Questions — Edexcel C34 (197 questions)

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Edexcel C34 2015 June Q8
  1. (a) Prove by differentiation that
$$\frac { \mathrm { d } } { \mathrm {~d} y } ( \ln \tan 2 y ) = \frac { 4 } { \sin 4 y } , \quad 0 < y < \frac { \pi } { 4 }$$ (b) Given that \(y = \frac { \pi } { 6 }\) when \(x = 0\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \cos x \sin 4 y , \quad 0 < y < \frac { \pi } { 4 }$$ Give your answer in the form \(\tan 2 y = A \mathrm { e } ^ { B \sin x }\), where \(A\) and \(B\) are constants to be determined.
Edexcel C34 2015 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c08fbab-283e-4c92-89a4-10f68f37e133-14_709_824_118_559} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with parametric equations $$x = t ^ { 2 } + 2 t , \quad y = t ^ { 3 } - 9 t , \quad t \in \mathbb { R }$$ The curve cuts the \(x\)-axis at the origin and at the points \(A\) and \(B\) as shown in Figure 3 .
  1. Find the coordinates of point \(A\) and show that point \(B\) has coordinates ( 15,0 ).
  2. Show that the equation of the tangent to the curve at \(B\) is \(9 x - 4 y - 135 = 0\) The tangent to the curve at \(B\) cuts the curve again at the point \(X\).
  3. Find the coordinates of \(X\).
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2015 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c08fbab-283e-4c92-89a4-10f68f37e133-16_319_508_237_719} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a right circular cylindrical rod which is expanding as it is heated.
At time \(t\) seconds the radius of the rod is \(x \mathrm {~cm}\) and the length of the rod is \(6 x \mathrm {~cm}\).
Given that the cross-sectional area of the rod is increasing at a constant rate of \(\frac { \pi } { 20 } \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\), find the rate of increase of the volume of the rod when \(x = 2\) Write your answer in the form \(k \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) where \(k\) is a rational number.
Edexcel C34 2015 June Q11
11. (a) Express \(1.5 \sin \theta - 1.2 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the value of \(R\) and the value of \(\alpha\) to 3 decimal places. The height, \(H\) metres, of sea water at the entrance to a harbour on a particular day, is modelled by the equation $$H = 3 + 1.5 \sin \left( \frac { \pi t } { 6 } \right) - 1.2 \cos \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 12$$ where \(t\) is the number of hours after midday.
(b) Using your answer to part (a), calculate the minimum value of \(H\) predicted by this model and the value of \(t\), to 2 decimal places, when this minimum occurs.
(c) Find, to the nearest minute, the times when the height of sea water at the entrance to the harbour is predicted by this model to be 4 metres.
Edexcel C34 2015 June Q12
  1. (i) Relative to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation
$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } - 5
1
6 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 3
1 \end{array} \right) \text { where } \lambda \text { is a scalar parameter. }$$ The point \(P\) lies on \(l _ { 1 }\). Given that \(\overrightarrow { O P }\) is perpendicular to \(l _ { 1 }\), calculate the coordinates of \(P\).
(ii) Relative to a fixed origin \(O\), the line \(l _ { 2 }\) is given by the equation $$l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 4
- 3
12 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 3
4 \end{array} \right) \text { where } \mu \text { is a scalar parameter. }$$ The point \(A\) does not lie on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O A }\) is parallel to the line \(l _ { 2 }\) and \(| \overrightarrow { O A } | = \sqrt { 2 }\) units, calculate the possible position vectors of the point \(A\).
Edexcel C34 2015 June Q13
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c08fbab-283e-4c92-89a4-10f68f37e133-22_536_929_223_504} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of part of the curve with equation \(y = 2 - \ln x , x > 0\) The finite region \(R\), shown shaded in Figure 5, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \mathrm { e }\). The table below shows corresponding values of \(x\) and \(y\) for \(y = 2 - \ln x\)
\(x\)e\(\frac { \mathrm { e } + \mathrm { e } ^ { 2 } } { 2 }\)\(\mathrm { e } ^ { 2 }\)
\(y\)10
  1. Complete the table giving the 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 by parts to show that \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x = x ( \ln x ) ^ { 2 } - 2 x \ln x + 2 x + c\) The area \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  4. Use calculus to find the exact volume of the solid generated. Write your answer in the form \(\pi \mathrm { e } ( p \mathrm { e } + q )\), where \(p\) and \(q\) are integers to be found.
