Questions C34 (197 questions)

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Edexcel C34 Specimen Q3
  1. Using the substitution \(u = \cos x + 1\), or otherwise, show that
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { ( \cos x + 1 ) } \sin x \mathrm {~d} x = \mathrm { e } ( \mathrm { e } - 1 )$$
Edexcel C34 Specimen Q4
4. (a) Use the binomial theorem to expand $$( 2 - 3 x ) ^ { - 2 } , \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction. $$\mathrm { f } ( x ) = \frac { a + b x } { ( 2 - 3 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 3 } , \quad \text { where } a \text { and } b \text { are constants. }$$ In the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), the coefficient of \(x\) is 0 and the coefficient of \(x ^ { 2 }\) is \(\frac { 9 } { 16 }\) Find
(b) the value of \(a\) and the value of \(b\),
(c) the coefficient of \(x ^ { 3 }\), giving your answer as a simplified fraction.
Edexcel C34 Specimen Q5
  1. The functions \(f\) and \(g\) are defined by
$$\begin{array} { l l } \mathrm { f } : x \mapsto \mathrm { e } ^ { - x } + 2 , & x \in \mathbb { R }
\mathrm {~g} : x \mapsto 2 \ln x , & x > 0 \end{array}$$
  1. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
  2. Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\)
  3. Find \(\mathrm { f } ^ { - 1 }\), stating its domain.
  4. On the same axes, sketch the curves with equation \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), giving the coordinates of all the points where the curves cross the axes.
Edexcel C34 Specimen Q6
6. The curve \(C\) has equation $$16 y ^ { 3 } + 9 x ^ { 2 } y - 54 x = 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Find the coordinates of the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
Edexcel C34 Specimen Q7
  1. (a) Show that
$$\cot x - \cot 2 x \equiv \operatorname { cosec } 2 x , \quad x \neq \frac { n \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve for \(0 \leqslant \theta \leqslant \pi\) $$\operatorname { cosec } \left( 3 \theta + \frac { \pi } { 3 } \right) + \cot \left( 3 \theta + \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 3 } }$$ You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 Specimen Q8
8. $$\mathrm { h } ( x ) = \frac { 2 } { x + 2 } + \frac { 4 } { x ^ { 2 } + 5 } - \frac { 18 } { \left( x ^ { 2 } + 5 \right) ( x + 2 ) } , \quad x \geqslant 0$$
  1. Show that \(\mathrm { h } ( x ) = \frac { 2 x } { x ^ { 2 } + 5 }\)
  2. Hence, or otherwise, find \(\mathrm { h } ^ { \prime } ( x )\) in its simplest form. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-26_679_1168_733_390} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a graph of the curve with equation \(y = \mathrm { h } ( x )\).
  3. Calculate the range of \(\mathrm { h } ( x )\).
Edexcel C34 Specimen Q9
  1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2
    3
    - 4 \end{array} \right) + \lambda \left( \begin{array} { l } 1
    2
    1 \end{array} \right)\), where \(\lambda\) is a scalar parameter.
The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 0
9
- 3 \end{array} \right) + \mu \left( \begin{array} { l } 5
0
2 \end{array} \right)\), where \(\mu\) is a scalar parameter.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(C\), find
  1. the coordinates of \(C\). The point \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 0\) and the point \(B\) is the point on \(l _ { 2 }\) where \(\mu = - 1\)
  2. Find the size of the angle \(A C B\). Give your answer in degrees to 2 decimal places.
  3. Hence, or otherwise, find the area of the triangle \(A B C\).
Edexcel C34 Specimen Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-34_599_923_322_571} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = \sin \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( \sqrt { 3 } , \frac { 1 } { 2 } \sqrt { 3 } \right)\)
  1. Find the value of \(\theta\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
  2. Show that \(Q\) has coordinates \(( k \sqrt { 3 } , 0 )\), giving the value of the constant \(k\). The finite shaded region \(S\) shown in Figure 3 is bounded by the curve \(C\), the line \(x = \sqrt { 3 }\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  3. Find the volume of the solid of revolution, giving your answer in the form \(p \pi \sqrt { 3 } + q \pi ^ { 2 }\), where \(p\) and \(q\) are constants. \includegraphics[max width=\textwidth, alt={}, center]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-39_61_29_2608_1886}
Edexcel C34 Specimen Q11
11. A team of conservationists is studying the population of meerkats on a nature reserve. The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 15 } P ( 5 - P ) , \quad t \geqslant 0$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that when \(t = 0 , P = 1\),
  1. solve the differential equation, giving your answer in the form $$P = \frac { a } { b + c \mathrm { e } ^ { - \frac { 1 } { 3 } t } }$$ where \(a\), \(b\) and \(c\) are integers.
