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Edexcel C4 2016 June Q8
15 marks Standard +0.3
8. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation $$\mathbf { r } = \left( \begin{array} { r } 8 \\ 1 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { r } - 5 \\ 4 \\ 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) lies on \(l _ { 1 }\) where \(\mu = 1\)
  1. Find the coordinates of \(A\). The point \(P\) has position vector \(\left( \begin{array} { l } 1 \\ 5 \\ 2 \end{array} \right)\).
    The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
  2. Write down a vector equation for the line \(l _ { 2 }\)
  3. Find the exact value of the distance \(A P\). Give your answer in the form \(k \sqrt { 2 }\), where \(k\) is a constant to be determined. The acute angle between \(A P\) and \(l _ { 2 }\) is \(\theta\).
  4. Find the value of \(\cos \theta\) A point \(E\) lies on the line \(l _ { 2 }\) Given that \(A P = P E\),
  5. find the area of triangle \(A P E\),
  6. find the coordinates of the two possible positions of \(E\).
Edexcel C4 2017 June Q1
8 marks Moderate -0.3
  1. The curve \(C\) has parametric equations
$$x = 3 t - 4 , y = 5 - \frac { 6 } { t } , \quad t > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) The point \(P\) lies on \(C\) where \(t = \frac { 1 } { 2 }\)
  2. Find the equation of the tangent to \(C\) at the point \(P\). Give your answer in the form \(y = p x + q\), where \(p\) and \(q\) are integers to be determined.
  3. Show that the cartesian equation for \(C\) can be written in the form $$y = \frac { a x + b } { x + 4 } , \quad x > - 4$$ where \(a\) and \(b\) are integers to be determined.
Edexcel C4 2017 June Q2
6 marks Standard +0.3
2. \(\quad \mathrm { f } ( x ) = ( 2 + k x ) ^ { - 3 } , \quad | k x | < 2\), where \(k\) is a positive constant The binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is $$A + B x + \frac { 243 } { 16 } x ^ { 2 }$$ where \(A\) and \(B\) are constants.
  1. Write down the value of \(A\).
  2. Find the value of \(k\).
  3. Find the value of \(B\).
Edexcel C4 2017 June Q3
12 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-08_560_1082_242_438} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \frac { 6 } { \left( \mathrm { e } ^ { x } + 2 \right) } , x \in \mathbb { R }\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(y\)-axis, the \(x\)-axis and the line with equation \(x = 1\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 6 } { \left( \mathrm { e } ^ { x } + 2 \right) }\)
\(x\)00.20.40.60.81
\(y\)21.718301.569811.419941.27165
  1. Complete the table above by giving the missing value of \(y\) to 5 decimal places.
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to find an estimate for the area of \(R\), giving your answer to 4 decimal places.
  3. Use the substitution \(u = \mathrm { e } ^ { x }\) to show that the area of \(R\) can be given by $$\int _ { a } ^ { b } \frac { 6 } { u ( u + 2 ) } \mathrm { d } u$$ where \(a\) and \(b\) are constants to be determined.
  4. Hence use calculus to find the exact area of \(R\). [Solutions based entirely on graphical or numerical methods are not acceptable.]
Edexcel C4 2017 June Q4
9 marks Standard +0.3
4. The curve \(C\) has equation $$4 x ^ { 2 } - y ^ { 3 } - 4 x y + 2 ^ { y } = 0$$ The point \(P\) with coordinates \(( - 2,4 )\) lies on \(C\).
  1. Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(P\). The normal to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\).
  2. Find the \(y\) coordinate of \(A\), giving your answer in the form \(p + q \ln 2\), where \(p\) and \(q\) are constants to be determined.
    (3)
Edexcel C4 2017 June Q5
7 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-16_589_540_248_705} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale The finite region \(S\), shown shaded in Figure 2, is bounded by the \(y\)-axis, the \(x\)-axis, the line with equation \(x = \ln 4\) and the curve with equation $$y = \mathrm { e } ^ { x } + 2 \mathrm { e } ^ { - x } , \quad x \geqslant 0$$ The region \(S\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Use integration to find the exact value of the volume of the solid generated. Give your answer in its simplest form.
[0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]
Edexcel C4 2017 June Q6
13 marks Standard +0.3
6. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 4 \\ 28 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ - 5 \\ 1 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5 \\ 3 \\ 1 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ 0 \\ - 4 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(X\).
