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Edexcel C34 2019 June Q1
7 marks Standard +0.3
1. $$f ( x ) = 2 x ^ { 3 } + x - 20$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be rewritten as $$x = \sqrt [ 3 ] { a - b x }$$ where \(a\) and \(b\) are positive constants to be determined.
  2. Starting with \(x _ { 1 } = 2.1\) use the iteration formula \(x _ { n + 1 } = \sqrt [ 3 ] { a - b x _ { n } }\), with the numerical values of \(a\) and \(b\), to calculate the values of \(x _ { 2 }\) and \(x _ { 3 }\) giving your answers to 3 decimal places.
  3. Using a suitable interval, show that 2.077 is a root of the equation \(\mathrm { f } ( x ) = 0\) correct to 3 decimal places.
  4. Hence state a root, to 3 decimal places, of the equation $$2 ( x + 2 ) ^ { 3 } + x - 18 = 0$$
    VIIIV SIHI NI JIIYM ION OCVIIV SIHI NI JIIIM ION OCVIIV SIHI NI JIIYM ION OC
Edexcel C34 2019 June Q2
7 marks Moderate -0.3
2. (a) Find \(\int \frac { 4 x + 3 } { x } \mathrm {~d} x , \quad x > 0\) (b) Given that \(y = 25\) at \(x = 1\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 4 x + 3 ) y ^ { \frac { 1 } { 2 } } } { x } \quad x > 0 , y > 0$$ giving your answer in the form \(y = [ \mathrm { g } ( x ) ] ^ { 2 }\).
VJYV SIHI NITIIYIM ION OC
VI4V SIHI NI JAHMA ION OC
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\includegraphics[max width=\textwidth, alt={}, center]{a9870c94-0910-46ec-a54a-44a431cb324e-05_52_49_2777_1886}
Edexcel C34 2019 June Q3
6 marks Moderate -0.3
3. A curve \(C\) has parametric equations $$x = \sqrt { 3 } \tan \theta , \quad y = \sec ^ { 2 } \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 3 }$$ The cartesian equation of \(C\) is $$y = \mathrm { f } ( x ) , \quad 0 \leqslant x \leqslant k , \quad \text { where } k \text { is a constant }$$
  1. State the value of \(k\).
  2. Find \(\mathrm { f } ( x )\) in its simplest form.
  3. Hence, or otherwise, find the gradient of the curve at the point where \(\theta = \frac { \pi } { 6 }\)
Edexcel C34 2019 June Q4
8 marks Standard +0.3
4. The curve \(C\) has equation $$3 y \mathrm { e } ^ { - 2 x } = 4 x ^ { 2 } + y ^ { 2 } + 2$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) on \(C\) has coordinates \(( 0,2 )\).
  2. Find the equation of the normal to \(C\) at \(P\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
    (3)
    VIIIV SIHI NI JIIYM ION OCVIUV SIHI NI JIIIM ION OOVI4V SIHI NI JIIYM ION OO
Edexcel C34 2019 June Q5
9 marks Moderate -0.3
5. A bath is filled with hot water. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of the water in the bath, \(t\) minutes after the bath has been filled, is given by $$\theta = 20 + A \mathrm { e } ^ { - k t }$$ where \(A\) and \(k\) are positive constants. Given that the temperature of the water in the bath is initially \(38 ^ { \circ } \mathrm { C }\),
  1. find the value of \(A\). The temperature of the water in the bath 16 minutes after the bath has been filled is \(24.5 ^ { \circ } \mathrm { C }\).
  2. Show that \(k = \frac { 1 } { 8 } \ln 2\) Using the values for \(k\) and \(A\),
  3. find the temperature of the water 40 minutes after the bath has been filled, giving your answer to 3 significant figures.
  4. Explain why the temperature of the water in the bath cannot fall to \(19 ^ { \circ } \mathrm { C }\).
Edexcel C34 2019 June Q6
7 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9870c94-0910-46ec-a54a-44a431cb324e-14_988_1120_123_395} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the graph with equation \(y = | 4 x + 10 a |\), where \(a\) is a positive constant. The graph cuts the \(y\)-axis at the point \(P\) and meets the \(x\)-axis at the point \(Q\) as shown.
    1. State the coordinates of \(P\).
    2. State the coordinates of \(Q\).
  2. A copy of Figure 1 is shown on page 15. On this copy, sketch the graph with equation $$y = | x | - a$$ Show on the sketch the coordinates of each point where your graph cuts or meets the coordinate axes.