Edexcel C34 2017 June Q1
  1. A curve \(C\) has equation
$$3 x ^ { 2 } + 2 x y - 2 y ^ { 2 } + 4 = 0$$ Find an equation for the tangent to \(C\) at the point ( 2,4 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
(6)
Edexcel C34 2017 June Q2
  1. Use integration by parts to find the exact value of \(\int _ { 1 } ^ { \mathrm { e } } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x\)
Write your answer in the form \(a + \frac { b } { \mathrm { e } }\), where \(a\) and \(b\) are integers.
Edexcel C34 2017 June Q3
3. The function g is defined by $$g ( x ) = \frac { 6 x } { 2 x + 3 } \quad x > 0$$
  1. Find the range of g .
  2. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain.
  3. Find the function \(\operatorname { gg } ( x )\), writing your answer as a single fraction in its simplest form.
Edexcel C34 2017 June Q4
4. $$f ( x ) = \frac { 27 } { ( 3 - 5 x ) ^ { 2 } } \quad | x | < \frac { 3 } { 5 }$$
  1. Find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient in its simplest form.
    (5) Use your answer to part (a) to find the series expansion in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), of
  2. \(g ( x ) = \frac { 27 } { ( 3 + 5 x ) ^ { 2 } } \quad | x | < \frac { 3 } { 5 }\)
  3. \(\mathrm { h } ( x ) = \frac { 27 } { ( 3 - x ) ^ { 2 } } \quad | x | < 3\)
Edexcel C34 2017 June Q5
5. $$\frac { 6 - 5 x - 4 x ^ { 2 } } { ( 2 - x ) ( 1 + 2 x ) } \equiv A + \frac { B } { ( 2 - x ) } + \frac { C } { ( 1 + 2 x ) }$$
  1. Find the values of the constants \(A , B\) and \(C\). $$f ( x ) = \frac { 6 - 5 x - 4 x ^ { 2 } } { ( 2 - x ) ( 1 + 2 x ) } \quad x > 2$$
  2. Using part (a), find \(\mathrm { f } ^ { \prime } ( x )\).
  3. Prove that \(\mathrm { f } ( x )\) is a decreasing function.
Edexcel C34 2017 June Q6
  1. The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { r } 5
    - 2
    4 \end{array} \right) + \lambda \left( \begin{array} { r } 6
    3
    - 1 \end{array} \right)\), where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { r } 10
    5
    - 3 \end{array} \right) + \mu \left( \begin{array} { l } 3
    1
    2 \end{array} \right)\), where \(\mu\) is a scalar parameter.
Justify, giving reasons in each case, whether the lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, intersecting or skew.
(6)
Edexcel C34 2017 June Q7
  1. (a) Prove that
$$\frac { 1 - \cos 2 x } { 1 + \cos 2 x } \equiv \tan ^ { 2 } x , \quad x \neq ( 2 n + 1 ) 90 ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), $$\frac { 2 - 2 \cos 2 \theta } { 1 + \cos 2 \theta } - 2 = 7 \sec \theta$$ Give your answers in degrees to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2017 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{29b56d51-120a-4275-a761-8b8aed7bca54-24_560_1029_219_463} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt { \frac { x } { x ^ { 2 } + 1 } } , \quad x \geqslant 0\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 7\)
The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { \frac { x } { x ^ { 2 } + 1 } }\)
\(x\)234567
\(y\)0.63250.54770.48510.43850.40270.3742
  1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for the area of \(R\), giving your answer to 3 decimal places. The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution.
  2. Use calculus to find the exact volume of the solid of revolution formed. Write your answer in its simplest form.
    \includegraphics[max width=\textwidth, alt={}, center]{29b56d51-120a-4275-a761-8b8aed7bca54-24_2255_47_314_1979}
Edexcel C34 2017 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{29b56d51-120a-4275-a761-8b8aed7bca54-28_615_328_210_808} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of a triangle \(A B C\).
