Questions C3 (1200 questions)

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Edexcel C3 2006 January Q1
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5cd53af1-bac9-4ed9-ac45-59ad2e372423-02_689_766_276_594}
\end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x ) , - 5 \leqslant x \leqslant 5\).
The point \(M ( 2,4 )\) is the maximum turning point of the graph.
Sketch, on separate diagrams, the graphs of
  1. \(y = \mathrm { f } ( x ) + 3\),
  2. \(y = | \mathrm { f } ( x ) |\),
  3. \(y = \mathrm { f } ( | x | )\). Show on each graph the coordinates of any maximum turning points.
Edexcel C3 2006 January Q2
  1. Express
$$\frac { 2 x ^ { 2 } + 3 x } { ( 2 x + 3 ) ( x - 2 ) } - \frac { 6 } { x ^ { 2 } - x - 2 }$$ as a single fraction in its simplest form.
Edexcel C3 2006 January Q3
3. The point \(P\) lies on the curve with equation \(y = \ln \left( \frac { 1 } { 3 } x \right)\). The \(x\)-coordinate of \(P\) is 3 . Find an equation of the normal to the curve at the point \(P\) in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
(5)
Edexcel C3 2006 January Q4
4. (a) Differentiate with respect to \(x\)
  1. \(x ^ { 2 } \mathrm { e } ^ { 3 x + 2 }\),
  2. \(\frac { \cos \left( 2 x ^ { 3 } \right) } { 3 x }\).
    (b) Given that \(x = 4 \sin ( 2 y + 6 )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).
Edexcel C3 2006 January Q5
5. $$f ( x ) = 2 x ^ { 3 } - x - 4$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt { \left( \frac { 2 } { x } + \frac { 1 } { 2 } \right) }$$ The equation \(2 x ^ { 3 } - x - 4 = 0\) has a root between 1.35 and 1.4.
  2. Use the iteration formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 2 } { x _ { n } } + \frac { 1 } { 2 } \right)$$ with \(x _ { 0 } = 1.35\), to find, to 2 decimal places, the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\). The only real root of \(\mathrm { f } ( x ) = 0\) is \(\alpha\).
  3. By choosing a suitable interval, prove that \(\alpha = 1.392\), to 3 decimal places.
Edexcel C3 2006 January Q6
6. $$f ( x ) = 12 \cos x - 4 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x + \alpha )\), where \(R \geqslant 0\) and \(0 \leqslant \alpha \leqslant 90 ^ { \circ }\),
  1. find the value of \(R\) and the value of \(\alpha\).
  2. Hence solve the equation $$12 \cos x - 4 \sin x = 7$$ for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers to one decimal place.
    1. Write down the minimum value of \(12 \cos x - 4 \sin x\).
    2. Find, to 2 decimal places, the smallest positive value of \(x\) for which this minimum value occurs.
      \includegraphics[max width=\textwidth, alt={}, center]{5cd53af1-bac9-4ed9-ac45-59ad2e372423-09_60_35_2669_1853}
Edexcel C3 2006 January Q7
7. (a) Show that
  1. \(\frac { \cos 2 x } { \cos x + \sin x } \equiv \cos x - \sin x , \quad x \neq \left( n - \frac { 1 } { 4 } \right) \pi , n \in \mathbb { Z }\),
  2. \(\frac { 1 } { 2 } ( \cos 2 x - \sin 2 x ) \equiv \cos ^ { 2 } x - \cos x \sin x - \frac { 1 } { 2 }\).
    (b) Hence, or otherwise, show that the equation $$\cos \theta \left( \frac { \cos 2 \theta } { \cos \theta + \sin \theta } \right) = \frac { 1 } { 2 }$$ can be written as $$\sin 2 \theta = \cos 2 \theta$$ (c) Solve, for \(0 \leqslant \theta < 2 \pi\), $$\sin 2 \theta = \cos 2 \theta$$ giving your answers in terms of \(\pi\).
Edexcel C3 2006 January Q8
8. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \rightarrow 2 x + \ln 2 , & x \in \mathbb { R } ,
\mathrm {~g} : x \rightarrow \mathrm { e } ^ { 2 x } , & x \in \mathbb { R } . \end{array}$$
  1. Prove that the composite function gf is $$\operatorname { gf } : x \rightarrow 4 \mathrm { e } ^ { 4 x } , \quad x \in \mathbb { R }$$
  2. In the space provided on page 19, sketch the curve with equation \(y = \operatorname { gf } ( x )\), and show the coordinates of the point where the curve cuts the \(y\)-axis.
