Questions C3 (1200 questions)

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Edexcel C3 2012 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3fbdfb55-5dd5-44ab-b031-d39e64bdfc3b-06_560_1145_210_386} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows part of the curve with equation \(y = \mathrm { f } ( x )\)
The curve passes through the points \(P ( - 1.5,0 )\) and \(Q ( 0,5 )\) as shown.
On separate diagrams, sketch the curve with equation
  1. \(y = | f ( x ) |\)
  2. \(y = \mathrm { f } ( | x | )\)
  3. \(y = 2 f ( 3 x )\) Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.
Edexcel C3 2012 June Q5
  1. (a) Express \(4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
    (b) Hence show that
$$4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta = \sec ^ { 2 } \theta$$ (c) Hence or otherwise solve, for \(0 < \theta < \pi\), $$4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta = 4$$ giving your answers in terms of \(\pi\).
Edexcel C3 2012 June Q6
6. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto \mathrm { e } ^ { x } + 2 , \quad x \in \mathbb { R }
& \mathrm {~g} : x \mapsto \ln x , \quad x > 0 \end{aligned}$$
  1. State the range of f.
  2. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
  3. Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\)
  4. Find \(\mathrm { f } ^ { - 1 }\), the inverse function of f , stating its domain.
  5. 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 C3 2012 June Q7
  1. Differentiate with respect to \(x\),
    1. \(x ^ { \frac { 1 } { 2 } } \ln ( 3 x )\)
    2. \(\frac { 1 - 10 x } { ( 2 x - 1 ) ^ { 5 } }\), giving your answer in its simplest form.
  2. Given that \(x = 3 \tan 2 y\) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).
Edexcel C3 2013 June Q1
  1. Express
$$\frac { 3 x + 5 } { x ^ { 2 } + x - 12 } - \frac { 2 } { x - 3 }$$ as a single fraction in its simplest form.
Edexcel C3 2013 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a80a71cb-42e0-4587-8f8e-bacd69b8d07a-03_499_1099_210_443} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x > 0\), where f is an increasing function of \(x\). The curve crosses the \(x\)-axis at the point \(( 1,0 )\) and the line \(x = 0\) is an asymptote to the curve. On separate diagrams, sketch the curve with equation
  1. \(y = \mathrm { f } ( 2 x ) , x > 0\)
  2. \(y = | \mathrm { f } ( x ) | , x > 0\) Indicate clearly on each sketch the coordinates of the point at which the curve crosses or meets the \(x\)-axis.
Edexcel C3 2013 June Q3
3. $$f ( x ) = 7 \cos x + \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),
  1. find the exact value of \(R\) and the value of \(\alpha\) to one decimal place.
  2. Hence solve the equation $$7 \cos x + \sin x = 5$$ for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers to one decimal place.
  3. State the values of \(k\) for which the equation $$7 \cos x + \sin x = k$$ has only one solution in the interval \(0 \leqslant x < 360 ^ { \circ }\)
Edexcel C3 2013 June Q4
  1. The functions f and g are defined by
$$\begin{array} { l l } \mathrm { f } : x \mapsto 2 | x | + 3 , & x \in \mathbb { R } ,
\mathrm {~g} : x \mapsto 3 - 4 x , & x \in \mathbb { R } \end{array}$$
  1. State the range of f.
  2. Find \(\mathrm { fg } ( 1 )\).
  3. Find \(\mathrm { g } ^ { - 1 }\), the inverse function of g .
  4. Solve the equation $$\operatorname { gg } ( x ) + [ \mathrm { g } ( x ) ] ^ { 2 } = 0$$
Edexcel C3 2013 June Q5
5. (a) Differentiate $$\frac { \cos 2 x } { \sqrt { x } }$$ with respect to \(x\).
(b) Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { 2 } 3 x \right)\) can be written in the form $$\mu \left( \tan 3 x + \tan ^ { 3 } 3 x \right)$$ where \(\mu\) is a constant.
(c) Given \(x = 2 \sin \left( \frac { y } { 3 } \right)\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\), simplifying your answer.
Edexcel C3 2013 June Q6
  1. (i) Use an appropriate double angle formula to show that
$$\operatorname { cosec } 2 x = \lambda \operatorname { cosec } x \sec x$$ and state the value of the constant \(\lambda\).
(ii) Solve, for \(0 \leqslant \theta < 2 \pi\), the equation $$3 \sec ^ { 2 } \theta + 3 \sec \theta = 2 \tan ^ { 2 } \theta$$ You must show all your working. Give your answers in terms of \(\pi\).