  1. Find the coordinates of the point \(X\).
  2. Find the size of the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in degrees to 2 decimal places. The point \(A\) lies on \(l _ { 1 }\) and has position vector \(\left( \begin{array} { r } 2 \\ 18 \\ 6 \end{array} \right)\)
  3. Find the distance \(A X\), giving your answer as a surd in its simplest form. The point \(Y\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { Y A }\) is perpendicular to the line \(l _ { 1 }\)
  4. find the distance \(Y A\), giving your answer to one decimal place. The point \(B\) lies on \(l _ { 1 }\) where \(| \overrightarrow { A X } | = 2 | \overrightarrow { A B } |\).
  5. Find the two possible position vectors of \(B\).
Edexcel C4 2017 June Q7
8 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-24_835_1160_255_529} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a vertical cylindrical tank of height 200 cm containing water. Water is leaking from a hole \(P\) on the side of the tank. At time \(t\) minutes after the leaking starts, the height of water in the tank is \(h \mathrm {~cm}\). The height \(h \mathrm {~cm}\) of the water in the tank satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = k ( h - 9 ) ^ { \frac { 1 } { 2 } } , \quad 9 < h \leqslant 200$$ where \(k\) is a constant. Given that, when \(h = 130\), the height of the water is falling at a rate of 1.1 cm per minute,
  1. find the value of \(k\). Given that the tank was full of water when the leaking started,
  2. solve the differential equation with your value of \(k\), to find the value of \(t\) when \(h = 50\)
Edexcel C4 2017 June Q8
12 marks Standard +0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-28_721_714_255_616} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Diagram not drawn to scale Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3 \theta \sin \theta , \quad y = \sec ^ { 3 } \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P ( k , 8 )\) lies on \(C\), where \(k\) is a constant.
  1. Find the exact value of \(k\). The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = k\).
  2. Show that the area of \(R\) can be expressed in the form $$\lambda \int _ { \alpha } ^ { \beta } \left( \theta \sec ^ { 2 } \theta + \tan \theta \sec ^ { 2 } \theta \right) \mathrm { d } \theta$$ where \(\lambda , \alpha\) and \(\beta\) are constants to be determined.
  3. Hence use integration to find the exact value of the area of \(R\).
Edexcel C4 2018 June Q1
8 marks Standard +0.3
  1. (a) Find the binomial series expansion of
$$\sqrt { 4 - 9 x } , | x | < \frac { 4 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) Give each coefficient in its simplest form.
(b) Use the expansion from part (a), with a suitable value of \(x\), to find an approximate value for \(\sqrt { 310 }\) Show all your working and give your answer to 3 decimal places.
Edexcel C4 2018 June Q2
10 marks Standard +0.3
  1. The curve \(C\) has equation
$$x ^ { 2 } + x y + y ^ { 2 } - 4 x - 5 y + 1 = 0$$
  1. Use implicit differentiation to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Find the \(x\) coordinates of the two points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) Give exact answers in their simplest form.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C4 2018 June Q3
14 marks Standard +0.3
3. (i) Given that $$\frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \equiv \frac { A } { ( 2 x + 1 ) } + \frac { B } { ( 2 x + 1 ) ^ { 2 } } + \frac { C } { ( x + 3 ) }$$
  1. find the values of the constants \(A , B\) and \(C\).
  2. Hence find $$\int \frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \mathrm { d } x , \quad x > - \frac { 1 } { 2 }$$ (ii) Find $$\int \left( \mathrm { e } ^ { x } + 1 \right) ^ { 3 } \mathrm {~d} x$$ (iii) Using the substitution \(u ^ { 3 } = x\), or otherwise, find $$\int \frac { 1 } { 4 x + 5 x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x , \quad x > 0$$
Edexcel C4 2018 June Q4
6 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-12_978_1264_121_411} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A water container is made in the shape of a hollow inverted right circular cone with semi-vertical angle of \(30 ^ { \circ }\), as shown in Figure 1. The height of the container is 50 cm . When the depth of the water in the container is \(h \mathrm {~cm}\), the surface of the water has radius \(r \mathrm {~cm}\) and the volume of water is \(V \mathrm {~cm} ^ { 3 }\).
