Questions — Edexcel C3 (377 questions)

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Edexcel C3 2012 January Q3
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
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
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
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
  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
8. (a) Starting from the formulae for \(\sin ( A + B )\) and \(\cos ( A + B )\), prove that
(b) Deduce that $$\tan ( A + B ) = \frac { \tan A + \tan B } { 1 - \tan A \tan B }$$ (c) 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
  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
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
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
  1. (a) 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.
(b) $$\mathrm { p } ( \theta ) = \frac { 4 } { 12 + 6 \cos \theta + 8 \sin \theta } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ Calculate
  1. the maximum value of \(\mathrm { p } ( \theta )\),
  2. the value of \(\theta\) at which the maximum occurs.
Edexcel C3 2013 January Q5
5. (i) Differentiate with respect to \(x\)
  1. \(y = x ^ { 3 } \ln 2 x\)
  2. \(y = ( x + \sin 2 x ) ^ { 3 }\) Given that \(x = \cot y\),
    (ii) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 1 } { 1 + x ^ { 2 } }\)
Edexcel C3 2013 January Q6
6. (i) Without using a calculator, find the exact value of $$\left( \sin 22.5 ^ { \circ } + \cos 22.5 ^ { \circ } \right) ^ { 2 }$$ You must show each stage of your working.
(ii) (a) Show that \(\cos 2 \theta + \sin \theta = 1\) may be written in the form $$k \sin ^ { 2 } \theta - \sin \theta = 0 , \text { stating the value of } k$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$\cos 2 \theta + \sin \theta = 1$$
Edexcel C3 2013 January Q7
7. $$\mathrm { h } ( x ) = \frac { 2 } { x + 2 } + \frac { 4 } { x ^ { 2 } + 5 } - \frac { 18 } { \left( x ^ { 2 } + 5 \right) ( x + 2 ) } , \quad x \geqslant 0$$
  1. Show that \(\mathrm { h } ( x ) = \frac { 2 x } { x ^ { 2 } + 5 }\)
  2. Hence, or otherwise, find \(\mathrm { h } ^ { \prime } ( x )\) in its simplest form. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c78b0245-5c5a-407f-ad8a-602949a76e05-10_729_1235_644_351} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a graph of the curve with equation \(y = \mathrm { h } ( x )\).
  3. Calculate the range of \(\mathrm { h } ( x )\).
Edexcel C3 2013 January Q8
  1. The value of Bob's car can be calculated from the formula
$$V = 17000 \mathrm { e } ^ { - 0.25 t } + 2000 \mathrm { e } ^ { - 0.5 t } + 500$$ where \(V\) is the value of the car in pounds \(( \pounds )\) and \(t\) is the age in years.
  1. Find the value of the car when \(t = 0\)
  2. Calculate the exact value of \(t\) when \(V = 9500\)
  3. Find the rate at which the value of the car is decreasing at the instant when \(t = 8\). Give your answer in pounds per year to the nearest pound.
Edexcel C3 2014 January Q1
1. $$f ( x ) = \sec x + 3 x - 2 , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$
  1. Show that there is a root of \(\mathrm { f } ( x ) = 0\) in the interval \([ 0.2,0.4 ]\)
  2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form $$x = \frac { 2 } { 3 } - \frac { 1 } { 3 \cos x }$$ The solution of \(\mathrm { f } ( x ) = 0\) is \(\alpha\), where \(\alpha = 0.3\) to 1 decimal place.
  3. Starting with \(x _ { 0 } = 0.3\), use the iterative formula $$x _ { n + 1 } = \frac { 2 } { 3 } - \frac { 1 } { 3 \cos x _ { n } }$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
  4. State the value of \(\alpha\) correct to 3 decimal places.
Edexcel C3 2014 January Q2
2. $$f ( x ) = \frac { 15 } { 3 x + 4 } - \frac { 2 x } { x - 1 } + \frac { 14 } { ( 3 x + 4 ) ( x - 1 ) } , \quad x > 1$$
  1. Express \(\mathrm { f } ( x )\) as a single fraction in its simplest form.
  2. Hence, or otherwise, find \(\mathrm { f } ^ { \prime } ( x )\), giving your answer as a single fraction in its simplest form.
Edexcel C3 2014 January Q3
  1. (a) By writing \(\operatorname { cosec } x\) as \(\frac { 1 } { \sin x }\), show that
$$\frac { \mathrm { d } ( \operatorname { cosec } x ) } { \mathrm { d } x } = - \operatorname { cosec } x \cot x$$ Given that \(y = \mathrm { e } ^ { 3 x } \operatorname { cosec } 2 x , 0 < x < \frac { \pi } { 2 }\),
(b) find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The curve with equation \(y = \mathrm { e } ^ { 3 x } \operatorname { cosec } 2 x , 0 < x < \frac { \pi } { 2 }\), has a single turning point.
