Questions — Edexcel C3 (377 questions)

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Edexcel C3 2009 January Q2
2. $$f ( x ) = \frac { 2 x + 2 } { x ^ { 2 } - 2 x - 3 } - \frac { x + 1 } { x - 3 }$$
  1. Express \(\mathrm { f } ( x )\) as a single fraction in its simplest form.
  2. Hence show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { ( x - 3 ) ^ { 2 } }\)
Edexcel C3 2009 January Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{502d98be-7013-4ce6-816b-27c671944503-04_767_913_246_511} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x ) , \quad 1 < x < 9\).
The points \(T ( 3,5 )\) and \(S ( 7,2 )\) are turning points on the graph.
Sketch, on separate diagrams, the graphs of
  1. \(y = 2 \mathrm { f } ( x ) - 4\),
  2. \(y = | \mathrm { f } ( x ) |\). Indicate on each diagram the coordinates of any turning points on your sketch.
Edexcel C3 2009 January Q4
4. Find the equation of the tangent to the curve \(x = \cos ( 2 y + \pi )\) at \(\left( 0 , \frac { \pi } { 4 } \right)\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants to be found.
Edexcel C3 2009 January Q5
5. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + \ln x , \quad x > 0 , \quad x \in \mathbb { R }
& \mathrm {~g} : x \mapsto \mathrm { e } ^ { x ^ { 2 } } , \quad x \in \mathbb { R } \end{aligned}$$
  1. Write down the range of g.
  2. Show that the composite function fg is defined by $$\mathrm { fg } : x \mapsto x ^ { 2 } + 3 \mathrm { e } ^ { x ^ { 2 } } , \quad x \in \mathbb { R } .$$
  3. Write down the range of fg.
  4. Solve the equation \(\frac { \mathrm { d } } { \mathrm { d } x } [ \mathrm { fg } ( x ) ] = x \left( x \mathrm { e } ^ { x ^ { 2 } } + 2 \right)\).
Edexcel C3 2009 January Q6
6. (a) (i) By writing \(3 \theta = ( 2 \theta + \theta )\), show that $$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$ (ii) Hence, or otherwise, for \(0 < \theta < \frac { \pi } { 3 }\), solve $$8 \sin ^ { 3 } \theta - 6 \sin \theta + 1 = 0 .$$ Give your answers in terms of \(\pi\).
(b) Using \(\sin ( \theta - \alpha ) = \sin \theta \cos \alpha - \cos \theta \sin \alpha\), or otherwise, show that $$\sin 15 ^ { \circ } = \frac { 1 } { 4 } ( \sqrt { } 6 - \sqrt { } 2 )$$
Edexcel C3 2009 January Q7
7. $$f ( x ) = 3 x e ^ { x } - 1$$ The curve with equation \(y = \mathrm { f } ( x )\) has a turning point \(P\).
  1. Find the exact coordinates of \(P\). The equation \(\mathrm { f } ( x ) = 0\) has a root between \(x = 0.25\) and \(x = 0.3\)
  2. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \mathrm { e } ^ { - x _ { n } }$$ with \(x _ { 0 } = 0.25\) to find, to 4 decimal places, the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\).
  3. By choosing a suitable interval, show that a root of \(\mathrm { f } ( x ) = 0\) is \(x = 0.2576\) correct to 4 decimal places.
Edexcel C3 2009 January Q8
8. (a) Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\).
(b) Hence find the maximum value of \(3 \cos \theta + 4 \sin \theta\) and the smallest positive value of \(\theta\) for which this maximum occurs. The temperature, \(\mathrm { f } ( t )\), of a warehouse is modelled using the equation $$f ( t ) = 10 + 3 \cos ( 15 t ) ^ { \circ } + 4 \sin ( 15 t ) ^ { \circ }$$ where \(t\) is the time in hours from midday and \(0 \leqslant t < 24\).
(c) Calculate the minimum temperature of the warehouse as given by this model.
(d) Find the value of \(t\) when this minimum temperature occurs.
Edexcel C3 2010 January Q1
  1. Express
$$\frac { x + 1 } { 3 x ^ { 2 } - 3 } - \frac { 1 } { 3 x + 1 }$$ as a single fraction in its simplest form.
