Questions — Edexcel C3 (377 questions)

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Edexcel C3 2013 June Q5
  1. Given that
$$x = \sec ^ { 2 } 3 y , \quad 0 < y < \frac { \pi } { 6 }$$
  1. find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
  2. Hence show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 6 x ( x - 1 ) ^ { \frac { 1 } { 2 } } }$$
  3. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(x\). Give your answer in its simplest form.
Edexcel C3 2013 June Q6
6. Find algebraically the exact solutions to the equations
  1. \(\ln ( 4 - 2 x ) + \ln ( 9 - 3 x ) = 2 \ln ( x + 1 ) , \quad - 1 < x < 2\)
  2. \(2 ^ { x } \mathrm { e } ^ { 3 x + 1 } = 10\) Give your answer to (b) in the form \(\frac { a + \ln b } { c + \ln d }\) where \(a , b , c\) and \(d\) are integers.
Edexcel C3 2013 June Q7
7. The function \(f\) has domain \(- 2 \leqslant x \leqslant 6\) and is linear from \(( - 2,10 )\) to \(( 2,0 )\) and from \(( 2,0 )\) to (6, 4). A sketch of the graph of \(y = \mathrm { f } ( x )\) is shown in Figure 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e29d66c-c3c6-4e4b-acfb-c73c60604d93-09_906_965_367_566} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Write down the range of f .
  2. Find \(\mathrm { ff } ( 0 )\). The function \(g\) is defined by $$\mathrm { g } : x \rightarrow \frac { 4 + 3 x } { 5 - x } , \quad x \in \mathbb { R } , \quad x \neq 5$$
  3. Find \(\mathrm { g } ^ { - 1 } ( x )\)
  4. Solve the equation \(\operatorname { gf } ( x ) = 16\)
Edexcel C3 2013 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e29d66c-c3c6-4e4b-acfb-c73c60604d93-11_453_1225_255_369} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Kate crosses a road, of constant width 7 m , in order to take a photograph of a marathon runner, John, approaching at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Kate is 24 m ahead of John when she starts to cross the road from the fixed point \(A\). John passes her as she reaches the other side of the road at a variable point \(B\), as shown in Figure 2.
Kate's speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and she moves in a straight line, which makes an angle \(\theta\), \(0 < \theta < 150 ^ { \circ }\), with the edge of the road, as shown in Figure 2. You may assume that \(V\) is given by the formula $$V = \frac { 21 } { 24 \sin \theta + 7 \cos \theta } , \quad 0 < \theta < 150 ^ { \circ }$$
  1. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants and where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) to 2 decimal places. Given that \(\theta\) varies,
  2. find the minimum value of \(V\). Given that Kate's speed has the value found in part (b),
  3. find the distance \(A B\). Given instead that Kate's speed is \(1.68 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  4. find the two possible values of the angle \(\theta\), given that \(0 < \theta < 150 ^ { \circ }\).
Edexcel C3 2014 June Q1
  1. Express
$$\frac { 3 } { 2 x + 3 } - \frac { 1 } { 2 x - 3 } + \frac { 6 } { 4 x ^ { 2 } - 9 }$$ as a single fraction in its simplest form.
Edexcel C3 2014 June Q2
2. A curve \(C\) has equation \(y = \mathrm { e } ^ { 4 x } + x ^ { 4 } + 8 x + 5\)
  1. Show that the \(x\) coordinate of any turning point of \(C\) satisfies the equation $$x ^ { 3 } = - 2 - \mathrm { e } ^ { 4 x }$$
  2. On the axes given on page 5, sketch, on a single diagram, the curves with equations
    1. \(y = x ^ { 3 }\),
    2. \(y = - 2 - e ^ { 4 x }\) On your diagram give the coordinates of the points where each curve crosses the \(y\)-axis and state the equation of any asymptotes.
  3. Explain how your diagram illustrates that the equation \(x ^ { 3 } = - 2 - e ^ { 4 x }\) has only one root. The iteration formula $$x _ { n + 1 } = \left( - 2 - \mathrm { e } ^ { 4 x _ { n } } \right) ^ { \frac { 1 } { 3 } } , \quad x _ { 0 } = - 1$$ can be used to find an approximate value for this root.
