out below. $$\begin{gathered} 3 \tan x - 5 \sin x = 0
      3 \frac { \sin x } { \cos x } - 5 \sin x = 0
      3 \sin x - 5 \sin x \cos x = 0
      3 - 5 \cos x = 0
      \cos x = \frac { 3 } { 5 }
      x = 53.1 ^ { \circ } \end{gathered}$$ Identify two errors or omissions made by this student, giving a brief explanation of each. The first four positive solutions, in order of size, of the equation $$\cos \left( 5 \alpha + 40 ^ { \circ } \right) = \frac { 3 } { 5 }$$ are \(\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 }\) and \(\alpha _ { 4 }\)
      (b) Find, to the nearest degree, the value of \(\alpha _ { 4 }\)
    Edexcel AS Paper 1 2021 November Q13
    13. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{235cd1dc-a3ab-473a-bf77-3e41b274dfd8-30_549_709_251_621} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The resting heart rate, \(h\), of a mammal, measured in beats per minute, is modelled by the equation $$h = p m ^ { q }$$ where \(p\) and \(q\) are constants and \(m\) is the mass of the mammal measured in kg .
    Figure 2 illustrates the linear relationship between \(\log _ { 10 } h\) and \(\log _ { 10 } m\)
    The line meets the vertical \(\log _ { 10 } h\) axis at 2.25 and has a gradient of - 0.235
    1. Find, to 3 significant figures, the value of \(p\) and the value of \(q\). A particular mammal has a mass of 5 kg and a resting heart rate of 119 beats per minute.
    2. Comment on the suitability of the model for this mammal.
    3. With reference to the model, interpret the value of the constant \(p\).
    Edexcel AS Paper 1 2021 November Q14
    1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where
    $$f ( x ) = - 3 x ^ { 2 } + 12 x + 8$$
    1. Write \(\mathrm { f } ( x )\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. The curve \(C\) has a maximum turning point at \(M\).
    2. Find the coordinates of \(M\). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{235cd1dc-a3ab-473a-bf77-3e41b274dfd8-34_735_841_913_612} \captionsetup{labelformat=empty} \caption{Figure 3}
      \end{figure} Figure 3 shows a sketch of the curve \(C\).
      The line \(l\) passes through \(M\) and is parallel to the \(x\)-axis.
      The region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(y\)-axis.
    3. Using algebraic integration, find the area of \(R\).
    Edexcel AS Paper 1 2021 November Q15
    15. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{235cd1dc-a3ab-473a-bf77-3e41b274dfd8-38_655_929_248_568} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of a circle \(C\) with centre \(N ( 7,4 )\)
    The line \(l\) with equation \(y = \frac { 1 } { 3 } x\) is a tangent to \(C\) at the point \(P\).
    Find
    1. the equation of line \(P N\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
    2. an equation for \(C\). The line with equation \(y = \frac { 1 } { 3 } x + k\), where \(k\) is a non-zero constant, is also a tangent to \(C\).
    3. Find the value of \(k\).
    Edexcel AS Paper 1 2021 November Q16
    1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where
    $$f ( x ) = a x ^ { 3 } + 15 x ^ { 2 } - 39 x + b$$ and \(a\) and \(b\) are constants.
    Given
    • the point \(( 2,10 )\) lies on \(C\)
    • the gradient of the curve at \(( 2,10 )\) is - 3
      1. (i) show that the value of \(a\) is - 2
        (ii) find the value of \(b\).
      2. Hence show that \(C\) has no stationary points.
      3. Write \(\mathrm { f } ( x )\) in the form \(( x - 4 ) \mathrm { Q } ( x )\) where \(\mathrm { Q } ( x )\) is a quadratic expression to be found.
      4. Hence deduce the coordinates of the points of intersection of the curve with equation
    $$y = \mathrm { f } ( 0.2 x )$$ and the coordinate axes.
    Edexcel AS Paper 1 Specimen Q1
    1. The line \(l\) passes through the points \(A ( 3,1 )\) and \(B ( 4 , - 2 )\).
    Find an equation for \(l\).
    Edexcel AS Paper 1 Specimen Q2
    2. The curve \(C\) has equation $$y = 2 x ^ { 2 } - 12 x + 16$$ Find the gradient of the curve at the point \(P ( 5,6 )\).
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
    Edexcel AS Paper 1 Specimen Q3
    3. Given that the point \(A\) has position vector \(3 \mathbf { i } - 7 \mathbf { j }\) and the point \(B\) has position vector \(8 \mathbf { i } + 3 \mathbf { j }\),
    1. find the vector \(\overrightarrow { A B }\)
    2. Find \(| \overrightarrow { A B } |\). Give your answer as a simplified surd.