Questions — Edexcel (10514 questions)

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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.
  1. Sketch the curve with equation \(y = \ln x\).
  2. Show that the tangent to the curve with equation \(y = \ln x\) at the point ( \(\mathrm { e } , 1\) ) passes through the origin.
  3. 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 \(x _ { n + 1 } = \mathrm { e } ^ { \frac { 1 } { 3 } x _ { n } }\),
  4. 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 \(\alpha\).
  5. 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.
  1. Using the identity for \(\cos ( A + B )\), prove that \(\cos \theta \equiv 1 - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)\).
  2. Prove that \(1 + \sin \theta - \cos \theta \equiv 2 \sin \left( \frac { 1 } { 2 } \theta \right) \left[ \cos \left( \frac { 1 } { 2 } \theta \right) + \sin \left( \frac { 1 } { 2 } \theta \right) \right]\).
  3. 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.
  1. Given that \(y = \tan x + 2 \cos x\), find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { \pi } { 4 }\).
  2. Given that \(x = \tan \frac { 1 } { 2 } y\), prove that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 1 + x ^ { 2 } }\).
  3. 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. 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),
  2. 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 .
    ᄀ
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Edexcel P4 2021 January Q5
8 marks Standard +0.3
5. In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable.
Using the substitution \(u = 3 + \sqrt { 2 x - 1 }\) find the exact value of $$\int _ { 1 } ^ { 13 } \frac { 4 } { 3 + \sqrt { 2 x - 1 } } d x$$ giving your answer in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers to be found.
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Edexcel P4 2021 January Q6
9 marks Standard +0.8
6. A curve has equation $$4 y ^ { 2 } + 3 x = 6 y \mathrm { e } ^ { - 2 x }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The curve crosses the \(y\)-axis at the origin and at the point \(P\).
  2. Find the equation of the normal to the curve at \(P\), writing your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found.
Edexcel P4 2021 January Q7
7 marks Standard +0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{216f5735-a7ad-4d70-9da9-ae1f098a97d9-14_620_615_278_662} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure}
  1. Find \(\int \mathrm { e } ^ { 2 x } \sin x \mathrm {~d} x\) Figure 2 shows a sketch of part of the curve with equation $$y = \mathrm { e } ^ { 2 x } \sin x \quad x \geqslant 0$$ The finite region \(R\) is bounded by the curve and the \(x\)-axis and is shown shaded in Figure 2.
  2. Show that the exact area of \(R\) is \(\frac { \mathrm { e } ^ { 2 \pi } + 1 } { 5 }\) (Solutions relying on calculator technology are not acceptable.)
    Question 7 continue
Edexcel P4 2021 January Q8
6 marks Standard +0.8
8. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } - 1 \\ 5 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 1 \\ 5 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 2 \\ - 2 \\ - 5 \end{array} \right) + \mu \left( \begin{array} { r } 4 \\ - 3 \\ b \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(b\) is a constant.
Prove that for all values of \(b \neq 7\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
Edexcel P4 2021 January Q9
10 marks Challenging +1.2
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{216f5735-a7ad-4d70-9da9-ae1f098a97d9-20_714_714_269_616} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with parametric equations $$x = \tan \theta \quad y = 2 \sin 2 \theta \quad \theta \geqslant 0$$ The finite region, shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \sqrt { 3 }\) The region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  1. Show that the exact volume of this solid of revolution is given by $$\int _ { 0 } ^ { k } p ( 1 - \cos 2 \theta ) d \theta$$ where \(p\) and \(k\) are constants to be found.
  2. Hence find, by algebraic integration, the exact volume of this solid of revolution.
Edexcel P4 2021 January Q10
14 marks Standard +0.3
10.
  1. Write \(\frac { 1 } { ( H - 5 ) ( H + 3 ) }\) in partial fraction form. The depth of water in a storage tank is being monitored.
    The depth of water in the tank, \(H\) metres, is modelled by the differential equation $$\frac { \mathrm { d } H } { \mathrm {~d} t } = - \frac { ( H - 5 ) ( H + 3 ) } { 40 }$$ where \(t\) is the time, in days, from when monitoring began.
    Given that the initial depth of water in the tank was 13 m ,
  2. solve the differential equation to show that $$H = \frac { 10 + 3 \mathrm { e } ^ { - 0.2 t } } { 2 - \mathrm { e } ^ { - 0.2 t } }$$
  3. Hence find the time taken for the depth of water in the tank to fall to 8 m .
    (Solutions relying entirely on calculator technology are not acceptable.) According to the model, the depth of water in the tank will eventually fall to \(k\) metres.
  4. State the value of the constant \(k\).
    Leave
    blank
    Q10
Edexcel P4 2022 January Q1
7 marks Moderate -0.3
  1. Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$\frac { 2 } { \sqrt { 9 - 2 x } } \quad | x | < \frac { 9 } { 2 }$$ giving each coefficient as a simplified fraction. By substituting \(x = 1\) into the answer to part (a),
  2. find an approximation for \(\sqrt { 7 }\), giving your answer to 4 decimal places.
Edexcel P4 2022 January Q2
4 marks Standard +0.3
2. The curve \(C\) has parametric equations $$x = \frac { t ^ { 4 } } { 2 t + 1 } \quad y = \frac { t ^ { 3 } } { 2 t + 1 } \quad t > 0$$
  1. Write down \(\frac { x } { y }\) in terms of \(t\), giving your answer in simplest form.
  2. Hence show that all points on \(C\) satisfy the equation $$x ^ { 3 } - 2 x y ^ { 3 } - y ^ { 4 } = 0$$
Edexcel P4 2022 January Q3
10 marks Standard +0.3
3. The curve \(C\) has equation $$3 y ^ { 2 } - 11 x ^ { 2 } + 11 x y = 20 y - 36 x + 28$$
  1. Find, in simplest form, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P ( 4 , k )\), where \(k\) is a constant, lies on \(C\).
    Given that \(k < 0\)
  2. find the value of the gradient of \(C\) at \(P\)
Edexcel P4 2022 January Q4
9 marks Standard +0.3
4. $$\mathrm { f } ( x ) = \frac { 4 - 4 x } { x ( x - 2 ) ^ { 2 } } \quad x > 2$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\)
  3. Find $$\int _ { 3 } ^ { 5 } f ( x ) d x$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are rational numbers to be found.