Edexcel C34 (Core Mathematics 3 & 4) 2015 January

1. The curve \(C\) has equation

$$y = \frac { 3 x - 2 } { ( x - 2 ) ^ { 2 } } , \quad x \neq 2$$ The point \(P\) on \(C\) has \(x\) coordinate 3
Find an equation of the normal to \(C\) at the point \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

2. Solve, for \(0 \leqslant \theta < 2 \pi\), $$2 \cos 2 \theta = 5 - 13 \sin \theta$$ Give your answers in radians to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

3. The function \(g\) is defined by $$\mathrm { g } : x \mapsto | 8 - 2 x | , \quad x \in \mathbb { R } , \quad x \geqslant 0$$

1. Sketch the graph with equation \(y = \mathrm { g } ( x )\), showing the coordinates of the points where the graph cuts or meets the axes.
2. Solve the equation $$| 8 - 2 x | = x + 5$$ The function f is defined by $$\mathrm { f } : x \mapsto x ^ { 2 } - 3 x + 1 , \quad x \in \mathbb { R } , \quad 0 \leqslant x \leqslant 4$$
3. Find fg(5).
4. Find the range of f. You must make your method clear.

4. Use the substitution \(x = 2 \sin \theta\) to find the exact value of $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$

1. (a) Use the binomial expansion, in ascending powers of \(x\), of \(\frac { 1 } { \sqrt { } ( 1 - 2 x ) }\) to show that

$$\frac { 2 + 3 x } { \sqrt { } ( 1 - 2 x ) } \approx 2 + 5 x + 6 x ^ { 2 } , \quad | x | < 0.5$$ (b) Substitute \(x = \frac { 1 } { 20 }\) into $$\frac { 2 + 3 x } { \sqrt { } ( 1 - 2 x ) } = 2 + 5 x + 6 x ^ { 2 }$$ to obtain an approximation to \(\sqrt { 10 }\)
Give your answer as a fraction in its simplest form.

6. (i) Given \(x = \tan ^ { 2 } 4 y , 0 < y < \frac { \pi } { 8 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(x\). Write your answer in the form \(\frac { 1 } { A \left( x ^ { p } + x ^ { q } \right) }\), where \(A , p\) and \(q\) are constants to
be found.
(ii) The volume \(V\) of a cube is increasing at a constant rate of \(2 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate at which the length of the edge of the cube is increasing when the volume of the cube is \(64 \mathrm {~cm} ^ { 3 }\).

7. (a) Given that $$2 \cos ( x + 30 ) ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ without using a calculator, show that $$\tan x ^ { \circ } = 3 \sqrt { 3 } - 4$$ (b) Hence or otherwise solve, for \(0 \leqslant \theta < 180\), $$2 \cos ( 2 \theta + 40 ) ^ { \circ } = \sin ( 2 \theta - 20 ) ^ { \circ }$$ Give your answers to one decimal place.

8. \begin{figure}[h] \end{figure} The value of Lin's car is modelled by the formula $$V = 18000 \mathrm { e } ^ { - 0.2 t } + 4000 \mathrm { e } ^ { - 0.1 t } + 1000 , \quad t \geqslant 0$$ where the value of the car is \(V\) pounds when the age of the car is \(t\) years.
A sketch of \(t\) against \(V\) is shown in Figure 1.

1. State the range of \(V\). According to this model,
2. find the rate at which the value of the car is decreasing when \(t = 10\) Give your answer in pounds per year.
3. Calculate the exact value of \(t\) when \(V = 15000\)

9. \begin{figure}[h] \end{figure} The curve \(C\) has parametric equations $$x = \ln ( t + 2 ) , \quad y = \frac { 4 } { t ^ { 2 } } \quad t > 0$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = \ln 3\) and \(x = \ln 5\)

1. Show that the area of \(R\) is given by the integral $$\int _ { 1 } ^ { 3 } \frac { 4 } { t ^ { 2 } ( t + 2 ) } \mathrm { d } t$$
2. Hence find an exact value for the area of \(R\). Write your answer in the form ( \(a + \ln b\) ), where \(a\) and \(b\) are rational numbers.
3. Find a cartesian equation of the curve \(C\) in the form \(y = \mathrm { f } ( x )\).

10. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 4 , \quad x > 0$$ Point \(A\) is the minimum turning point on the curve.

1. Show, by using calculus, that the \(x\) coordinate of point \(A\) is a solution of $$x = \frac { 6 } { 1 + \ln \left( x ^ { 2 } \right) }$$
2. Starting with \(x _ { 0 } = 2.27\), use the iteration $$x _ { n + 1 } = \frac { 6 } { 1 + \ln \left( x _ { n } ^ { 2 } \right) }$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
3. Use your answer to part (b) to deduce the coordinates of point \(A\) to one decimal place.

11. 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 }

12. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 4 , \quad x > 0$$ The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 3\)

1. Complete the table below with the value of \(y\) corresponding to \(x = 2\). Give your answer to 4 decimal places.

   \(x\)	1	1.5	2	2.5	3
   \(y\)	2	1.3041		0.9089	1.2958

2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(S\), giving your answer to 3 decimal places.
3. Use calculus to find the exact area of \(S\). Give your answer in the form \(\frac { a } { b } + \ln c\), where \(a , b\) and \(c\) are integers.
4. Hence calculate the percentage error in using your answer to part (b) to estimate the area of \(S\). Give your answer to one decimal place.
5. Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(S\).

14
- 6
- 13 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
1
4 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } p
- 7
4 \end{array} \right) + \mu \left( \begin{array} { l } q
2
1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) and \(q\) are constants. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular,

1. show that \(q = 3\) Given further that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at point \(X\), find
2. the value of \(p\),
3. the coordinates of \(X\). The point \(A\) lies on \(l _ { 1 }\) and has position vector \(\left( \begin{array} { r } 6
   - 2
   3 \end{array} \right)\)
   Given that point \(B\) also lies on \(l _ { 1 }\) and that \(A B = 2 A X\)
4. find the two possible position vectors of \(B\).
   12. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{03548211-79cb-4629-b6ca-aa9dfcc77a33-21_615_732_233_605} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 4 , \quad x > 0$$ The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 3\)
5. Complete the table below with the value of \(y\) corresponding to \(x = 2\). Give your answer to 4 decimal places.

   \(x\)	1	1.5	2	2.5	3
   \(y\)	2	1.3041		0.9089	1.2958

6. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(S\), giving your answer to 3 decimal places.
7. Use calculus to find the exact area of \(S\). Give your answer in the form \(\frac { a } { b } + \ln c\), where \(a , b\) and \(c\) are integers.
8. Hence calculate the percentage error in using your answer to part (b) to estimate the area of \(S\). Give your answer to one decimal place.
9. Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(S\). 13. (a) Express \(10 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\) Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places. Alana models the height above the ground of a passenger on a Ferris wheel by the equation $$H = 12 - 10 \cos ( 30 t ) ^ { \circ } + 3 \sin ( 30 t ) ^ { \circ }$$ where the height of the passenger above the ground is \(H\) metres at time \(t\) minutes after the wheel starts turning.
   \includegraphics[max width=\textwidth, alt={}, center]{03548211-79cb-4629-b6ca-aa9dfcc77a33-23_419_567_516_1160}
10. Calculate
    1. the maximum value of \(H\) predicted by this model,
    2. the value of \(t\) when this maximum first occurs. Give each answer to 2 decimal places.
11. Calculate the value of \(t\) when the passenger is 18 m above the ground for the first time. Give your answer to 2 decimal places.
12. Determine the time taken for the Ferris wheel to complete two revolutions.