Questions — Edexcel PURE (39 questions)

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Edexcel PURE 2024 October Q6
Standard +0.3
  1. The functions f and g are defined by
$$\begin{array} { l l } \mathrm { f } ( x ) = 6 - \frac { 21 } { 2 x + 3 } & x \geqslant 0 \\ \mathrm {~g} ( x ) = x ^ { 2 } + 5 & x \in \mathbb { R } \end{array}$$
  1. Find \(\mathrm { gf } ( 2 )\)
  2. Find \(f ^ { - 1 }\)
  3. Solve the equation $$\operatorname { gg } ( x ) = 126$$
Edexcel PURE 2024 October Q7
Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-20_554_559_264_753} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x ^ { 3 } \sqrt { 4 x + 7 } \quad x \geqslant - \frac { 7 } { 4 }$$
  1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { k x ^ { 2 } ( 2 x + 3 ) } { \sqrt { 4 x + 7 } }$$ where \(k\) is a constant to be found. The point \(P\), shown in Figure 3, is the minimum turning point on \(C\).
  2. Find the coordinates of \(P\).
  3. Hence find the range of the function g defined by $$g ( x ) = - 4 f ( x ) \quad x \geqslant - \frac { 7 } { 4 }$$ The point \(Q\) with coordinates \(\left( \frac { 1 } { 2 } , \frac { 3 } { 8 } \right)\) lies on \(C\).
  4. Find the coordinates of the point to which \(Q\) is mapped when \(C\) is transformed to the curve with equation $$y = 40 \mathrm { f } \left( x - \frac { 3 } { 2 } \right) - 8$$
Edexcel PURE 2024 October Q8
Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-24_472_595_246_735} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The heart rate of a horse is being monitored.
The heart rate \(H\), measured in beats per minute (bpm), is modelled by the equation $$H = 32 + 40 \mathrm { e } ^ { - 0.2 t } - 20 \mathrm { e } ^ { - 0.9 t }$$ where \(t\) minutes is the time after monitoring began.
Figure 4 is a sketch of \(H\) against \(t\). \section*{Use the equation of the model to answer parts (a) to (e).}
  1. State the initial heart rate of the horse. In the long term, the heart rate of the horse approaches \(L \mathrm { bpm }\).
  2. State the value of \(L\). The heart rate of the horse reaches its maximum value after \(T\) minutes.
  3. Find the value of \(T\), giving your answer to 3 decimal places.
    (Solutions based entirely on calculator technology are not acceptable.) The heart rate of the horse is 37 bpm after \(M\) minutes.
  4. Show that \(M\) is a solution of the equation $$t = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t } } \right)$$ Using the iteration formula $$t _ { n + 1 } = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t _ { n } } } \right) \quad \text { with } \quad t _ { 1 } = 10$$
    1. find, to 4 decimal places, the value of \(t _ { 2 }\)
    2. find, to 4 decimal places, the value of \(M\)
Edexcel PURE 2024 October Q9
Standard +0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-28_753_1111_248_477} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 6 x ^ { 2 } + 4 x - 2 } { 2 x + 1 } \quad x > - \frac { 1 } { 2 }$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\), giving the answer in simplest form. The line \(l\) is the normal to \(C\) at the point \(P ( 2,6 )\)
  2. Show that an equation for \(l\) is $$16 y + 5 x = 106$$
  3. Write \(\mathrm { f } ( x )\) in the form \(A x + B + \frac { D } { 2 x + 1 }\) where \(A , B\) and \(D\) are constants. The region \(R\), shown shaded in Figure 5, is bounded by \(C , l\) and the \(x\)-axis.
  4. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(P + Q \ln 3\), where \(P\) and \(Q\) are rational constants.
    (Solutions based entirely on calculator technology are not acceptable.)
Edexcel PURE 2024 October Q1
Moderate -0.3
  1. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of
$$( 8 - 3 x ) ^ { - \frac { 1 } { 3 } } \quad | x | < \frac { 8 } { 3 }$$ giving each coefficient as a simplified fraction.
(b) Use the answer from part (a) with \(x = \frac { 2 } { 3 }\) to find a rational approximation to \(\sqrt [ 3 ] { 6 }\)
Edexcel PURE 2024 October Q2
Standard +0.8
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
The curve \(C _ { 1 }\) has equation $$y = x ^ { 4 } + 10 x ^ { 2 } + 8 \quad x \in \mathbb { R }$$ The curve \(C _ { 2 }\) has equation $$y = 2 x ^ { 2 } - 7 \quad x \in \mathbb { R }$$ Use algebra to prove by contradiction that \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
Edexcel PURE 2024 October Q3
Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-06_549_750_251_660} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = 3 \sin ^ { 3 } \theta \quad y = 1 + \cos 2 \theta \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$
  1. Show that $$\frac { d y } { d x } = k \operatorname { cosec } \theta \quad \theta \neq 0$$ where \(k\) is a constant to be found. The point \(P\) lies on \(C\) where \(\theta = \frac { \pi } { 6 }\)
  2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
  3. Show that \(C\) has Cartesian equation $$8 x ^ { 2 } = 9 ( 2 - y ) ^ { 3 } \quad - q \leqslant x \leqslant q$$ where \(q\) is a constant to be found.