  3. Hence, or otherwise, solve the equation $$| 4 x + 10 a | = | x | - a$$ giving your answers in terms of \(a\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a9870c94-0910-46ec-a54a-44a431cb324e-15_860_1128_447_392} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} \(\_\_\_\_\) 7
Edexcel C34 2019 June Q7
9 marks Standard +0.3
7. (a) Express \(5 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 4 decimal places. The height of sea water, \(H\) metres, on a harbour wall is modelled by the equation $$H = 6 + 2.5 \cos \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \sin \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t < 12$$ where \(t\) is the number of hours after midday.
(b) Calculate the times at which the model predicts that the height of sea water on the harbour wall will be 4.6 metres. Give your answers to the nearest minute. \includegraphics[max width=\textwidth, alt={}, center]{a9870c94-0910-46ec-a54a-44a431cb324e-18_2257_54_314_1977}
Edexcel C34 2019 June Q8
11 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9870c94-0910-46ec-a54a-44a431cb324e-22_524_1443_260_246} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { 6 x + 2 } { 3 x ^ { 2 } + 5 } , \quad x \in \mathbb { R }$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\), writing your answer as a single fraction in its simplest form. The curve has two turning points, a maximum at point \(A\) and a minimum at point \(B\), as shown in Figure 2.
  2. Using part (a), find the coordinates of point \(A\) and the coordinates of point \(B\).
  3. State the coordinates of the maximum turning point of the function with equation $$y = \mathrm { f } ( 2 x ) + 4 \quad x \in \mathbb { R }$$
  4. Find the range of the function $$\operatorname { g } ( x ) = \frac { 6 x + 2 } { 3 x ^ { 2 } + 5 } , \quad x \leqslant 0$$
Edexcel C34 2019 June Q9
8 marks Standard +0.3
9. (a) Using the formula for \(\sin ( A + B )\) and the relevant double angle formulae, find an
identity for \(\sin 3 x\), giving your answer in the form $$\sin ( 3 x ) \equiv P \sin x + Q \sin ^ { 3 } x$$ where \(P\) and \(Q\) are constants to be determined.
(b) Hence, showing each step of your working, evaluate $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \sin 3 x \cos x d x$$ (Solutions based entirely on graphical or numerical methods are not acceptable.)
VIIIV SIHI NI III M LON OCVIIV SIHI NI JIIIM ION OCVI4V SIHIL NI JIIYM ION OC
Edexcel C34 2019 June Q10
9 marks Standard +0.3
  1. (a) Use the binomial series to find the expansion of
$$\frac { 1 } { ( 2 + 3 x ) ^ { 3 } } \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each term as a simplified fraction.
(b) Hence or otherwise, find the coefficient of \(x ^ { 2 }\) in the series expansion of
  1. \(\frac { 1 } { ( 2 + 6 x ) ^ { 3 } } \quad | x | < \frac { 1 } { 3 }\)
  2. \(\frac { 4 - x } { ( 2 + 3 x ) ^ { 3 } } \quad | x | < \frac { 2 } { 3 }\)
Edexcel C34 2019 June Q11
12 marks Standard +0.3
11. (a) Given $$\frac { 9 } { t ^ { 2 } ( t - 3 ) } \equiv \frac { A } { t } + \frac { B } { t ^ { 2 } } + \frac { C } { ( t - 3 ) }$$ find the value of the constants \(A , B\) and \(C\).
(b) $$I = \int _ { 4 } ^ { 12 } \frac { 9 } { t ^ { 2 } ( t - 3 ) } \mathrm { d } t$$ Find the exact value of \(I\), giving your answer in the form \(\ln ( a ) - b\), where \(a\) and \(b\) are positive constants. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9870c94-0910-46ec-a54a-44a431cb324e-34_535_880_959_525} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve \(C\) with parametric equations $$x = 2 \ln ( t - 3 ) , \quad y = \frac { 6 } { t } \quad t > 3$$ The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = 2 \ln 9\) The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution.
(c) Show that the exact volume of the solid generated is $$k \times I$$ where \(k\) is a constant to be found.
Edexcel C34 2019 June Q12
13 marks Standard +0.3
  1. Relative to a fixed origin \(O\),
    the point \(A\) has position vector \(( 2 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k } )\) the point \(B\) has position vector \(( 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } )\) the point \(C\) has position vector ( \(2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\) )
The line \(l\) passes through the points \(A\) and \(B\).
  1. Find the vector \(\overrightarrow { A B }\).
  2. Find a vector equation for the line \(l\).
  3. Show that the size of the angle \(C A B\) is \(62.8 ^ { \circ }\), to one decimal place.
  4. Hence find the area of triangle \(C A B\), giving your answer to 3 significant figures. The point \(D\) lies on the line \(l\). Given that the area of triangle \(C A D\) is twice the area of triangle \(C A B\),
  5. find the two possible position vectors of point \(D\).
