Questions — Edexcel C34 (197 questions)

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Edexcel C34 2019 January Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae871952-f525-44e6-8bac-09308aa1964f-38_570_671_310_680} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Diagram not drawn to scale Figure 3 shows a container in the shape of an inverted right circular cone which contains some water. The cone has an internal radius of 3 m and a vertical height of 5 m as shown in Figure 3. At time \(t\) seconds,the height of the water is \(h\) metres,the volume of the water is \(V \mathrm {~m} ^ { 3 }\) and water is leaking from a hole in the bottom of the container at a constant rate of \(0.02 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
[The volume of a cone of radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) .]
  1. Show that,while the water is leaking, $$h ^ { 2 } \frac { \mathrm {~d} h } { \mathrm {~d} t } = - \frac { 1 } { \mathrm { k } \pi }$$ where \(k\) is a constant to be found. Given that the container is initially full of water,
  2. express \(h\) in terms of \(t\) .
  3. Find the time taken for the container to empty,giving your answer to the nearest minute.
Edexcel C34 2019 January Q11
11. (a) Given that \(0 \leqslant \mathrm { f } ( x ) \leqslant \pi\), sketch the graph of \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \arccos ( x - 1 ) , \quad 0 \leqslant x \leqslant 2$$ The equation \(\arccos ( x - 1 ) - \tan x = 0\) has a single root \(\alpha\).
(b) Show that \(0.9 < \alpha < 1.1\) The iteration formula $$x _ { n + 1 } = \arctan \left( \arccos \left( x _ { n } - 1 \right) \right)$$ can be used to find an approximation for \(\alpha\).
(c) Taking \(x _ { 0 } = 1.1\) find, to 3 decimal places, the values of \(x _ { 1 }\) and \(x _ { 2 }\)
Edexcel C34 2019 January Q12
12. Given that \(k\) is a positive constant,
  1. sketch the graph with equation $$y = 2 | x | - k$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and the \(y\)-axis.
  2. Find, in terms of \(k\), the values of \(x\) for which $$2 | x | - k = \frac { 1 } { 2 } x + \frac { 1 } { 4 } k$$
Edexcel C34 2019 January Q13
13. A scientist is studying a population of insects. The number of insects, \(N\), in the population, \(t\) days after the start of the study is modelled by the equation $$N = \frac { 240 } { 1 + k \mathrm { e } ^ { - \frac { t } { 16 } } }$$ where \(k\) is a constant.
Given that there were 50 insects at the start of the study,
  1. find the value of \(k\)
  2. use the model to find the value of \(t\) when \(N = 100\)
  3. Show that $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { 1 } { p } N - \frac { 1 } { q } N ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
    END
Edexcel C34 2014 June Q1
1. $$f ( x ) = 2 x ^ { 3 } + x - 10$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 1.5,2 ]\) The only real root of \(\mathrm { f } ( x ) = 0\) is \(\alpha\) The iterative formula $$x _ { n + 1 } = \left( 5 - \frac { 1 } { 2 } x _ { n } \right) ^ { \frac { 1 } { 3 } } , \quad x _ { 0 } = 1.5$$ can be used to find an approximate value for \(\alpha\)
  2. Calculate \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
  3. By choosing a suitable interval, show that \(\alpha = 1.6126\) correct to 4 decimal places.
Edexcel C34 2014 June Q2
2. A curve \(C\) has the equation $$x ^ { 3 } - 3 x y - x + y ^ { 3 } - 11 = 0$$ Find an equation of the tangent to \(C\) at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C34 2014 June Q3
3. Given that $$y = \frac { \cos 2 \theta } { 1 + \sin 2 \theta } , \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \frac { a } { 1 + \sin 2 \theta } , \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ where \(a\) is a constant to be determined.
Edexcel C34 2014 June Q4
4. Find
  1. \(\int ( 2 x + 3 ) ^ { 12 } \mathrm {~d} x\)
  2. \(\int \frac { 5 x } { 4 x ^ { 2 } + 1 } \mathrm {~d} x\)
Edexcel C34 2014 June Q5
5. $$f ( x ) = \left( 8 + 27 x ^ { 3 } \right) ^ { \frac { 1 } { 3 } } , \quad | x | < \frac { 2 } { 3 }$$ Find the first three non-zero terms of the binomial expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\). Give each coefficient as a simplified fraction.
Edexcel C34 2014 June Q6
6. (a) Express \(\frac { 5 - 4 x } { ( 2 x - 1 ) ( x + 1 ) }\) in partial fractions.
