Questions C4 (1162 questions)

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Edexcel C4 Q4
  1. Show that the volume of the solid formed is \(\frac { 1 } { 4 } \pi ( \pi + 2 )\).
  2. Find a cartesian equation for the curve.
OCR MEI C4 Q6
6 Use the Insert provided for this question. The graph of \(y = \tan x\) is given on the Insert.
On this graph sketch the graph of \(y = \operatorname { cotx }\).
Show clearly where your graph crosses the graph of \(y = \tan x\) and indicate the asymptotes.
Edexcel C4 Q1
  1. Use the binomial theorem to expand
$$\sqrt { } ( 4 - 9 x ) , \quad | x | < \frac { 4 } { 9 } ,$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying each term.
Edexcel C4 Q3
3. (a) Express \(\frac { 5 x + 3 } { ( 2 x - 3 ) ( x + 2 ) }\) in partial fractions.
(b) Hence find the exact value of \(\int _ { 2 } ^ { 6 } \frac { 5 x + 3 } { ( 2 x - 3 ) ( x + 2 ) } \mathrm { d } x\), giving your answer as a single logarithm.
Edexcel C4 Q5
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5a77103e-73c1-4f32-af93-f2d5627e2290-02_467_783_317_303}
\end{figure} Figure 1 shows the graph of the curve with equation $$y = x \mathrm { e } ^ { 2 x } , \quad x \geq 0 .$$ The finite region \(R\) bounded by the lines \(x = 1\), the \(x\)-axis and the curve is shown shaded in Figure 1.
  1. Use integration to find the exact value of the area for \(R\).
  2. Complete the table with the values of \(y\) corresponding to \(x = 0.4\) and 0.8 .
    \(x\)00.20.40.60.81
    \(y = x \mathrm { e } ^ { 2 x }\)00.298361.992077.38906
  3. Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures.
Edexcel C4 Q6
6. A curve has parametric equations $$x = 2 \cot t , \quad y = 2 \sin ^ { 2 } t , \quad 0 < t \leq \frac { \pi } { 2 } .$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of the parameter \(t\).
  2. Find an equation of the tangent to the curve at the point where \(t = \frac { \pi } { 4 }\).
  3. Find a cartesian equation of the curve in the form \(y = \mathrm { f } ( x )\). State the domain on which the curve is defined.
Edexcel C4 Q7
7. The line \(l _ { 1 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { l } 3
1
2 \end{array} \right) + \lambda \left( \begin{array} { r } 1
- 1
4 \end{array} \right)$$ and the line \(l _ { 2 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } 0
4
- 2 \end{array} \right) + \mu \left( \begin{array} { r } 1
- 1
0 \end{array} \right) ,$$ where \(\lambda\) and \(\mu\) are parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(B\) and the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\).
  1. Find the coordinates of \(B\).
  2. Find the value of \(\cos \theta\), giving your answer as a simplified fraction. The point \(A\), which lies on \(l _ { 1 }\), has position vector \(\mathbf { a } = 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\).
    The point \(C\), which lies on \(l _ { 2 }\), has position vector \(\mathbf { c } = 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k }\).
    The point \(D\) is such that \(A B C D\) is a parallelogram.
  3. Show that \(| \overrightarrow { A B } | = | \overrightarrow { B C } |\).
  4. Find the position vector of the point \(D\).
Edexcel C4 Q8
8. Liquid is pouring into a container at a constant rate of \(20 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) and is leaking out at a rate proportional to the volume of the liquid already in the container.
  1. Explain why, at time \(t\) seconds, the volume, \(V \mathrm {~cm} ^ { 3 }\), of liquid in the container satisfies the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = 20 - k V$$ where \(k\) is a positive constant. The container is initially empty.
  2. By solving the differential equation, show that $$V = A + B \mathrm { e } ^ { - k t }$$ giving the values of \(A\) and \(B\) in terms of \(k\). Given also that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = 10\) when \(t = 5\),
  3. find the volume of liquid in the container at 10 s after the start. Materials required for examination
    Mathematical Formulae (Green) Items included with question papers
    Nil Paper Reference(s)
    6666 \section*{Advanced Level} \section*{Monday 23 January 2006 - Afternoon} Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
Edexcel C4 Q1
  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 \(M\). \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-01_738_1088_1316_380}
\end{figure} Figure 1 shows part of a curve \(C\) with equation \(y = x ^ { 2 } + 3\). The shaded region is bounded by \(C\), the \(x\)-axis and the lines \(x = 1\) and \(x = 3\). The shaded region is rotated \(360 ^ { \circ }\) about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer as an exact multiple of \(\pi\).
Edexcel C4 Q3
3. (a) Given that \(a ^ { x } = \mathrm { e } ^ { k x }\), where \(a\) and \(k\) are constants, \(a > 0\) and \(x \in \mathbb { R }\), prove that \(k = \ln a\).
