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Edexcel Paper 1 2020 October Q3
  1. Relative to a fixed origin \(O\)
  • point \(A\) has position vector \(2 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k }\)
  • point \(B\) has position vector \(3 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }\)
  • point \(C\) has position vector \(2 \mathbf { i } - 16 \mathbf { j } + 4 \mathbf { k }\)
    1. Find \(\overrightarrow { A B }\)
    2. Show that quadrilateral \(O A B C\) is a trapezium, giving reasons for your answer.
Edexcel Paper 1 2020 October Q4
  1. The function f is defined by
$$f ( x ) = \frac { 3 x - 7 } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$
  1. Find \(f ^ { - 1 } ( 7 )\)
  2. Show that \(\operatorname { ff } ( x ) = \frac { a x + b } { x - 3 }\) where \(a\) and \(b\) are integers to be found.
    VI4V SIHI NI ILIUM ION OCVIAV SIHI NI III IM I O N OOVJAV SIHI NI III M M ION OC
Edexcel Paper 1 2020 October Q5
  1. A car has six forward gears.
The fastest speed of the car
  • in \(1 ^ { \text {st } }\) gear is \(28 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
  • in \(6 ^ { \text {th } }\) gear is \(115 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence,
  1. find the fastest speed of the car in \(3 { } ^ { \text {rd } }\) gear. Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,
  2. find the fastest speed of the car in \(5 ^ { \text {th } }\) gear.
Edexcel Paper 1 2020 October Q6
  1. (a) Express \(\sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
    Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.
The temperature, \(\theta ^ { \circ } \mathrm { C }\), inside a room on a given day is modelled by the equation $$\theta = 5 + \sin \left( \frac { \pi t } { 12 } - 3 \right) + 2 \cos \left( \frac { \pi t } { 12 } - 3 \right) \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight.
Using the equation of the model and your answer to part (a),
(b) deduce the maximum temperature of the room during this day,
(c) find the time of day when the maximum temperature occurs, giving your answer to the nearest minute.
Edexcel Paper 1 2020 October Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dc0ac5df-24a7-41b5-8410-f0e9b332ba64-16_868_805_242_632} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) and a straight line \(l\).
The curve \(C\) meets \(l\) at the points \(( - 2,13 )\) and \(( 0,25 )\) as shown.
The shaded region \(R\) is bounded by \(C\) and \(l\) as shown in Figure 1.
Given that
  • \(\mathrm { f } ( x )\) is a quadratic function in \(x\)
  • ( \(- 2,13\) ) is the minimum turning point of \(y = \mathrm { f } ( x )\)
    use inequalities to define \(R\).
Edexcel Paper 1 2020 October Q8
  1. A new smartphone was released by a company.
The company monitored the total number of phones sold, \(n\), at time \(t\) days after the phone was released. The company observed that, during this time,
the rate of increase of \(n\) was proportional to \(n\)
Use this information to write down a suitable equation for \(n\) in terms of \(t\).
(You do not need to evaluate any unknown constants in your equation.)
Edexcel Paper 1 2020 October Q9
9.
\includegraphics[max width=\textwidth, alt={}]{dc0ac5df-24a7-41b5-8410-f0e9b332ba64-22_602_752_246_657}
\section*{Figure 2} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 4 \left( x ^ { 2 } - 2 \right) \mathrm { e } ^ { - 2 x } \quad x \in \mathbb { R }$$
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) = 8 \left( 2 + x - x ^ { 2 } \right) \mathrm { e } ^ { - 2 x }\)
  2. Hence find, in simplest form, the exact coordinates of the stationary points of \(C\). The function g and the function h are defined by $$\begin{array} { l l } \mathrm { g } ( x ) = 2 \mathrm { f } ( x ) & x \in \mathbb { R }
    \mathrm {~h} ( x ) = 2 \mathrm { f } ( x ) - 3 & x \geqslant 0 \end{array}$$
  3. Find (i) the range of \(g\)
    (ii) the range of h
Edexcel Paper 1 2020 October Q10
  1. (a) Use the substitution \(x = u ^ { 2 } + 1\) to show that
$$\int _ { 5 } ^ { 10 } \frac { 3 \mathrm {~d} x } { ( x - 1 ) ( 3 + 2 \sqrt { x - 1 } ) } = \int _ { p } ^ { q } \frac { 6 \mathrm {~d} u } { u ( 3 + 2 u ) }$$ where \(p\) and \(q\) are positive constants to be found.
(b) Hence, using algebraic integration, show that $$\int _ { 5 } ^ { 10 } \frac { 3 \mathrm {~d} x } { ( x - 1 ) ( 3 + 2 \sqrt { x - 1 } ) } = \ln a$$ where \(a\) is a rational constant to be found.
Edexcel Paper 1 2020 October Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dc0ac5df-24a7-41b5-8410-f0e9b332ba64-30_738_837_242_614} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Circle \(C _ { 1 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 100\)
Circle \(C _ { 2 }\) has equation \(( x - 15 ) ^ { 2 } + y ^ { 2 } = 40\)
The circles meet at points \(A\) and \(B\) as shown in Figure 3.
