Questions — Edexcel C4 (386 questions)

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Edexcel C4 2014 June Q6
10 marks Standard +0.2
6. With respect to a fixed origin, the point \(A\) with position vector \(\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) lies on the line \(l _ { 1 }\) with equation $$\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 2 \\ - 1 \end{array} \right) , \quad \text { where } \lambda \text { is a scalar parameter, }$$ and the point \(B\) with position vector \(4 \mathbf { i } + p \mathbf { j } + 3 \mathbf { k }\), where \(p\) is a constant, lies on the line \(l _ { 2 }\) with equation $$\mathbf { r } = \left( \begin{array} { l } 7 \\ 0 \\ 7 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 5 \\ 4 \end{array} \right) , \quad \text { where } \mu \text { is a scalar parameter. }$$
  1. Find the value of the constant \(p\).
  2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the position vector of their point of intersection, \(C\).
  3. Find the size of the angle \(A C B\), giving your answer in degrees to 3 significant figures.
  4. Find the area of the triangle \(A B C\), giving your answer to 3 significant figures. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 6 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
Edexcel C4 2014 June Q7
10 marks Challenging +1.2
7. The rate of increase of the number, \(N\), of fish in a lake is modelled by the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { ( k t - 1 ) ( 5000 - N ) } { t } \quad t > 0 , \quad 0 < N < 5000$$ In the given equation, the time \(t\) is measured in years from the start of January 2000 and \(k\) is a positive constant.
  1. By solving the differential equation, show that $$N = 5000 - A t \mathrm { e } ^ { - k t }$$ where \(A\) is a positive constant. After one year, at the start of January 2001, there are 1200 fish in the lake. After two years, at the start of January 2002, there are 1800 fish in the lake.
  2. Find the exact value of the constant \(A\) and the exact value of the constant \(k\).
  3. Hence find the number of fish in the lake after five years. Give your answer to the nearest hundred fish. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 7 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
Edexcel C4 2014 June Q8
12 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e14881c1-5ba5-4868-92ee-8bc58d4884dc-13_808_965_248_502} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The curve shown in Figure 3 has parametric equations $$x = t - 4 \sin t , y = 1 - 2 \cos t , \quad - \frac { 2 \pi } { 3 } \leqslant t \leqslant \frac { 2 \pi } { 3 }$$ The point \(A\), with coordinates ( \(k , 1\) ), lies on the curve. Given that \(k > 0\)
  1. find the exact value of \(k\),
  2. find the gradient of the curve at the point \(A\). There is one point on the curve where the gradient is equal to \(- \frac { 1 } { 2 }\)
  3. Find the value of \(t\) at this point, showing each step in your working and giving your answer to 4 decimal places.
    [0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]
Edexcel C4 2016 June Q1
6 marks Moderate -0.3
  1. Use the binomial series to find the expansion of
$$\frac { 1 } { ( 2 + 5 x ) ^ { 3 } } , \quad | x | < \frac { 2 } { 5 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Give each coefficient as a fraction in its simplest form.
(6)
Edexcel C4 2016 June Q2
9 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cbfbb690-bc85-46e5-a97f-35df4b6f1c84-03_712_1091_248_470} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = x ^ { 2 } \ln x , x \geqslant 1\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line \(x = 2\) The table below shows corresponding values of \(x\) and \(y\) for \(y = x ^ { 2 } \ln x\)
\(x\)11.21.41.61.82
\(y\)00.26251.20321.90442.7726
  1. Complete the table above, giving the missing value of \(y\) to 4 decimal places.
  2. Use the trapezium rule with all the values of \(y\) in the completed table to obtain an estimate for the area of \(R\), giving your answer to 3 decimal places.
  3. Use integration to find the exact value for the area of \(R\).
Edexcel C4 2016 June Q3
9 marks Standard +0.3
  1. The curve \(C\) has equation
$$2 x ^ { 2 } y + 2 x + 4 y - \cos ( \pi y ) = 17$$
  1. Use implicit differentiation to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) with coordinates \(\left( 3 , \frac { 1 } { 2 } \right)\) lies on \(C\).