Edexcel C34 2019 June Q13
12 marks Standard +0.3
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9870c94-0910-46ec-a54a-44a431cb324e-42_649_709_242_614} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve with equation \(y = 12 x ^ { 2 } \ln \left( 2 x ^ { 2 } \right) , x > 0\) The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = 2\) The table below shows corresponding values of \(x\) and \(y\) for \(y = 12 x ^ { 2 } \ln \left( 2 x ^ { 2 } \right)\), with the values of \(y\) given to 3 significant figures.
\(x\)11.251.51.752
\(y\)8.3221.440.666.699.8
  1. Use the trapezium rule, with all the values of \(y\), to obtain an estimate for the area of \(R\), giving your answer to 2 significant figures.
  2. Use the substitution \(u = x ^ { 2 }\) to show that the area of \(R\) is given by $$\int _ { 1 } ^ { 4 } 6 u ^ { \frac { 1 } { 2 } } \ln ( 2 u ) \mathrm { d } u$$
  3. Hence, using calculus, find the exact area of \(R\), writing your answer in the form \(a + b \ln 2\), where \(a\) and \(b\) are constants to be found.
Edexcel C34 2019 June Q14
7 marks Moderate -0.5
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9870c94-0910-46ec-a54a-44a431cb324e-46_524_855_255_539} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of the curves \(C _ { 1 }\) and \(C _ { 2 }\) $$\begin{aligned} & C _ { 1 } \text { has equation } y = 3 + \mathrm { e } ^ { x + 1 } \quad x \in \mathbb { R } \\ & C _ { 2 } \text { has equation } y = 10 - \mathrm { e } ^ { x } \quad x \in \mathbb { R } \end{aligned}$$ Given that \(C _ { 1 }\) and \(C _ { 2 }\) cut the \(y\)-axis at the points \(P\) and \(Q\) respectively,
  1. find the exact distance \(P Q\). \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the point \(R\).
  2. Find the exact coordinates of \(R\).
    VIIIV SIHI NI IAIUM ION OCVIIV SIHI NI JIIIM ION OCVIIV SIHI NI JIIYM ION OC
Edexcel C34 2017 October Q1
8 marks Standard +0.3
1. $$f ( x ) = x ^ { 5 } + x ^ { 3 } - 12 x ^ { 2 } - 8 , \quad x \in \mathbb { R }$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt [ 3 ] { \frac { 4 \left( 3 x ^ { 2 } + 2 \right) } { x ^ { 2 } + 1 } }$$
  2. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { \frac { 4 \left( 3 x _ { n } ^ { 2 } + 2 \right) } { x _ { n } ^ { 2 } + 1 } }$$ with \(x _ { 0 } = 2\), to find \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) giving your answers to 3 decimal places. The equation \(\mathrm { f } ( x ) = 0\) has a single root, \(\alpha\).
  3. By choosing a suitable interval, prove that \(\alpha = 2.247\) to 3 decimal places.
Edexcel C34 2017 October Q2
11 marks Standard +0.3
2. The curve \(C\) has equation $$y ^ { 3 } + x ^ { 2 } y - 6 x = 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Hence find the exact coordinates of the points on \(C\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
Edexcel C34 2017 October Q3
8 marks Moderate -0.8
3. The number of bacteria in a liquid culture is modelled by the formula $$N = 3500 ( 1.035 ) ^ { t } , \quad t \geqslant 0$$ where \(N\) is the number of bacteria \(t\) hours after the start of a scientific study.
  1. State the number of bacteria at the start of the scientific study.
    (1)
  2. Find the time taken from the start of the study for the number of bacteria to reach 10000
    Give your answer in hours and minutes, to the nearest minute.
  3. Use calculus to find the rate of increase in the number of bacteria when \(t = 8\) Give your answer, in bacteria per hour, to the nearest whole number.
Edexcel C34 2017 October Q4
9 marks Standard +0.3
4. (a) Prove that $$\frac { 1 - \cos 2 x } { \sin 2 x } \equiv \tan x , \quad x \neq \frac { n \pi } { 2 }$$ (b) Hence solve, for \(0 \leqslant \theta < 2 \pi\), $$3 \sec ^ { 2 } \theta - 7 = \frac { 1 - \cos 2 \theta } { \sin 2 \theta }$$ Give your answers in radians to 3 decimal places, as appropriate.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2017 October Q5
8 marks Moderate -0.8
5. (i) Find $$\int \left( ( 3 x + 5 ) ^ { 9 } + \mathrm { e } ^ { 5 x } \right) \mathrm { d } x$$ (ii) Given that \(b\) is a constant greater than 2 , and $$\int _ { 2 } ^ { b } \frac { x } { x ^ { 2 } + 5 } \mathrm {~d} x = \ln ( \sqrt { 6 } )$$ use integration to find the value of \(b\).