  1. Show that angle \(A O B = 0.635\) radians to 3 significant figures, where \(O\) is the origin. The region shown shaded in Figure 3 is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
  2. Find the perimeter of the shaded region, giving your answer to one decimal place.
Edexcel Paper 1 2020 October Q12
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Show that $$\operatorname { cosec } \theta - \sin \theta \equiv \cos \theta \cot \theta \quad \theta \neq ( 180 n ) ^ { \circ } \quad n \in \mathbb { Z }$$
  2. Hence, or otherwise, solve for \(0 < x < 180 ^ { \circ }\) $$\operatorname { cosec } x - \sin x = \cos x \cot \left( 3 x - 50 ^ { \circ } \right)$$
Edexcel Paper 1 2020 October Q13
  1. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$a _ { n + 1 } = \frac { k \left( a _ { n } + 2 \right) } { a _ { n } } \quad n \in \mathbb { N }$$ where \(k\) is a constant.
Given that
  • the sequence is a periodic sequence of order 3
  • \(a _ { 1 } = 2\)
    1. show that
$$k ^ { 2 } + k - 2 = 0$$
  • For this sequence explain why \(k \neq 1\)
  • Find the value of $$\sum _ { r = 1 } ^ { 80 } a _ { r }$$
    •
    Edexcel Paper 1 2020 October Q14
    1. A large spherical balloon is deflating.
    At time \(t\) seconds the balloon has radius \(r \mathrm {~cm}\) and volume \(V \mathrm {~cm} ^ { 3 }\)
    The volume of the balloon is modelled as decreasing at a constant rate.
    1. Using this model, show that $$\frac { \mathrm { d } r } { \mathrm {~d} t } = - \frac { k } { r ^ { 2 } }$$ where \(k\) is a positive constant. Given that
      • the initial radius of the balloon is 40 cm
      • after 5 seconds the radius of the balloon is 20 cm
      • the volume of the balloon continues to decrease at a constant rate until the balloon is empty
      • solve the differential equation to find a complete equation linking \(r\) and \(t\).
      • Find the limitation on the values of \(t\) for which the equation in part (b) is valid.
    Edexcel Paper 1 2020 October Q15
    1. The curve \(C\) has equation
    $$x ^ { 2 } \tan y = 9 \quad 0 < y < \frac { \pi } { 2 }$$
    1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 18 x } { x ^ { 4 } + 81 }$$
    2. Prove that \(C\) has a point of inflection at \(x = \sqrt [ 4 ] { 27 }\)
    Edexcel Paper 1 2020 October Q16
    1. Prove by contradiction that there are no positive integers \(p\) and \(q\) such that
    $$4 p ^ { 2 } - q ^ { 2 } = 25$$
    Edexcel Paper 1 2021 October Q1
    1. $$f ( x ) = a x ^ { 3 } + 10 x ^ { 2 } - 3 a x - 4$$ Given that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\), find the value of the constant \(a\).
    You must make your method clear.
    Edexcel Paper 1 2021 October Q2
    1. Given that
    $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 5 \quad x \in \mathbb { R }$$
    1. express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers to be found. The curve with equation \(y = \mathrm { f } ( x )\)
      • meets the \(y\)-axis at the point \(P\)
      • has a minimum turning point at the point \(Q\)
      • Write down
        1. the coordinates of \(P\)
        2. the coordinates of \(Q\)
    Edexcel Paper 1 2021 October Q3
    1. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
    $$u _ { n + 1 } = k - \frac { 24 } { u _ { n } } \quad u _ { 1 } = 2$$ where \(k\) is an integer.
    Given that \(u _ { 1 } + 2 u _ { 2 } + u _ { 3 } = 0\)
    1. show that $$3 k ^ { 2 } - 58 k + 240 = 0$$
    2. Find the value of \(k\), giving a reason for your answer.
    3. Find the value of \(u _ { 3 }\)
    Edexcel Paper 1 2021 October Q4
    1. The curve with equation \(y = \mathrm { f } ( x )\) where
    $$f ( x ) = x ^ { 2 } + \ln \left( 2 x ^ { 2 } - 4 x + 5 \right)$$ has a single turning point at \(x = \alpha\)
    1. Show that \(\alpha\) is a solution of the equation $$2 x ^ { 3 } - 4 x ^ { 2 } + 7 x - 2 = 0$$ The iterative formula $$x _ { n + 1 } = \frac { 1 } { 7 } \left( 2 + 4 x _ { n } ^ { 2 } - 2 x _ { n } ^ { 3 } \right)$$ is used to find an approximate value for \(\alpha\).
      Starting with \(x _ { 1 } = 0.3\)
    2. calculate, giving each answer to 4 decimal places,
      1. the value of \(x _ { 2 }\)
      2. the value of \(x _ { 4 }\) Using a suitable interval and a suitable function that should be stated,
    3. show that \(\alpha\) is 0.341 to 3 decimal places.