    The normal to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
  2. Find the \(x\) coordinate of \(A\), giving your answer in the form \(\frac { a \pi + b } { c \pi + d }\), where \(a , b , c\) and \(d\) are integers to be determined.
Edexcel C4 2016 June Q4
7 marks Moderate -0.8
4. The rate of decay of the mass of a particular substance is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { 5 } { 2 } x , \quad t \geqslant 0$$ where \(x\) is the mass of the substance measured in grams and \(t\) is the time measured in days.
Given that \(x = 60\) when \(t = 0\),
  1. solve the differential equation, giving \(x\) in terms of \(t\). You should show all steps in your working and give your answer in its simplest form.
  2. Find the time taken for the mass of the substance to decay from 60 grams to 20 grams. Give your answer to the nearest minute.
Edexcel C4 2016 June Q5
6 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cbfbb690-bc85-46e5-a97f-35df4b6f1c84-09_605_1131_248_466} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \tan t , \quad y = 5 \sqrt { 3 } \sin 2 t , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( 4 \sqrt { 3 } , \frac { 15 } { 2 } \right)\).
  1. Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(P\). Give your answer as a simplified surd. The point \(Q\) lies on the curve \(C\), where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
  2. Find the exact coordinates of the point \(Q\).
Edexcel C4 2016 June Q6
15 marks Standard +0.8
6. (i) Given that \(y > 0\), find $$\int \frac { 3 y - 4 } { y ( 3 y + 2 ) } d y$$ (ii) (a) Use the substitution \(x = 4 \sin ^ { 2 } \theta\) to show that $$\int _ { 0 } ^ { 3 } \sqrt { \left( \frac { x } { 4 - x } \right) } \mathrm { d } x = \lambda \int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 2 } \theta \mathrm {~d} \theta$$ where \(\lambda\) is a constant to be determined.
(b) Hence use integration to find $$\int _ { 0 } ^ { 3 } \sqrt { \left( \frac { x } { 4 - x } \right) } d x$$ giving your answer in the form \(a \pi + b\), where \(a\) and \(b\) are exact constants.
Edexcel C4 2016 June Q7
8 marks Standard +0.3
7. (a) Find $$\int ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } d x$$ giving your answer in its simplest form. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cbfbb690-bc85-46e5-a97f-35df4b6f1c84-13_727_1177_596_370} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = ( 2 x - 1 ) ^ { \frac { 3 } { 4 } } , \quad x \geqslant \frac { 1 } { 2 }$$ The curve \(C\) cuts the line \(y = 8\) at the point \(P\) with coordinates \(( k , 8 )\), where \(k\) is a constant.
(b) Find the value of \(k\). The finite region \(S\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(y = 8\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
(c) Find the exact value of the volume of the solid generated.
Edexcel C4 2016 June Q8
15 marks Standard +0.3
8. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation $$\mathbf { r } = \left( \begin{array} { r } 8 \\ 1 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { r } - 5 \\ 4 \\ 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) lies on \(l _ { 1 }\) where \(\mu = 1\)
  1. Find the coordinates of \(A\). The point \(P\) has position vector \(\left( \begin{array} { l } 1 \\ 5 \\ 2 \end{array} \right)\).
    The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
  2. Write down a vector equation for the line \(l _ { 2 }\)
  3. Find the exact value of the distance \(A P\). Give your answer in the form \(k \sqrt { 2 }\), where \(k\) is a constant to be determined. The acute angle between \(A P\) and \(l _ { 2 }\) is \(\theta\).
  4. Find the value of \(\cos \theta\) A point \(E\) lies on the line \(l _ { 2 }\) Given that \(A P = P E\),
  5. find the area of triangle \(A P E\),
  6. find the coordinates of the two possible positions of \(E\).
Edexcel C4 2017 June Q1
8 marks Moderate -0.3
  1. The curve \(C\) has parametric equations
$$x = 3 t - 4 , y = 5 - \frac { 6 } { t } , \quad t > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) The point \(P\) lies on \(C\) where \(t = \frac { 1 } { 2 }\)
  2. Find the equation of the tangent to \(C\) at the point \(P\). Give your answer in the form \(y = p x + q\), where \(p\) and \(q\) are integers to be determined.