  2. Hence show that the population cannot exceed 5000
Edexcel C34 2016 June Q1
  1. (a) Express \(3 \cos \theta + 5 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places.
    (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation
$$3 \cos \theta + 5 \sin \theta = 2$$ Give your answers to one decimal place.
(c) Use your solutions to parts (a) and (b) to deduce the smallest positive value of \(\theta\) for which $$3 \cos \theta - 5 \sin \theta = 2$$
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Edexcel C34 2016 June Q2
2. The point \(P\) with coordinates \(\left( \frac { \pi } { 2 } , 1 \right)\) lies on the curve with equation $$4 x \sin x = \pi y ^ { 2 } + 2 x , \quad \frac { \pi } { 6 } \leqslant x \leqslant \frac { 5 \pi } { 6 }$$ Find an equation of the normal to the curve at \(P\).
Edexcel C34 2016 June Q3
3. (a) Find the binomial expansion of $$( 1 + a x ) ^ { - 3 } , \quad | a x | < 1$$ 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 \(a\). $$f ( x ) = \frac { 2 + 3 x } { ( 1 + a x ) ^ { 3 } } , \quad | a x | < 1$$ In the series expansion of \(\mathrm { f } ( x )\), the coefficient of \(x ^ { 2 }\) is 3
Given that \(a < 0\)
(b) find the value of the constant \(a\),
(c) find the coefficient of \(x ^ { 3 }\) in the series expansion of \(\mathrm { f } ( x )\), giving your answer as a simplified fraction.
Edexcel C34 2016 June Q4
4. $$\mathrm { g } ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 7 x ^ { 2 } + 8 x - 48 } { x ^ { 2 } + x - 12 } , \quad x > 3 , \quad x \in \mathbb { R }$$
  1. Given that $$\frac { x ^ { 4 } + x ^ { 3 } - 7 x ^ { 2 } + 8 x - 48 } { x ^ { 2 } + x - 12 } \equiv x ^ { 2 } + A + \frac { B } { x - 3 }$$ find the values of the constants \(A\) and \(B\).
  2. Hence, or otherwise, find the equation of the tangent to the curve with equation \(y = \mathrm { g } ( x )\) at the point where \(x = 4\). Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be determined.
    (5)
Edexcel C34 2016 June Q5
5．Use integration by parts to find the exact value of $$\int _ { 0 } ^ { 2 } x 2 ^ { x } \mathrm {~d} x$$ Write your answer as a single simplified fraction．
Edexcel C34 2016 June Q6
6. Given that \(a\) and \(b\) are constants and that \(a > b > 0\)
  1. on separate diagrams, sketch the graph with equation
    1. \(y = | x - a |\)
    2. \(y = | x - a | - b\) Show on each sketch the coordinates of each point at which the graph crosses or meets the \(x\)-axis and the \(y\)-axis.
  2. Hence or otherwise find the complete set of values of \(x\) for which $$| x - a | - b < \frac { 1 } { 2 } x$$ giving your answer in terms of \(a\) and \(b\).
Edexcel C34 2016 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d67f716-c8af-4460-8a6b-62073ba9b825-13_695_986_121_497} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Diagram not drawn to scale Figure 1 shows a sketch of part of the curve with equation \(y = \frac { 1 } { \sqrt { 2 x + 5 } } , x > - 2.5\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines with equations \(x = 2\) and \(x = 5\)
  1. Use the trapezium rule with three strips of equal width to find an estimate for the area of \(R\), giving your answer to 3 decimal places.
  2. Use calculus to find the exact area of \(R\).
  3. Hence calculate the magnitude of the error of the estimate found in part (a), giving your answer to one significant figure.
Edexcel C34 2016 June Q8
8. (a) Prove that $$\sin 2 x - \tan x \equiv \tan x \cos 2 x , \quad x \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta < \frac { \pi } { 2 }\)
  1. \(\sin 2 \theta - \tan \theta = \sqrt { 3 } \cos 2 \theta\)
  2. \(\tan ( \theta + 1 ) \cos ( 2 \theta + 2 ) - \sin ( 2 \theta + 2 ) = 2\) Give your answers in radians to 3 significant figures, as appropriate.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2016 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d67f716-c8af-4460-8a6b-62073ba9b825-17_574_1333_260_303} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The population of a species of animal is being studied. The population \(P\), at time \(t\) years from the start of the study, is assumed to be $$P = \frac { 9000 \mathrm { e } ^ { k t } } { 3 \mathrm { e } ^ { k t } + 7 } , \quad t \geqslant 0$$ where \(k\) is a positive constant.
