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Edexcel C3 2018 June Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{42aff260-e734-48ff-a92a-674032cb0377-12_595_930_219_603} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \mathrm { e } ^ { - 2 x } + x ^ { 2 } - 3$$ The curve \(C\) crosses the \(y\)-axis at the point \(A\). The line \(l\) is the normal to \(C\) at the point \(A\).
  1. Find the equation of \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. The line \(l\) meets \(C\) again at the point \(B\), as shown in Figure 1 .
  2. Show that the \(x\) coordinate of \(B\) is a solution of $$x = \sqrt { 1 + \frac { 1 } { 2 } x - \mathrm { e } ^ { - 2 x } }$$ Using the iterative formula $$x _ { n + 1 } = \sqrt { 1 + \frac { 1 } { 2 } x _ { n } - \mathrm { e } ^ { - 2 x _ { n } } }$$ with \(x _ { 1 } = 1\)
  3. find \(x _ { 2 }\) and \(x _ { 3 }\) to 3 decimal places.
Edexcel C3 2018 June Q5
8 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{42aff260-e734-48ff-a92a-674032cb0377-16_561_848_214_699} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 5 - x | + 3 , \quad x \geqslant 0$$ Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly one root,
  1. state the set of possible values of \(k\).
  2. Solve the equation \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x + 10\) The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation \(y = 4 \mathrm { f } ( x - 1 )\). The vertex on the graph with equation \(y = 4 \mathrm { f } ( x - 1 )\) has coordinates \(( p , q )\).
  3. State the value of \(p\) and the value of \(q\).
Edexcel C3 2018 June Q6
11 marks Standard +0.3
  1. (i) Using the identity for \(\tan ( A \pm B )\), solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\),
$$\frac { \tan 2 x + \tan 32 ^ { \circ } } { 1 - \tan 2 x \tan 32 ^ { \circ } } = 5$$ Give your answers, in degrees, to 2 decimal places.
(ii) (a) Using the identity for \(\tan ( A \pm B )\), show that $$\tan \left( 3 \theta - 45 ^ { \circ } \right) \equiv \frac { \tan 3 \theta - 1 } { 1 + \tan 3 \theta } , \quad \theta \neq ( 60 n + 45 ) ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 < \theta < 180 ^ { \circ }\), $$( 1 + \tan 3 \theta ) \tan \left( \theta + 28 ^ { \circ } \right) = \tan 3 \theta - 1$$
Edexcel C3 2018 June Q7
9 marks Challenging +1.2
  1. The curve \(C\) has equation \(y = \frac { \ln \left( x ^ { 2 } + 1 \right) } { x ^ { 2 } + 1 } , \quad x \in \mathbb { R }\)
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a single fraction, simplifying your answer.
    2. Hence find the exact coordinates of the stationary points of \(C\).
Edexcel C3 2018 June Q8
7 marks Standard +0.3
  1. (a) By writing \(\sec \theta = \frac { 1 } { \cos \theta }\), show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \sec \theta ) = \sec \theta \tan \theta\) (b) Given that
$$x = \mathrm { e } ^ { \sec y } \quad x > \mathrm { e } , \quad 0 < y < \frac { \pi } { 2 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \sqrt { \mathrm {~g} ( x ) } } , \quad x > \mathrm { e }$$ where \(\mathrm { g } ( x )\) is a function of \(\ln x\).
