Questions — Edexcel (10514 questions)

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Edexcel C3 2013 January Q5
11 marks Moderate -0.3
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
11 marks Standard +0.3
6.
  1. 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.
  2. (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
12 marks Standard +0.3
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
9 marks Standard +0.3
  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
7 marks Moderate -0.3
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
7 marks Standard +0.3
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
8 marks Standard +0.3
  1. 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 \(y = \mathrm { e } ^ { 3 x } \operatorname { cosec } 2 x , 0 < x < \frac { \pi } { 2 }\),
  2. 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.
  3. 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
8 marks Moderate -0.3
  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
9 marks Standard +0.3
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
10 marks Moderate -0.3
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
13 marks Standard +0.3
7.
  1. (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.
  2. 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
13 marks Standard +0.8
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
8 marks Moderate -0.3
  1. Given that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(1 + \tan ^ { 2 } \theta \equiv \sec ^ { 2 } \theta\).
  2. 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
12 marks Moderate -0.3
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
10 marks Standard +0.3
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 \(\operatorname { fg } ( x ) = \frac { 1 } { 4 }\).
Edexcel C3 2005 June Q4
9 marks Moderate -0.3
4. $$\mathrm { f } ( x ) = 3 \mathrm { e } ^ { x } - \frac { 1 } { 2 } \ln x - 2 , \quad x > 0 .$$
  1. Differentiate to find \(\mathrm { f } ^ { \prime } ( x )\). The curve with equation \(y = \mathrm { f } ( x )\) has a turning point at \(P\). The \(x\)-coordinate of \(P\) is \(\alpha\).
  2. Show that \(\alpha = \frac { 1 } { 6 } \mathrm { e } ^ { - \alpha }\). The iterative formula $$x _ { n + 1 } = \frac { 1 } { 6 } \mathrm { e } ^ { - x _ { n } } , x _ { 0 } = 1$$ is used to find an approximate value for \(\alpha\).
  3. Calculate the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 decimal places.
  4. By considering the change of sign of \(\mathrm { f } ^ { \prime } ( x )\) in a suitable interval, prove that \(\alpha = 0.1443\) correct to 4 decimal places.
Edexcel C3 2005 June Q5
15 marks Standard +0.3
5.
  1. Using the identity \(\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B\), prove that $$\cos 2 A \equiv 1 - 2 \sin ^ { 2 } A$$
  2. Show that $$2 \sin 2 \theta - 3 \cos 2 \theta - 3 \sin \theta + 3 \equiv \sin \theta ( 4 \cos \theta + 6 \sin \theta - 3 )$$
  3. Express \(4 \cos \theta + 6 \sin \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
  4. Hence, for \(0 \leqslant \theta < \pi\), solve $$2 \sin 2 \theta = 3 ( \cos 2 \theta + \sin \theta - 1 )$$ giving your answers in radians to 3 significant figures, where appropriate.
Edexcel C3 2005 June Q6
11 marks Moderate -0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5af2eea6-bac1-455b-b25a-487d113e44ca-08_458_876_285_539}
\end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(( 1 , a ) , a < 0\). One line meets the \(x\)-axis at \(( 3,0 )\). The other line meets the \(x\)-axis at \(( - 1,0 )\) and the \(y\)-axis at \(( 0 , b ) , b < 0\). In separate diagrams, sketch the graph with equation
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = \mathrm { f } ( | x | )\). Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that \(\mathrm { f } ( x ) = | x - 1 | - 2\), find
  3. the value of \(a\) and the value of \(b\),
  4. the value of \(x\) for which \(\mathrm { f } ( x ) = 5 x\).
Edexcel C3 2005 June Q7
10 marks Standard +0.3
  1. A particular species of orchid is being studied. The population \(p\) at time \(t\) years after the study started is assumed to be
$$p = \frac { 2800 a \mathrm { e } ^ { 0.2 t } } { 1 + a \mathrm { e } ^ { 0.2 t } } , \text { where } a \text { is a constant. }$$ Given that there were 300 orchids when the study started,
  1. show that \(a = 0.12\),
  2. use the equation with \(a = 0.12\) to predict the number of years before the population of orchids reaches 1850.