  3. Show that the cartesian equation for \(C\) can be written in the form $$y = \frac { a x + b } { x + 4 } , \quad x > - 4$$ where \(a\) and \(b\) are integers to be determined.
Edexcel C4 2017 June Q2
6 marks Standard +0.3
2. \(\quad \mathrm { f } ( x ) = ( 2 + k x ) ^ { - 3 } , \quad | k x | < 2\), where \(k\) is a positive constant The binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is $$A + B x + \frac { 243 } { 16 } x ^ { 2 }$$ where \(A\) and \(B\) are constants.
  1. Write down the value of \(A\).
  2. Find the value of \(k\).
  3. Find the value of \(B\).
Edexcel C4 2017 June Q3
12 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-08_560_1082_242_438} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \frac { 6 } { \left( \mathrm { e } ^ { x } + 2 \right) } , x \in \mathbb { R }\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(y\)-axis, the \(x\)-axis and the line with equation \(x = 1\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 6 } { \left( \mathrm { e } ^ { x } + 2 \right) }\)
\(x\)00.20.40.60.81
\(y\)21.718301.569811.419941.27165
  1. Complete the table above by giving the missing value of \(y\) to 5 decimal places.
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to find an estimate for the area of \(R\), giving your answer to 4 decimal places.
  3. Use the substitution \(u = \mathrm { e } ^ { x }\) to show that the area of \(R\) can be given by $$\int _ { a } ^ { b } \frac { 6 } { u ( u + 2 ) } \mathrm { d } u$$ where \(a\) and \(b\) are constants to be determined.
  4. Hence use calculus to find the exact area of \(R\). [Solutions based entirely on graphical or numerical methods are not acceptable.]
Edexcel C4 2017 June Q4
9 marks Standard +0.3
4. The curve \(C\) has equation $$4 x ^ { 2 } - y ^ { 3 } - 4 x y + 2 ^ { y } = 0$$ The point \(P\) with coordinates \(( - 2,4 )\) lies on \(C\).
  1. Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(P\). The normal to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\).
  2. Find the \(y\) coordinate of \(A\), giving your answer in the form \(p + q \ln 2\), where \(p\) and \(q\) are constants to be determined.
    (3)
Edexcel C4 2017 June Q5
7 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-16_589_540_248_705} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale The finite region \(S\), shown shaded in Figure 2, is bounded by the \(y\)-axis, the \(x\)-axis, the line with equation \(x = \ln 4\) and the curve with equation $$y = \mathrm { e } ^ { x } + 2 \mathrm { e } ^ { - x } , \quad x \geqslant 0$$ The region \(S\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Use integration to find the exact value of the volume of the solid generated. Give your answer in its simplest form.
[0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]
Edexcel C4 2017 June Q6
13 marks Standard +0.3
6. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 4 \\ 28 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ - 5 \\ 1 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5 \\ 3 \\ 1 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ 0 \\ - 4 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(X\).
  1. Find the coordinates of the point \(X\).
  2. Find the size of the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in degrees to 2 decimal places. The point \(A\) lies on \(l _ { 1 }\) and has position vector \(\left( \begin{array} { r } 2 \\ 18 \\ 6 \end{array} \right)\)
  3. Find the distance \(A X\), giving your answer as a surd in its simplest form. The point \(Y\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { Y A }\) is perpendicular to the line \(l _ { 1 }\)
  4. find the distance \(Y A\), giving your answer to one decimal place. The point \(B\) lies on \(l _ { 1 }\) where \(| \overrightarrow { A X } | = 2 | \overrightarrow { A B } |\).