  4. Calculate the values of \(x _ { 1 }\) and \(x _ { 2 }\), giving your answers to 5 decimal places.
  5. Hence deduce the coordinates, to 2 decimal places, of the turning point of the curve \(C\).
    \includegraphics[max width=\textwidth, alt={}, center]{be00fdaa-2fe3-4f06-a710-08ec67fb911e-04_1285_1294_308_331}
Edexcel C3 2014 June Q3
3. (i) (a) Show that \(2 \tan x - \cot x = 5 \operatorname { cosec } x\) may be written in the form $$a \cos ^ { 2 } x + b \cos x + c = 0$$ stating the values of the constants \(a , b\) and \(c\).
(b) Hence solve, for \(0 \leqslant x < 2 \pi\), the equation $$2 \tan x - \cot x = 5 \operatorname { cosec } x$$ giving your answers to 3 significant figures.
(ii) Show that $$\tan \theta + \cot \theta \equiv \lambda \operatorname { cosec } 2 \theta , \quad \theta \neq \frac { n \pi } { 2 } , \quad n \in \mathbb { Z }$$ stating the value of the constant \(\lambda\).
Edexcel C3 2014 June Q4
  1. (i) Given that
$$x = \sec ^ { 2 } 2 y , \quad 0 < y < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 4 x \sqrt { ( x - 1 ) } }$$ (ii) Given that $$y = \left( x ^ { 2 } + x ^ { 3 } \right) \ln 2 x$$ find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { \mathrm { e } } { 2 }\), giving your answer in its simplest form.
(iii) Given that $$f ( x ) = \frac { 3 \cos x } { ( x + 1 ) ^ { \frac { 1 } { 3 } } } , \quad x \neq - 1$$ show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { g } ( x ) } { ( x + 1 ) ^ { \frac { 4 } { 3 } } } , \quad x \neq - 1$$ where \(\mathrm { g } ( x )\) is an expression to be found.
Edexcel C3 2014 June Q5
5. (a) Sketch the graph with equation $$y = | 4 x - 3 |$$ stating the coordinates of any points where the graph cuts or meets the axes. Find the complete set of values of \(x\) for which
(b) $$| 4 x - 3 | > 2 - 2 x$$ (c) $$| 4 x - 3 | > \frac { 3 } { 2 } - 2 x$$
Edexcel C3 2014 June Q6
6. The function f is defined by $$\mathrm { f } : x \rightarrow \mathrm { e } ^ { 2 x } + k ^ { 2 } , \quad x \in \mathbb { R } , \quad k \text { is a positive constant. }$$
  1. State the range of f .
  2. Find \(\mathrm { f } ^ { - 1 }\) and state its domain. The function g is defined by $$g : x \rightarrow \ln ( 2 x ) , \quad x > 0$$
  3. Solve the equation $$\mathrm { g } ( x ) + \mathrm { g } \left( x ^ { 2 } \right) + \mathrm { g } \left( x ^ { 3 } \right) = 6$$ giving your answer in its simplest form.
  4. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
  5. Find, in terms of the constant \(k\), the solution of the equation $$\mathrm { fg } ( x ) = 2 k ^ { 2 }$$
Edexcel C3 2014 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be00fdaa-2fe3-4f06-a710-08ec67fb911e-13_456_881_214_534} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(C\), with equation \(y = 6 \cos x + 2.5 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\)
  1. Express \(6 \cos x + 2.5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\) to 3 decimal places.
  2. Find the coordinates of the points on the graph where the curve \(C\) crosses the coordinate axes. A student records the number of hours of daylight each Sunday throughout the year. She starts on the last Sunday in May with a recording of 18 hours, and continues until her final recording 52 weeks later. She models her results with the continuous function given by $$H = 12 + 6 \cos \left( \frac { 2 \pi t } { 52 } \right) + 2.5 \sin \left( \frac { 2 \pi t } { 52 } \right) , \quad 0 \leqslant t \leqslant 52$$ where \(H\) is the number of hours of daylight and \(t\) is the number of weeks since her first recording. Use this function to find
  3. the maximum and minimum values of \(H\) predicted by the model,
  4. the values for \(t\) when \(H = 16\), giving your answers to the nearest whole number.
