Questions — Edexcel (10514 questions)

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Edexcel F2 2023 June Q6
9 marks Challenging +1.2
Given that \(y = \sec x\)
  1. show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \sec x \tan x \left( p \sec ^ { 2 } x + q \right)$$ where \(p\) and \(q\) are integers to be determined.
  2. Hence determine the Taylor series expansion about \(\frac { \pi } { 3 }\) of sec \(x\) in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\), giving each coefficient in simplest form.
  3. Use the answer to part (b) to determine, to four significant figures, an approximate value of \(\sec \left( \frac { 7 \pi } { 24 } \right)\)
Edexcel F2 2023 June Q7
11 marks Challenging +1.2
  1. Show that the substitution \(z = y ^ { - 2 }\) transforms the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y + 4 x ^ { 2 } y ^ { 3 } \ln x = 0 \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - \frac { 2 z } { x } = 8 x \ln x \quad x > 0$$
  2. By solving differential equation (II), determine the general solution of differential equation (I), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\)
Edexcel F2 2023 June Q8
13 marks Challenging +1.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{709ed2f1-f81c-4820-ac31-e1f86baae9d7-28_552_759_246_660} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$r = 6 ( 1 + \cos \theta ) \quad 0 \leqslant \theta \leqslant \pi$$ Given that \(C\) meets the initial line at the point \(A\), as shown in Figure 1,
  1. write down the polar coordinates of \(A\). The line \(l _ { 1 }\), also shown in Figure 1, is the tangent to \(C\) at the point \(B\) and is parallel to the initial line.
  2. Use calculus to determine the polar coordinates of \(B\). The line \(l _ { 2 }\), also shown in Figure 1, is the tangent to \(C\) at \(A\) and is perpendicular to the initial line. The region \(R\), shown shaded in Figure 1, is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\)
  3. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(p \sqrt { 3 } + q \pi\) where \(p\) and \(q\) are constants to be determined.
Edexcel F2 2024 June Q1
4 marks Moderate -0.3
  1. The complex number \(z = x + i y\) satisfies the equation
$$| z - 3 - 4 i | = | z + 1 + i |$$
  1. Determine an equation for the locus of \(z\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
  2. Shade, on an Argand diagram, the region defined by $$| z - 3 - 4 i | \leqslant | z + 1 + i |$$ You do not need to determine the coordinates of any intercepts on the coordinate axes.
Edexcel F2 2024 June Q2
7 marks Challenging +1.8
2. $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y ^ { 3 } = 4$$
  1. Show that $$x \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = a y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + \left( b y ^ { 2 } + c \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are integers to be determined. Given that \(y = 1\) at \(x = 2\)
  2. determine the Taylor series expansion for \(y\) in ascending powers of \(( x - 2 )\), up to and including the term in \(( x - 2 ) ^ { 3 }\), giving each coefficient in simplest form.
Edexcel F2 2024 June Q3
8 marks Standard +0.3
  1. Express $$\frac { 1 } { ( n + 3 ) ( n + 5 ) }$$ in partial fractions.
  2. Hence, using the method of differences, show that for all positive integer values of \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 3 ) ( r + 5 ) } = \frac { n ( p n + q ) } { 40 ( n + 4 ) ( n + 5 ) }$$ where \(p\) and \(q\) are integers to be determined.
  3. Use the answer to part (b) to determine, as a simplified fraction, the value of $$\frac { 1 } { 9 \times 11 } + \frac { 1 } { 10 \times 12 } + \ldots + \frac { 1 } { 24 \times 26 }$$
Edexcel F2 2024 June Q4
9 marks Challenging +1.2
  1. Show that the substitution \(y ^ { 2 } = \frac { 1 } { t }\) transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + y = x y ^ { 3 }$$ into the differential equation $$\frac { \mathrm { d } t } { \mathrm {~d} x } - 2 t = - 2 x$$
  2. Solve differential equation (II) and determine \(y ^ { 2 }\) in terms of \(x\).
Edexcel F2 2024 June Q5
6 marks Standard +0.8
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
Use algebra to determine the values of \(x\) for which $$\frac { x + 1 } { ( x - 3 ) ( x + 2 ) } \leqslant 1 - \frac { 2 } { x - 3 }$$
Edexcel F2 2024 June Q6
7 marks Challenging +1.2
  1. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { z - \mathrm { i } } { z + 1 } \quad z \neq - 1$$ Given that \(T\) maps the imaginary axis in the \(z\)-plane to the circle \(C\) in the \(w\)-plane, determine (i) the coordinates of the centre of \(C\) (ii) the radius of \(C\)
Edexcel F2 2024 June Q7
7 marks Challenging +1.2
Given that \(y = \mathrm { e } ^ { x } \sin x\)
  1. show that $$\frac { \mathrm { d } ^ { 6 } y } { \mathrm {~d} x ^ { 6 } } = k \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(k\) is a constant to be determined.
