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 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.
Edexcel AS Paper 1 Specimen Q4
4. $$f ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } + 2 x - 6$$
  1. Use the factor theorem to show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Hence show that 3 is the only real root of the equation \(\mathrm { f } ( x ) = 0\)
Edexcel AS Paper 1 Specimen Q5
5. Given that
show that \(\int _ { 1 } ^ { 2 \sqrt { 2 } } \mathrm { f } ( x ) \mathrm { d } x = 16 + 3 \sqrt { 2 }\) $$\mathrm { f } ( x ) = 2 x + 3 + \frac { 12 } { x ^ { 2 } } , \quad x > 0$$
Edexcel AS Paper 1 Specimen Q6
  1. Prove, from first principles, that the derivative of \(3 x ^ { 2 }\) is \(6 x\).
  2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(\left( 2 - \frac { x } { 2 } \right) ^ { 7 }\), giving each term in its simplest form.
    (b) Explain how you would use your expansion to give an estimate for the value of \(1.995 ^ { 7 }\)
Edexcel AS Paper 1 Specimen Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b084faa-a680-4f35-bb5c-a4edf5171b5f-10_609_675_262_753} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A triangular lawn is modelled by the triangle \(A B C\), shown in Figure 1. The length \(A B\) is to be 30 m long. Given that angle \(B A C = 70 ^ { \circ }\) and angle \(A B C = 60 ^ { \circ }\),
  1. calculate the area of the lawn to 3 significant figures.
  2. Why is your answer unlikely to be accurate to the nearest square metre?
Edexcel AS Paper 1 Specimen Q9
  1. Solve, for \(360 ^ { \circ } \leqslant x < 540 ^ { \circ }\),
$$12 \sin ^ { 2 } x + 7 \cos x - 13 = 0$$ Give your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(5)
Edexcel AS Paper 1 Specimen Q10
  1. The equation \(k x ^ { 2 } + 4 k x + 3 = 0\), where \(k\) is a constant, has no real roots.
Prove that $$0 \leqslant k < \frac { 3 } { 4 }$$
Edexcel AS Paper 1 Specimen Q11
  1. (a) Prove that for all positive values of \(x\) and \(y\)
$$\sqrt { x y } \leqslant \frac { x + y } { 2 }$$ (b) Prove by counter example that this is not true when \(x\) and \(y\) are both negative.
Edexcel AS Paper 1 Specimen Q12
12. A student was asked to give the exact solution to the equation $$2 ^ { 2 x + 4 } - 9 \left( 2 ^ { x } \right) = 0$$ The student's attempt is shown below: $$\begin{aligned} & 2 ^ { 2 x + 4 } - 9 \left( 2 ^ { x } \right) = 0
& 2 ^ { 2 x } + 2 ^ { 4 } - 9 \left( 2 ^ { x } \right) = 0
& \text { Let } \quad 2 ^ { x } = y
& y ^ { 2 } - 9 y + 8 = 0
& ( y - 8 ) ( y - 1 ) = 0
& y = 8 \text { or } y = 1
& \text { So } x = 3 \text { or } x = 0 \end{aligned}$$
  1. Identify the two errors made by the student.
  2. Find the exact solution to the equation.
Edexcel AS Paper 1 Specimen Q13
  1. (a) Factorise completely \(x ^ { 3 } + 10 x ^ { 2 } + 25 x\)
    (b) Sketch the curve with equation
$$y = x ^ { 3 } + 10 x ^ { 2 } + 25 x$$ showing the coordinates of the points at which the curve cuts or touches the \(x\)-axis. The point with coordinates \(( - 3,0 )\) lies on the curve with equation $$y = ( x + a ) ^ { 3 } + 10 ( x + a ) ^ { 2 } + 25 ( x + a )$$ where \(a\) is a constant.
(c) Find the two possible values of \(a\).
Edexcel AS Paper 1 Specimen Q14
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b084faa-a680-4f35-bb5c-a4edf5171b5f-20_777_1319_315_370} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A town's population, \(P\), is modelled by the equation \(P = a b ^ { t }\), where \(a\) and \(b\) are constants and \(t\) is the number of years since the population was first recorded. The line \(l\) shown in Figure 2 illustrates the linear relationship between \(t\) and \(\log _ { 10 } P\) for the population over a period of 100 years.
The line \(l\) meets the vertical axis at \(( 0,5 )\) as shown. The gradient of \(l\) is \(\frac { 1 } { 200 }\).
  1. Write down an equation for \(l\).
  2. Find the value of \(a\) and the value of \(b\).
  3. With reference to the model interpret
    1. the value of the constant \(a\),
    2. the value of the constant \(b\).
  4. Find
    1. the population predicted by the model when \(t = 100\), giving your answer to the nearest hundred thousand,
    2. the number of years it takes the population to reach 200000 , according to the model.
  5. State two reasons why this may not be a realistic population model.
Edexcel AS Paper 1 Specimen Q15
15. Diagram not drawn to scale \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b084faa-a680-4f35-bb5c-a4edf5171b5f-22_725_844_251_623} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The curve \(C _ { 1 }\), shown in Figure 3, has equation \(y = 4 x ^ { 2 } - 6 x + 4\).
The point \(P \left( \frac { 1 } { 2 } , 2 \right)\) lies on \(C _ { 1 }\)
The curve \(C _ { 2 }\), also shown in Figure 3, has equation \(y = \frac { 1 } { 2 } x + \ln ( 2 x )\).
The normal to \(C _ { 1 }\) at the point \(P\) meets \(C _ { 2 }\) at the point \(Q\). Find the exact coordinates of \(Q\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel AS Paper 1 Specimen Q16
16. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b084faa-a680-4f35-bb5c-a4edf5171b5f-24_458_604_285_751} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows the plan view of the design for a swimming pool.
The shape of this pool \(A B C D E A\) consists of a rectangular section \(A B D E\) joined to a semicircular section \(B C D\) as shown in Figure 4. Given that \(A E = 2 x\) metres, \(E D = y\) metres and the area of the pool is \(250 \mathrm {~m} ^ { 2 }\),
  1. show that the perimeter, \(P\) metres, of the pool is given by $$P = 2 x + \frac { 250 } { x } + \frac { \pi x } { 2 }$$
  2. Explain why \(0 < x < \sqrt { \frac { 500 } { \pi } }\)
  3. Find the minimum perimeter of the pool, giving your answer to 3 significant figures.
Edexcel AS Paper 1 Specimen Q17
  1. A circle \(C\) with centre at ( \(- 2,6\) ) passes through the point ( 10,11 ).
    1. Show that the circle \(C\) also passes through the point \(( 10,1 )\).
    The tangent to the circle \(C\) at the point \(( 10,11 )\) meets the \(y\) axis at the point \(P\) and the tangent to the circle \(C\) at the point \(( 10,1 )\) meets the \(y\) axis at the point \(Q\).
  2. Show that the distance \(P Q\) is 58 explaining your method clearly.
Edexcel AS Paper 1 Specimen Q1
  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
  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
  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.