Questions — Edexcel C4 (360 questions)

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Edexcel C4 Q1
  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
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
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
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
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
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
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
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 \(t = \frac { \pi } { 4 }\).
Edexcel C4 Specimen Q5
5. The vector equations of two straight lines are $$\begin{aligned} & \mathbf { r } = 5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and }
& \mathbf { r } = 2 \mathbf { i } - 11 \mathbf { j } + a \mathbf { k } + \mu ( - 3 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) . \end{aligned}$$ Given that the two lines intersect, find
  1. the coordinates of the point of intersection,
  2. the value of the constant \(a\),
  3. the acute angle between the two lines.
Edexcel C4 Specimen Q6
6. Given that $$\frac { 11 x - 1 } { ( 1 - x ) ^ { 2 } ( 2 + 3 x ) } \equiv \frac { A } { ( 1 - x ) ^ { 2 } } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 2 + 3 x ) }$$
  1. find the values of \(A , B\) and \(C\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 11 x - 1 } { ( 1 - x ) ^ { 2 } ( 2 + 3 x ) } \mathrm { d } x\), giving your answer in the form \(k + \ln a\), where \(k\) is an integer and \(a\) is a simplified fraction.
Edexcel C4 Specimen Q7
7. (a) Given that \(u = \frac { x } { 2 } - \frac { 1 } { 8 } \sin 4 x\), show that \(\frac { \mathrm { d } u } { \mathrm {~d} x } = \sin ^ { 2 } 2 x\). \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{0191bf56-a59e-44fe-af8c-bad796156f63-5_697_1239_587_367}
\end{figure} Figure 2 shows the finite region bounded by the curve \(y = x ^ { \frac { 1 } { 2 } } \sin 2 x\), the line \(x = \frac { \pi } { 4 }\) and the \(x\)-axis. This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
(b) Using the result in part (a), or otherwise, find the exact value of the volume generated.
(8)
Edexcel C4 Specimen Q8
8. A circular stain grows in such a way that the rate of increase of its radius is inversely proportional to the square of the radius. Given that the area of the stain at time \(t\) seconds is \(A \mathrm {~cm} ^ { 2 }\),
  1. show that \(\frac { \mathrm { d } A } { \mathrm {~d} t } \propto \frac { 1 } { \sqrt { A } }\).
    (6) Another stain, which is growing more quickly, has area \(S \mathrm {~cm} ^ { 2 }\) at time \(t\) seconds. It is given that $$\frac { \mathrm { d } S } { \mathrm {~d} t } = \frac { 2 \mathrm { e } ^ { 2 t } } { \sqrt { S } }$$ Given that, for this second stain, \(S = 9\) at time \(t = 0\),
  2. solve the differential equation to find the time at which \(S = 16\). Give your answer to 2 significant figures. \section*{END}
Edexcel C4 2009 January Q2
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a5579938-e202-4543-8513-6483ede49850-03_410_552_205_694} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve \(y = \frac { 3 } { \sqrt { } ( 1 + 4 x ) }\). The region \(R\) is bounded by the curve, the \(x\)-axis, and the lines \(x = 0\) and \(x = 2\), as shown shaded in Figure 1.
  1. Use integration to find the area of \(R\). The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis.
  2. Use integration to find the exact value of the volume of the solid formed.
Edexcel C4 2013 June Q2
  1. Use the binomial expansion to show that $$\left. \sqrt { ( } \frac { 1 + x } { 1 - x } \right) \approx 1 + x + \frac { 1 } { 2 } x ^ { 2 } , \quad | x | < 1$$
  2. Substitute \(x = \frac { 1 } { 26 }\) into $$\sqrt { \left( \frac { 1 + x } { 1 - x } \right) = 1 + x + \frac { 1 } { 2 } x ^ { 2 } }$$ to obtain an approximation to \(\sqrt { } 3\)
    Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers.
Edexcel C4 Specimen Q2
The curve \(C\) has equation $$13 x ^ { 2 } + 13 y ^ { 2 } - 10 x y = 52$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(x\) and \(y\), simplifying your answer.
(6)
Edexcel C4 Q1
  1. The function \(f\) is given by
$$f ( x ) = \frac { 3 ( x + 1 ) } { ( x + 2 ) ( x - 1 ) } , x \in \mathbb { R } , x \neq - 2 , x \neq 1$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence, or otherwise, prove that \(\mathrm { f } ^ { \prime } ( x ) < 0\) for all values of \(x\) in the domain.