  2. Hence determine the first 5 non-zero terms in the Maclaurin series expansion for \(y\), giving each coefficient in simplest form.
Edexcel F2 2024 June Q8
10 marks Challenging +1.3
  1. Given that \(t = \ln x\), where \(x > 0\), show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { - 2 t } \left( \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } \right)$$
  2. Hence show that the transformation \(t = \ln x\), where \(x > 0\), transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 y = 1 + 4 \ln x - 2 ( \ln x ) ^ { 2 }$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } - 2 y = 1 + 4 t - 2 t ^ { 2 }$$
  3. Solve differential equation (II) to determine \(y\) in terms of \(t\).
  4. Hence determine the general solution of differential equation (I).
Edexcel F2 2024 June Q9
8 marks Challenging +1.2
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Use De Moivre's theorem to show that $$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
  2. Hence determine the smallest positive root of the equation $$48 x ^ { 6 } - 72 x ^ { 4 } + 27 x ^ { 2 } - 1 = 0$$ giving your answer to 3 decimal places.
Edexcel F2 2024 June Q10
9 marks Challenging +1.2
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{09582a82-cd57-4c2f-aefa-8412d4f4cb64-32_497_919_292_573} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with polar equation $$r = 1 + \cos \theta \quad 0 \leqslant \theta \leqslant \pi$$ and the line \(l\) with polar equation $$r = k \sec \theta \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ where \(k\) is a positive constant.
Given that
  • \(\quad C\) and \(l\) intersect at the point \(P\)
  • \(O P = 1 + \frac { \sqrt { 3 } } { 2 }\)
    1. determine the exact value of \(k\).
The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\), the initial line and \(l\).
  • Use algebraic integration to show that the area of \(R\) is $$p \pi + q \sqrt { 3 } + r$$ where \(p , q\) and \(r\) are simplified rational numbers to be determined.
    •
    Edexcel P4 2021 June Q1
    7 marks Standard +0.3
    1. Given that \(k\) is a constant and the binomial expansion of
    $$\sqrt { 1 + k x } \quad | k x | < 1$$ in ascending powers of \(x\) up to the term in \(x ^ { 3 }\) is $$1 + \frac { 1 } { 8 } x + A x ^ { 2 } + B x ^ { 3 }$$
      1. find the value of \(k\),
      2. find the value of the constant \(A\) and the constant \(B\).
    2. Use the expansion to find an approximate value to \(\sqrt { 1.15 }\) Show your working and give your answer to 6 decimal places.
    Edexcel P4 2021 June Q2
    7 marks Standard +0.3
    2. \includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-06_974_1088_116_548} \section*{Figure 1} Figure 1 shows a sketch of part of the curve with equation $$y = \frac { 9 } { ( 2 x - 3 ) ^ { 1.25 } } \quad x > \frac { 3 } { 2 }$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(y = 9\) and the line with equation \(x = 6\) This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution. Find, by algebraic integration, the exact volume of the solid generated.
    Edexcel P4 2021 June Q3
    7 marks Standard +0.3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{960fe82f-c180-422c-b409-a5cdc5fae924-08_524_878_255_532} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A bowl with circular cross section and height 20 cm is shown in Figure 2.
    The bowl is initially empty and water starts flowing into the bowl.
    When the depth of water is \(h \mathrm {~cm}\), the volume of water in the bowl, \(V \mathrm {~cm} ^ { 3 }\), is modelled by the equation $$V = \frac { 1 } { 3 } h ^ { 2 } ( h + 4 ) \quad 0 \leqslant h \leqslant 20$$ Given that the water flows into the bowl at a constant rate of \(160 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\), find, according to the model,
    1. the time taken to fill the bowl,
    2. the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 5\)
    Edexcel P4 2021 June Q4
    8 marks Standard +0.8
    4. Use algebraic integration and the substitution \(u = \sqrt { x }\) to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 10 } { 5 x + 2 x \sqrt { x } } \mathrm {~d} x$$ Write your answer in the form \(4 \ln \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers to be found.
    (Solutions relying entirely on calculator technology are not acceptable.)
    Edexcel P4 2021 June Q5
    9 marks Standard +0.8
    5. A curve has equation $$y ^ { 2 } = y \mathrm { e } ^ { - 2 x } - 3 x$$
    1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y \mathrm { e } ^ { - 2 x } + 3 } { \mathrm { e } ^ { - 2 x } - 2 y }$$ The curve crosses the \(y\)-axis at the origin and at the point \(P\).
      The tangent to the curve at the origin and the tangent to the curve at \(P\) meet at the point \(R\).
