Questions C34 (197 questions)

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Edexcel C34 2017 October Q3
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
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
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
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
  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. 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
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
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
  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.
Edexcel C34 2017 October Q12
  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
  1. (a) 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.
    (b) 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
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
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
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
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
6. (a) Use binomial expansions to show that, for \(| x | < \frac { 1 } { 2 }\)
(b) 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.
(c) 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
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
8. (i) Find \(\int x \sin x d x\)
(ii) (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. 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.
Edexcel C34 2018 October Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c6bde466-61ec-437d-a3b4-84511a98d788-32_492_636_260_660} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = \frac { 3 x - 4 } { x - 3 } , \quad x \in \mathbb { R } , \quad x < 3$$ The graph cuts the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\), as shown in Figure 2 .
  1. State the range of g .
  2. State the coordinates of
    1. point \(A\)
    2. point \(B\)
  3. Find \(\operatorname { gg } ( x )\) in its simplest form.
  4. Sketch the graph with equation \(y = | \mathrm { g } ( x ) |\) On your sketch, show the coordinates of each point at which the graph meets or cuts the axes and state the equation of each asymptote.
  5. Find the exact solution of the equation \(| \mathrm { g } ( x ) | = 8\)
Edexcel C34 2018 October Q11
11. Relative to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation $$l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 2
3
- 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
4
3 \end{array} \right)$$ where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) passes through the origin and is parallel to \(l _ { 1 }\)
  1. Find a vector equation for \(l _ { 2 }\) The point \(A\) and the point \(B\) both lie on \(l _ { 1 }\) with parameters \(\lambda = 0\) and \(\lambda = 3\) respectively.
    Write down
    1. the coordinates of \(A\),
    2. the coordinates of \(B\).
  3. Find the size of the acute angle between \(O A\) and \(l _ { 1 }\) Give your answer in degrees to one decimal place. The point \(D\) lies on \(l _ { 2 }\) such that \(O A B D\) is a parallelogram.
  4. Find the area of \(O A B D\), giving your answer to the nearest whole number.
Edexcel C34 2018 October Q12
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c6bde466-61ec-437d-a3b4-84511a98d788-40_520_663_255_644} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve \(C\) with parametric equations $$x = 7 t ^ { 2 } - 5 , \quad y = t \left( 9 - t ^ { 2 } \right) , \quad t \in \mathbb { R }$$
  1. Find an equation of the tangent to \(C\) at the point where \(t = 1\) Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The curve \(C\) cuts the \(x\)-axis at the points \(A\) and \(B\), as shown in Figure 3
    1. Find the \(x\) coordinate of the point \(A\).
    2. Find the \(x\) coordinate of the point \(B\). The region \(R\), shown shaded in Figure 3, is enclosed by the loop of the curve \(C\).
  3. Use integration to find the area of \(R\).
Edexcel C34 2018 October Q13
13. The volume of a spherical balloon of radius \(r \mathrm {~m}\) is \(V \mathrm {~m} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\)
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\) Given that the volume of the balloon increases with time \(t\) seconds according to the formula $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 20 } { V ( 0.05 t + 1 ) ^ { 3 } } , \quad t \geqslant 0$$
  2. find an expression in terms of \(r\) and \(t\) for \(\frac { \mathrm { d } r } { \mathrm {~d} t }\) Given that \(V = 1\) when \(t = 0\)
  3. solve the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 20 } { V ( 0.05 t + 1 ) ^ { 3 } }$$ giving your answer in the form \(V ^ { 2 } = \mathrm { f } ( t )\).
  4. Hence find the radius of the balloon at time \(t = 20\), giving your answer to 3 significant figures.
    \includegraphics[max width=\textwidth, alt={}]{c6bde466-61ec-437d-a3b4-84511a98d788-48_2632_1828_121_121}
Edexcel C34 Specimen Q1
  1. (a) Express \(5 \cos 2 \theta - 12 \sin 2 \theta\) in the form \(R \cos ( 2 \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\) Give the value of \(\alpha\) to 2 decimal places.
    (b) Hence solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation
$$5 \cos 2 \theta - 12 \sin 2 \theta = 10$$ giving your answers to 1 decimal place.
Edexcel C34 Specimen Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-04_479_855_310_566} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { e } ^ { x } \sqrt { \sin x } , 0 \leqslant x \leqslant \pi\). 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 values of \(y\) corresponding to \(x = \frac { \pi } { 4 }\) and \(x = \frac { \pi } { 2 }\), giving your answers to 5 decimal places.
    \(x\)0\(\frac { \pi } { 4 }\)\(\frac { \pi } { 2 }\)\(\frac { 3 \pi } { 4 }\)\(\pi\)
    \(y\)08.872070
  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. The curve \(y = \mathrm { e } ^ { x } \sqrt { \sin x } , 0 \leqslant x \leqslant \pi\), has a maximum turning point at \(Q\), shown in Figure 1.
  3. Find the \(x\) coordinate of \(Q\).