Edexcel C4 Q2
2. The curve \(C\) is described by the parametric equations $$x = 3 \cos t , \quad y = \cos 2 t , \quad 0 \leq t \leq \pi .$$
  1. Find a cartesian equation of the curve \(C\).
  2. Draw a sketch of the curve \(C\).
Edexcel C4 Q3
3. Use the substitution \(x = \sin \theta\) to show that, for \(| x | \leq 1\), $$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { x } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } + c \text {, where } c \text { is an arbitrary constant. }$$
Edexcel C4 Q4
  1. A measure of the effective voltage, \(M\) volts, in an electrical circuit is given by
$$M ^ { 2 } = \int _ { 0 } ^ { 1 } V ^ { 2 } \mathrm {~d} t$$ where \(V\) volts is the voltage at time \(t\) seconds. Pairs of values of \(V\) and \(t\) are given in the following table.
\(t\)00.250.50.751
\(V\)- 4820737- 161- 29
\(V ^ { 2 }\)
Use the trapezium rule with five values of \(V ^ { 2 }\) to estimate the value of \(M\).
(6)
Edexcel C4 Q5
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{964070ca-a2c0-4935-8a5b-f1f656495f2e-3_771_1049_251_477}
\end{figure} Figure 1 shows part of the curve with equation \(y = 1 + \frac { 1 } { 2 \sqrt { x } }\). The shaded region \(R\), bounded by the curve, that \(x\)-axis and the lines \(x = 1\) and \(x = 4\), is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Using integration, show that the volume of the solid generated is \(\pi \left( 5 + \frac { 1 } { 2 } \ln 2 \right)\).
(8)
Edexcel C4 Q6
6. Liquid is poured into a container at a constant rate of \(30 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds liquid is leaking from the container at a rate of \(\frac { 2 } { 15 } V \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\), where \(V \mathrm {~cm} ^ { 3 }\) is the volume of liquid in the container at that time.
  1. Show that $$- 15 \frac { \mathrm {~d} V } { \mathrm {~d} t } = 2 V - 450$$ Given that \(V = 1000\) when \(t = 0\),
  2. find the solution of the differential equation, in the form \(V = \mathrm { f } ( t )\).
  3. Find the limiting value of \(V\) as \(t \rightarrow \infty\).
Edexcel C4 Q7
7. The curve \(C\) has equation \(y = \frac { x } { 4 + x ^ { 2 } }\).
  1. Use calculus to find the coordinates of the turning points of \(C\). Using the result \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 x \left( x ^ { 2 } - 12 \right) } { \left( 4 + x ^ { 2 } \right) ^ { 3 } }\), or otherwise,
  2. determine the nature of each of the turning points.
  3. Sketch the curve \(C\).
Edexcel C4 Q8
8. (i) Given that \(\cos ( x + 30 ) ^ { \circ } = 3 \cos ( x - 30 ) ^ { \circ }\), prove that tan \(x ^ { \circ } = - \frac { \sqrt { 3 } } { 2 }\).
(ii) (a) Prove that \(\frac { 1 - \cos 2 \theta } { \sin 2 \theta } \equiv \tan \theta\).
(b) Verify that \(\theta = 180 ^ { \circ }\) is a solution of the equation \(\sin 2 \theta = 2 - 2 \cos 2 \theta\).
(c) Using the result in part (a), or otherwise, find the other two solutions, \(0 < \theta < 360 ^ { \circ }\), of the equation using \(\sin 2 \theta = 2 - 2 \cos 2 \theta\).
Edexcel C4 Q9
9. The equations of the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) ,
l _ { 2 } : & \mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ) , \end{array}$$ where \(\lambda\) and \(\mu\) are parameters.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of \(Q\), their point of intersection.
  2. Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\). The point \(P\) with \(x\)-coordinate 3 lies on the line \(l _ { 1 }\) and the point \(R\) with \(x\)-coordinate 4 lies on the line \(l _ { 2 }\).
  3. Find, in its simplest form, the exact area of the triangle \(P Q R\). END
Edexcel C4 Q1
  1. Use integration by parts to find the exact value of \(\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x \mathrm {~d} x\).
    (6)
  2. Fluid flows out of a cylindrical tank with constant cross section. At time \(t\) minutes, \(t \geq 0\), the volume of fluid remaining in the tank is \(V \mathrm {~m} ^ { 3 }\). The rate at which the fluid flows, in \(\mathrm { m } ^ { 3 } \mathrm {~min} ^ { - 1 }\), is proportional to the square root of \(V\).
    1. Show that the depth \(h\) metres of fluid in the tank satisfies the differential equation
    $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - k \sqrt { } h , \quad \text { where } k \text { is a positive constant. }$$
  3. Show that the general solution of the differential equation may be written as $$h = ( A - B t ) ^ { 2 } , \quad \text { where } A \text { and } B \text { are constants. }$$ Given that at time \(t = 0\) the depth of fluid in the tank is 1 m , and that 5 minutes later the depth of fluid has reduced to 0.5 m ,
  4. find the time, \(T\) minutes, which it takes for the tank to empty.
  5. Find the depth of water in the tank at time \(0.5 T\) minutes.