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Edexcel AS Paper 1 2021 November Q5
6 marks Moderate -0.8
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
6 marks Standard +0.3
  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
5 marks Standard +0.3
  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
7 marks Moderate -0.3
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
4 marks Moderate -0.8
  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
5 marks Moderate -0.3
  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
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
3 marks Easy -1.3
  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
4 marks Easy -1.8
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
4 marks Easy -1.3
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
6 marks Moderate -0.8
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 marks Moderate -0.5
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
4 marks Moderate -0.8
  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
5 marks Moderate -0.3
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
5 marks Standard +0.3
  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
4 marks Moderate -0.3
  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
3 marks Moderate -0.5
  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
4 marks Standard +0.3
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
7 marks Standard +0.3
  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
13 marks Moderate -0.3
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.