Edexcel C3 2018 June Q9
9 marks Standard +0.3
  1. (a) Express \(\sin \theta - 2 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
Give the exact value of \(R\) and the value of \(\alpha\), in radians, to 3 decimal places. $$\mathrm { M } ( \theta ) = 40 + ( 3 \sin \theta - 6 \cos \theta ) ^ { 2 }$$ (b) Find
  1. the maximum value of \(\mathrm { M } ( \theta )\),
  2. the smallest value of \(\theta\), in the range \(0 < \theta \leqslant 2 \pi\), at which the maximum value of \(\mathrm { M } ( \theta )\) occurs. $$N ( \theta ) = \frac { 30 } { 5 + 2 ( \sin 2 \theta - 2 \cos 2 \theta ) ^ { 2 } }$$ (c) Find
  3. the maximum value of \(\mathrm { N } ( \theta )\),
  4. the largest value of \(\theta\), in the range \(0 < \theta \leqslant 2 \pi\), at which the maximum value of \(\mathrm { N } ( \theta )\) occurs.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
    END
Edexcel C3 Q1
4 marks Moderate -0.8
  1. Express
$$\frac { 3 x ^ { 2 } } { \left( 2 x ^ { 2 } + 7 x + 6 \right) } \times \frac { 7 ( 3 + 2 x ) } { 3 x ^ { 5 } }$$ as a single fraction in its simplest form.
Edexcel C3 Q2
5 marks Easy -1.2
2. The function \(f\) is defined by $$\mathrm { f } : x \mapsto 2 x , \quad x \in \mathbb { R }$$
  1. Find \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\). The function g is defined by $$\mathrm { g } : x \mapsto 3 x ^ { 2 } + 2 , \quad x \in \mathbb { R }$$
  2. Find \(\mathrm { gf } ^ { - 1 } ( x )\).
  3. State the range of \(\mathrm { gf } ^ { - 1 } ( x )\).
Edexcel C3 Q3
6 marks Easy -1.2
3. Find the exact solutions of
  1. \(\mathrm { e } ^ { 2 x + 3 } = 6\),
  2. \(\ln ( 3 x + 2 ) = 4\).
Edexcel C3 Q4
12 marks Moderate -0.3
4. Differentiate with respect to \(x\)
  1. \(x ^ { 3 } \mathrm { e } ^ { 3 x }\),
  2. \(\frac { 2 x } { \cos x }\),
  3. \(\tan ^ { 2 } x\). Given that \(x = \cos y ^ { 2 }\),
  4. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).
Edexcel C3 Q5
11 marks Moderate -0.5
5. (a) Using the formulae $$\begin{gathered} \sin ( A \pm B ) = \sin A \cos B \pm \cos A \sin B \\ \cos ( A \pm B ) = \cos A \cos B \mp \sin A \sin B \end{gathered}$$ show that
  1. \(\sin ( A + B ) - \sin ( A - B ) = 2 \cos A \sin B\),
  2. \(\cos ( A - B ) - \cos ( A + B ) = 2 \sin A \sin B\).
    (b) Use the above results to show that $$\frac { \sin ( A + B ) - \sin ( A - B ) } { \cos ( A - B ) - \cos ( A + B ) } = \cot A$$ Using the result of part (b) and the exact values of \(\sin 60 ^ { \circ }\) and \(\cos 60 ^ { \circ }\),
    (c) find an exact value for \(\cot 75 ^ { \circ }\) in its simplest form.
    5. continuedLeave blank
Edexcel C3 Q6
12 marks Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ddc10fc0-f3f2-4c5f-b152-eba68a21990f-08_871_1495_286_273}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve has a minimum point at \(( - 0.5 , - 2 )\) and a maximum point at \(( 0.4 , - 4 )\). The lines \(x = 1\), the \(x\)-axis and the \(y\)-axis are asymptotes of the curve, as shown in Fig. 1. On a separate diagram sketch the graphs of
  1. \(y = | \mathrm { f } ( x ) |\),
  2. \(y = \mathrm { f } ( x - 3 )\),
  3. \(y = \mathrm { f } ( | x | )\). In each case show clearly
    1. the coordinates of any points at which the curve has a maximum or minimum point,
    2. how the curve approaches the asymptotes of the curve.
      6. continued
Edexcel C3 Q7
13 marks Standard +0.3
7. (a) Sketch the curve with equation \(y = \ln x\).
(b) Show that the tangent to the curve with equation \(y = \ln x\) at the point ( \(\mathrm { e } , 1\) ) passes through the origin.
