Questions — Edexcel C3 (377 questions)

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Edexcel C3 2005 June Q4
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
5. (a) Using the identity \(\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B\), prove that $$\cos 2 A \equiv 1 - 2 \sin ^ { 2 } A$$ (b) Show that $$2 \sin 2 \theta - 3 \cos 2 \theta - 3 \sin \theta + 3 \equiv \sin \theta ( 4 \cos \theta + 6 \sin \theta - 3 )$$ (c) Express \(4 \cos \theta + 6 \sin \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
(d) 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
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
  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
  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
  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
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
  1. (a) Using \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(\operatorname { cosec } ^ { 2 } \theta - \cot ^ { 2 } \theta \equiv 1\).
    (b) Hence, or otherwise, prove that
$$\operatorname { cosec } ^ { 4 } \theta - \cot ^ { 4 } \theta \equiv \operatorname { cosec } ^ { 2 } \theta + \cot ^ { 2 } \theta$$ (c) Solve, for \(90 ^ { \circ } < \theta < 180 ^ { \circ }\), $$\operatorname { cosec } ^ { 4 } \theta - \cot ^ { 4 } \theta = 2 - \cot \theta$$
Edexcel C3 2006 June Q7
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
  1. (a) Given that \(\cos A = \frac { 3 } { 4 }\), where \(270 ^ { \circ } < A < 360 ^ { \circ }\), find the exact value of \(\sin 2 A\).
    (b) (i) Show that \(\cos \left( 2 x + \frac { \pi } { 3 } \right) + \cos \left( 2 x - \frac { \pi } { 3 } \right) \equiv \cos 2 x\).
Given that $$y = 3 \sin ^ { 2 } x + \cos \left( 2 x + \frac { \pi } { 3 } \right) + \cos \left( 2 x - \frac { \pi } { 3 } \right)$$ (ii) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x\).
Edexcel C3 2007 June Q1
Find the exact solutions to the equations
  1. \(\ln x + \ln 3 = \ln 6\),
  2. \(\mathrm { e } ^ { x } + 3 \mathrm { e } ^ { - x } = 4\).
Edexcel C3 2007 June Q3
3. A curve \(C\) has equation $$y = x ^ { 2 } \mathrm { e } ^ { x }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), using the product rule for differentiation.
  2. Hence find the coordinates of the turning points of \(C\).
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  4. Determine the nature of each turning point of the curve \(C\).
Edexcel C3 2007 June Q4
4. $$f ( x ) = - x ^ { 3 } + 3 x ^ { 2 } - 1$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be rewritten as $$x = \sqrt { } \left( \frac { 1 } { 3 - x } \right)$$
  2. Starting with \(x _ { 1 } = 0.6\), use the iteration $$\left. x _ { n + 1 } = \sqrt { ( } \frac { 1 } { 3 - x _ { n } } \right)$$ to calculate the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving all your answers to 4 decimal places.
  3. Show that \(x = 0.653\) is a root of \(\mathrm { f } ( x ) = 0\) correct to 3 decimal places.
Edexcel C3 2007 June Q5
5. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto \ln ( 2 x - 1 ) , & x \in \mathbb { R } , x > \frac { 1 } { 2 }
\mathrm {~g} : x \mapsto \frac { 2 } { x - 3 } , & x \in \mathbb { R } , x \neq 3 \end{array}$$
  1. Find the exact value of fg(4).
  2. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\), stating its domain.
  3. Sketch the graph of \(y = | \mathrm { g } ( x ) |\). Indicate clearly the equation of the vertical asymptote and the coordinates of the point at which the graph crosses the \(y\)-axis.
  4. Find the exact values of \(x\) for which \(\left| \frac { 2 } { x - 3 } \right| = 3\).
Edexcel C3 2007 June Q6
  1. (a) Express \(3 \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
    (b) Hence find the greatest value of \(( 3 \sin x + 2 \cos x ) ^ { 4 }\).
    (c) Solve, for \(0 < x < 2 \pi\), the equation
$$3 \sin x + 2 \cos x = 1$$ giving your answers to 3 decimal places.
Edexcel C3 2007 June Q7
  1. (a) Prove that
$$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta } = 2 \operatorname { cosec } 2 \theta , \quad \theta \neq 90 n ^ { \circ }$$ (b) On the axes on page 20, sketch the graph of \(y = 2 \operatorname { cosec } 2 \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
(c) Solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation $$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta } = 3 ,$$ giving your answers to 1 decimal place.
\includegraphics[max width=\textwidth, alt={}, center]{f3c3c777-7808-4d82-a1f4-2dee6674be1e-11_899_1253_315_347}
Edexcel C3 2007 June Q8
8. The amount of a certain type of drug in the bloodstream \(t\) hours after it has been taken is given by the formula $$x = D \mathrm { e } ^ { - \frac { 1 } { 8 } t } ,$$ where \(x\) is the amount of the drug in the bloodstream in milligrams and \(D\) is the dose given in milligrams. A dose of 10 mg of the drug is given.