    [0pt] [You must show your working. Answers based entirely on graphical or numerical methods are not acceptable.]
    \includegraphics[max width=\textwidth, alt={}, center]{be00fdaa-2fe3-4f06-a710-08ec67fb911e-14_40_58_2460_1893}
Edexcel C3 2014 June Q1
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where
$$f ( x ) = \frac { 4 x + 1 } { x - 2 } , \quad x > 2$$
  1. Show that $$f ^ { \prime } ( x ) = \frac { - 9 } { ( x - 2 ) ^ { 2 } }$$ Given that \(P\) is a point on \(C\) such that \(\mathrm { f } ^ { \prime } ( x ) = - 1\),
  2. find the coordinates of \(P\).
Edexcel C3 2014 June Q2
2. Find the exact solutions, in their simplest form, to the equations
  1. \(2 \ln ( 2 x + 1 ) - 10 = 0\)
  2. \(3 ^ { x } \mathrm { e } ^ { 4 x } = \mathrm { e } ^ { 7 }\)
Edexcel C3 2014 June Q3
3. The curve \(C\) has equation \(x = 8 y \tan 2 y\) The point \(P\) has coordinates \(\left( \pi , \frac { \pi } { 8 } \right)\)
  1. Verify that \(P\) lies on \(C\).
  2. Find the equation of the tangent to \(C\) at \(P\) in the form \(a y = x + b\), where the constants \(a\) and \(b\) are to be found in terms of \(\pi\).
Edexcel C3 2014 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{16c69ee4-255e-4d77-955a-92e1eb2f7d3e-05_665_776_233_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the graph with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(Q ( 6 , - 1 )\). The graph crosses the \(y\)-axis at the point \(P ( 0,11 )\). Sketch, on separate diagrams, the graphs of
  1. \(y = | f ( x ) |\)
  2. \(y = 2 f ( - x ) + 3\) On each diagram, show the coordinates of the points corresponding to \(P\) and \(Q\).
    Given that \(\mathrm { f } ( x ) = a | x - b | - 1\), where \(a\) and \(b\) are constants,
  3. state the value of \(a\) and the value of \(b\).
Edexcel C3 2014 June Q5
5. $$\mathrm { g } ( x ) = \frac { x } { x + 3 } + \frac { 3 ( 2 x + 1 ) } { x ^ { 2 } + x - 6 } , \quad x > 3$$
  1. Show that \(\mathrm { g } ( x ) = \frac { x + 1 } { x - 2 } , \quad x > 3\)
  2. Find the range of g.
  3. Find the exact value of \(a\) for which \(\mathrm { g } ( a ) = \mathrm { g } ^ { - 1 } ( a )\).
Edexcel C3 2014 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{16c69ee4-255e-4d77-955a-92e1eb2f7d3e-09_458_1164_239_383} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = 2 \cos \left( \frac { 1 } { 2 } x ^ { 2 } \right) + x ^ { 3 } - 3 x - 2$$ The curve crosses the \(x\)-axis at the point \(Q\) and has a minimum turning point at \(R\).
  1. Show that the \(x\) coordinate of \(Q\) lies between 2.1 and 2.2
  2. Show that the \(x\) coordinate of \(R\) is a solution of the equation $$x = \sqrt { 1 + \frac { 2 } { 3 } x \sin \left( \frac { 1 } { 2 } x ^ { 2 } \right) }$$ Using the iterative formula $$x _ { n + 1 } = \sqrt { 1 + \frac { 2 } { 3 } x _ { n } \sin \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } \right) } , \quad x _ { 0 } = 1.3$$
  3. find the values of \(x _ { 1 }\) and \(x _ { 2 }\) to 3 decimal places.
Edexcel C3 2014 June Q7
7. (a) Show that $$\operatorname { cosec } 2 x + \cot 2 x = \cot x , \quad x \neq 90 n ^ { \circ } , \quad n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), $$\operatorname { cosec } \left( 4 \theta + 10 ^ { \circ } \right) + \cot \left( 4 \theta + 10 ^ { \circ } \right) = \sqrt { 3 }$$ You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C3 2014 June Q8
8. A rare species of primrose is being studied. The population, \(P\), of primroses at time \(t\) years after the study started is modelled by the equation $$P = \frac { 800 \mathrm { e } ^ { 0.1 t } } { 1 + 3 \mathrm { e } ^ { 0.1 t } } , \quad t \geqslant 0 , \quad t \in \mathbb { R }$$
  1. Calculate the number of primroses at the start of the study.
  2. Find the exact value of \(t\) when \(P = 250\), giving your answer in the form \(a \ln ( b )\) where \(a\) and \(b\) are integers.