(c) Show that the \(x\)-coordinate of this turning point is at \(x = \frac { 1 } { 2 } \arctan k\) where the value
of the constant \(k\) should be found. of the constant \(k\) should be found.
Edexcel C3 2014 January Q4
  1. A pot of coffee is delivered to a meeting room at 11 am . At a time \(t\) minutes after 11 am the temperature, \(\theta ^ { \circ } \mathrm { C }\), of the coffee in the pot is given by the equation
$$\theta = A + 60 \mathrm { e } ^ { - k t }$$ where \(A\) and \(k\) are positive constants. Given also that the temperature of the coffee at 11 am is \(85 ^ { \circ } \mathrm { C }\) and that 15 minutes later it is \(58 ^ { \circ } \mathrm { C }\),
  1. find the value of \(A\).
  2. Show that \(k = \frac { 1 } { 15 } \ln \left( \frac { 20 } { 11 } \right)\)
  3. Find, to the nearest minute, the time at which the temperature of the coffee reaches \(50 ^ { \circ } \mathrm { C }\).
Edexcel C3 2014 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f22eb1b-21de-45f1-9a8a-deac7ac8d0b0-14_646_1013_207_532} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve shown in Figure 1 has equation $$x = 3 \sin y + 3 \cos y , \quad - \frac { \pi } { 4 } < y < \frac { \pi } { 4 }$$
  1. Express the equation of the curve in the form
    \(x = R \sin ( y + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
  2. Find the coordinates of the point on the curve where the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(\frac { 1 } { 2 }\). Give your answers to 3 decimal places.
Edexcel C3 2014 January Q6
  1. Given that \(a\) and \(b\) are constants and that \(0 < a < b\),
    1. on separate diagrams, sketch the graph with equation
      1. \(y = | 2 x + a |\),
      2. \(y = | 2 x + a | - b\).
    Show on each sketch the coordinates of each point at which the graph crosses or meets the axes.
  2. Solve, for \(x\), the equation $$| 2 x + a | - b = \frac { 1 } { 3 } x$$ giving any answers in terms of \(a\) and \(b\).
Edexcel C3 2014 January Q7
7. (i) (a) Prove that $$\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$$ (You may use the double angle formulae and the identity $$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B )$$ (b) Hence solve the equation $$2 \cos 3 \theta + \cos 2 \theta + 1 = 0$$ giving answers in the interval \(0 \leqslant \theta \leqslant \pi\).
Solutions based entirely on graphical or numerical methods are not acceptable.
(ii) Given that \(\theta = \arcsin x\) and that \(0 < \theta < \frac { \pi } { 2 }\), show that $$\cot \theta = \frac { \sqrt { \left( 1 - x ^ { 2 } \right) } } { x } , \quad 0 < x < 1$$
Edexcel C3 2014 January Q8
8. The function \(f\) is defined by $$\mathrm { f } : x \rightarrow 3 - 2 \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$
  1. Find the inverse function, \(\mathrm { f } ^ { - 1 } ( x )\) and give its domain.
  2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \ln x\). The equation \(\mathrm { f } ( t ) = k \mathrm { e } ^ { t }\), where \(k\) is a positive constant, has exactly one real solution.
  3. Find the value of \(k\).
Edexcel C3 2005 June Q1
  1. (a) Given that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(1 + \tan ^ { 2 } \theta \equiv \sec ^ { 2 } \theta\).
    (b) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation
$$2 \tan ^ { 2 } \theta + \sec \theta = 1 ,$$ giving your answers to 1 decimal place.
Edexcel C3 2005 June Q2
2. (a) Differentiate with respect to \(x\)
  1. \(3 \sin ^ { 2 } x + \sec 2 x\),
  2. \(\{ x + \ln ( 2 x ) \} ^ { 3 }\). Given that \(y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } , \quad x \neq 1\),
    (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 } { ( x - 1 ) ^ { 3 } }\).
Edexcel C3 2005 June Q3
3. The function \(f\) is defined by $$f : x \rightarrow \frac { 5 x + 1 } { x ^ { 2 } + x - 2 } - \frac { 3 } { x + 2 } , x > 1$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 2 } { x - 1 } , x > 1\).
  2. Find \(\mathrm { f } ^ { - 1 } ( x )\). The function \(g\) is defined by $$\mathrm { g } : x \rightarrow x ^ { 2 } + 5 , \quad x \in \mathbb { R }$$
  3. Solve \(\operatorname { fg } ( x ) = \frac { 1 } { 4 }\).