Edexcel PURE 2024 October Q4
Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-10_634_638_255_717} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$3 x ^ { 2 } + 2 y ^ { 2 } - 4 x y + 8 ^ { x } - 11 = 0$$ The point \(P\) has coordinates ( 1,2 ).
  1. Verify that \(P\) lies on \(C\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The normal to \(C\) at \(P\) crosses the \(x\)-axis at a point \(Q\).
  3. Find the \(x\) coordinate of \(Q\), giving your answer in the form \(a + b \ln 2\) where \(a\) and \(b\) are integers.
Edexcel PURE 2024 October Q5
Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-14_569_616_242_785} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a container in the shape of a hollow, inverted, right circular cone.
The height of the container is 30 cm and the radius is 12 cm , as shown in Figure 3.
The container is initially empty when water starts flowing into it.
When the height of water is \(h \mathrm {~cm}\), the surface of the water has radius \(r \mathrm {~cm}\) and the volume of water is \(V \mathrm {~cm} ^ { 3 }\)
  1. Show that $$V = \frac { 4 \pi h ^ { 3 } } { 75 }$$ [The volume \(V\) of a right circular cone with vertical height \(h\) and base radius \(r\) is given by the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) ] Given that water flows into the container at a constant rate of \(2 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
  2. find, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), the rate at which \(h\) is changing, exactly 1.5 minutes after water starts flowing into the container.
Edexcel PURE 2024 October Q6
Challenging +1.2
  1. Use the substitution \(u = \sqrt { x ^ { 3 } + 1 }\) to show that
$$\int \frac { 9 x ^ { 5 } } { \sqrt { x ^ { 3 } + 1 } } \mathrm {~d} x = 2 \left( x ^ { 3 } + 1 \right) ^ { k } \left( x ^ { 3 } - A \right) + c$$ where \(k\) and \(A\) are constants to be found and \(c\) is an arbitrary constant.
Edexcel PURE 2024 October Q7
Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-18_510_680_251_696} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = \frac { 3 x - 1 } { x + 2 } \quad x > - 2$$
  1. Show that $$\frac { 3 x - 1 } { x + 2 } \equiv A + \frac { B } { x + 2 }$$ where \(A\) and \(B\) are constants to be found. The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 1\) This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  2. Use the answer to part (a) and algebraic integration to find the exact volume of the solid generated, giving your answer in the form $$\pi ( p + q \ln 2 )$$ where \(p\) and \(q\) are rational constants.
Edexcel PURE 2024 October Q8
Standard +0.3
  1. Relative to a fixed origin \(O\)
  • the point \(A\) has coordinates \(( - 10,5 , - 4 )\)
  • the point \(B\) has coordinates \(( - 6,4 , - 1 )\)
The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).
  1. Find a vector equation for \(l _ { 1 }\) The line \(l _ { 2 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3 \\ p \\ q \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 4 \\ 1 \end{array} \right)$$ where \(p\) and \(q\) are constants and \(\mu\) is a scalar parameter.
    Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at \(B\),
  2. find the value of \(p\) and the value of \(q\). The acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\)
  3. Find the exact value of \(\cos \theta\) Given that the point \(C\) lies on \(l _ { 2 }\) such that \(A C\) is perpendicular to \(l _ { 2 }\)
  4. find the exact length of \(A C\), giving your answer as a surd.
Edexcel PURE 2024 October Q9
Challenging +1.2
  1. (a) Express \(\frac { 1 } { x ( 2 x - 1 ) }\) in partial fractions.
The height above ground, \(h\) metres, of a carriage on a fairground ride is modelled by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 1 } { 50 } h ( 2 h - 1 ) \cos \left( \frac { t } { 10 } \right)$$ where \(t\) seconds is the time after the start of the ride.
Given that, at the start of the ride, the carriage is 2.5 m above ground,
(b) solve the differential equation to show that, according to the model, $$h = \frac { 5 } { 10 - 8 \mathrm { e } ^ { k \sin \left( \frac { t } { 10 } \right) } }$$ where \(k\) is a constant to be found.
(c) Hence find, according to the model, the time taken for the carriage to reach its maximum height above ground for the 3rd time.
Give your answer to the nearest second.
(Solutions relying entirely on calculator technology are not acceptable.)
Edexcel PURE 2024 October Q10
Standard +0.8
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-30_563_602_255_735} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of the curve with parametric equations $$x = 3 t ^ { 2 } \quad y = \sin t \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve and the \(x\)-axis.
  1. Show that the area of \(R\) is $$k \int _ { 0 } ^ { \frac { \pi } { 2 } } t \sin ^ { 2 } t \cos t \mathrm {~d} t$$ where \(k\) is a constant to be found.
  2. Hence, using algebraic integration, find the exact area of \(R\), giving your answer in the form $$p \pi + q$$ where \(p\) and \(q\) are constants.