  3. Find the exact value of \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) when \(t = 10\). Give your answer in its simplest form.
  4. Explain why the population of primroses can never be 270
Edexcel C3 2014 June Q9
9. (a) Express \(2 \sin \theta - 4 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the value of \(\alpha\) to 3 decimal places. $$H ( \theta ) = 4 + 5 ( 2 \sin 3 \theta - 4 \cos 3 \theta ) ^ { 2 }$$ Find
(b) (i) the maximum value of \(\mathrm { H } ( \theta )\),
(ii) the smallest value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this maximum value occurs. Find
(c) (i) the minimum value of \(\mathrm { H } ( \theta )\),
(ii) the largest value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this minimum value occurs.
Edexcel C3 2015 June Q1
  1. Given that
$$\tan \theta ^ { \circ } = p , \text { where } p \text { is a constant, } p \neq \pm 1$$ use standard trigonometric identities, to find in terms of \(p\),
  1. \(\tan 2 \theta ^ { \circ }\)
  2. \(\cos \theta ^ { \circ }\)
  3. \(\cot ( \theta - 45 ) ^ { \circ }\) Write each answer in its simplest form.
Edexcel C3 2015 June Q2
2. Given that $$\mathrm { f } ( x ) = 2 \mathrm { e } ^ { x } - 5 , \quad x \in \mathbb { R }$$
  1. sketch, on separate diagrams, the curve with equation
    1. \(y = \mathrm { f } ( x )\)
    2. \(y = | \mathrm { f } ( x ) |\) On each diagram, show the coordinates of each point at which the curve meets or cuts the axes. On each diagram state the equation of the asymptote.
  2. Deduce the set of values of \(x\) for which \(\mathrm { f } ( x ) = | \mathrm { f } ( x ) |\)
  3. Find the exact solutions of the equation \(| \mathrm { f } ( x ) | = 2\)
Edexcel C3 2015 June Q3
3. $$g ( \theta ) = 4 \cos 2 \theta + 2 \sin 2 \theta$$ Given that \(\mathrm { g } ( \theta ) = R \cos ( 2 \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),
  1. find the exact value of \(R\) and the value of \(\alpha\) to 2 decimal places.
  2. Hence solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), $$4 \cos 2 \theta + 2 \sin 2 \theta = 1$$ giving your answers to one decimal place. Given that \(k\) is a constant and the equation \(\mathrm { g } ( \theta ) = k\) has no solutions,
  3. state the range of possible values of \(k\).
Edexcel C3 2015 June Q4
  1. Water is being heated in an electric kettle. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of the water \(t\) seconds after the kettle is switched on, is modelled by the equation
$$\theta = 120 - 100 \mathrm { e } ^ { - \lambda t } , \quad 0 \leqslant t \leqslant T$$
  1. State the value of \(\theta\) when \(t = 0\) Given that the temperature of the water in the kettle is \(70 ^ { \circ } \mathrm { C }\) when \(t = 40\),
  2. find the exact value of \(\lambda\), giving your answer in the form \(\frac { \ln a } { b }\), where \(a\) and \(b\) are integers. When \(t = T\), the temperature of the water reaches \(100 ^ { \circ } \mathrm { C }\) and the kettle switches off.
  3. Calculate the value of \(T\) to the nearest whole number.
Edexcel C3 2015 June Q5
5. The point \(P\) lies on the curve with equation $$x = ( 4 y - \sin 2 y ) ^ { 2 }$$ Given that \(P\) has \(( x , y )\) coordinates \(\left( p , \frac { \pi } { 2 } \right)\), where \(p\) is a constant,
  1. find the exact value of \(p\). The tangent to the curve at \(P\) cuts the \(y\)-axis at the point \(A\).
  2. Use calculus to find the coordinates of \(A\).