    Edexcel Paper 1 2021 October Q5
    1. In this question you should show all stages of your working.
    \section*{Solutions relying entirely on calculator technology are not acceptable.} A company made a profit of \(\pounds 20000\) in its first year of trading, Year 1
    A model for future trading predicts that the yearly profit will increase by \(8 \%\) each year, so that the yearly profits will form a geometric sequence. According to the model,
    1. show that the profit for Year 3 will be \(\pounds 23328\)
    2. find the first year when the yearly profit will exceed £65000
    3. find the total profit for the first 20 years of trading, giving your answer to the nearest £1000
    Edexcel Paper 1 2021 October Q6
    6. Figure 1 Figure 1 shows a sketch of triangle \(A B C\).
    Given that
    • \(\overrightarrow { A B } = - 3 \mathbf { i } - 4 \mathbf { j } - 5 \mathbf { k }\)
    • \(\overrightarrow { B C } = \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\)
      1. find \(\overrightarrow { A C }\)
      2. show that \(\cos A B C = \frac { 9 } { 10 }\)
    Edexcel Paper 1 2021 October Q7
    1. The circle \(C\) has equation
    $$x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 11 = 0$$
    1. Find
      1. the coordinates of the centre of \(C\),
      2. the exact radius of \(C\), giving your answer as a simplified surd. The line \(l\) has equation \(y = 3 x + k\) where \(k\) is a constant.
        Given that \(l\) is a tangent to \(C\),
    2. find the possible values of \(k\), giving your answers as simplified surds.
    Edexcel Paper 1 2021 October Q8
    1. A scientist is studying the growth of two different populations of bacteria.
    The number of bacteria, \(N\), in the first population is modelled by the equation $$N = A \mathrm { e } ^ { k t } \quad t \geqslant 0$$ where \(A\) and \(k\) are positive constants and \(t\) is the time in hours from the start of the study.
    Given that
    • there were 1000 bacteria in this population at the start of the study
    • it took exactly 5 hours from the start of the study for this population to double
      1. find a complete equation for the model.
      2. Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures.
    The number of bacteria, \(M\), in the second population is modelled by the equation $$M = 500 \mathrm { e } ^ { 1.4 k t } \quad t \geqslant 0$$ where \(k\) has the value found in part (a) and \(t\) is the time in hours from the start of the study.
    Given that \(T\) hours after the start of the study, the number of bacteria in the two different populations was the same,
  • find the value of \(T\).
    •
    Edexcel Paper 1 2021 October Q9
    9. $$f ( x ) = \frac { 50 x ^ { 2 } + 38 x + 9 } { ( 5 x + 2 ) ^ { 2 } ( 1 - 2 x ) } \quad x \neq - \frac { 2 } { 5 } \quad x \neq \frac { 1 } { 2 }$$ Given that \(\mathrm { f } ( x )\) can be expressed in the form $$\frac { A } { 5 x + 2 } + \frac { B } { ( 5 x + 2 ) ^ { 2 } } + \frac { C } { 1 - 2 x }$$ where \(A\), \(B\) and \(C\) are constants
      1. find the value of \(B\) and the value of \(C\)
      2. show that \(A = 0\)
      1. Use binomial expansions to show that, in ascending powers of \(x\) $$f ( x ) = p + q x + r x ^ { 2 } + \ldots$$ where \(p , q\) and \(r\) are simplified fractions to be found.
      2. Find the range of values of \(x\) for which this expansion is valid.
    Edexcel Paper 1 2021 October Q10
    1. In this question you should show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable.
    1. Given that \(1 + \cos 2 \theta + \sin 2 \theta \neq 0\) prove that $$\frac { 1 - \cos 2 \theta + \sin 2 \theta } { 1 + \cos 2 \theta + \sin 2 \theta } \equiv \tan \theta$$
    2. Hence solve, for \(0 < x < 180 ^ { \circ }\) $$\frac { 1 - \cos 4 x + \sin 4 x } { 1 + \cos 4 x + \sin 4 x } = 3 \sin 2 x$$ giving your answers to one decimal place where appropriate.
    Edexcel Paper 1 2021 October Q11
    11. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{08ede5ea-85e9-44eb-be6a-5878096734e2-34_705_837_248_614} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = ( \ln x ) ^ { 2 } \quad x > 0$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 4\) The table below shows corresponding values of \(x\) and \(y\), with the values of \(y\) given to 4 decimal places.
    \(x\)22.533.54
    \(y\)0.48050.83961.20691.56941.9218
    1. Use the trapezium rule, with all the values of \(y\) in the table, to obtain an estimate for the area of \(R\), giving your answer to 3 significant figures.
    2. Use algebraic integration to find the exact area of \(R\), giving your answer in the form $$y = a ( \ln 2 ) ^ { 2 } + b \ln 2 + c$$ where \(a\), \(b\) and \(c\) are integers to be found.