Edexcel C34 2017 October Q6
10 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a6d0dba-d948-4124-9740-a88c17b0be65-16_618_1018_228_456} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = 2 \mathrm { e } ^ { - x } \sqrt { \sin x } , 0 \leqslant x \leqslant \pi\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve and the \(x\)-axis.
  1. Complete the table below with the value of \(y\) corresponding to \(x = \frac { \pi } { 2 }\), giving your answer to 5 decimal places.
    \(x\)0\(\frac { \pi } { 4 }\)\(\frac { \pi } { 2 }\)\(\frac { 3 \pi } { 4 }\)\(\pi\)
    \(y\)00.766790.159400
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of the region \(R\). Give your answer to 4 decimal places.
  3. Given \(y = 2 \mathrm { e } ^ { - x } \sqrt { \sin x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) for \(0 < x < \pi\). The curve \(C\) has a maximum turning point when \(x = a\).
  4. Use your answer to part (c) to find the value of \(a\), giving your answer to 3 decimal places.
Edexcel C34 2017 October Q7
9 marks Standard +0.3
  1. (a) Use the binomial series to expand
$$\frac { 1 } { ( 2 - 3 x ) ^ { 3 } } \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each term as a simplified fraction. $$f ( x ) = \frac { 4 + k x } { ( 2 - 3 x ) ^ { 3 } } \quad \text { where } k \text { is a constant and } | x | < \frac { 2 } { 3 }$$ Given that the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), is $$\frac { 1 } { 2 } + A x + \frac { 81 } { 16 } x ^ { 2 } + \cdots$$ where \(A\) is a constant,
(b) find the value of \(k\),
(c) find the value of \(A\).
Edexcel C34 2017 October Q8
8 marks Standard +0.3
8. Use partial fractions, and integration, to find the exact value of \(\int _ { 3 } ^ { 4 } \frac { 2 x ^ { 2 } - 3 } { x ( x - 1 ) } \mathrm { d } x\) Write your answer in the form \(a + \ln b\), where \(a\) is an integer and \(b\) is a rational constant.
Edexcel C34 2017 October Q9
13 marks Standard +0.3
9. $$\mathrm { f } ( x ) = 2 \ln ( x ) - 4 , \quad x > 0 , \quad x \in \mathbb { R }$$
  1. Sketch, on separate diagrams, the curve with equation
    1. \(y = \mathrm { f } ( x )\)
    2. \(y = | \mathrm { f } ( x ) |\) On each diagram, show the coordinates of each point at which the curve meets or cuts the axes. On each diagram state the equation of the asymptote.
  2. Find the exact solutions of the equation \(| \mathrm { f } ( x ) | = 4\) $$\mathrm { g } ( x ) = \mathrm { e } ^ { x + 5 } - 2 , \quad x \in \mathbb { R }$$
  3. Find \(\mathrm { gf } ( x )\), giving your answer in its simplest form.
  4. Hence, or otherwise, state the range of gf.
Edexcel C34 2017 October Q10
13 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a6d0dba-d948-4124-9740-a88c17b0be65-32_556_716_237_607} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = \frac { 20 t } { 2 t + 1 } \quad y = t ( t - 4 ) , \quad t > 0$$ The curve cuts the \(x\)-axis at the point \(P\).
  1. Find the \(x\) coordinate of \(P\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( t - A ) ( 2 t + 1 ) ^ { 2 } } { B }\) where \(A\) and \(B\) are constants to be found.
    1. Make \(t\) the subject of the formula $$x = \frac { 20 t } { 2 t + 1 }$$
    2. Hence find a cartesian equation of the curve \(C\). Write your answer in the form $$y = \mathrm { f } ( x ) , \quad 0 < x < k$$ where \(\mathrm { f } ( x )\) is a single fraction and \(k\) is a constant to be found.
Edexcel C34 2017 October Q11
14 marks Standard +0.8
  1. (a) Given \(0 \leqslant h < 25\), use the substitution \(u = 5 - \sqrt { h }\) to show that
$$\int \frac { \mathrm { d } h } { 5 - \sqrt { h } } = - 10 \ln ( 5 - \sqrt { h } ) - 2 \sqrt { h } + k$$ where \(k\) is a constant.
(6) A team of scientists is studying a species of tree.
The rate of change in height of a tree of this species is modelled by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { t ^ { 0.2 } ( 5 - \sqrt { h } ) } { 5 }$$ where \(h\) is the height of the tree in metres and \(t\) is the time in years after the tree is planted.
One of these trees is 2 metres high when it is planted.
(b) Use integration to calculate the time it would take for this tree to reach a height of 15 metres, giving your answer to one decimal place.
(c) Hence calculate the rate of change in height of this tree when its height is 15 metres. Write your answer in centimetres per year to the nearest centimetre.