Edexcel C3 2013 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a80a71cb-42e0-4587-8f8e-bacd69b8d07a-11_481_858_228_552} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 x + 1 \right) \mathrm { e } ^ { x ^ { 2 } }$$ The curve cuts the \(x\)-axis at points \(A\) and \(B\) as shown in Figure 2 .
  1. Calculate the \(x\) coordinate of \(A\) and the \(x\) coordinate of \(B\), giving your answers to 3 decimal places.
  2. Find \(\mathrm { f } ^ { \prime } ( x )\). The curve has a minimum turning point at the point \(P\) as shown in Figure 2.
  3. Show that the \(x\) coordinate of \(P\) is the solution of $$x = - \frac { 3 \left( 2 x ^ { 2 } + 1 \right) } { 2 \left( x ^ { 2 } + 2 \right) }$$
  4. Use the iteration formula $$x _ { n + 1 } = - \frac { 3 \left( 2 x _ { n } ^ { 2 } + 1 \right) } { 2 \left( x _ { n } ^ { 2 } + 2 \right) } , \quad \text { with } x _ { 0 } = - 2.4$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places. The \(x\) coordinate of \(P\) is \(\alpha\).
  5. By choosing a suitable interval, prove that \(\alpha = - 2.43\) to 2 decimal places.
Edexcel C3 2013 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a80a71cb-42e0-4587-8f8e-bacd69b8d07a-13_721_1227_116_322} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The population of a town is being studied. The population \(P\), at time \(t\) years from the start of the study, is assumed to be $$P = \frac { 8000 } { 1 + 7 \mathrm { e } ^ { - k t } } , \quad t \geqslant 0$$ where \(k\) is a positive constant.
The graph of \(P\) against \(t\) is shown in Figure 3. Use the given equation to
  1. find the population at the start of the study,
  2. find a value for the expected upper limit of the population. Given also that the population reaches 2500 at 3 years from the start of the study,
  3. calculate the value of \(k\) to 3 decimal places. Using this value for \(k\),
  4. find the population at 10 years from the start of the study, giving your answer to 3 significant figures.
  5. Find, using \(\frac { \mathrm { d } P } { \mathrm {~d} t }\), the rate at which the population is growing at 10 years from the start of the study.
Edexcel C3 2013 June Q1
1. $$g ( x ) = \frac { 6 x + 12 } { x ^ { 2 } + 3 x + 2 } - 2 , \quad x \geqslant 0$$
  1. Show that \(\mathrm { g } ( x ) = \frac { 4 - 2 x } { x + 1 } , x \geqslant 0\)
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-02_494_922_628_511} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { g } ( x ) , x \geqslant 0\) The curve meets the \(y\)-axis at \(( 0,4 )\) and crosses the \(x\)-axis at \(( 2,0 )\). On separate diagrams sketch the graph with equation
    1. \(y = 2 \mathrm {~g} ( 2 x )\),
    2. \(y = \mathrm { g } ^ { - 1 } ( x )\). Show on each sketch the coordinates of each point at which the graph meets or crosses the axes.
Edexcel C3 2013 June Q2
2. Given that \(\tan 40 ^ { \circ } = p\), find in terms of \(p\)
  1. \(\cot 40 ^ { \circ }\)
  2. \(\sec 40 ^ { \circ }\)
  3. \(\tan 85 ^ { \circ }\)
Edexcel C3 2013 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-05_654_967_244_507} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the graph with equation \(y = 2 | x | - 5\).
The graph intersects the positive \(x\)-axis at the point \(P\) and the negative \(y\)-axis at the point \(Q\).
  1. State the coordinates of \(P\) and the coordinates of \(Q\).
  2. Solve the equation $$2 | x | - 5 = 3 - x$$
Edexcel C3 2013 June Q4
  1. (a) On the same diagram, sketch and clearly label the graphs with equations
$$y = \mathrm { e } ^ { x } \quad \text { and } \quad y = 10 - x$$ Show on your sketch the coordinates of each point at which the graphs cut the axes.
(b) Explain why the equation \(\mathrm { e } ^ { x } - 10 + x = 0\) has only one solution.
(c) Show that the solution of the equation $$\mathrm { e } ^ { x } - 10 + x = 0$$ lies between \(x = 2\) and \(x = 3\)
(d) Use the iterative formula $$x _ { n + 1 } = \ln \left( 10 - x _ { n } \right) , \quad x _ { 1 } = 2$$ to calculate the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
Give your answers to 4 decimal places.
Edexcel C3 2013 June Q5
  1. (i) (a) Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { \frac { 1 } { 2 } } \ln x \right) = \frac { \ln x } { 2 \sqrt { } x } + \frac { 1 } { \sqrt { } x }\)
The curve with equation \(y = x ^ { \frac { 1 } { 2 } } \ln x , x > 0\) has one turning point at the point \(P\).