  3. Write down the range of gf.
  4. Find the value of \(x\) for which \(\frac { \mathrm { d } } { \mathrm { d } x } [ \operatorname { gf } ( x ) ] = 3\), giving your answer to 3 significant figures.
Edexcel C3 2007 January Q1
  1. (a) By writing \(\sin 3 \theta\) as \(\sin ( 2 \theta + \theta )\), show that
$$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$ (b) Given that \(\sin \theta = \frac { \sqrt { } 3 } { 4 }\), find the exact value of \(\sin 3 \theta\).
Edexcel C3 2007 January Q2
2. $$f ( x ) = 1 - \frac { 3 } { x + 2 } + \frac { 3 } { ( x + 2 ) ^ { 2 } } , x \neq - 2$$
  1. Show that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + x + 1 } { ( x + 2 ) ^ { 2 } } , x \neq - 2\).
  2. Show that \(x ^ { 2 } + x + 1 > 0\) for all values of \(x\).
  3. Show that \(\mathrm { f } ( x ) > 0\) for all values of \(x , x \neq - 2\).
Edexcel C3 2007 January Q3
3. The curve \(C\) has equation $$x = 2 \sin y .$$
  1. Show that the point \(P \left( \sqrt { } 2 , \frac { \pi } { 4 } \right)\) lies on \(C\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 2 } }\) at \(P\).
  3. Find an equation of the normal to \(C\) at \(P\). Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are exact constants.
Edexcel C3 2007 January Q4
4. (i) The curve \(C\) has equation $$y = \frac { x } { 9 + x ^ { 2 } }$$ Use calculus to find the coordinates of the turning points of \(C\).
(ii) Given that $$y = \left( 1 + \mathrm { e } ^ { 2 x } \right) ^ { \frac { 3 } { 2 } }$$ find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { 1 } { 2 } \ln 3\).
Edexcel C3 2007 January Q5
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a4ad749b-181b-4680-8771-94d9b581125a-07_865_926_301_516}
\end{figure} Figure 1 shows an oscilloscope screen. The curve shown on the screen satisfies the equation $$y = \sqrt { 3 } \cos x + \sin x$$
  1. Express the equation of the curve in the form \(y = R \sin ( x + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
  2. Find the values of \(x , 0 \leqslant x < 2 \pi\), for which \(y = 1\).
Edexcel C3 2007 January Q6
  1. The function \(f\) is defined by
$$\mathrm { f } : x \mapsto \ln ( 4 - 2 x ) , \quad x < 2 \quad \text { and } \quad x \in \mathbb { R } .$$
  1. Show that the inverse function of f is defined by $$\mathrm { f } ^ { - 1 } : x \mapsto 2 - \frac { 1 } { 2 } \mathrm { e } ^ { x }$$ and write down the domain of \(\mathrm { f } ^ { - 1 }\).
  2. Write down the range of \(\mathrm { f } ^ { - 1 }\).
  3. In the space provided on page 16, sketch the graph of \(y = f ^ { - 1 } ( x )\). State the coordinates of the points of intersection with the \(x\) and \(y\) axes. The graph of \(y = x + 2\) crosses the graph of \(y = f ^ { - 1 } ( x )\) at \(x = k\). The iterative formula $$x _ { n + 1 } = - \frac { 1 } { 2 } e ^ { x _ { n } } , x _ { 0 } = - 0.3$$ is used to find an approximate value for \(k\).
  4. Calculate the values of \(x _ { 1 }\) and \(x _ { 2 }\), giving your answers to 4 decimal places.
  5. Find the value of \(k\) to 3 decimal places.
Edexcel C3 2007 January Q7
7. $$f ( x ) = x ^ { 4 } - 4 x - 8$$
  1. Show that there is a root of \(\mathrm { f } ( x ) = 0\) in the interval \([ - 2 , - 1 ]\).
  2. Find the coordinates of the turning point on the graph of \(y = \mathrm { f } ( x )\).
  3. Given that \(\mathrm { f } ( x ) = ( x - 2 ) \left( x ^ { 3 } + a x ^ { 2 } + b x + c \right)\), find the values of the constants, \(a , b\) and \(c\).
  4. In the space provided on page 21, sketch the graph of \(y = \mathrm { f } ( x )\).
  5. Hence sketch the graph of \(y = | \mathrm { f } ( x ) |\).
Edexcel C3 2007 January Q8
    1. Prove that
$$\sec ^ { 2 } x - \operatorname { cosec } ^ { 2 } x \equiv \tan ^ { 2 } x - \cot ^ { 2 } x$$ (ii) Given that $$y = \arccos x , \quad - 1 \leqslant x \leqslant 1 \text { and } 0 \leqslant y \leqslant \pi ,$$
  1. express arcsin \(x\) in terms of \(y\).