(b) (i) Find a general solution of the differential equation $$( 2 x - 1 ) ( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = ( 5 - 4 x ) y , \quad x > \frac { 1 } { 2 }$$ Given that \(y = 4\) when \(x = 2\),
(ii) find the particular solution of this differential equation. Give your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel C34 2014 June Q7
7. The function f is defined by $$\mathrm { f } : x \mapsto \frac { 3 x - 5 } { x + 1 } , \quad x \in \mathbb { R } , x \neq - 1$$
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\)
  2. Show that $$\operatorname { ff } ( x ) = \frac { x + a } { x - 1 } , \quad x \in \mathbb { R } , x \neq - 1 , x \neq 1$$ where \(a\) is an integer to be determined. The function \(g\) is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } - 3 x , \quad x \in \mathbb { R } , 0 \leqslant x \leqslant 5$$
  3. Find the value of fg(2)
  4. Find the range of g
Edexcel C34 2014 June Q8
8. The volume \(V\) of a spherical balloon is increasing at a constant rate of \(250 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of the radius of the balloon, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), at the instant when the volume of the balloon is \(12000 \mathrm {~cm} ^ { 3 }\).
Give your answer to 2 significant figures.
[0pt] [You may assume that the volume \(V\) of a sphere of radius \(r\) is given by the formula \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\).]
Edexcel C34 2014 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{423eb549-0873-4185-8faf-12dedafcd108-13_849_841_214_571} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { e } ^ { \sqrt { x } } , x > 0\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines \(x = 4\) and \(x = 9\)
  1. Use the trapezium rule, with 5 strips of equal width, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places.
  2. Use the substitution \(u = \sqrt { x }\) to find, by integrating, the exact value for the area of \(R\).
Edexcel C34 2014 June Q10
10. (a) Use the identity for \(\sin ( A + B )\) to prove that $$\sin 2 A \equiv 2 \sin A \cos A$$ (b) Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ \ln \left( \tan \left( \frac { 1 } { 2 } x \right) \right) \right] = \operatorname { cosec } x$$ A curve \(C\) has the equation $$y = \ln \left( \tan \left( \frac { 1 } { 2 } x \right) \right) - 3 \sin x , \quad 0 < x < \pi$$ (c) Find the \(x\) coordinates of the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) Give your answers to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2014 June Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{423eb549-0873-4185-8faf-12dedafcd108-17_600_1024_221_470} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = \mathrm { e } ^ { a - 3 x } - 3 \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$ where \(a\) is a constant and \(a > \ln 4\) The curve \(C\) has a turning point \(P\) and crosses the \(x\)-axis at the point \(Q\) as shown in Figure 2.
  1. Find, in terms of \(a\), the coordinates of the point \(P\).
  2. Find, in terms of \(a\), the \(x\) coordinate of the point \(Q\).
  3. Sketch the curve with equation $$y = \left| \mathrm { e } ^ { a - 3 x } - 3 \mathrm { e } ^ { - x } \right| , \quad x \in \mathbb { R } , \quad a > \ln 4$$ Show on your sketch the exact coordinates, in terms of \(a\), of the points at which the curve meets or cuts the coordinate axes.
Edexcel C34 2014 June Q12
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{423eb549-0873-4185-8faf-12dedafcd108-19_568_956_221_502} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve \(C\) with parametric equations $$x = \tan t , \quad y = 2 \sin ^ { 2 } t , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The finite region \(S\), shown shaded 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.
  1. Show that the volume of the solid of revolution formed is given by $$4 \pi \int _ { 0 } ^ { \frac { \pi } { 3 } } \left( \tan ^ { 2 } t - \sin ^ { 2 } t \right) \mathrm { d } t$$
  2. Hence use integration to find the exact value for this volume.
Edexcel C34 2014 June Q13
13. (a) Express \(2 \sin \theta + \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give your value of \(\alpha\) to 2 decimal places.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{423eb549-0873-4185-8faf-12dedafcd108-21_467_1365_870_301} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows the design for a logo that is to be displayed on the side of a large building. The logo consists of three rectangles, \(C , D\) and \(E\), each of which is in contact with two horizontal parallel lines \(l _ { 1 }\) and \(l _ { 2 }\). Rectangle \(D\) touches rectangles \(C\) and \(E\) as shown in Figure 4. Rectangles \(C , D\) and \(E\) each have length 4 m and width 2 m . The acute angle \(\theta\) between the line \(l _ { 2 }\) and the longer edge of each rectangle is shown in Figure 4. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are 4 m apart,
(b) show that $$2 \sin \theta + \cos \theta = 2$$ Given also that \(0 < \theta < 45 ^ { \circ }\),
(c) solve the equation $$2 \sin \theta + \cos \theta = 2$$ giving the value of \(\theta\) to 1 decimal place. Rectangles \(C\) and \(D\) and rectangles \(D\) and \(E\) touch for a distance \(h \mathrm {~m}\) as shown in Figure 4. Using your answer to part (c), or otherwise,
(d) find the value of \(h\), giving your answer to 2 significant figures.