  1. Show that \(V = \frac { 1 } { 9 } \pi h ^ { 3 }\) [0pt] [You may assume the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.] Given that the volume of water in the container increases at a constant rate of \(200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\),
  2. find the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 15\) Give your answer in its simplest form in terms of \(\pi\).
Edexcel C4 2018 June Q5
7 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-16_938_1257_125_486} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 1 + t - 5 \sin t , \quad y = 2 - 4 \cos t , \quad - \pi \leqslant t \leqslant \pi$$ The point \(A\) lies on the curve \(C\). Given that the coordinates of \(A\) are ( \(k , 2\) ), where \(k > 0\)
  1. find the exact value of \(k\), giving your answer in a fully simplified form.
  2. Find the equation of the tangent to \(C\) at the point \(A\). Give your answer in the form \(y = p x + q\), where \(p\) and \(q\) are exact real values.
Edexcel C4 2018 June Q6
6 marks Standard +0.3
  1. Given that \(y = 2\) when \(x = - \frac { \pi } { 8 }\), solve the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } } { 3 \cos ^ { 2 } 2 x } \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel C4 2018 June Q7
15 marks Standard +0.8
7. The point \(A\) with coordinates ( \(- 3,7,2\) ) lies on a line \(l _ { 1 }\) The point \(B\) also lies on the line \(l _ { 1 }\) Given that \(\quad \overrightarrow { A B } = \left( \begin{array} { r } 4 \\ - 6 \\ 2 \end{array} \right)\),
  1. find the coordinates of point \(B\). The point \(P\) has coordinates ( \(9,1,8\) )
  2. Find the cosine of the angle \(P A B\), giving your answer as a simplified surd.
  3. Find the exact area of triangle \(P A B\), giving your answer in its simplest form. The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
  4. Find a vector equation for the line \(l _ { 2 }\) The point \(Q\) lies on the line \(l _ { 2 }\) Given that the line segment \(A P\) is perpendicular to the line segment \(B Q\),
  5. find the coordinates of the point \(Q\).
Edexcel C4 2018 June Q8
9 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-28_680_1266_118_482} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
  1. Find \(\int x \cos 4 x d x\) Figure 3 shows part of the curve with equation \(y = \sqrt { x } \sin 2 x , \quad x \geqslant 0\) The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \frac { \pi } { 4 }\) The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  2. Find the exact value of the volume of this solid of revolution, giving your answer in its simplest form.
    (Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-32_2630_1828_121_121}
Edexcel C4 Q1
6 marks Moderate -0.3
  1. Use the substitution \(u = 4 + 3 x ^ { 2 }\) to find the exact value of
$$\int _ { 0 } ^ { 2 } \frac { 2 x } { \left( 4 + 3 x ^ { 2 } \right) ^ { 2 } } d x$$
\includegraphics[max width=\textwidth, alt={}]{a5902f63-b19f-4e37-94b8-35c3b47ab9de-03_2546_1815_88_158}
\begin{center} \begin{tabular}{|l|l|} \hline \multirow[b]{2}{*}{\begin{tabular}{l}
Edexcel C4 Q3
13 marks Standard +0.3
3. \(f ( x ) = \frac { 1 + 14 x } { ( 1 - x ) ( 1 + 2 x ) } , \quad | x | < \frac { 1 } { 2 } .\)
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
    (3)
  2. Hence find the exact value of \(\int _ { \frac { 1 } { 6 } } ^ { \frac { 1 } { 3 } } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(\ln p\), where \(p\) is rational.
    (5)
  3. Use the binomial theorem to expand \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying each term.
    (5)
    \end{tabular}} & Leave blank
    \hline &
    \hline \end{tabular} \end{center}
    1. continued
    2. The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { c } 11 \\ 5 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { l } 4 \\ 2 \\ 4 \end{array} \right)\), where \(\lambda\) is a parameter.
    The line \(l _ { 2 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { c } 24 \\ 4 \\ 13 \end{array} \right) + \mu \left( \begin{array} { l } 7 \\ 1 \\ 5 \end{array} \right)\), where \(\mu\) is a parameter.
  4. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
  5. Find the coordinates of their point of intersection. Given that \(\theta\) is the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\),
  6. find the value of \(\cos \theta\). Give your answer in the form \(k \sqrt { } 3\), where \(k\) is a simplified fraction.
