Questions — Edexcel (10514 questions)

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Edexcel C3 2010 January Q4
9 marks Moderate -0.3
4.
  1. Given that \(y = \frac { \ln \left( x ^ { 2 } + 1 \right) } { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Given that \(x = \tan y\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).
Edexcel C3 2010 January Q5
3 marks Moderate -0.3
5. Sketch the graph of \(y = \ln | x |\), stating the coordinates of any points of intersection with the axes.
Edexcel C3 2010 January Q6
9 marks Moderate -0.3
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
11 marks Standard +0.3
  1. 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 \(y = \mathrm { e } ^ { 2 x } \sec 3 x\),
  2. 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 \(( a , b )\).
  3. Find the values of the constants \(a\) and \(b\), giving your answers to 3 significant figures.
Edexcel C3 2010 January Q8
7 marks Standard +0.3
8. Solve $$\operatorname { cosec } ^ { 2 } 2 x - \cot 2 x = 1$$ for \(0 \leqslant x \leqslant 180 ^ { \circ }\).
Edexcel C3 2010 January Q9
15 marks Moderate -0.3
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 $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R } \\ \mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , x > 1 \end{array}$$
    1. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
    2. Find fg and state its range.
Edexcel C3 2011 January Q1
9 marks Standard +0.3
  1. 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.
  2. Hence write down the minimum value of \(7 \cos x - 24 \sin x\).
  3. 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
9 marks Moderate -0.3
2.
  1. 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 $$f ( x ) = \frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) } - 2 , \quad x > 1$$
  2. show that $$f ( x ) = \frac { 3 } { 2 x - 1 }$$
  3. Hence differentiate \(\mathrm { f } ( x )\) and find \(\mathrm { f } ^ { \prime } ( 2 )\).
Edexcel C3 2011 January Q3
6 marks Moderate -0.3
  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
8 marks Moderate -0.3
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
13 marks Standard +0.3
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
13 marks Moderate -0.3
  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
8 marks Standard +0.3
  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
9 marks Moderate -0.3
8.
  1. 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 $$x = \sec 2 y$$
  2. find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
  3. 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
9 marks Moderate -0.3
Differentiate with respect to \(x\), giving your answer in its simplest form,
  1. \(x ^ { 2 } \ln ( 3 x )\)
  2. \(\frac { \sin 4 x } { x ^ { 3 } }\)
Edexcel C3 2012 January Q3
6 marks Moderate -0.8
3. The area, \(A \mathrm {~mm} ^ { 2 }\), of a bacterial culture growing in milk, \(t\) hours after midday, is given by $$A = 20 \mathrm { e } ^ { 1.5 t } , \quad t \geqslant 0$$
  1. Write down the area of the culture at midday.
  2. Find the time at which the area of the culture is twice its area at midday. Give your answer to the nearest minute.
Edexcel C3 2012 January Q4
7 marks Standard +0.3
4. The point \(P\) is the point on the curve \(x = 2 \tan \left( y + \frac { \pi } { 12 } \right)\) with \(y\)-coordinate \(\frac { \pi } { 4 }\). Find an equation of the normal to the curve at \(P\).
Edexcel C3 2012 January Q5
10 marks Standard +0.3
5. Solve, for \(0 \leqslant \theta \leqslant 180 ^ { \circ }\), $$2 \cot ^ { 2 } 3 \theta = 7 \operatorname { cosec } 3 \theta - 5$$ Give your answers in degrees to 1 decimal place.
Edexcel C3 2012 January Q6
12 marks Standard +0.3
6. $$f ( x ) = x ^ { 2 } - 3 x + 2 \cos \left( \frac { 1 } { 2 } x \right) , \quad 0 \leqslant x \leqslant \pi$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a solution in the interval \(0.8 < x < 0.9\) The curve with equation \(y = \mathrm { f } ( x )\) has a minimum point \(P\).