Edexcel C34 2014 June Q14
14. Relative to a fixed origin \(O\), the line \(l\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 1
- 4
6 \end{array} \right) + \lambda \left( \begin{array} { r } 2
1
- 1 \end{array} \right)$$ where \(\lambda\) is a scalar parameter. Points \(A\) and \(B\) lie on the line \(l\), where \(A\) has coordinates ( \(1 , a , 5\) ) and \(B\) has coordinates ( \(b , - 1,3\) ).
  1. Find the value of the constant \(a\) and the value of the constant \(b\).
  2. Find the vector \(\overrightarrow { A B }\). The point \(C\) has coordinates ( \(4 , - 3,2\) )
  3. Show that the size of the angle \(C A B\) is \(30 ^ { \circ }\)
  4. Find the exact area of the triangle \(C A B\), giving your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined. The point \(D\) lies on the line \(l\) so that the area of the triangle \(C A D\) is twice the area of the triangle \(C A B\).
  5. Find the coordinates of the two possible positions of \(D\).
Edexcel C34 2015 June Q1
  1. A curve has equation
$$4 x ^ { 2 } - y ^ { 2 } + 2 x y + 5 = 0$$ The points \(P\) and \(Q\) lie on the curve.
Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\) at \(P\) and at \(Q\),
  1. use implicit differentiation to show that \(y - 6 x = 0\) at \(P\) and at \(Q\).
  2. Hence find the coordinates of \(P\) and \(Q\).
Edexcel C34 2015 June Q2
2. Given that $$\frac { 4 \left( x ^ { 2 } + 6 \right) } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } } \equiv \frac { A } { ( 1 - 2 x ) } + \frac { B } { ( 2 + x ) } + \frac { C } { ( 2 + x ) ^ { 2 } }$$
  1. find the values of the constants \(A\) and \(C\) and show that \(B = 0\)
    (4)
  2. Hence, or otherwise, find the series expansion of $$\frac { 4 \left( x ^ { 2 } + 6 \right) } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } } , \quad | x | < \frac { 1 } { 2 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying each term.
    (5)
Edexcel C34 2015 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c08fbab-283e-4c92-89a4-10f68f37e133-05_799_885_118_534} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = ( 2 x - 5 ) \mathrm { e } ^ { x } , \quad x \in \mathbb { R }$$ The curve has a minimum turning point at \(A\).
  1. Use calculus to find the exact coordinates of \(A\). Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly two roots,
  2. state the range of possible values of \(k\).
  3. Sketch the curve with equation \(y = | \mathrm { f } ( x ) |\). Indicate clearly on your sketch the coordinates of the points at which the curve crosses or meets the axes.
Edexcel C34 2015 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c08fbab-283e-4c92-89a4-10f68f37e133-07_330_494_210_724} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the points \(A\) and \(B\) with position vectors \(\mathbf { a }\) and \(\mathbf { b }\) respectively, relative to a fixed origin \(O\). Given that \(| \mathbf { a } | = 5 , | \mathbf { b } | = 6\) and a.b \(= 20\)
  1. find the cosine of angle \(A O B\),
  2. find the exact length of \(A B\).
  3. Show that the area of triangle \(O A B\) is \(5 \sqrt { 5 }\)
Edexcel C34 2015 June Q5
5. (i) Find the \(x\) coordinate of each point on the curve \(y = \frac { x } { x + 1 } , x \neq - 1\), at which the gradient is \(\frac { 1 } { 4 }\)
(ii) Given that $$\int _ { a } ^ { 2 a } \frac { t + 1 } { t } \mathrm {~d} t = \ln 7 \quad a > 0$$ find the exact value of the constant \(a\).
Edexcel C34 2015 June Q6
6. The mass, \(m\) grams, of a radioactive substance \(t\) years after first being observed, is modelled by the equation $$m = 25 \mathrm { e } ^ { 1 - k t }$$ where \(k\) is a positive constant.
  1. State the value of \(m\) when the radioactive substance was first observed. Given that the mass is 50 grams, 10 years after first being observed,
  2. show that \(k = \frac { 1 } { 10 } \ln \left( \frac { 1 } { 2 } \mathrm { e } \right)\)
  3. Find the value of \(t\) when \(m = 20\), giving your answer to the nearest year.
Edexcel C34 2015 June Q7
7. (a) Use the substitution \(t = \tan x\) to show that the equation $$4 \tan 2 x - 3 \cot x \sec ^ { 2 } x = 0$$ can be written in the form $$3 t ^ { 4 } + 8 t ^ { 2 } - 3 = 0$$ (b) Hence solve, for \(0 \leqslant x < 2 \pi\), $$4 \tan 2 x - 3 \cot x \sec ^ { 2 } x = 0$$ Give each answer in terms of \(\pi\). You must make your method clear.