Questions — Edexcel C4 (386 questions)

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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 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 \(t = \frac { \pi } { 4 }\).
Edexcel C4 Specimen Q5
11 marks Standard +0.3
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
11 marks Standard +0.3
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
12 marks Challenging +1.2
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
13 marks Standard +0.3
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
9 marks Moderate -0.3
\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
9 marks Moderate -0.3
  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
6 marks Standard +0.3
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
6 marks Standard +0.3
  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
4 marks Moderate -0.3
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
6 marks Standard +0.3
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
6 marks Moderate -0.3
  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
8 marks Standard +0.3
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
11 marks Moderate -0.3
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
11 marks Standard +0.3
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
13 marks Standard +0.3
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
14 marks Standard +0.8
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
6 marks Moderate -0.8
  1. The following is a table of values for \(y = \sqrt { } ( 1 + \sin x )\), where \(x\) is in radians.
\(x\)00.511.52
\(y\)11.216\(p\)1.413\(q\)
  1. Find the value of \(p\) and the value of \(q\).
    (2)
  2. Use the trapezium rule and all the values of \(y\) in the completed table to obtain an estimate of \(I\), where $$I = \int _ { 0 } ^ { 2 } \sqrt { } ( 1 + \sin x ) \mathrm { d } x$$ (4)
Edexcel C4 Q2
7 marks Moderate -0.3
2. (a) Use integration by parts to find $$\int x \cos 2 x d x$$ (b) Prove that the answer to part (a) may be expressed as $$\frac { 1 } { 2 } \sin x ( 2 x \cos x - \sin x ) + C ,$$ where \(C\) is an arbitrary constant.
Edexcel C4 Q3
8 marks Standard +0.3
3. (a) Expand \(( 1 + 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each term.
(b) Hence, or otherwise, find the first three terms in the expansion of \(\frac { x + 4 } { ( 1 + 3 x ) ^ { 2 } }\) as a series in ascending powers of \(x\).
Edexcel C4 Q4
12 marks Standard +0.3
4. Relative to a fixed origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } - \mathbf { k }\), and the point \(B\) has position vector \(7 \mathbf { i } + 14 \mathbf { j } + 5 \mathbf { k }\).
  1. Find the vector \(\overrightarrow { A B }\).
  2. Calculate the cosine of \(\angle O A B\).
  3. Show that, for all values of \(\lambda\), the point P with position vector \(\lambda \mathbf { i } + 2 \lambda \mathbf { j } + ( 2 \lambda - 9 ) \mathbf { k }\) lies on the line through \(A\) and \(B\).
  4. Find the value of \(\lambda\) for which \(O P\) is perpendicular to \(A B\).
  5. Hence find the coordinates of the foot of the perpendicular from \(O\) to \(A B\).