Edexcel C34 (Core Mathematics 3 & 4) 2017 June

1. A curve \(C\) has equation

$$3 x ^ { 2 } + 2 x y - 2 y ^ { 2 } + 4 = 0$$ Find an equation for the tangent to \(C\) at the point ( 2,4 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
(6)

1. Use integration by parts to find the exact value of \(\int _ { 1 } ^ { \mathrm { e } } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x\)

Write your answer in the form \(a + \frac { b } { \mathrm { e } }\), where \(a\) and \(b\) are integers.

3. The function g is defined by $$g ( x ) = \frac { 6 x } { 2 x + 3 } \quad x > 0$$

1. Find the range of g .
2. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain.
3. Find the function \(\operatorname { gg } ( x )\), writing your answer as a single fraction in its simplest form.

4. $$f ( x ) = \frac { 27 } { ( 3 - 5 x ) ^ { 2 } } \quad | x | < \frac { 3 } { 5 }$$

1. Find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient in its simplest form.
   (5) Use your answer to part (a) to find the series expansion in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), of
2. \(g ( x ) = \frac { 27 } { ( 3 + 5 x ) ^ { 2 } } \quad | x | < \frac { 3 } { 5 }\)
3. \(\mathrm { h } ( x ) = \frac { 27 } { ( 3 - x ) ^ { 2 } } \quad | x | < 3\)

5. $$\frac { 6 - 5 x - 4 x ^ { 2 } } { ( 2 - x ) ( 1 + 2 x ) } \equiv A + \frac { B } { ( 2 - x ) } + \frac { C } { ( 1 + 2 x ) }$$

1. Find the values of the constants \(A , B\) and \(C\). $$f ( x ) = \frac { 6 - 5 x - 4 x ^ { 2 } } { ( 2 - x ) ( 1 + 2 x ) } \quad x > 2$$
2. Using part (a), find \(\mathrm { f } ^ { \prime } ( x )\).
3. Prove that \(\mathrm { f } ( x )\) is a decreasing function.

1. The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { r } 5
   - 2
   4 \end{array} \right) + \lambda \left( \begin{array} { r } 6
   3
   - 1 \end{array} \right)\), where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { r } 10
   5
   - 3 \end{array} \right) + \mu \left( \begin{array} { l } 3
   1
   2 \end{array} \right)\), where \(\mu\) is a scalar parameter.

Justify, giving reasons in each case, whether the lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, intersecting or skew.
(6)

1. (a) Prove that

$$\frac { 1 - \cos 2 x } { 1 + \cos 2 x } \equiv \tan ^ { 2 } x , \quad x \neq ( 2 n + 1 ) 90 ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), $$\frac { 2 - 2 \cos 2 \theta } { 1 + \cos 2 \theta } - 2 = 7 \sec \theta$$ Give your answers in degrees to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt { \frac { x } { x ^ { 2 } + 1 } } , \quad x \geqslant 0\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 7\)
The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { \frac { x } { x ^ { 2 } + 1 } }\)

1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for the area of \(R\), giving your answer to 3 decimal places. The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution.
2. Use calculus to find the exact volume of the solid of revolution formed. Write your answer in its simplest form.
   \includegraphics[max width=\textwidth, alt={}, center]{29b56d51-120a-4275-a761-8b8aed7bca54-24_2255_47_314_1979}

9. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of a triangle \(A B C\).
Given \(\overrightarrow { A B } = 2 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }\) and \(\overrightarrow { A C } = 5 \mathbf { i } - 6 \mathbf { j } + \mathbf { k }\),

1. find the size of angle \(C A B\), giving your answer in degrees to 2 decimal places,
2. find the area of triangle \(A B C\), giving your answer to 2 decimal places. Using your answer to part (b), or otherwise,
3. find the shortest distance from \(A\) to \(B C\), giving your answer to 2 decimal places.