Edexcel C3 2010 January Q2
2. $$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 3 x - 11$$
  1. Show that \(\mathrm { f } ( x ) = 0\) can be rearranged as $$x = \sqrt { } \left( \frac { 3 x + 11 } { x + 2 } \right) , \quad x \neq - 2 .$$ The equation \(\mathrm { f } ( x ) = 0\) has one positive root \(\alpha\). The iterative formula \(x _ { n + 1 } = \sqrt { } \left( \frac { 3 x _ { n } + 11 } { x _ { n } + 2 } \right)\) is used to find an approximation to \(\alpha\).
  2. Taking \(x _ { 1 } = 0\), find, to 3 decimal places, the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
  3. Show that \(\alpha = 2.057\) correct to 3 decimal places.
Edexcel C3 2010 January Q3
3. (a) Express \(5 \cos x - 3 \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
(b) Hence, or otherwise, solve the equation $$5 \cos x - 3 \sin x = 4$$ for \(0 \leqslant x < 2 \pi\), giving your answers to 2 decimal places.
Edexcel C3 2010 January Q4
4. (i) Given that \(y = \frac { \ln \left( x ^ { 2 } + 1 \right) } { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
(ii) Given that \(x = \tan y\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).
Edexcel C3 2010 January Q5
5. Sketch the graph of \(y = \ln | x |\), stating the coordinates of any points of intersection with the axes.
Edexcel C3 2010 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2f133cc-1723-4512-a351-c319daf80fca-07_380_574_269_722} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the graph of \(y = \mathrm { f } ( x )\).
The graph intersects the \(y\)-axis at the point \(( 0,1 )\) and the point \(A ( 2,3 )\) is the maximum turning point. Sketch, on separate axes, the graphs of
  1. \(y = \mathrm { f } ( - x ) + 1\),
  2. \(y = \mathrm { f } ( x + 2 ) + 3\),
  3. \(y = 2 \mathrm { f } ( 2 x )\). On each sketch, show the coordinates of the point at which your graph intersects the \(y\)-axis and the coordinates of the point to which \(A\) is transformed.
Edexcel C3 2010 January Q7
  1. (a) By writing \(\sec x\) as \(\frac { 1 } { \cos x }\), show that \(\frac { \mathrm { d } ( \sec x ) } { \mathrm { d } x } = \sec x \tan x\).
Given that \(y = \mathrm { e } ^ { 2 x } \sec 3 x\),
(b) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The curve with equation \(y = \mathrm { e } ^ { 2 x } \sec 3 x , - \frac { \pi } { 6 } < x < \frac { \pi } { 6 }\), has a minimum turning point at \(( a , b )\).
(c) Find the values of the constants \(a\) and \(b\), giving your answers to 3 significant figures.
Edexcel C3 2010 January Q8
8. Solve $$\operatorname { cosec } ^ { 2 } 2 x - \cot 2 x = 1$$ for \(0 \leqslant x \leqslant 180 ^ { \circ }\).
Edexcel C3 2010 January Q9
9. (i) Find the exact solutions to the equations
  1. \(\ln ( 3 x - 7 ) = 5\)
  2. \(3 ^ { x } \mathrm { e } ^ { 7 x + 2 } = 15\)
    (ii) The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R }
    \mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , x > 1 \end{array}$$
  3. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
  4. Find fg and state its range.
Edexcel C3 2011 January Q1
  1. (a) Express \(7 \cos x - 24 \sin x\) in the form \(R \cos ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 3 decimal places.
    (b) Hence write down the minimum value of \(7 \cos x - 24 \sin x\).
    (c) Solve, for \(0 \leqslant x < 2 \pi\), the equation
$$7 \cos x - 24 \sin x = 10$$ giving your answers to 2 decimal places.
Edexcel C3 2011 January Q2
2. (a) Express $$\frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) }$$ as a single fraction in its simplest form. Given that $$f ( x ) = \frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) } - 2 , \quad x > 1$$ (b) show that $$f ( x ) = \frac { 3 } { 2 x - 1 }$$ (c) Hence differentiate \(\mathrm { f } ( x )\) and find \(\mathrm { f } ^ { \prime } ( 2 )\).
Edexcel C3 2011 January Q3
  1. Find all the solutions of
$$2 \cos 2 \theta = 1 - 2 \sin \theta$$ in the interval \(0 \leqslant \theta < 360 ^ { \circ }\).