(c) Use your sketch to explain why the line \(y = m x\) cuts the curve \(y = \ln x\) between \(x = 1\) and \(x = \mathrm { e }\) if \(0 < m < \frac { 1 } { \mathrm { e } }\). Taking \(x _ { 0 } = 1.86\) and using the iteration \(x _ { n + 1 } = \mathrm { e } ^ { \frac { 1 } { 3 } x _ { n } }\),
(d) calculate \(x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 }\) and \(x _ { 5 }\), giving your answer to \(x _ { 5 }\) to 3 decimal places. The root of \(\ln x - \frac { 1 } { 3 } x = 0\) is \(\alpha\).
(e) By considering the change of sign of \(\ln x - \frac { 1 } { 3 } x\) over a suitable interval, show that your answer for \(x _ { 5 }\) is an accurate estimate of \(\alpha\), correct to 3 decimal places.
7. continuedLeave blank
Edexcel C3 Q8
12 marks Standard +0.3
  1. In a particular circuit the current, \(I\) amperes, is given by
$$I = 4 \sin \theta - 3 \cos \theta , \quad \theta > 0$$ where \(\theta\) is an angle related to the voltage. Given that \(I = R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 \leqslant \alpha < 360 ^ { \circ }\),
  1. find the value of \(R\), and the value of \(\alpha\) to 1 decimal place.
  2. Hence solve the equation \(4 \sin \theta - 3 \cos \theta = 3\) to find the values of \(\theta\) between 0 and \(360 ^ { \circ }\).
  3. Write down the greatest value for \(I\).
  4. Find the value of \(\theta\) between 0 and \(360 ^ { \circ }\) at which the greatest value of \(I\) occurs.
    8. continued
Edexcel C3 Specimen Q1
8 marks Moderate -0.8
  1. The function f is defined by
$$\mathrm { f } : x \mapsto | x - 2 | - 3 , x \in \mathbb { R }$$
  1. Solve the equation \(\mathrm { f } ( x ) = 1\). The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } - 4 x + 11 , x \geq 0$$
  2. Find the range of g .
  3. Find \(g f ( - 1 )\).
Edexcel C3 Specimen Q2
8 marks Moderate -0.3
2. \(\quad \mathrm { f } ( x ) = x ^ { 3 } - 2 x - 5\).
  1. Show that there is a root \(\alpha\) of \(\mathrm { f } ( x ) = 0\) for \(x\) in the interval \([ 2,3 ]\). The root \(\alpha\) is to be estimated using the iterative formula $$x _ { n + 1 } = \sqrt { \left( 2 + \frac { 5 } { x _ { n } } \right) } , \quad x _ { 0 } = 2$$
  2. Calculate the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 significant figures.
  3. Prove that, to 5 significant figures, \(\alpha\) is 2.0946.
Edexcel C3 Specimen Q3
10 marks Standard +0.3
3. (a) Using the identity for \(\cos ( A + B )\), prove that \(\cos \theta \equiv 1 - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)\).
(b) Prove that \(1 + \sin \theta - \cos \theta \equiv 2 \sin \left( \frac { 1 } { 2 } \theta \right) \left[ \cos \left( \frac { 1 } { 2 } \theta \right) + \sin \left( \frac { 1 } { 2 } \theta \right) \right]\).
(c) Hence, or otherwise, solve the equation $$1 + \sin \theta - \cos \theta = 0 , \quad 0 \leq \theta < 2 \pi$$
Edexcel C3 Specimen Q4
10 marks Standard +0.3
4. $$\mathrm { f } ( x ) = x + \frac { 3 } { x - 1 } - \frac { 12 } { x ^ { 2 } + 2 x - 3 } , x \in \mathbb { R } , x > 1$$
  1. Show that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 x + 3 } { x + 3 }\).
  2. Solve the equation \(\mathrm { f } ^ { \prime } ( x ) = \frac { 22 } { 25 }\).
Edexcel C3 Specimen Q5
12 marks Moderate -0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{937edb48-ef4c-4974-a571-60b0fded841b-3_394_680_982_680}
\end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve meets the \(x\)-axis at \(P ( p , 0 )\) and meets the \(y\)-axis at \(Q ( 0 , q )\).