  5. Find the two possible position vectors of \(B\).
Edexcel C4 2017 June Q7
8 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-24_835_1160_255_529} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a vertical cylindrical tank of height 200 cm containing water. Water is leaking from a hole \(P\) on the side of the tank. At time \(t\) minutes after the leaking starts, the height of water in the tank is \(h \mathrm {~cm}\). The height \(h \mathrm {~cm}\) of the water in the tank satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = k ( h - 9 ) ^ { \frac { 1 } { 2 } } , \quad 9 < h \leqslant 200$$ where \(k\) is a constant. Given that, when \(h = 130\), the height of the water is falling at a rate of 1.1 cm per minute,
  1. find the value of \(k\). Given that the tank was full of water when the leaking started,
  2. solve the differential equation with your value of \(k\), to find the value of \(t\) when \(h = 50\)
Edexcel C4 2017 June Q8
12 marks Standard +0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-28_721_714_255_616} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Diagram not drawn to scale Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3 \theta \sin \theta , \quad y = \sec ^ { 3 } \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P ( k , 8 )\) lies on \(C\), where \(k\) is a constant.
  1. Find the exact value of \(k\). The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = k\).
  2. Show that the area of \(R\) can be expressed in the form $$\lambda \int _ { \alpha } ^ { \beta } \left( \theta \sec ^ { 2 } \theta + \tan \theta \sec ^ { 2 } \theta \right) \mathrm { d } \theta$$ where \(\lambda , \alpha\) and \(\beta\) are constants to be determined.
  3. Hence use integration to find the exact value of the area of \(R\).
Edexcel C4 2018 June Q1
8 marks Standard +0.3
  1. (a) Find the binomial series expansion of
$$\sqrt { 4 - 9 x } , | x | < \frac { 4 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) Give each coefficient in its simplest form.
(b) Use the expansion from part (a), with a suitable value of \(x\), to find an approximate value for \(\sqrt { 310 }\) Show all your working and give your answer to 3 decimal places.
Edexcel C4 2018 June Q2
10 marks Standard +0.3
  1. The curve \(C\) has equation
$$x ^ { 2 } + x y + y ^ { 2 } - 4 x - 5 y + 1 = 0$$
  1. Use implicit differentiation to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Find the \(x\) coordinates of the two points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) Give exact answers in their simplest form.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C4 2018 June Q3
14 marks Standard +0.3
3. (i) Given that $$\frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \equiv \frac { A } { ( 2 x + 1 ) } + \frac { B } { ( 2 x + 1 ) ^ { 2 } } + \frac { C } { ( x + 3 ) }$$
  1. find the values of the constants \(A , B\) and \(C\).
  2. Hence find $$\int \frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \mathrm { d } x , \quad x > - \frac { 1 } { 2 }$$ (ii) Find $$\int \left( \mathrm { e } ^ { x } + 1 \right) ^ { 3 } \mathrm {~d} x$$ (iii) Using the substitution \(u ^ { 3 } = x\), or otherwise, find $$\int \frac { 1 } { 4 x + 5 x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x , \quad x > 0$$
Edexcel C4 2018 June Q4
6 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-12_978_1264_121_411} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A water container is made in the shape of a hollow inverted right circular cone with semi-vertical angle of \(30 ^ { \circ }\), as shown in Figure 1. The height of the container is 50 cm . When the depth of the water in the container 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 \(V = \frac { 1 } { 9 } \pi h ^ { 3 }\) [0pt] [You may assume the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.] Given that the volume of water in the container increases at a constant rate of \(200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\),
  2. find the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 15\) Give your answer in its simplest form in terms of \(\pi\).
Edexcel C4 2018 June Q5
7 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-16_938_1257_125_486} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 1 + t - 5 \sin t , \quad y = 2 - 4 \cos t , \quad - \pi \leqslant t \leqslant \pi$$ The point \(A\) lies on the curve \(C\). Given that the coordinates of \(A\) are ( \(k , 2\) ), where \(k > 0\)
  1. find the exact value of \(k\), giving your answer in a fully simplified form.
  2. Find the equation of the tangent to \(C\) at the point \(A\). Give your answer in the form \(y = p x + q\), where \(p\) and \(q\) are exact real values.
Edexcel C4 2018 June Q6
6 marks Standard +0.3
  1. Given that \(y = 2\) when \(x = - \frac { \pi } { 8 }\), solve the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } } { 3 \cos ^ { 2 } 2 x } \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).