Given \(\overrightarrow { A B } = 2 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }\) and \(\overrightarrow { A C } = 5 \mathbf { i } - 6 \mathbf { j } + \mathbf { k }\),
  1. find the size of angle \(C A B\), giving your answer in degrees to 2 decimal places,
  2. find the area of triangle \(A B C\), giving your answer to 2 decimal places. Using your answer to part (b), or otherwise,
  3. find the shortest distance from \(A\) to \(B C\), giving your answer to 2 decimal places.
Edexcel C34 2017 June Q10
  1. (a) Write \(2 \sin \theta - \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha \leqslant 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to one decimal place.
    (3)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{29b56d51-120a-4275-a761-8b8aed7bca54-32_513_1194_404_374} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the graph with equation \(y = 2 \sin \theta - \cos \theta , \quad 0 \leqslant \theta < 360 ^ { \circ }\)
(b) Sketch the graph with equation $$y = | 2 \sin \theta - \cos \theta | , \quad 0 \leqslant \theta < 360 ^ { \circ }$$ stating the coordinates of all points at which the graph meets or cuts the coordinate axes. The temperature of a warehouse is modelled by the equation $$f ( t ) = 5 + \left| 2 \sin ( 15 t ) ^ { \circ } - \cos ( 15 t ) ^ { \circ } \right| , \quad 0 \leqslant t < 24$$ where \(\mathrm { f } ( t )\) is the temperature of the warehouse in degrees Celsius and \(t\) is the time measured in hours from midnight. State
(c) (i) the maximum value of \(f ( t )\),
(ii) the largest value of \(t\), for \(0 \leqslant t < 24\), at which this maximum value occurs. Give your answer to one decimal place.
Edexcel C34 2017 June Q11
11. $$y = \left( 2 x ^ { 2 } - 3 \right) \tan \left( \frac { 1 } { 2 } x \right) , \quad 0 < x < \pi$$
  1. Find the exact value of \(x\) when \(y = 0\) Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = \alpha\),
  2. show that $$2 \alpha ^ { 2 } - 3 + 4 \alpha \sin \alpha = 0$$ The iterative formula $$x _ { n + 1 } = \frac { 3 } { \left( 2 x _ { n } + 4 \sin x _ { n } \right) }$$ can be used to find an approximation for \(\alpha\).
  3. Taking \(x _ { 1 } = 0.7\), find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving each answer to 4 decimal places.
  4. By choosing a suitable interval, show that \(\alpha = 0.7283\) to 4 decimal places.
    \includegraphics[max width=\textwidth, alt={}]{29b56d51-120a-4275-a761-8b8aed7bca54-38_2253_50_314_1977}
Edexcel C34 2017 June Q12
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{29b56d51-120a-4275-a761-8b8aed7bca54-40_471_949_219_493} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a right cylindrical water tank. The diameter of the circular cross section of the tank is 4 m and the height is 2.25 m . Water is flowing into the tank at a constant rate of \(0.4 \pi \mathrm {~m} ^ { 3 } \mathrm {~min} ^ { - 1 }\). There is a tap at a point \(T\) at the base of the tank. When the tap is open, water leaves the tank at a rate of \(0.2 \pi \sqrt { h } \mathrm {~m} ^ { 3 } \mathrm {~min} ^ { - 1 }\), where \(h\) is the height of the water in metres.
  1. Show that at time \(t\) minutes after the tap has been opened, the height \(h \mathrm {~m}\) of the water in the tank satisfies the differential equation $$20 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 2 - \sqrt { h }$$ At the instant when the tap is opened, \(t = 0\) and \(h = 0.16\)
  2. Use the differential equation to show that the time taken to fill the tank to a height of 2.25 m is given by $$\int _ { 0.16 } ^ { 2.25 } \frac { 20 } { 2 - \sqrt { h } } \mathrm {~d} h$$ Using the substitution \(h = ( 2 - x ) ^ { 2 }\), or otherwise,
  3. find the time taken to fill the tank to a height of 2.25 m . Give your answer in minutes to the nearest minute.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2017 June Q13
13. Figure 5 A colony of ants is being studied. The number of ants in the colony is modelled by the equation $$P = 200 - \frac { 160 \mathrm { e } ^ { 0.6 t } } { 15 + \mathrm { e } ^ { 0.8 t } } \quad t \in \mathbb { R } , t \geqslant 0$$ where \(P\) is the number of ants, measured in thousands, \(t\) years after the study started. A sketch of the graph of \(P\) against \(t\) is shown in Figure 5