(b) Hence, using the derivative of \(\mathrm { e } ^ { k x }\), prove that when \(y = 2 ^ { x }\), $$\frac { \mathrm { d } y } { \mathrm { dx } } = 2 ^ { x } \ln 2 .$$ (c) Hence deduce that the gradient of the curve with equation \(y = 2 ^ { x }\) at the point \(( 2,4 )\) is \(\ln 16\).
Edexcel C4 Q4
4. $$\mathrm { f } ( x ) = ( 1 + 3 x ) ^ { - 1 } , | x | < \frac { 1 } { 3 }$$
  1. Expand \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
  2. Hence show that, for small \(x\), $$\frac { 1 + x } { 1 + 3 x } \approx 1 - 2 x + 6 x ^ { 2 } - 18 x ^ { 3 }$$
  3. Taking a suitable value for \(x\), which should be stated, use the series expansion in part (b) to find an approximate value for \(\frac { 101 } { 103 }\), giving your answer to 5 decimal places.
Edexcel C4 Q5
5. (a) Use integration by parts to show that $$\int _ { 0 } ^ { \frac { \pi } { 4 } } x \sec ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 4 } \pi - \frac { 1 } { 2 } \ln 2$$ \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-03_793_1138_524_354}
\end{figure} The finite region \(R\), bounded by the equation \(y = x ^ { \frac { 1 } { 2 } } \sec x\), the line \(x = \frac { \pi } { 4 }\) and the \(x\)-axis is shown in Fig. 1. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
(b) Find the volume of the solid of revolution generated.
(c) Find the gradient of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } \sec x\) at the point where \(x = \frac { \pi } { 4 }\).
Edexcel C4 Q6
6. Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, \(l _ { 1 }\) and \(l _ { 2 }\), along which they travel are $$\begin{aligned} \mathbf { r } & = 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )
\text { and } \mathbf { r } & = 9 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + \mu ( 4 \mathbf { i } + \mathbf { j } - \mathbf { k } ) , \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalars.
  1. Show that the submarines are moving in perpendicular directions.
  2. Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\), find the position vector of \(A\). The point \(b\) has position vector \(10 \mathbf { j } - 11 \mathbf { k }\).
  3. Show that only one of the submarines passes through the point \(B\).
  4. Given that 1 unit on each coordinate axis represents 100 m , find, in km , the distance \(A B\).
Edexcel C4 Q7
7. In a chemical reaction two substances combine to form a third substance. At time \(t , t \geq 0\), the concentration of this third substance is \(x\) and the reaction is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 1 - 2 x ) ( 1 - 4 x ) , \text { where } k \text { is a positive constant. }$$
  1. Solve this differential equation and hence show that $$\ln \left| \frac { 1 - 2 x } { 1 - 4 x } \right| = 2 k t + c , \text { where } c \text { is an arbitrary constant. }$$
  2. Given that \(x = 0\) when \(t = 0\), find an expression for \(x\) in terms of \(k\) and \(t\).
  3. Find the limiting value of the concentration \(x\) as \(t\) becomes very large. \section*{8.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-05_716_1026_326_468}
    \end{figure} Part of the design of a stained glass window is shown in Fig. 2. The two loops enclose an area of blue glass. The remaining area within the rectangle \(A B C D\) is red glass. The loops are described by the curve with parametric equations $$x = 3 \cos t , \quad y = 9 \sin 2 t , \quad 0 \leq t < 2 \pi .$$
  4. Find the cartesian equation of the curve in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
  5. Show that the shaded area in Fig. 2, enclosed by the curve and the \(x\)-axis, is given by $$\int _ { 0 } ^ { \frac { \pi } { 2 } } A \sin 2 t \sin t \mathrm {~d} t , \text { stating the value of the constant } A \text {. }$$
  6. Find the value of this integral. The sides of the rectangle \(A B C D\), in Fig. 2, are the tangents to the curve that are parallel to the coordinate axes. Given that 1 unit on each axis represents 1 cm ,
  7. find the total area of the red glass.
Edexcel C4 Q9
9. 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. Use the trapezium rule and all the values of \(y\) in the completed table to obtain an estimate of \(I\), where $$I = \int _ { 0 } ^ { 2 } \sqrt { } ( 1 + \sin x ) \mathrm { d } x$$
Edexcel C4 Q10
10. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-07_624_828_335_584}
\end{figure} In Fig. 1, the curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x + \frac { 2 } { x ^ { 2 } } , \quad x > 0$$ The shaded region is bounded by \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 2\). The shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer in the form \(\pi ( a + \ln b )\), where \(a\) and \(b\) are constants.
[0pt] [P2 January 2002 Question 4]
Edexcel C4 Q11
11. Given that $$\frac { 10 ( 2 - 3 x ) } { ( 1 - 2 x ) ( 2 + x ) } \equiv \frac { A } { 1 - 2 x } + \frac { B } { 2 + x } ,$$
  1. find the values of the constants \(A\) and \(B\).
