Questions — Edexcel (10514 questions)

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Edexcel C34 2019 June Q11
12 marks Standard +0.3
11.
  1. 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 \(C\).
  2. $$I = \int _ { 4 } ^ { 12 } \frac { 9 } { t ^ { 2 } ( t - 3 ) } \mathrm { d } t$$ 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.
  3. 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.
  1. Prove that $$\frac { 1 - \cos 2 x } { \sin 2 x } \equiv \tan x , \quad x \neq \frac { n \pi } { 2 }$$
  2. 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.
  1. Find $$\int \left( ( 3 x + 5 ) ^ { 9 } + \mathrm { e } ^ { 5 x } \right) \mathrm { d } x$$
  2. 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. 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,
  2. find the value of \(k\),
  3. 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. 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.
  2. 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.
  3. 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.
Edexcel C34 2017 October Q12
14 marks Standard +0.8
  1. Relative to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations
$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 2 \\ 1 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 7 \end{array} \right) + \mu \left( \begin{array} { l } 8 \\ 4 \\ 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\).
  1. Write down the coordinates of \(A\). Given that the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\),
  2. show that \(\sin \theta = k \sqrt { 2 }\), where \(k\) is a rational number to be found. The point \(B\) lies on \(l _ { 1 }\) where \(\lambda = 4\) The point \(C\) lies on \(l _ { 2 }\) such that \(A C = 2 A B\).
  3. Find the exact area of triangle \(A B C\).
  4. Find the coordinates of the two possible positions of \(C\).
Edexcel C34 2018 October Q1
8 marks Standard +0.3
  1. Write \(\cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), 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.
  2. Hence solve, for \(0 \leqslant \theta < \pi\), the equation $$\cos 2 \theta + 4 \sin 2 \theta = 1.2$$ giving your answers to 2 decimal places.
Edexcel C34 2018 October Q2
7 marks Standard +0.3
2. A curve \(C\) has equation $$x ^ { 3 } - 4 x y + 2 x + 3 y ^ { 2 } - 3 = 0$$ Find an equation of the normal to \(C\) at the point ( \(- 3,2\) ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{c6bde466-61ec-437d-a3b4-84511a98d788-05_108_166_2612_1781}
Edexcel C34 2018 October Q3
6 marks Moderate -0.8
3. Given \(\cos \theta ^ { \circ } = p\), where \(p\) is a constant and \(\theta ^ { \circ }\) is acute use standard trigonometric identities to find, in terms of \(p\),
  1. \(\sec \theta ^ { \circ }\)
  2. \(\sin ( \theta - 90 ) ^ { \circ }\)
  3. \(\sin 2 \theta ^ { \circ }\) Write each answer in its simplest form.
Edexcel C34 2018 October Q4
10 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c6bde466-61ec-437d-a3b4-84511a98d788-08_510_783_260_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = 8 x - x \mathrm { e } ^ { 3 x } , x \geqslant 0\) The curve meets the \(x\)-axis at the origin and cuts the \(x\)-axis at the point \(A\).
  1. Find the exact \(x\) coordinate of \(A\), giving your answer in its simplest form. The curve has a maximum turning point at the point \(M\).
  2. Show, by using calculus, that the \(x\) coordinate of \(M\) is a solution of $$x = \frac { 1 } { 3 } \ln \left( \frac { 8 } { 1 + 3 x } \right)$$
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \ln \left( \frac { 8 } { 1 + 3 x _ { n } } \right)$$ with \(x _ { 0 } = 0.4\) to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
Edexcel C34 2018 October Q5
10 marks Standard +0.3
5. $$f ( x ) = \frac { 4 x ^ { 2 } + 5 x + 3 } { ( x + 2 ) ( 1 - x ) ^ { 2 } } \equiv \frac { A } { ( x + 2 ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }$$
  1. Find the values of the constants \(A\), \(B\) and \(C\).
    1. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
    2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\), writing your answer in the form \(p + \ln q\), where \(p\) and \(q\) are constants.
      "
      ◯
      VGHV SIHIN NI III M I ION OCVIIV SIHI NI JIIIM ION OCVI4V SIHIL NI JIIYM ION OC
Edexcel C34 2018 October Q6
9 marks Standard +0.3
6.
  1. Use binomial expansions to show that, for \(| x | < \frac { 1 } { 2 }\)
  2. Find the exact value of \(\sqrt { \frac { 1 + 2 x } { 1 - x } }\) when \(x = \frac { 1 } { 10 }\) Give your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined.
  3. Substitute \(x = \frac { 1 } { 10 }\) into the expansion given in part (a) and hence find an approximate value for \(\sqrt { 3 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers. $$\sqrt { \frac { 1 + 2 x } { 1 - x } } \approx 1 + \frac { 3 } { 2 } x + \frac { 3 } { 8 } x ^ { 2 }$$
Edexcel C34 2018 October Q7
8 marks Standard +0.3
7. A curve has equation $$y = \ln ( 1 - \cos 2 x ) , \quad x \in \mathbb { R } , 0 < x < \pi$$ Show that
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k \cot x\), where \(k\) is a constant to be found. Hence find the exact coordinates of the point on the curve where
  2. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sqrt { 3 }\)
Edexcel C34 2018 October Q8
10 marks Standard +0.3
8.
  1. Find \(\int x \sin x d x\)
  2. (a) Use the substitution \(x = \sec \theta\) to show that
    (b) Hence find the exact value of $$\int _ { 1 } ^ { 2 } \sqrt { 1 - \frac { 1 } { x ^ { 2 } } } \mathrm {~d} x = \int _ { 0 } ^ { \frac { \pi } { 3 } } \tan ^ { 2 } \theta \mathrm {~d} \theta$$ Hence find the exact value of $$\int _ { 1 } ^ { 2 } \sqrt { 1 - \frac { 1 } { x ^ { 2 } } } \mathrm {~d} x$$
Edexcel C34 2018 October Q9
9 marks Standard +0.3
9. A rare species of mammal is being studied. The population \(P\), \(t\) years after the study started, is modelled by the formula $$P = \frac { 900 \mathrm { e } ^ { \frac { 1 } { 4 } t } } { 3 \mathrm { e } ^ { \frac { 1 } { 4 } t } - 1 } , \quad t \in \mathbb { R } , \quad t \geqslant 0$$ Using the model,
  1. calculate the number of mammals at the start of the study,
  2. calculate the exact value of \(t\) when \(P = 315\) Give your answer in the form \(a \ln k\), where \(a\) and \(k\) are integers to be determined.
    1. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\)
    2. Hence find the value of \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) when \(t = 8\), giving your answer to 2 decimal places.