A sketch of the graph of \(P\) against \(t\) is shown in Figure 2 .
Use the given equation to
  1. find the population at the start of the study,
  2. find the value for the upper limit of the population. Given that \(P = 2500\) when \(t = 4\)
  3. calculate the value of \(k\), giving your answer to 3 decimal places. Using this value for \(k\),
  4. find, using \(\frac { \mathrm { d } P } { \mathrm {~d} t }\), the rate at which the population is increasing when \(t = 10\) Give your answer to the nearest integer.
Edexcel C34 2016 June Q10
10. (a) Given that \(- \frac { \pi } { 2 } < \mathrm { g } ( x ) < \frac { \pi } { 2 }\), sketch the graph of \(y = \mathrm { g } ( x )\) where $$\mathrm { g } ( x ) = \arctan x , \quad x \in \mathbb { R }$$ (b) Find the exact value of \(x\) for which $$3 g ( x + 1 ) - \pi = 0$$ The equation \(\arctan x - 4 + \frac { 1 } { 2 } x = 0\) has a positive root at \(x = \alpha\) radians.
(c) Show that \(5 < \alpha < 6\) The iteration formula $$x _ { n + 1 } = 8 - 2 \arctan x _ { n }$$ can be used to find an approximation for \(\alpha\)
(d) Taking \(x _ { 0 } = 5\), use this formula to find \(x _ { 1 }\) and \(x _ { 2 }\), giving each answer to 3 decimal places.
Edexcel C34 2016 June Q11
11. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 7
4
9 \end{array} \right) + \lambda \left( \begin{array} { l } 1
1
4 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 6
- 7
3 \end{array} \right) + \mu \left( \begin{array} { l } 5
4
b \end{array} \right) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(b\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(X\),
  1. show that \(b = - 3\) and find the coordinates of \(X\). The point \(A\) lies on \(l _ { 1 }\) and has coordinates (6, 3, 5)
    The point \(B\) lies on \(l _ { 2 }\) and has coordinates \(( 14,9 , - 9 )\)
  2. Show that angle \(A X B = \arccos \left( - \frac { 1 } { 10 } \right)\)
  3. Using the result obtained in part (b), find the exact area of triangle \(A X B\). Write your answer in the form \(p \sqrt { q }\) where \(p\) and \(q\) are integers to be determined.
Edexcel C34 2016 June Q12
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d67f716-c8af-4460-8a6b-62073ba9b825-23_503_1333_267_301} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve with parametric equations $$x = 3 \sin t , \quad y = 2 \sin 2 t , \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The finite region \(S\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \frac { 3 } { 2 }\) The shaded region \(S\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  1. Show that the volume of the solid of revolution is given by $$k \int _ { 0 } ^ { a } \sin ^ { 2 } t \cos ^ { 3 } t \mathrm {~d} t$$ where \(k\) and \(a\) are constants to be given in terms of \(\pi\).
  2. Use the substitution \(u = \sin t\), or otherwise, to find the exact value of this volume, giving your answer in the form \(\frac { p \pi } { q }\) where \(p\) and \(q\) are integers. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2016 June Q13
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d67f716-c8af-4460-8a6b-62073ba9b825-25_362_697_246_612} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a hemispherical bowl containing some water.
At \(t\) seconds, the height of the water is \(h \mathrm {~cm}\) and the volume of the water is \(V \mathrm {~cm} ^ { 3 }\), where $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 30 - h ) , \quad 0 < h \leqslant 10$$ The water is leaking from a hole in the bottom of the bowl. Given that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = - \frac { 1 } { 10 } V\)
  1. show that \(\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { h ( 30 - h ) } { 30 ( 20 - h ) }\)
  2. Write \(\frac { 30 ( 20 - h ) } { h ( 30 - h ) }\) in partial fraction form. Given that \(h = 10\) when \(t = 0\),
  3. use your answers to parts (a) and (b) to find the time taken for the height of the water to fall to 5 cm . Give your answer in seconds to 2 decimal places.