  1. On separate diagrams, sketch the curve with equation
    1. \(y = | \mathrm { f } ( x ) |\),
    2. \(y = 3 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). In each case show, in terms of \(p\) or \(q\), the coordinates of points at which the curve meets the axes. Given that \(\mathrm { f } ( x ) = 3 \ln ( 2 x + 3 )\),
  2. state the exact value of \(q\),
  3. find the value of \(p\),
  4. find an equation for the tangent to the curve at \(P\).
Edexcel C3 Specimen Q6
13 marks Moderate -0.3
6. As a substance cools its temperature, \(T ^ { \circ } \mathrm { C }\), is related to the time ( \(t\) minutes) for which it has been cooling. The relationship is given by the equation $$T = 20 + 60 \mathrm { e } ^ { - 0.1 t } , t \geq 0$$
  1. Find the value of \(T\) when the substance started to cool.
  2. Explain why the temperature of the substance is always above \(20 ^ { \circ } \mathrm { C }\).
  3. Sketch the graph of \(T\) against \(t\).
  4. Find the value, to 2 significant figures, of \(t\) at the instant \(T = 60\).
  5. Find \(\frac { \mathrm { d } T } { \mathrm {~d} t }\).
  6. Hence find the value of \(T\) at which the temperature is decreasing at a rate of \(1.8 ^ { \circ } \mathrm { C }\) per minute.
Edexcel C3 Specimen Q7
14 marks Moderate -0.3
7. (i) Given that \(y = \tan x + 2 \cos x\), find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { \pi } { 4 }\).
(ii) Given that \(x = \tan \frac { 1 } { 2 } y\), prove that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 1 + x ^ { 2 } }\).
(iii) Given that \(y = \mathrm { e } ^ { - x } \sin 2 x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed in the form \(R \mathrm { e } ^ { - x } \cos ( 2 x + \alpha )\). Find, to 3 significant figures, the values of \(R\) and \(\alpha\), where \(0 < \alpha < \frac { \pi } { 2 }\).
Edexcel P4 2021 January Q1
7 marks Standard +0.3
  1. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( \frac { 1 } { 4 } - 5 x \right) ^ { \frac { 1 } { 2 } } \quad | x | < \frac { 1 } { 20 }$$ giving each coefficient in its simplest form. By substituting \(x = \frac { 1 } { 100 }\) into the answer for (a),
(b) find an approximation for \(\sqrt { 5 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers to be found.
Edexcel P4 2021 January Q2
5 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{216f5735-a7ad-4d70-9da9-ae1f098a97d9-04_511_506_264_721} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of parallelogram \(A B C D\).
Given that \(\overrightarrow { A B } = 6 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { B C } = 2 \mathbf { i } + 5 \mathbf { j } + 8 \mathbf { k }\)
  1. find the size of angle \(A B C\), giving your answer in degrees, to 2 decimal places.
  2. Find the area of parallelogram \(A B C D\), giving your answer to one decimal place.
Edexcel P4 2021 January Q3
2 marks Moderate -0.8
3. Prove by contradiction that there is no greatest odd integer.
Edexcel P4 2021 January Q4
7 marks Standard +0.3
4. The curve \(C\) is defined by the parametric equations $$x = \frac { 1 } { t } + 2 \quad y = \frac { 1 - 2 t } { 3 + t } \quad t > 0$$
  1. Show that the equation of \(C\) can be written in the form \(y = \mathrm { g } ( x )\) where g is the function $$\mathrm { g } ( x ) = \frac { a x + b } { c x + d } \quad x > k$$ where \(a , b , c , d\) and \(k\) are integers to be found.
  2. Hence, or otherwise, state the range of g .
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