Questions C3 (1200 questions)

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Edexcel C3 2015 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{57ea7a94-6939-4c12-a6fd-01cd6af73310-10_1004_1120_260_420} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 is a sketch showing part of the curve with equation \(y = 2 ^ { x + 1 } - 3\) and part of the line with equation \(y = 17 - x\). The curve and the line intersect at the point \(A\).
  1. Show that the \(x\) coordinate of \(A\) satisfies the equation $$x = \frac { \ln ( 20 - x ) } { \ln 2 } - 1$$
  2. Use the iterative formula $$x _ { n + 1 } = \frac { \ln \left( 20 - x _ { n } \right) } { \ln 2 } - 1 , \quad x _ { 0 } = 3$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
  3. Use your answer to part (b) to deduce the coordinates of the point \(A\), giving your answers to one decimal place.
Edexcel C3 2015 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{57ea7a94-6939-4c12-a6fd-01cd6af73310-12_632_873_294_532} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation $$\mathrm { g } ( x ) = x ^ { 2 } ( 1 - x ) \mathrm { e } ^ { - 2 x } , \quad x \geqslant 0$$
  1. Show that \(\mathrm { g } ^ { \prime } ( x ) = \mathrm { f } ( x ) \mathrm { e } ^ { - 2 x }\), where \(\mathrm { f } ( x )\) is a cubic function to be found.
  2. Hence find the range of g .
  3. State a reason why the function \(\mathrm { g } ^ { - 1 } ( x )\) does not exist.
Edexcel C3 2015 June Q8
  1. (a) Prove that
$$\sec 2 A + \tan 2 A \equiv \frac { \cos A + \sin A } { \cos A - \sin A } , \quad A \neq \frac { ( 2 n + 1 ) \pi } { 4 } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta < 2 \pi\), $$\sec 2 \theta + \tan 2 \theta = \frac { 1 } { 2 }$$ Give your answers to 3 decimal places.
Edexcel C3 2015 June Q9
9. Given that \(k\) is a negative constant and that the function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 2 - \frac { ( x - 5 k ) ( x - k ) } { x ^ { 2 } - 3 k x + 2 k ^ { 2 } } , \quad x \geqslant 0$$
  1. show that \(\mathrm { f } ( x ) = \frac { x + k } { x - 2 k }\)
  2. Hence find \(\mathrm { f } ^ { \prime } ( x )\), giving your answer in its simplest form.
  3. State, with a reason, whether \(\mathrm { f } ( x )\) is an increasing or a decreasing function. Justify your answer.
Edexcel C3 2016 June Q1
  1. The functions \(f\) and \(g\) are defined by
$$\begin{aligned} & \mathrm { f } : x \rightarrow 7 x - 1 , \quad x \in \mathbb { R }
& \mathrm {~g} : x \rightarrow \frac { 4 } { x - 2 } , \quad x \neq 2 , x \in \mathbb { R } \end{aligned}$$
  1. Solve the equation \(\operatorname { fg } ( x ) = x\)
  2. Hence, or otherwise, find the largest value of \(a\) such that \(\mathrm { g } ( a ) = \mathrm { f } ^ { - 1 } ( a )\)
Edexcel C3 2016 June Q2
2.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), writing your answer as a single fraction in its simplest form.
  2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } < 0\)
    2. $$y = \frac { 4 x } { x ^ { 2 } + 5 }$$
Edexcel C3 2016 June Q3
  1. (a) Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places.
    (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\),
$$\frac { 2 } { 2 \cos \theta - \sin \theta - 1 } = 15$$ Give your answers to one decimal place.
(c) Use your solutions to parts (a) and (b) to deduce the smallest positive value of \(\theta\) for which $$\frac { 2 } { 2 \cos \theta + \sin \theta - 1 } = 15$$ Give your answer to one decimal place.
Edexcel C3 2016 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d3ba2776-eedb-48f0-834f-41aa454afba3-06_675_1118_205_406} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = g ( x )\), where $$\mathrm { g } ( x ) = \left| 4 \mathrm { e } ^ { 2 x } - 25 \right| , \quad x \in \mathbb { R }$$ The curve cuts the \(y\)-axis at the point \(A\) and meets the \(x\)-axis at the point \(B\). The curve has an asymptote \(y = k\), where \(k\) is a constant, as shown in Figure 1
  1. Find, giving each answer in its simplest form,
    1. the \(y\) coordinate of the point \(A\),
    2. the exact \(x\) coordinate of the point \(B\),
    3. the value of the constant \(k\). The equation \(\mathrm { g } ( x ) = 2 x + 43\) has a positive root at \(x = \alpha\)
  2. Show that \(\alpha\) is a solution of \(x = \frac { 1 } { 2 } \ln \left( \frac { 1 } { 2 } x + 17 \right)\) The iteration formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 1 } { 2 } x _ { n } + 17 \right)$$ can be used to find an approximation for \(\alpha\)
  3. Taking \(x _ { 0 } = 1.4\) find the values of \(x _ { 1 }\) and \(x _ { 2 }\) Give each answer to 4 decimal places.
  4. By choosing a suitable interval, show that \(\alpha = 1.437\) to 3 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-07_2258_47_315_37}
Edexcel C3 2016 June Q5
5. (i) Find, using calculus, the \(x\) coordinate of the turning point of the curve with equation $$y = \mathrm { e } ^ { 3 x } \cos 4 x , \quad \frac { \pi } { 4 } \leqslant x < \frac { \pi } { 2 }$$ Give your answer to 4 decimal places.