Edexcel C3 2011 January Q4
4. Joan brings a cup of hot tea into a room and places the cup on a table. At time \(t\) minutes after Joan places the cup on the table, the temperature, \(\theta ^ { \circ } \mathrm { C }\), of the tea is modelled by the equation $$\theta = 20 + A \mathrm { e } ^ { - k t } ,$$ where \(A\) and \(k\) are positive constants. Given that the initial temperature of the tea was \(90 ^ { \circ } \mathrm { C }\),
  1. find the value of \(A\). The tea takes 5 minutes to decrease in temperature from \(90 ^ { \circ } \mathrm { C }\) to \(55 ^ { \circ } \mathrm { C }\).
  2. Show that \(k = \frac { 1 } { 5 } \ln 2\).
  3. Find the rate at which the temperature of the tea is decreasing at the instant when \(t = 10\). Give your answer, in \({ } ^ { \circ } \mathrm { C }\) per minute, to 3 decimal places.
Edexcel C3 2011 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3ff6824f-9fbf-4b5b-8bab-91332c549b36-08_624_1054_274_447} \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 $$\mathrm { f } ( x ) = ( 8 - x ) \ln x , \quad x > 0$$ The curve cuts the \(x\)-axis at the points \(A\) and \(B\) and has a maximum turning point at \(Q\), as shown in Figure 1.
  1. Write down the coordinates of \(A\) and the coordinates of \(B\).
  2. Find f'(x).
  3. Show that the \(x\)-coordinate of \(Q\) lies between 3.5 and 3.6
  4. Show that the \(x\)-coordinate of \(Q\) is the solution of $$x = \frac { 8 } { 1 + \ln x }$$ To find an approximation for the \(x\)-coordinate of \(Q\), the iteration formula $$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } }$$ is used.
  5. Taking \(x _ { 0 } = 3.55\), find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\). Give your answers to 3 decimal places.
Edexcel C3 2011 January Q6
  1. The function \(f\) is defined by
$$\mathrm { f } : x \mapsto \frac { 3 - 2 x } { x - 5 } , \quad x \in \mathbb { R } , x \neq 5$$
  1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
    (3) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3ff6824f-9fbf-4b5b-8bab-91332c549b36-10_901_1091_593_429} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The function g has domain \(- 1 \leqslant x \leqslant 8\), and is linear from \(( - 1 , - 9 )\) to \(( 2,0 )\) and from \(( 2,0 )\) to \(( 8,4 )\). Figure 2 shows a sketch of the graph of \(y = \mathrm { g } ( x )\).
  2. Write down the range of g.
  3. Find \(\operatorname { gg } ( 2 )\).
  4. Find \(\mathrm { fg } ( 8 )\).
  5. On separate diagrams, sketch the graph with equation
    1. \(y = | \mathrm { g } ( x ) |\),
    2. \(y = \mathrm { g } ^ { - 1 } ( x )\). Show on each sketch the coordinates of each point at which the graph meets or cuts the axes.
  6. State the domain of the inverse function \(\mathrm { g } ^ { - 1 }\).
Edexcel C3 2011 January Q7
  1. The curve \(C\) has equation
$$y = \frac { 3 + \sin 2 x } { 2 + \cos 2 x }$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 \sin 2 x + 4 \cos 2 x + 2 } { ( 2 + \cos 2 x ) ^ { 2 } }$$
  2. Find an equation of the tangent to \(C\) at the point on \(C\) where \(x = \frac { \pi } { 2 }\). Write your answer in the form \(y = a x + b\), where \(a\) and \(b\) are exact constants.
Edexcel C3 2011 January Q8
8. (a) Given that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \cos x ) = - \sin x$$ show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x ) = \sec x \tan x\). Given that $$x = \sec 2 y$$ (b) find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
(c) Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). \includegraphics[max width=\textwidth, alt={}, center]{3ff6824f-9fbf-4b5b-8bab-91332c549b36-14_102_93_2473_1804}
Edexcel C3 2012 January Q1
Differentiate with respect to \(x\), giving your answer in its simplest form,
  1. \(x ^ { 2 } \ln ( 3 x )\)
  2. \(\frac { \sin 4 x } { x ^ { 3 } }\)