  2. Hence evaluate \(\arccos x + \arcsin x\). Give your answer in terms of \(\pi\).
Edexcel C3 2008 January Q1
  1. Given that
$$\frac { 2 x ^ { 4 } - 3 x ^ { 2 } + x + 1 } { \left( x ^ { 2 } - 1 \right) } \equiv \left( a x ^ { 2 } + b x + c \right) + \frac { d x + e } { \left( x ^ { 2 } - 1 \right) }$$ find the values of the constants \(a , b , c , d\) and \(e\).
(4)
Edexcel C3 2008 January Q2
2. A curve \(C\) has equation $$y = \mathrm { e } ^ { 2 x } \tan x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$
  1. Show that the turning points on \(C\) occur where \(\tan x = - 1\).
  2. Find an equation of the tangent to \(C\) at the point where \(x = 0\).
Edexcel C3 2008 January Q3
3. $$\mathrm { f } ( x ) = \ln ( x + 2 ) - x + 1 , \quad x > - 2 , x \in \mathbb { R } .$$
  1. Show that there is a root of \(\mathrm { f } ( x ) = 0\) in the interval \(2 < x < 3\).
  2. Use the iterative formula $$x _ { n + 1 } = \ln \left( x _ { n } + 2 \right) + 1 , x _ { 0 } = 2.5$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) giving your answers to 5 decimal places.
  3. Show that \(x = 2.505\) is a root of \(\mathrm { f } ( x ) = 0\) correct to 3 decimal places.
Edexcel C3 2008 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a15db39c-d54b-4cf4-8da7-01f3db223415-05_735_1171_223_390} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve passes through the origin \(O\) and the points \(A ( 5,4 )\) and \(B ( - 5 , - 4 )\).
In separate diagrams, sketch the graph with equation
  1. \(y = | f ( x ) |\),
  2. \(y = \mathrm { f } ( | x | )\),
  3. \(y = 2 f ( x + 1 )\). On each sketch, show the coordinates of the points corresponding to \(A\) and \(B\).
Edexcel C3 2008 January Q5
5. The radioactive decay of a substance is given by $$R = 1000 \mathrm { e } ^ { - c t } , \quad t \geqslant 0 .$$ where \(R\) is the number of atoms at time \(t\) years and \(c\) is a positive constant.
  1. Find the number of atoms when the substance started to decay. It takes 5730 years for half of the substance to decay.
  2. Find the value of \(c\) to 3 significant figures.
  3. Calculate the number of atoms that will be left when \(t = 22920\).
  4. In the space provided on page 13, sketch the graph of \(R\) against \(t\).
Edexcel C3 2008 January Q6
6. (a) Use the double angle formulae and the identity $$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$ to obtain an expression for \(\cos 3 x\) in terms of powers of \(\cos x\) only.
(b) (i) Prove that $$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } \equiv 2 \sec x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$ (ii) Hence find, for \(0 < x < 2 \pi\), all the solutions of $$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } = 4$$
Edexcel C3 2008 January Q7
  1. A curve \(C\) has equation
$$y = 3 \sin 2 x + 4 \cos 2 x , - \pi \leqslant x \leqslant \pi$$ The point \(A ( 0,4 )\) lies on \(C\).
  1. Find an equation of the normal to the curve \(C\) at \(A\).
  2. Express \(y\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 3 significant figures.
  3. Find the coordinates of the points of intersection of the curve \(C\) with the \(x\)-axis. Give your answers to 2 decimal places.
Edexcel C3 2008 January Q8
  1. The functions \(f\) and \(g\) are defined by
$$\begin{aligned} & \mathrm { f } : x \mapsto 1 - 2 x ^ { 3 } , x \in \mathbb { R }
& \mathrm {~g} : x \mapsto \frac { 3 } { x } - 4 , x > 0 , x \in \mathbb { R } \end{aligned}$$
  1. Find the inverse function \(\mathrm { f } ^ { - 1 }\).
  2. Show that the composite function gf is $$\text { gf } : x \mapsto \frac { 8 x ^ { 3 } - 1 } { 1 - 2 x ^ { 3 } }$$
  3. Solve \(\operatorname { gf } ( x ) = 0\).
  4. Use calculus to find the coordinates of the stationary point on the graph of \(y = \operatorname { gf } ( x )\).
Edexcel C3 2009 January Q1
  1. (a) Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point where \(x = 2\) on the curve with equation
$$y = x ^ { 2 } \sqrt { } ( 5 x - 1 )$$ (b) Differentiate \(\frac { \sin 2 x } { x ^ { 2 } }\) with respect to \(x\).