    2. Find the coordinates of \(R\). \includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-17_2644_1838_121_116}
    Edexcel P4 2021 June Q6
    10 marks Standard +0.3
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{960fe82f-c180-422c-b409-a5cdc5fae924-18_563_844_255_552} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 2 \cos 2 t \quad y = 4 \sin t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.
      1. Show, making your working clear, that the area of \(R = \int _ { 0 } ^ { \frac { \pi } { 4 } } 32 \sin ^ { 2 } t \cos t d t\)
      2. Hence find, by algebraic integration, the exact value of the area of \(R\).
    2. Show that all points on \(C\) satisfy \(y = \sqrt { a x + b }\), where \(a\) and \(b\) are constants to be found. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where f is the function $$f ( x ) = \sqrt { a x + b } \quad - 2 \leqslant x \leqslant 2$$ and \(a\) and \(b\) are the constants found in part (b).
    3. State the range of f.
    Edexcel P4 2021 June Q7
    10 marks Standard +0.3
    1. Relative to a fixed origin \(O\), the line \(l\) has equation
    $$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 10 \\ - 9 \end{array} \right) + \lambda \left( \begin{array} { l } 4 \\ 4 \\ 2 \end{array} \right) \quad \text { where } \lambda \text { is a scalar parameter }$$ Given that \(\overrightarrow { O A }\) is a unit vector parallel to \(l\),
    1. find \(\overrightarrow { O A }\) The point \(X\) lies on \(l\).
      Given that \(X\) is the point on \(l\) that is closest to the origin,
    2. find the coordinates of \(X\). The points \(O , X\) and \(A\) form the triangle \(O X A\).
    3. Find the exact area of triangle \(O X A\).
    Edexcel P4 2021 June Q8
    9 marks Standard +0.3
    8.
    1. Given that \(y = 1\) at \(x = 0\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x y ^ { \frac { 1 } { 3 } } } { \mathrm { e } ^ { 2 x } } \quad y \geqslant 0$$ giving your answer in the form \(y ^ { 2 } = \mathrm { g } ( x )\).
    2. Hence find the equation of the horizontal asymptote to the curve with equation \(y ^ { 2 } = \mathrm { g } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-27_2644_1840_118_111} \includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-29_2646_1838_121_116}
    Edexcel P4 2021 June Q9
    8 marks Standard +0.3
    9.
    1. Relative to a fixed origin \(O\), the points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively. Points \(A , B\) and \(C\) lie in a straight line, with \(B\) lying between \(A\) and \(C\).
      Given \(A B : A C = 1 : 3\) show that $$\mathbf { c } = 3 \mathbf { b } - 2 \mathbf { a }$$
    2. Given that \(n \in \mathbb { N }\), prove by contradiction that if \(n ^ { 2 }\) is a multiple of 3 then \(n\) is a multiple of 3
      \includegraphics[max width=\textwidth, alt={}]{960fe82f-c180-422c-b409-a5cdc5fae924-32_2644_1837_118_114}
    Edexcel P4 2022 June Q1
    7 marks Standard +0.3
    1. The binomial expansion of
    $$( 3 + k x ) ^ { - 2 } \quad | k x | < 3$$ where \(k\) is a non-zero constant, may be written in the form $$A + B x + C x ^ { 2 } + D x ^ { 3 } + \ldots$$ where \(A\), \(B\), \(C\) and \(D\) are constants.
    1. Find the value of \(A\) Given that \(C = 3 B\)
    2. show that $$k ^ { 2 } + 6 k = 0$$
    3. Hence (i) find the value of \(k\) (ii) find the value of \(D\)
    Edexcel P4 2022 June Q2
    9 marks Standard +0.8
    1. Express \(\frac { 1 } { ( 1 + 3 x ) ( 1 - x ) }\) in partial fractions.
    2. Hence find the solution of the differential equation $$( 1 + 3 x ) ( 1 - x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = \tan y \quad - \frac { 1 } { 3 } < x \leqslant \frac { 1 } { 2 }$$ for which \(x = \frac { 1 } { 2 }\) when \(y = \frac { \pi } { 2 }\) Give your answer in the form \(\sin ^ { n } y = \mathrm { f } ( x )\) where \(n\) is an integer to be found.
    Edexcel P4 2022 June Q3
    8 marks Standard +0.3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2dffe245-b18a-4486-af8e-bad598ceb6e8-08_401_652_246_708} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A tablet is dissolving in water.
    The tablet is modelled as a cylinder, shown in Figure 1.
    At \(t\) seconds after the tablet is dropped into the water, the radius of the tablet is \(x \mathrm {~mm}\) and the length of the tablet is \(3 x \mathrm {~mm}\). The cross-sectional area of the tablet is decreasing at a constant rate of \(0.5 \mathrm {~mm} ^ { 2 } \mathrm {~s} ^ { - 1 }\)
    1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when \(x = 7\)
    2. Find, according to the model, the rate of decrease of the volume of the tablet when \(x = 4\)