  1. Calculate the number of ants in the colony at the start of the study.
  2. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) The population of ants initially decreases, reaching a minimum value after \(T\) years, as shown in Figure 5
  3. Using your answer to part (b), calculate the value of \(T\) to 2 decimal places.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2017 June Q14
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{29b56d51-120a-4275-a761-8b8aed7bca54-48_506_812_219_571} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 8 \cos ^ { 3 } \theta , \quad y = 6 \sin ^ { 2 } \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ Given that the point \(P\) lies on \(C\) and has parameter \(\theta = \frac { \pi } { 3 }\)
  1. find the coordinates of \(P\). The line \(l\) is the normal to \(C\) at \(P\).
  2. Show that an equation of \(l\) is \(y = x + 3.5\) The finite region \(S\), shown shaded in Figure 6, is bounded by the curve \(C\), the line \(l\), the \(y\)-axis and the \(x\)-axis.
  3. Show that the area of \(S\) is given by $$4 + 144 \int _ { 0 } ^ { \frac { \pi } { 3 } } \left( \sin \theta \cos ^ { 2 } \theta - \sin \theta \cos ^ { 4 } \theta \right) d \theta$$
  4. Hence, by integration, find the exact area of \(S\).
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
    END
Edexcel C34 2018 June Q1
  1. (i) Find
$$\int \frac { 2 x ^ { 2 } + 5 x + 1 } { x ^ { 2 } } \mathrm {~d} x , \quad x > 0$$ (ii) Find $$\int x \cos 2 x \mathrm {~d} x$$
Edexcel C34 2018 June Q2
2. A curve \(C\) has parametric equations $$x = \frac { 3 } { 2 } t - 5 , \quad y = 4 - \frac { 6 } { t } \quad t \neq 0$$
  1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(t = 3\), giving your answer as a fraction in its simplest form.
  2. Show that a cartesian equation of \(C\) can be expressed in the form $$y = \frac { a x + b } { x + 5 } \quad x \neq k$$ where \(a , b\) and \(k\) are integers to be found.
Edexcel C34 2018 June Q3
3. $$f ( x ) = 2 ^ { x - 1 } - 4 + 1.5 x \quad x \in \mathbb { R }$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \frac { 1 } { 3 } \left( 8 - 2 ^ { x } \right)$$ The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\), where \(\alpha = 1.6\) to one decimal place.
  2. Starting with \(x _ { 0 } = 1.6\), use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( 8 - 2 ^ { x _ { n } } \right)$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
  3. By choosing a suitable interval, prove that \(\alpha = 1.633\) to 3 decimal places.
Edexcel C34 2018 June Q4
4. (a) Find the binomial expansion of $$( 1 + p x ) ^ { - 4 } , \quad | p x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient as simply as possible in terms of the constant \(p\). $$f ( x ) = \frac { 3 + 4 x } { ( 1 + p x ) ^ { 4 } } \quad | p x | < 1$$ where \(p\) is a positive constant. In the series expansion of \(\mathrm { f } ( x )\), the coefficient of \(x ^ { 2 }\) is twice the coefficient of \(x\).
(b) Find the value of \(p\).
(c) Hence find the coefficient of \(x ^ { 3 }\) in the series expansion of \(\mathrm { f } ( x )\), giving your answer as a simplified fraction.
Edexcel C34 2018 June Q5
    1. The functions \(f\) and \(g\) are defined by
$$\begin{array} { l l } \mathrm { f } : x \rightarrow \mathrm { e } ^ { 2 x } - 5 , & x \in \mathbb { R }
\mathrm {~g} : x \rightarrow \ln ( 3 x - 1 ) , & x \in \mathbb { R } , x > \frac { 1 } { 3 } \end{array}$$
  1. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
  2. Find \(\mathrm { fg } ( 3 )\), giving your answer in its simplest form.
    (ii) (a) Sketch the graph with equation $$y = | 4 x - a |$$ where \(a\) is a positive constant. State the coordinates of each point where the graph cuts or meets the coordinate axes. Given that $$| 4 x - a | = 9 a$$ where \(a\) is a positive constant,
  3. find the possible values of $$| x - 6 a | + 3 | x |$$ giving your answers, in terms of \(a\), in their simplest form.