(ii) Given \(x = \sin ^ { 2 } 2 y , \quad 0 < y < \frac { \pi } { 4 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(y\). Write your answer in the form $$\frac { \mathrm { d } y } { \mathrm {~d} x } = p \operatorname { cosec } ( q y ) , \quad 0 < y < \frac { \pi } { 4 }$$ where \(p\) and \(q\) are constants to be determined. \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-09_2258_47_315_37}
Edexcel C3 2016 June Q6
6. $$f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 3 x ^ { 2 } + 7 x - 6 } { x ^ { 2 } + x - 6 } , \quad x > 2 , x \in \mathbb { R }$$
  1. Given that $$\frac { x ^ { 4 } + x ^ { 3 } - 3 x ^ { 2 } + 7 x - 6 } { x ^ { 2 } + x - 6 } \equiv x ^ { 2 } + A + \frac { B } { x - 2 }$$ find the values of the constants \(A\) and \(B\).
  2. Hence or otherwise, using calculus, find an equation of the normal to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 3\)
Edexcel C3 2016 June Q7
7. (a) For \(- \frac { \pi } { 2 } \leqslant y \leqslant \frac { \pi } { 2 }\), sketch the graph of \(y = \mathrm { g } ( x )\) where $$g ( x ) = \arcsin x \quad - 1 \leqslant x \leqslant 1$$ (b) Find the exact value of \(x\) for which $$3 g ( x + 1 ) + \pi = 0$$
Edexcel C3 2016 June Q8
  1. (a) Prove that
$$2 \cot 2 x + \tan x \equiv \cot x \quad x \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(- \pi \leqslant x < \pi\), $$6 \cot 2 x + 3 \tan x = \operatorname { cosec } ^ { 2 } x - 2$$ Give your answers to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-14_2258_47_315_37}
Edexcel C3 2016 June Q9
9. The amount of an antibiotic in the bloodstream, from a given dose, is modelled by the formula $$x = D \mathrm { e } ^ { - 0.2 t }$$ where \(x\) is the amount of the antibiotic in the bloodstream in milligrams, \(D\) is the dose given in milligrams and \(t\) is the time in hours after the antibiotic has been given. A first dose of 15 mg of the antibiotic is given.
  1. Use the model to find the amount of the antibiotic in the bloodstream 4 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 15 mg is given 5 hours after the first dose has been given. Using the same model for the second dose,
  2. show that the total amount of the antibiotic in the bloodstream 2 hours after the second dose is given is 13.754 mg to 3 decimal places. No more doses of the antibiotic are given. At time \(T\) hours after the second dose is given, the total amount of the antibiotic in the bloodstream is 7.5 mg .
  3. Show that \(T = a \ln \left( b + \frac { b } { \mathrm { e } } \right)\), where \(a\) and \(b\) are integers to be determined.
    VIIIV SIHI NITIIIM I I N O CVI4V SIHI NI IHIHM ION OCVI4V SIHI NI JIIIM ION OO
    \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-16_2258_47_315_37}
Edexcel C3 2017 June Q1
  1. Express \(\frac { 4 x } { x ^ { 2 } - 9 } - \frac { 2 } { x + 3 }\) as a single fraction in its simplest form.
Edexcel C3 2017 June Q2
2. Find the exact solutions, in their simplest form, to the equations
  1. \(\mathrm { e } ^ { 3 x - 9 } = 8\)
  2. \(\ln ( 2 y + 5 ) = 2 + \ln ( 4 - y )\)
    \includegraphics[max width=\textwidth, alt={}, center]{f0a633e3-5c63-4d21-8ffa-d4e7dc43a536-05_37_1813_0_6}
Edexcel C3 2017 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0a633e3-5c63-4d21-8ffa-d4e7dc43a536-06_476_1107_221_422} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the graph of \(y = \mathrm { g } ( x )\), where $$g ( x ) = 3 + \sqrt { x + 2 } , \quad x \geqslant - 2$$
  1. State the range of g .
  2. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain.
  3. Find the exact value of \(x\) for which $$\mathrm { g } ( x ) = x$$
  4. Hence state the value of \(a\) for which $$\mathrm { g } ( a ) = \mathrm { g } ^ { - 1 } ( a )$$
Edexcel C3 2017 June Q4
  1. (a) Write \(5 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 \leqslant \alpha < \frac { \pi } { 2 }\)
Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.
(b) Show that the equation $$5 \cot 2 x - 3 \operatorname { cosec } 2 x = 2$$ can be rewritten in the form $$5 \cos 2 x - 2 \sin 2 x = c$$ where \(c\) is a positive constant to be determined.
