Edexcel C4 (Core Mathematics 4)

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. Use the trapezium rule with five values of \(V ^ { 2 }\) to estimate the value of \(M\). \begin{figure}[h] \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 \(\pi\).

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 \(\ln 16\).

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.

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] \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 \(x = \frac { \pi } { 4 }\).

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 \(A B\).

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.

9. The following is a table of values for \(y = \sqrt { } ( 1 + \sin x )\), where \(x\) is in radians.

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 $$I = \int _ { 0 } ^ { 2 } \sqrt { } ( 1 + \sin x ) \mathrm { d } x$$

10. \begin{figure}[h] \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]

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]

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.

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 \(A B\).

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.

15. \begin{figure}[h] \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]

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.

   \(t\)	0	5	10	15	20	25
   \(\boldsymbol { v }\)		1.56	7.23	17.36

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.

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 \(15500 \mathrm {~m} ^ { 3 }\).

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]

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).

20. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { x ^ { 2 } + 1 } { ( 1 + x ) ( 3 - x ) } , 0 \leq x < 3$$

1. Given that \(\mathrm { f } ( x ) = A + \frac { B } { 1 + x } + \frac { C } { 3 - x }\), find the values of the constants \(A , B\) and \(C\). The finite region \(R\), shown in Fig. 1, is bounded by the curve with equation \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).
2. Find the area of \(R\), giving your answer in the form \(p + q \ln r\), where \(p , q\) and \(r\) are rational constants to be found.
   21. (a) Prove that, when \(x = \frac { 1 } { 15 }\), the value of \(( 1 + 5 x ) ^ { - \frac { 1 } { 2 } }\) is exactly equal to \(\sin 60 ^ { \circ }\).
3. Expand \(( 1 + 5 x ) ^ { - \frac { 1 } { 2 } } , | x | < 0.2\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each term.
4. Use your answer to part (b) to find an approximation for \(\sin 60 ^ { \circ }\).
5. Find the difference between the exact value of \(\sin 60 ^ { \circ }\) and the approximation in part (c).
   22. A curve is given parametrically by the equations $$x = 5 \cos t , \quad y = - 2 + 4 \sin t , \quad 0 \leq t < 2 \pi$$
6. Find the coordinates of all the points at which \(C\) intersects the coordinate axes, giving your answers in surd form where appropriate.
7. Sketch the graph at \(C\).
   \(P\) is the point on \(C\) where \(t = \frac { 1 } { 6 } \pi\).
8. Show that the normal to \(C\) at \(P\) has equation $$8 \sqrt { 3 } y = 10 x - 25 \sqrt { 3 }$$ [P3 June 2002 Question 6]
   23. A Pancho car has value \(\pounds V\) at time \(t\) years. A model for \(V\) assumes that the rate of decrease of \(V\) at time \(t\) is proportional to \(V\).
9. By forming and solving an appropriate differential equation, show that \(V = A \mathrm { e } ^ { - k t }\), where \(A\) and \(k\) are positive constants. The value of a new Pancho car is \(\pounds 20000\), and when it is 3 years old its value is \(\pounds 11000\).
10. Find, to the nearest \(\pounds 100\), an estimate for the value of the Pancho when it is 10 years old. A Pancho car is regarded as 'scrap' when its value falls below \(\pounds 500\).
11. Find the approximate age of the Pancho when it becomes 'scrap'.
    [0pt] [P3 June 2002 Question 7]
    24. Referred to an origin \(O\), the points \(A , B\) and \(C\) have position vectors \(( 9 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) , ( 6 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k } )\) and ( \(3 \mathbf { i } + p \mathbf { j } + q \mathbf { k }\) ) respectively, where \(p\) and \(q\) are constants.
12. Find, in vector form, an equation of the line \(l\) which passes through \(A\) and \(B\). Given that \(C\) lies on \(l\),
13. find the value of \(p\) and the value of \(q\),
14. calculate, in degrees, the acute angle between \(O C\) and \(A B\). The point \(D\) lies on \(A B\) and is such that \(O D\) is perpendicular to \(A B\).
15. Find the position vector of \(D\).
    25. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-16_504_940_1213_584}
    \end{figure} Figure 2 shows part of the curve with equation \(y = x ^ { 2 } + 2\).
    The finite region \(R\) is bounded by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 2\).
16. Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\).
17. State, with a reason, whether your answer in part (a) is an under-estimate or over-estimate of the area of \(R\).
