Questions — Edexcel AS Paper 1 (152 questions)

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Edexcel AS Paper 1 2021 November Q11
6 marks Moderate -0.3
  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
9 marks Standard +0.3
  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
7 marks Standard +0.3
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
10 marks Standard +0.3
  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
9 marks Standard +0.3
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
11 marks Standard +0.3
  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
6 marks Moderate -0.8
  1. A curve has equation
$$y = 2 x ^ { 3 } - 2 x ^ { 2 } - 2 x + 8$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation.
    VIIIV SIHI NI JIIYM IONOOVIUV SIHI NI JIIAM ION OOVI4V SIHI NI JIIIM I ON OO
Edexcel AS Paper 1 Specimen Q2
4 marks Moderate -0.8
  1. The quadrilateral \(O A B C\) has \(\overrightarrow { O A } = 4 \mathbf { i } + 2 \mathbf { j } , \overrightarrow { O B } = 6 \mathbf { i } - 3 \mathbf { j }\) and \(\overrightarrow { O C } = 8 \mathbf { i } - 20 \mathbf { j }\).
    1. Find \(\overrightarrow { A B }\).
    2. Show that quadrilateral \(O A B C\) is a trapezium.
Edexcel AS Paper 1 Specimen Q3
8 marks Easy -1.3
  1. A tank, which contained water, started to leak from a hole in its base.
The volume of water in the tank 24 minutes after the leak started was \(4 \mathrm {~m} ^ { 3 }\) The volume of water in the tank 60 minutes after the leak started was \(2.8 \mathrm {~m} ^ { 3 }\) The volume of water, \(V \mathrm {~m} ^ { 3 }\), in the tank \(t\) minutes after the leak started, can be described by a linear model between \(V\) and \(t\).
  1. Find an equation linking \(V\) with \(t\). Use this model to find
    1. the initial volume of water in the tank,
    2. the time taken for the tank to empty.
  3. Suggest a reason why this linear model may not be suitable.
Edexcel AS Paper 1 Specimen Q4
4 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa7abe9f-f5c0-4578-afd1-73176c717536-08_755_775_248_662} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = g ( x )\).
The curve has a single turning point, a minimum, at the point \(M ( 4 , - 1.5 )\).
The curve crosses the \(x\)-axis at two points, \(P ( 2,0 )\) and \(Q ( 7,0 )\).
The curve crosses the \(y\)-axis at a single point \(R ( 0,5 )\).
  1. State the coordinates of the turning point on the curve with equation \(y = 2 \mathrm {~g} ( x )\).
  2. State the largest root of the equation $$g ( x + 1 ) = 0$$
  3. State the range of values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) \leqslant 0\) Given that the equation \(\mathrm { g } ( x ) + k = 0\), where \(k\) is a constant, has no real roots,
  4. state the range of possible values for \(k\).
Edexcel AS Paper 1 Specimen Q5
8 marks Moderate -0.3
5. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12$$
  1. Using the factor theorem, explain why \(\mathrm { f } ( x )\) is divisible by \(( x + 3 )\).
  2. Hence fully factorise \(\mathrm { f } ( x )\).
  3. Show that \(\frac { x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12 } { x ^ { 3 } + 5 x ^ { 2 } + 6 x }\) can be written in the form \(A + \frac { B } { x }\) where \(A\) and \(B\) are integers to be found.
Edexcel AS Paper 1 Specimen Q6
6 marks Moderate -0.8
  1. (i) Use a counter example to show that the following statement is false.
$$" n ^ { 2 } - n - 1 \text { is a prime number, for } 3 \leqslant n \leqslant 10 \text {." }$$ (ii) Prove that the following statement is always true.
"The difference between the cube and the square of an odd number is even."
For example \(5 ^ { 3 } - 5 ^ { 2 } = 100\) is even. \includegraphics[max width=\textwidth, alt={}, center]{fa7abe9f-f5c0-4578-afd1-73176c717536-12_2255_51_314_1978}
Edexcel AS Paper 1 Specimen Q7
8 marks Moderate -0.8
  1. (a) Expand \(\left( 1 + \frac { 3 } { x } \right) ^ { 2 }\) simplifying each term.
    (b) Use the binomial expansion to find, in ascending powers of \(x\), the first four terms in the expansion of
$$\left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$ simplifying each term.
(c) Hence find the coefficient of \(x\) in the expansion of $$\left( 1 + \frac { 3 } { x } \right) ^ { 2 } \left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$
Edexcel AS Paper 1 Specimen Q8
8 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa7abe9f-f5c0-4578-afd1-73176c717536-16_607_983_255_541} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with equation \(y = \sqrt { x } , x \geqslant 0\) The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = a\), where \(a\) is a constant. Given that the area of \(R\) is 10
  1. find, in simplest form, the value of
    1. \(\int _ { 1 } ^ { a } \sqrt { 8 x } \mathrm {~d} x\)
    2. \(\int _ { 0 } ^ { a } \sqrt { x } \mathrm {~d} x\)
  2. show that \(a = 2 ^ { k }\), where \(k\) is a rational constant to be found.