  3. Show that \(p = \frac { 336 } { 0.12 + \mathrm { e } ^ { - 0.2 t } }\).
  4. Hence show that the population cannot exceed 2800.
Edexcel C3 2006 June Q1
6 marks Moderate -0.3
  1. Simplify \(\frac { 3 x ^ { 2 } - x - 2 } { x ^ { 2 } - 1 }\).
  2. Hence, or otherwise, express \(\frac { 3 x ^ { 2 } - x - 2 } { x ^ { 2 } - 1 } - \frac { 1 } { x ( x + 1 ) }\) as a single fraction in its simplest form.
Edexcel C3 2006 June Q4
9 marks Moderate -0.8
  1. A heated metal ball is dropped into a liquid. As the ball cools, its temperature, \(T ^ { \circ } \mathrm { C }\), \(t\) minutes after it enters the liquid, is given by
$$T = 400 \mathrm { e } ^ { - 0.05 t } + 25 , \quad t \geqslant 0$$
  1. Find the temperature of the ball as it enters the liquid.
  2. Find the value of \(t\) for which \(T = 300\), giving your answer to 3 significant figures.
  3. Find the rate at which the temperature of the ball is decreasing at the instant when \(t = 50\). Give your answer in \({ } ^ { \circ } \mathrm { C }\) per minute to 3 significant figures.
  4. From the equation for temperature \(T\) in terms of \(t\), given above, explain why the temperature of the ball can never fall to \(20 ^ { \circ } \mathrm { C }\).
Edexcel C3 2006 June Q5
11 marks Standard +0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{f0f328ed-3550-4b8d-8b80-016df8773b21-07_465_565_296_701}
\end{figure} Figure 2 shows part of the curve with equation $$y = ( 2 x - 1 ) \tan 2 x , \quad 0 \leqslant x < \frac { \pi } { 4 }$$ The curve has a minimum at the point \(P\). The \(x\)-coordinate of \(P\) is \(k\).
  1. Show that \(k\) satisfies the equation $$4 k + \sin 4 k - 2 = 0$$ The iterative formula $$x _ { n + 1 } = \frac { 1 } { 4 } \left( 2 - \sin 4 x _ { n } \right) , x _ { 0 } = 0.3$$ is used to find an approximate value for \(k\).
  2. Calculate the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 decimal places.
  3. Show that \(k = 0.277\), correct to 3 significant figures.
Edexcel C3 2006 June Q6
10 marks Standard +0.3
  1. Using \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(\operatorname { cosec } ^ { 2 } \theta - \cot ^ { 2 } \theta \equiv 1\).
  2. Hence, or otherwise, prove that $$\operatorname { cosec } ^ { 4 } \theta - \cot ^ { 4 } \theta \equiv \operatorname { cosec } ^ { 2 } \theta + \cot ^ { 2 } \theta$$
  3. Solve, for \(90 ^ { \circ } < \theta < 180 ^ { \circ }\), $$\operatorname { cosec } ^ { 4 } \theta - \cot ^ { 4 } \theta = 2 - \cot \theta$$
Edexcel C3 2006 June Q7
12 marks Moderate -0.3
7. For the constant \(k\), where \(k > 1\), the functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto \ln ( x + k ) , \quad x > - k , \\ & \mathrm {~g} : x \mapsto | 2 x - k | , \quad x \in \mathbb { R } . \end{aligned}$$
  1. On separate axes, sketch the graph of f and the graph of g . On each sketch state, in terms of \(k\), the coordinates of points where the graph meets the coordinate axes.
  2. Write down the range of f.
  3. Find \(\mathrm { fg } \left( \frac { k } { 4 } \right)\) in terms of \(k\), giving your answer in its simplest form. The curve \(C\) has equation \(y = \mathrm { f } ( x )\). The tangent to \(C\) at the point with \(x\)-coordinate 3 is parallel to the line with equation \(9 y = 2 x + 1\).
  4. Find the value of \(k\).
Edexcel C3 2006 June Q8
12 marks Standard +0.3
  1. Given that \(\cos A = \frac { 3 } { 4 }\), where \(270 ^ { \circ } < A < 360 ^ { \circ }\), find the exact value of \(\sin 2 A\).
    1. Show that \(\cos \left( 2 x + \frac { \pi } { 3 } \right) + \cos \left( 2 x - \frac { \pi } { 3 } \right) \equiv \cos 2 x\). Given that $$y = 3 \sin ^ { 2 } x + \cos \left( 2 x + \frac { \pi } { 3 } \right) + \cos \left( 2 x - \frac { \pi } { 3 } \right)$$
    2. show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x\).