18. Using integration, find the volume of the solid generated when \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in terms of \(\pi\).
    26. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-17_723_987_287_655}
    \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)
    [0pt] [P2 January 2003 Question 4]
    27. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-18_757_1150_285_494}
    \end{figure} Figure 2 shows the cross-section of a road tunnel and its concrete surround. The curved section of the tunnel is modelled by the curve with equation \(y = 8 \sqrt { \left( \sin \frac { \pi x } { 10 } \right) }\), in the interval \(0 \leq x \leq 10\). The concrete surround is represented by the shaded area bounded by the curve, the \(x\)-axis and the lines \(x = - 2 , x = 12\) and \(y = 10\). The units on both axes are metres.
19. Using this model, copy and complete the table below, giving the values of \(y\) to 2 decimal places.
20. Use the trapezium rule, with all the values from your table, to calculate an estimate for the value of \(I\).
21. Use integration to calculate the exact value of \(I\).
22. Verify that the answer obtained by the trapezium rule is within 3\% of the exact value.
    47. When \(( 1 + a x ) ^ { n }\) is expanded as a series in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 2 }\) are - 6 and 27 respectively.
23. Find the value of \(a\) and the value of \(n\).
24. Find the coefficient of \(x ^ { 3 }\).
25. State the set of values of \(x\) for which the expansion is valid.
    48. The curve \(C\) has equation \(5 x ^ { 2 } + 2 x y - 3 y ^ { 2 } + 3 = 0\). The point \(P\) on the curve \(C\) has coordinates \(( 1,2 )\).
26. Find the gradient of the curve at \(P\).
27. Find the equation of the normal to the curve \(C\) at \(P\), in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
    49. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-32_545_824_850_837}
    \end{figure} Figure 1 shows a cross-section \(R\) of a dam. The line \(A C\) is the vertical face of the dam, \(A B\) is the horizontal base and the curve \(B C\) is the profile. Taking \(x\) and \(y\) to be the horizontal and vertical axes, then \(A , B\) and \(C\) have coordinates \(( 0,0 ) , \left( 3 \pi ^ { 2 } , 0 \right)\) and \(( 0,30 )\) respectively. The area of the cross-section is to be calculated. Initially the profile \(B C\) is approximated by a straight line.
28. Find an estimate for the area of the cross-section \(R\) using this approximation. The profile \(B C\) is actually described by the parametric equations. $$x = 16 t ^ { 2 } - \pi ^ { 2 } , \quad y = 30 \sin 2 t , \quad \frac { \pi } { 4 } \leq t \leq \frac { \pi } { 2 } .$$
29. Find the exact area of the cross-section \(R\).
30. Calculate the percentage error in the estimate of the area of the cross-section \(R\) that you found in part (a).
    50. (a) Express \(\frac { 13 - 2 x } { ( 2 x - 3 ) ( x + 1 ) }\) in partial fractions.
31. Given that \(y = 4\) at \(x = 2\), use your answer to part (a) to find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ( 13 - 2 x ) } { ( 2 x - 3 ) ( x + 1 ) } , \quad x > 1.5$$ Express your answer in the form \(y = \mathrm { f } ( x )\).
    [0pt] [P3 January 2004 Question 6]
    51. The curve \(C\) has equation \(y = \frac { x } { 4 + x ^ { 2 } }\).
32. 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,
33. determine the nature of each of the turning points.
34. Sketch the curve \(C\).
    52. 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.
35. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of \(Q\), their point of intersection.
36. 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 \(l _ { 2 }\).
37. Find, in its simplest form, the exact area of the triangle \(P Q R\).
    [0pt] [P3 January 2004 Question 8]
    53. Figure 1
    \includegraphics[max width=\textwidth, alt={}, center]{615ec68b-3a32-4309-bb54-acf39ed09f96-35_568_886_335_640} Figure 1 shows parts of the curve \(C\) with equation $$y = \frac { x + 2 } { \sqrt { } x } .$$ The shaded region \(R\) is bounded by \(C\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\). This region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid \(S\).
38. Find, by integration, the exact volume of \(S\). The solid \(S\) is used to model a wooden support with a circular base and a circular top.
39. Show that the base and the top have the same radius. Given that the actual radius of the base is 6 cm ,
40. show that the volume of the wooden support is approximately \(630 \mathrm {~cm} ^ { 3 }\).
    54. \(\mathrm { f } ( x ) = x + \frac { \mathrm { e } ^ { x } } { 5 } , \quad x \in \mathbb { R }\).
41. Find \(\mathrm { f } ^ { \prime } ( x )\). The curve \(C\), with equation \(y = \mathrm { f } ( x )\), crosses the \(y\)-axis at the point \(A\).
42. Find an equation for the tangent to \(C\) at \(A\).
43. Complete the table, giving the values of \(\sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) }\) to 2 decimal places.