Edexcel C4 Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a5902f63-b19f-4e37-94b8-35c3b47ab9de-08_497_919_270_635}
\end{figure} The curve shown in Fig. 1 has parametric equations $$x = \cos t , y = \sin 2 t , \quad 0 \leq t < 2 \pi .$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of the parameter \(t\).
  2. Find the values of the parameter \(t\) at the points where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).
  3. Hence give the exact values of the coordinates of the points on the curve where the tangents are parallel to the \(x\)-axis.
  4. Show that a cartesian equation for the part of the curve where \(0 \leq t < \pi\) is $$y = 2 x \sqrt { } \left( 1 - x ^ { 2 } \right)$$
  5. Write down a cartesian equation for the part of the curve where \(\pi \leq t < 2 \pi\).
    5. continued
Edexcel C4 Q6
13 marks
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{a5902f63-b19f-4e37-94b8-35c3b47ab9de-10_579_1326_268_423}
\end{figure} Figure 2 shows the curve with equation $$y = x ^ { 2 } \sin \left( \frac { 1 } { 2 } x \right) , \quad 0 < x \leq 2 \pi .$$ The finite region \(R\) bounded by the line \(x = \pi\), the \(x\)-axis, and the curve is shown shaded in Fig 2.
  1. Find the exact value of the area of \(R\), by integration. Give your answer in terms of \(\pi\). The table shows corresponding values of \(x\) and \(y\).
    \(x\)\(\pi\)\(\frac { 5 \pi } { 4 }\)\(\frac { 3 \pi } { 2 }\)\(\frac { 7 \pi } { 4 }\)\(2 \pi\)
    \(y\)9.869614.24715.702\(G\)0
  2. Find the value of \(G\).
  3. Use the trapezium rule with values of \(x ^ { 2 } \sin \left( \frac { 1 } { 2 } x \right)\)
    1. at \(x = \pi , x = \frac { 3 \pi } { 2 }\) and \(x = 2 \pi\) to find an approximate value for the area \(R\), giving your answer to 4 significant figures,
    2. at \(x = \pi , x = \frac { 5 \pi } { 4 } , x = \frac { 3 \pi } { 2 } , x = \frac { 7 \pi } { 4 }\) and \(x = 2 \pi\) to find an improved approximation for the area \(R\), giving your answer to 4 significant figures.
      6. continued
Edexcel C4 Q7
13 marks Standard +0.3
7. In an experiment a scientist considered the loss of mass of a collection of picked leaves. The mass \(M\) grams of a single leaf was measured at times \(t\) days after the leaf was picked. The scientist attempted to find a relationship between \(M\) and \(t\). In a preliminary model she assumed that the rate of loss of mass was proportional to the mass \(M\) grams of the leaf.
  1. Write down a differential equation for the rate of change of mass of the leaf, using this model.
  2. Show, by differentiation, that \(M = 10 ( 0.98 ) ^ { t }\) satisfies this differential equation. Further studies implied that the mass \(M\) grams of a certain leaf satisfied a modified differential equation $$10 \frac { \mathrm {~d} M } { \mathrm {~d} t } = - k ( 10 M - 1 )$$ where \(k\) is a positive constant and \(t \geq 0\).
    Given that the mass of this leaf at time \(t = 0\) is 10 grams, and that its mass at time \(t = 10\) is 8.5 grams,
  3. solve the modified differential equation (I) to find the mass of this leaf at time \(t = 15\).
    7. continued
Edexcel C4 Specimen Q1
5 marks Moderate -0.3
Use the binomial theorem to expand \(( 4 - 3 x ) ^ { - \frac { 1 } { 2 } }\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
Edexcel C4 Specimen Q3
8 marks Standard +0.3
3. Use the substitution \(x = \tan \theta\) to show that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x = \frac { \pi } { 8 } + \frac { 1 } { 4 }$$ (8)
Edexcel C4 Specimen Q4
9 marks Moderate -0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0191bf56-a59e-44fe-af8c-bad796156f63-3_458_1552_415_223}
\end{figure} Figure 1 shows part of the curve with parametric equations $$x = \tan t , \quad y = \sin 2 t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 } .$$
  1. Find the gradient of the curve at the point \(P\) where \(t = \frac { \pi } { 3 }\).
  2. Find an equation of the normal to the curve at \(P\).
  3. Find an equation of the normal to the curve at the point \(Q\) where \(t = \frac { \pi } { 4 }\).