  2. Show that the \(x\)-coordinate of \(P\) is the solution of the equation $$x = \frac { 3 + \sin \left( \frac { 1 } { 2 } x \right) } { 2 }$$
  3. Using the iteration formula $$x _ { n + 1 } = \frac { 3 + \sin \left( \frac { 1 } { 2 } x _ { n } \right) } { 2 } , \quad x _ { 0 } = 2$$ find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
  4. By choosing a suitable interval, show that the \(x\)-coordinate of \(P\) is 1.9078 correct to 4 decimal places.
Edexcel C3 2012 January Q7
12 marks Standard +0.3
  1. The function f is defined by
$$\mathrm { f } : x \mapsto \frac { 3 ( x + 1 ) } { 2 x ^ { 2 } + 7 x - 4 } - \frac { 1 } { x + 4 } , \quad x \in \mathbb { R } , x > \frac { 1 } { 2 }$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { 2 x - 1 }\)
  2. Find \(\mathrm { f } ^ { - 1 } ( x )\)
  3. Find the domain of \(\mathrm { f } ^ { - 1 }\) $$\mathrm { g } ( x ) = \ln ( x + 1 )$$
  4. Find the solution of \(\mathrm { fg } ( x ) = \frac { 1 } { 7 }\), giving your answer in terms of e .
Edexcel C3 2012 January Q8
13 marks Standard +0.3
8.
  1. Starting from the formulae for \(\sin ( A + B )\) and \(\cos ( A + B )\), prove that
  2. Deduce that $$\tan ( A + B ) = \frac { \tan A + \tan B } { 1 - \tan A \tan B }$$
  3. Hence, or otherwise, solve, for \(0 \leqslant \theta \leqslant \pi\), $$\tan \left( \theta + \frac { \pi } { 6 } \right) = \frac { 1 + \sqrt { } 3 \tan \theta } { \sqrt { } 3 - \tan \theta }$$
(c) Hence, or otherwise, solve, for \(0 \leqslant \theta \leqslant \pi\),
(c) $$1 + \sqrt { } 3 \tan \theta = ( \sqrt { } 3 - \tan \theta ) \tan ( \pi - \theta )$$ \section*{}
Edexcel C3 2013 January Q1
7 marks Moderate -0.8
  1. The curve \(C\) has equation
$$y = ( 2 x - 3 ) ^ { 5 }$$ The point \(P\) lies on \(C\) and has coordinates \(( w , - 32 )\).
Find
  1. the value of \(w\),
  2. the equation of the tangent to \(C\) at the point \(P\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C3 2013 January Q2
8 marks Standard +0.3
2. $$\mathrm { g } ( x ) = \mathrm { e } ^ { x - 1 } + x - 6$$
  1. Show that the equation \(\mathrm { g } ( x ) = 0\) can be written as $$x = \ln ( 6 - x ) + 1 , \quad x < 6$$ The root of \(\mathrm { g } ( x ) = 0\) is \(\alpha\).
    The iterative formula $$x _ { n + 1 } = \ln \left( 6 - x _ { n } \right) + 1 , \quad x _ { 0 } = 2$$ is used to find an approximate value for \(\alpha\).
  2. Calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 4 decimal places.
  3. By choosing a suitable interval, show that \(\alpha = 2.307\) correct to 3 decimal places.
Edexcel C3 2013 January Q3
9 marks Moderate -0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c78b0245-5c5a-407f-ad8a-602949a76e05-04_620_1095_223_420} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
The curve passes through the points \(Q ( 0,2 )\) and \(P ( - 3,0 )\) as shown.
  1. Find the value of ff(-3). On separate diagrams, sketch the curve with equation
  2. \(y = \mathrm { f } ^ { - 1 } ( x )\),
  3. \(y = \mathrm { f } ( | x | ) - 2\),
  4. \(y = 2 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.
Edexcel C3 2013 January Q4
8 marks Standard +0.3
  1. Express \(6 \cos \theta + 8 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 3 decimal places.
  2. $$\mathrm { p } ( \theta ) = \frac { 4 } { 12 + 6 \cos \theta + 8 \sin \theta } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ Calculate
    1. the maximum value of \(\mathrm { p } ( \theta )\),
    2. the value of \(\theta\) at which the maximum occurs.