(b) Find the exact coordinates of \(P\). Give your answer in its simplest form.
(ii) A curve \(C\) has equation \(y = \frac { x - k } { x + k }\), where \(k\) is a positive constant. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), and show that \(C\) has no turning points.
Edexcel C3 2013 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-10_775_1392_233_278} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the graph of \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left\{ \begin{array} { r r } 5 - 2 x , & x \leqslant 4
\mathrm { e } ^ { 2 x - 8 } - 4 , & x > 4 \end{array} \right.$$
  1. State the range of \(\mathrm { f } ( x )\).
  2. Determine the exact value of ff(0).
  3. Solve \(\mathrm { f } ( x ) = 21\) Give each answer as an exact answer.
  4. Explain why the function f does not have an inverse.
Edexcel C3 2013 June Q7
7. (a) Prove that $$\frac { \cos x } { 1 - \sin x } + \frac { 1 - \sin x } { \cos x } = 2 \sec x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence find, for \(0 < x < \frac { \pi } { 4 }\), the exact solution of $$\frac { \cos x } { 1 - \sin x } + \frac { 1 - \sin x } { \cos x } = 8 \sin x$$
Edexcel C3 2013 June Q8
8. (a) Express \(9 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 4 decimal places.
(b) (i) State the maximum value of \(9 \cos \theta - 2 \sin \theta\)
(ii) Find the value of \(\theta\), for \(0 < \theta < 2 \pi\), at which this maximum occurs. Ruth models the height \(H\) above the ground of a passenger on a Ferris wheel by the equation $$H = 10 - 9 \cos \left( \frac { \pi t } { 5 } \right) + 2 \sin \left( \frac { \pi t } { 5 } \right)$$ where \(H\) is measured in metres and \(t\) is the time in minutes after the wheel starts turning.
\includegraphics[max width=\textwidth, alt={}, center]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-14_572_458_719_1158}
(c) Calculate the maximum value of \(H\) predicted by this model, and the value of \(t\), when this maximum first occurs. Give your answers to 2 decimal places.
(d) Determine the time for the Ferris wheel to complete two revolutions.
Edexcel C3 2013 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-16_570_903_237_534} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve with equation \(x = \left( 9 + 16 y - 2 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\).
The curve crosses the \(x\)-axis at the point \(A\).
  1. State the coordinates of \(A\).
  2. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\), in terms of \(y\).
  3. Find an equation of the tangent to the curve at \(A\).
Edexcel C3 2013 June Q1
  1. Given that
$$\frac { 3 x ^ { 4 } - 2 x ^ { 3 } - 5 x ^ { 2 } - 4 } { x ^ { 2 } - 4 } \equiv a x ^ { 2 } + b x + c + \frac { d x + e } { x ^ { 2 } - 4 } , \quad x \neq \pm 2$$ find the values of the constants \(a , b , c , d\) and \(e\).
(4)
Edexcel C3 2013 June Q2
2. Given that $$\mathrm { f } ( x ) = \ln x , \quad x > 0$$ sketch on separate axes the graphs of
  1. \(\quad y = \mathrm { f } ( x )\),
  2. \(y = | \mathrm { f } ( x ) |\),
  3. \(y = - \mathrm { f } ( x - 4 )\). Show, on each diagram, the point where the graph meets or crosses the \(x\)-axis. In each case, state the equation of the asymptote.
Edexcel C3 2013 June Q3
3. Given that $$2 \cos ( x + 50 ) ^ { \circ } = \sin ( x + 40 ) ^ { \circ }$$
  1. Show, without using a calculator, that $$\tan x ^ { \circ } = \frac { 1 } { 3 } \tan 40 ^ { \circ }$$
  2. Hence solve, for \(0 \leqslant \theta < 360\), $$2 \cos ( 2 \theta + 50 ) ^ { \circ } = \sin ( 2 \theta + 40 ) ^ { \circ }$$ giving your answers to 1 decimal place.
Edexcel C3 2013 June Q4
4. $$\mathrm { f } ( x ) = 25 x ^ { 2 } \mathrm { e } ^ { 2 x } - 16 , \quad x \in \mathbb { R }$$
  1. Using calculus, find the exact coordinates of the turning points on the curve with equation \(y = \mathrm { f } ( x )\).
  2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as \(x = \pm \frac { 4 } { 5 } \mathrm { e } ^ { - x }\) The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\), where \(\alpha = 0.5\) to 1 decimal place.
  3. Starting with \(x _ { 0 } = 0.5\), use the iteration formula $$x _ { n + 1 } = \frac { 4 } { 5 } \mathrm { e } ^ { - x _ { n } }$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
  4. Give an accurate estimate for \(\alpha\) to 2 decimal places, and justify your answer.