1. (a) Write \(2 \sin \theta - \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha \leqslant 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to one decimal place.
   (3)

\begin{figure}[h] \end{figure} Figure 3 shows a sketch of the graph with equation \(y = 2 \sin \theta - \cos \theta , \quad 0 \leqslant \theta < 360 ^ { \circ }\)
(b) Sketch the graph with equation $$y = | 2 \sin \theta - \cos \theta | , \quad 0 \leqslant \theta < 360 ^ { \circ }$$ stating the coordinates of all points at which the graph meets or cuts the coordinate axes. The temperature of a warehouse is modelled by the equation $$f ( t ) = 5 + \left| 2 \sin ( 15 t ) ^ { \circ } - \cos ( 15 t ) ^ { \circ } \right| , \quad 0 \leqslant t < 24$$ where \(\mathrm { f } ( t )\) is the temperature of the warehouse in degrees Celsius and \(t\) is the time measured in hours from midnight. State
(c) (i) the maximum value of \(f ( t )\),
(ii) the largest value of \(t\), for \(0 \leqslant t < 24\), at which this maximum value occurs. Give your answer to one decimal place.

11. $$y = \left( 2 x ^ { 2 } - 3 \right) \tan \left( \frac { 1 } { 2 } x \right) , \quad 0 < x < \pi$$

1. Find the exact value of \(x\) when \(y = 0\) Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = \alpha\),
2. show that $$2 \alpha ^ { 2 } - 3 + 4 \alpha \sin \alpha = 0$$ The iterative formula $$x _ { n + 1 } = \frac { 3 } { \left( 2 x _ { n } + 4 \sin x _ { n } \right) }$$ can be used to find an approximation for \(\alpha\).
3. Taking \(x _ { 1 } = 0.7\), find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving each answer to 4 decimal places.
4. By choosing a suitable interval, show that \(\alpha = 0.7283\) to 4 decimal places.
   \includegraphics[max width=\textwidth, alt={}]{29b56d51-120a-4275-a761-8b8aed7bca54-38_2253_50_314_1977}

12. \begin{figure}[h] \end{figure} Figure 4 shows a right cylindrical water tank. The diameter of the circular cross section of the tank is 4 m and the height is 2.25 m . Water is flowing into the tank at a constant rate of \(0.4 \pi \mathrm {~m} ^ { 3 } \mathrm {~min} ^ { - 1 }\). There is a tap at a point \(T\) at the base of the tank. When the tap is open, water leaves the tank at a rate of \(0.2 \pi \sqrt { h } \mathrm {~m} ^ { 3 } \mathrm {~min} ^ { - 1 }\), where \(h\) is the height of the water in metres.

1. Show that at time \(t\) minutes after the tap has been opened, the height \(h \mathrm {~m}\) of the water in the tank satisfies the differential equation $$20 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 2 - \sqrt { h }$$ At the instant when the tap is opened, \(t = 0\) and \(h = 0.16\)
2. Use the differential equation to show that the time taken to fill the tank to a height of 2.25 m is given by $$\int _ { 0.16 } ^ { 2.25 } \frac { 20 } { 2 - \sqrt { h } } \mathrm {~d} h$$ Using the substitution \(h = ( 2 - x ) ^ { 2 }\), or otherwise,
3. find the time taken to fill the tank to a height of 2.25 m . Give your answer in minutes to the nearest minute.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

13. Figure 5 A colony of ants is being studied. The number of ants in the colony is modelled by the equation $$P = 200 - \frac { 160 \mathrm { e } ^ { 0.6 t } } { 15 + \mathrm { e } ^ { 0.8 t } } \quad t \in \mathbb { R } , t \geqslant 0$$ where \(P\) is the number of ants, measured in thousands, \(t\) years after the study started. A sketch of the graph of \(P\) against \(t\) is shown in Figure 5

1. Calculate the number of ants in the colony at the start of the study.
2. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) The population of ants initially decreases, reaching a minimum value after \(T\) years, as shown in Figure 5
3. Using your answer to part (b), calculate the value of \(T\) to 2 decimal places.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

14. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 8 \cos ^ { 3 } \theta , \quad y = 6 \sin ^ { 2 } \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ Given that the point \(P\) lies on \(C\) and has parameter \(\theta = \frac { \pi } { 3 }\)

1. find the coordinates of \(P\). The line \(l\) is the normal to \(C\) at \(P\).
2. Show that an equation of \(l\) is \(y = x + 3.5\) The finite region \(S\), shown shaded in Figure 6, is bounded by the curve \(C\), the line \(l\), the \(y\)-axis and the \(x\)-axis.
3. Show that the area of \(S\) is given by $$4 + 144 \int _ { 0 } ^ { \frac { \pi } { 3 } } \left( \sin \theta \cos ^ { 2 } \theta - \sin \theta \cos ^ { 4 } \theta \right) d \theta$$
4. Hence, by integration, find the exact area of \(S\).
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

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