(c) Hence or otherwise, solve, for \(0 \leqslant x < \pi\), $$5 \cot 2 x - 3 \operatorname { cosec } 2 x = 2$$ giving your answers to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C3 2017 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0a633e3-5c63-4d21-8ffa-d4e7dc43a536-14_549_958_221_493} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = 2 \ln ( 2 x + 5 ) - \frac { 3 x } { 2 } , \quad x > - 2.5$$ The point \(P\) with \(x\) coordinate - 2 lies on \(C\).
  1. Find an equation of the normal to \(C\) at \(P\). Write your answer in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. The normal to \(C\) at \(P\) cuts the curve again at the point \(Q\), as shown in Figure 2
  2. Show that the \(x\) coordinate of \(Q\) is a solution of the equation $$x = \frac { 20 } { 11 } \ln ( 2 x + 5 ) - 2$$ The iteration formula $$x _ { n + 1 } = \frac { 20 } { 11 } \ln \left( 2 x _ { n } + 5 \right) - 2$$ can be used to find an approximation for the \(x\) coordinate of \(Q\).
  3. Taking \(x _ { 1 } = 2\), find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving each answer to 4 decimal places.
Edexcel C3 2017 June Q6
  1. Given that \(a\) and \(b\) are positive constants,
    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. Given that the equation $$| 2 x - a | + b = \frac { 3 } { 2 } x + 8$$ has a solution at \(x = 0\) and a solution at \(x = c\),
  2. find \(c\) in terms of \(a\).
Edexcel C3 2017 June Q7
    1. Given \(y = 2 x \left( x ^ { 2 } - 1 \right) ^ { 5 }\), show that
      1. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { g } ( x ) \left( x ^ { 2 } - 1 \right) ^ { 4 }\) where \(\mathrm { g } ( x )\) is a function to be determined.
    2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } \geqslant 0\)
      (ii) Given
    $$x = \ln ( \sec 2 y ) , \quad 0 < y < \frac { \pi } { 4 }$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(x\) in its simplest form.
Edexcel C3 2017 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0a633e3-5c63-4d21-8ffa-d4e7dc43a536-26_663_1454_210_242} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The number of rabbits on an island is modelled by the equation $$P = \frac { 100 \mathrm { e } ^ { - 0.1 t } } { 1 + 3 \mathrm { e } ^ { - 0.9 t } } + 40 , \quad t \in \mathbb { R } , t \geqslant 0$$ where \(P\) is the number of rabbits, \(t\) years after they were introduced onto the island.
A sketch of the graph of \(P\) against \(t\) is shown in Figure 3.
  1. Calculate the number of rabbits that were introduced onto the island.
  2. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) The number of rabbits initially increases, reaching a maximum value \(P _ { T }\) when \(t = T\)
  3. Using your answer from part (b), calculate
    1. the value of \(T\) to 2 decimal places,
    2. the value of \(P _ { T }\) to the nearest integer.
      (Solutions based entirely on graphical or numerical methods are not acceptable.) For \(t > T\), the number of rabbits decreases, as shown in Figure 3, but never falls below \(k\), where \(k\) is a positive constant.
  4. Use the model to state the maximum value of \(k\).
Edexcel C3 2017 June Q9
  1. (a) Prove that
$$\sin 2 x - \tan x \equiv \tan x \cos 2 x , \quad x \neq ( 2 n + 1 ) 90 ^ { \circ } , \quad n \in \mathbb { Z }$$ (b) Given that \(x \neq 90 ^ { \circ }\) and \(x \neq 270 ^ { \circ }\), solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$\sin 2 x - \tan x = 3 \tan x \sin x$$ Give your answers in degrees to one decimal place where appropriate.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
\includegraphics[max width=\textwidth, alt={}]{f0a633e3-5c63-4d21-8ffa-d4e7dc43a536-32_2632_1826_121_121}
Edexcel C3 2018 June Q1
  1. Given \(y = 2 x ( 3 x - 1 ) ^ { 5 }\),
    1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer as a single fully factorised expression.
    2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } \leqslant 0\)
Edexcel C3 2018 June Q2
  1. The function f is defined by
$$\mathrm { f } ( x ) = \frac { 6 } { 2 x + 5 } + \frac { 2 } { 2 x - 5 } + \frac { 60 } { 4 x ^ { 2 } - 25 } , \quad x > 4$$
  1. Show that \(\mathrm { f } ( x ) = \frac { A } { B x + C }\) where \(A , B\) and \(C\) are constants to be found.
  2. Find \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
Edexcel C3 2018 June Q3
  1. The value of a car is modelled by the formula
$$V = 16000 \mathrm { e } ^ { - k t } + A , \quad t \geqslant 0 , t \in \mathbb { R }$$ where \(V\) is the value of the car in pounds, \(t\) is the age of the car in years, and \(k\) and \(A\) are positive constants. Given that the value of the car is \(\pounds 17500\) when new and \(\pounds 13500\) two years later,
  1. find the value of \(A\),
  2. show that \(k = \ln \left( \frac { 2 } { \sqrt { 3 } } \right)\)
  3. Find the age of the car, in years, when the value of the car is \(\pounds 6000\) Give your answer to 2 decimal places.