    \(x\)	0	0.5	1	1.5	2
    \(\sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) }\)	0.45	0.91

44. Use the trapezium rule, with all the values from your table, to find an approximation for the value of $$\int _ { 0 } ^ { 2 } \sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) } \mathrm { d } x$$
    1. The circle \(C\) has centre \(( 5,13 )\) and touches the \(x\)-axis.
    2. Find an equation of \(C\) in terms of \(x\) and \(y\).
    3. Find an equation of the tangent to \(C\) at the point (10, 1), giving your answer in the form \(a y + b x + c = 0\), where \(a , b\) and \(c\) are integers.
    4. Use the substitution \(u = 1 + \sin x\) and integration to show that
    $$\int \sin x \cos x ( 1 + \sin x ) ^ { 5 } \mathrm {~d} x = \frac { 1 } { 42 } ( 1 + \sin x ) ^ { 6 } [ 6 \sin x - 1 ] + \text { constant. }$$
    1. Given that
    $$\frac { 3 + 5 x } { ( 1 + 3 x ) ( 1 - x ) } \equiv \frac { A } { 1 + 3 x } + \frac { B } { 1 - x } ,$$
45. find the values of the constants \(A\) and \(B\).
46. Hence, or otherwise, find the series expansion in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), of $$\frac { 3 + 5 x } { ( 1 + 3 x ) ( 1 - x ) } .$$
47. State, with a reason, whether your series expansion in part (b) is valid for \(x = \frac { 1 } { 2 }\).
    [0pt] [P3 June 2004 Question 5]
    58. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-38_528_1028_300_311}
    \end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = 3 t \sin t , y = 2 \sec t , \quad 0 \leq t < \frac { \pi } { 2 } .$$ The point \(P ( a , 4 )\) lies on \(C\).
48. Find the exact value of \(a\). The region \(R\) is enclosed by \(C\), the axes and the line \(x = a\) as shown in Fig. 1 .
49. Show that the area of \(R\) is given by $$6 \int _ { 0 } ^ { \frac { \pi } { 3 } } ( \tan t + t ) \mathrm { d } t .$$
50. Find the exact value of the area of \(R\).
    59. A drop of oil is modelled as a circle of radius \(r\). At time \(t\) $$r = 4 \left( 1 - \mathrm { e } ^ { - \lambda t } \right) , \quad t > 0$$ where \(\lambda\) is a positive constant.
51. Show that the area \(A\) of the circle satisfies $$\frac { \mathrm { d } A } { \mathrm {~d} t } = 32 \pi \lambda \left( \mathrm { e } ^ { - \lambda t } - \mathrm { e } ^ { - 2 \lambda t } \right)$$ In an alternative model of the drop of oil its area \(A\) at time \(t\) satisfies $$\frac { \mathrm { d } A } { \mathrm {~d} t } = \frac { A ^ { \frac { 3 } { 2 } } } { t ^ { 2 } } , \quad \quad t > 0$$ Given that the area of the drop is 1 at \(t = 1\),
52. find an expression for \(A\) in terms of t for this alternative model.
53. Show that, in the alternative model, the value of \(A\) cannot exceed 4 .
    60. Relative to a fixed origin \(O\), the vector equations of the two lines \(l _ { 1 }\) and \(l _ { 2 }\) are $$l _ { 1 } : \mathbf { r } = 9 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } + t ( - 8 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k } ) ,$$ and $$l _ { 2 } : \mathbf { r } = - 16 \mathbf { i } + \alpha \mathbf { j } + 10 \mathbf { k } + s ( \mathbf { i } - 4 \mathbf { j } + 9 \mathbf { k } ) ,$$ where \(\alpha\) is a constant.
    The two lines intersect at the point \(A\).
54. Find the value of \(\alpha\).
55. Find the position vector of the point \(A\).
56. Prove that the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(60 ^ { \circ }\). Point \(B\) lies on \(l _ { 1 }\) and point \(C\) lies on \(l _ { 2 }\). The triangle \(A B C\) is equilateral with sides of length \(14 \sqrt { } 2\).
57. Find one of the possible position vectors for the point \(B\) and the corresponding position vector for the point \(C\).