  1. Find the amount of the drug in the bloodstream 5 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 10 mg is given after 5 hours.
  2. Show that the amount of the drug in the bloodstream 1 hour after the second dose is 13.549 mg to 3 decimal places. No more doses of the drug are given. At time \(T\) hours after the second dose is given, the amount of the drug in the bloodstream is 3 mg .
  3. Find the value of \(T\).
Edexcel C3 2008 June Q1
  1. The point \(P\) lies on the curve with equation
$$y = 4 \mathrm { e } ^ { 2 x + 1 }$$ The \(y\)-coordinate of \(P\) is 8 .
  1. Find, in terms of \(\ln 2\), the \(x\)-coordinate of \(P\).
  2. Find the equation of the tangent to the curve at the point \(P\) in the form \(y = a x + b\), where \(a\) and \(b\) are exact constants to be found.
Edexcel C3 2008 June Q2
2. $$f ( x ) = 5 \cos x + 12 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\),
  1. find the value of \(R\) and the value of \(\alpha\) to 3 decimal places.
  2. Hence solve the equation $$5 \cos x + 12 \sin x = 6$$ for \(0 \leqslant x < 2 \pi\).
    1. Write down the maximum value of \(5 \cos x + 12 \sin x\).
    2. Find the smallest positive value of \(x\) for which this maximum value occurs.
Edexcel C3 2008 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f47675f8-a2c2-4c4c-b878-ffe15a95c19d-05_623_977_207_479} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of \(y = f ( x ) , x \in \mathbb { R }\).
The graph consists of two line segments that meet at the point \(P\).
The graph cuts the \(y\)-axis at the point \(Q\) and the \(x\)-axis at the points \(( - 3,0 )\) and \(R\). Sketch, on separate diagrams, the graphs of
  1. \(y = | f ( x ) |\),
  2. \(y = \mathrm { f } ( - x )\). Given that \(\mathrm { f } ( x ) = 2 - | x + 1 |\),
  3. find the coordinates of the points \(P , Q\) and \(R\),
  4. solve \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x\).
Edexcel C3 2008 June Q4
4. The function \(f\) is defined by $$f : x \mapsto \frac { 2 ( x - 1 ) } { x ^ { 2 } - 2 x - 3 } - \frac { 1 } { x - 3 } , \quad x > 3$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { x + 1 } , \quad x > 3\).
  2. Find the range of f.
  3. Find \(\mathrm { f } ^ { - 1 } ( x )\). State the domain of this inverse function. The function \(g\) is defined by $$\mathrm { g } : x \mapsto 2 x ^ { 2 } - 3 , \quad x \in \mathbb { R }$$
  4. Solve \(\mathrm { fg } ( x ) = \frac { 1 } { 8 }\).
Edexcel C3 2008 June Q5
5. (a) Given that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(1 + \cot ^ { 2 } \theta \equiv \operatorname { cosec } ^ { 2 } \theta\).
(b) Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$2 \cot ^ { 2 } \theta - 9 \operatorname { cosec } \theta = 3$$ giving your answers to 1 decimal place.
Edexcel C3 2008 June Q6
6. (a) Differentiate with respect to \(x\),
  1. \(\mathrm { e } ^ { 3 x } ( \sin x + 2 \cos x )\),
  2. \(x ^ { 3 } \ln ( 5 x + 2 )\). Given that \(y = \frac { 3 x ^ { 2 } + 6 x - 7 } { ( x + 1 ) ^ { 2 } } , \quad x \neq - 1\),
    (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 20 } { ( x + 1 ) ^ { 3 } }\).
    (c) Hence find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and the real values of \(x\) for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 15 } { 4 }\).
Edexcel C3 2008 June Q7
7. $$f ( x ) = 3 x ^ { 3 } - 2 x - 6$$
  1. Show that \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), between \(x = 1.4\) and \(x = 1.45\)
  2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt { } \left( \frac { 2 } { x } + \frac { 2 } { 3 } \right) , \quad x \neq 0$$
  3. Starting with \(x _ { 0 } = 1.43\), use the iteration $$x _ { \mathrm { n } + 1 } = \sqrt { } \left( \frac { 2 } { x _ { \mathrm { n } } } + \frac { 2 } { 3 } \right)$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
  4. By choosing a suitable interval, show that \(\alpha = 1.435\) is correct to 3 decimal places.
Edexcel C3 2009 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bcb0c693-66ae-4b97-99f8-b10fb9396886-02_579_1240_251_383} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve with equation \(y = - x ^ { 3 } + 2 x ^ { 2 } + 2\), which intersects the \(x\)-axis at the point \(A\) where \(x = \alpha\). To find an approximation to \(\alpha\), the iterative formula $$x _ { n + 1 } = \frac { 2 } { \left( x _ { n } \right) ^ { 2 } } + 2$$ is used.
  1. Taking \(x _ { 0 } = 2.5\), find the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give your answers to 3 decimal places where appropriate.
  2. Show that \(\alpha = 2.359\) correct to 3 decimal places.