Edexcel AS Paper 1 Specimen Q9
6 marks Moderate -0.3
  1. Find any real values of \(x\) such that
$$2 \log _ { 4 } ( 2 - x ) - \log _ { 4 } ( x + 5 ) = 1$$
Edexcel AS Paper 1 Specimen Q10
8 marks Moderate -0.8
  1. A circle \(C\) has centre \(( 2,5 )\). Given that the point \(P ( - 2,3 )\) lies on \(C\).
    1. find an equation for \(C\).
    The line \(l\) is the tangent to \(C\) at the point \(P\). The point \(Q ( 2 , k )\) lies on \(l\).
  2. Find the value of \(k\).
Edexcel AS Paper 1 Specimen Q11
9 marks Standard +0.3
  1. (i) Solve, for \(- 90 ^ { \circ } \leqslant \theta < 270 ^ { \circ }\), the equation,
$$\sin \left( 2 \theta + 10 ^ { \circ } \right) = - 0.6$$ giving your answers to one decimal place.
(ii) (a) A student's attempt at the question
"Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation \(7 \tan x = 8 \sin x\) " is set out below. $$\begin{gathered} 7 \tan x = 8 \sin x \\ 7 \times \frac { \sin x } { \cos x } = 8 \sin x \\ 7 \sin x = 8 \sin x \cos x \\ 7 = 8 \cos x \\ \cos x = \frac { 7 } { 8 } \\ x = 29.0 ^ { \circ } \text { (to } 3 \text { sf) } \end{gathered}$$ Identify two mistakes made by this student, giving a brief explanation of each mistake.
(b) Find the smallest positive solution to the equation $$7 \tan \left( 4 \alpha + 199 ^ { \circ } \right) = 8 \sin \left( 4 \alpha + 199 ^ { \circ } \right)$$
Edexcel AS Paper 1 Specimen Q12
8 marks Moderate -0.3
12.
[diagram]
Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3 x - 2 \sqrt { x } , x \geqslant 0\) and the line \(l\) with equation \(y = 8 x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.
  1. Using algebra, find the \(x\) coordinate of point \(A\).
  2. [diagram]
    The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\).
Edexcel AS Paper 1 Specimen Q13
8 marks Moderate -0.3
  1. The growth of pond weed on the surface of a pond is being investigated.
The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), can be modelled by the equation $$A = 0.2 \mathrm { e } ^ { 0.3 t }$$ where \(t\) is the number of days after the start of the investigation.
  1. State the surface area of the pond covered by the weed at the start of the investigation.
  2. Find the rate of increase of the surface area of the pond covered by the weed, in \(\mathrm { m } ^ { 2 } /\) day, exactly 5 days after the start of the investigation. Given that the pond has a surface area of \(100 \mathrm {~m} ^ { 2 }\),
  3. find, to the nearest hour, the time taken, according to the model, for the surface of the pond to be fully covered by the weed. The pond is observed for one month and by the end of the month \(90 \%\) of the surface area of the pond was covered by the weed.
  4. Evaluate the model in light of this information, giving a reason for your answer.
Edexcel AS Paper 1 Specimen Q14
9 marks Standard +0.8
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa7abe9f-f5c0-4578-afd1-73176c717536-30_673_819_246_623} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of the curve \(C\) with equation \(y = ( x - 2 ) ^ { 2 } ( x + 3 )\) The region \(R\), shown shaded in Figure 5, is bounded by \(C\), the vertical line passing through the maximum turning point of \(C\) and the \(x\)-axis. Find the exact area of \(R\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel AS Paper 1 Specimen Q1
3 marks Easy -1.2
The line \(l\) passes through the points \(A (3, 1)\) and \(B (4, -2)\). Find an equation for \(l\). [3]
Edexcel AS Paper 1 Specimen Q2
4 marks Easy -1.2
The curve \(C\) has equation $$y = 2x^2 - 12x + 16$$ Find the gradient of the curve at the point \(P (5, 6)\). (Solutions based entirely on graphical or numerical methods are not acceptable.) [4]
Edexcel AS Paper 1 Specimen Q3
4 marks Easy -1.2
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 \(\overrightarrow{AB}\) [2]
  2. Find \(|\overrightarrow{AB}|\). Give your answer as a simplified surd. [2]
Edexcel AS Paper 1 Specimen Q4
6 marks Moderate -0.8
$$f(x) = 4x^3 - 12x^2 + 2x - 6$$
  1. Use the factor theorem to show that \((x - 3)\) is a factor of \(f(x)\). [2]
  2. Hence show that \(3\) is the only real root of the equation \(f(x) = 0\) [4]
Edexcel AS Paper 1 Specimen Q5
5 marks Moderate -0.3
Given that $$f(x) = 2x + 3 + \frac{12}{x^2}, \quad x > 0$$ show that \(\int_1^{2\sqrt{2}} f(x)\,dx = 16 + 3\sqrt{2}\) [5]