  2. Hence, or otherwise, find the series expansion in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), of \(\frac { 10 ( 2 - 3 x ) } { ( 1 - 2 x ) ( 2 + x ) }\), for \(| x | < \frac { 1 } { 2 }\).
    [0pt] [P3 January 2002 Question 3]
Edexcel C4 Q12
12. A radioactive isotope decays in such a way that the rate of change of the number \(N\) of radioactive atoms present after \(t\) days, is proportional to \(N\).
  1. Write down a differential equation relating \(N\) and \(t\).
  2. Show that the general solution may be written as \(N = A \mathrm { e } ^ { - k t }\), where \(A\) and \(k\) are positive constants. Initially the number of radioactive atoms present is \(7 \times 10 ^ { 18 }\) and 8 days later the number present is \(3 \times 10 ^ { 17 }\).
  3. Find the value of \(k\).
  4. Find the number of radioactive atoms present after a further 8 days.
Edexcel C4 Q13
13. 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 \(A B\).
Edexcel C4 Q14
14. (i) Use integration by parts to find the exact value of \(\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x \mathrm {~d} x\).
(ii) Use the substitution \(x = \sin \theta\) to show that, for \(| x | \leq 1\), \(\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { x } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } + c\), where \(c\) is an arbitrary constant.
Edexcel C4 Q15
15. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-10_573_1138_300_416}
\end{figure} A table top, in the shape of a parallelogram, is made from two types of wood. The design is shown in Fig. 1. The area inside the ellipse is made from one type of wood, and the surrounding area is made from a second type of wood. The ellipse has parametric equations, $$x = 5 \cos \theta , \quad y = 4 \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The parallelogram consists of four line segments, which are tangents to the ellipse at the points where \(\theta = \alpha , \theta = - \alpha , \theta = \pi - \alpha , \theta = - \pi + \alpha\).
  1. Find an equation of the tangent to the ellipse at ( \(5 \cos \alpha , 4 \sin \alpha\) ), and show that it can be written in the form $$5 y \sin \alpha + 4 x \cos \alpha = 20$$
  2. Find by integration the area enclosed by the ellipse.
  3. Hence show that the area enclosed between the ellipse and the parallelogram is $$\frac { 80 } { \sin 2 \alpha } - 20 \pi$$
  4. Given that \(0 < \alpha < \frac { \pi } { 4 }\), find the value of \(\alpha\) for which the areas of two types of wood are equal.
    [0pt] [P3 January 2002 Question 8]
Edexcel C4 Q16
16. The speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of a lorry at time \(t\) seconds is modelled by $$v = 5 \left( \mathrm { e } ^ { 0.1 t } - 1 \right) \sin ( 0.1 t ) , \quad 0 \leq t \leq 30 .$$
  1. Copy and complete the following table, showing the speed of the lorry at 5 second intervals. Use radian measure for \(0.1 t\) and give your values of \(v\) to 2 decimal places where appropriate.
    \(t\)0510152025
    \(\boldsymbol { v }\)1.567.2317.36
  2. Verify that, according to this model, the lorry is moving more slowly at \(t = 25\) than at \(t = 24.5\). The distance, \(s\) metres, travelled by the lorry during the first 25 seconds is given by \(s = \int _ { 0 } ^ { 25 } v \mathrm {~d} t\).
  3. Estimate \(s\) by using the trapezium rule with all the values from your table.
Edexcel C4 Q17
17. Figure 2
\includegraphics[max width=\textwidth, alt={}, center]{615ec68b-3a32-4309-bb54-acf39ed09f96-12_674_776_330_541} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac { 4 } { x - 3 } , x \neq 3\). The points \(A\) and \(B\) on the curve have \(x\)-coordinates 3.25 and 5 respectively.
  1. Write down the \(y\)-coordinates of \(A\) and \(B\).
  2. Show that an equation of \(C\) is \(\frac { 3 y + 4 } { y } , y \neq 0\).
    (1) The shaded region \(R\) is bounded by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis. The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis to form a solid shape \(S\).
  3. Find the volume of \(S\), giving your answer in the form \(\pi ( a + b \ln c )\), where \(a , b\) and \(c\) are integers. The solid shape \(S\) is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,
  4. show that the volume of the tower is approximately \(15500 \mathrm {~m} ^ { 3 }\).
Edexcel C4 Q18
18. (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.
[0pt] [P3 June 2002 Question 2]
Edexcel C4 Q19
19. The circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 16 y - 209 = 0\).
  1. Find the coordinates of the centre of \(C\) and the radius of \(C\). The point \(P ( x , y )\) lies on \(C\).
  2. Find, in terms of \(x\) and \(y\), the gradient of the tangent to \(C\) at \(P\).
  3. Hence or otherwise, find an equation of the tangent to \(C\) at the point (21, 8).