Edexcel C34 (Core Mathematics 3 & 4) 2019 January

1. (a) Express \(7 \sin 2 \theta - 2 \cos 2 \theta\) in the form \(R \sin ( 2 \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places.
   (b) Hence solve, for \(0 \leqslant \theta < 90 ^ { \circ }\), the equation

$$7 \sin 2 \theta - 2 \cos 2 \theta = 4$$ giving your answers in degrees to one decimal place.
(c) Express \(28 \sin \theta \cos \theta + 8 \sin ^ { 2 } \theta\) in the form \(a \sin 2 \theta + b \cos 2 \theta + c\), where \(a\), \(b\) and \(c\) are constants to be found.
(d) Use your answers to part (a) and part (c) to deduce the exact maximum value of \(28 \sin \theta \cos \theta + 8 \sin ^ { 2 } \theta\)

2. Given that $$\frac { 3 x ^ { 2 } + 4 x - 7 } { ( x + 1 ) ( x - 3 ) } \equiv A + \frac { B } { x + 1 } + \frac { C } { x - 3 }$$

1. find the values of the constants \(A , B\) and \(C\).
2. Hence, or otherwise, find the series expansion of $$\frac { 3 x ^ { 2 } + 4 x - 7 } { ( x + 1 ) ( x - 3 ) } \quad | x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\)
   Give each coefficient as a simplified fraction.

3. The function f is defined by $$f : x \mapsto 2 x ^ { 2 } + 3 k x + k ^ { 2 } \quad x \in \mathbb { R } , - 4 k \leqslant x \leqslant 0$$ where \(k\) is a positive constant.

1. Find, in terms of \(k\), the range of f . The function g is defined by $$\mathrm { g } : x \mapsto 2 k - 3 x \quad x \in \mathbb { R }$$ Given that \(\operatorname { gf } ( - 2 ) = - 12\)
2. find the possible values of \(k\).

1. The curve \(C\) has equation

$$81 y ^ { 3 } + 64 x ^ { 2 } y + 256 x = 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Hence find the coordinates of the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

5. The angle \(x\) and the angle \(y\) are such that $$\tan x = m \text { and } 4 \tan y = 8 m + 5$$ where \(m\) is a constant.
Given that \(16 \sec ^ { 2 } x + 16 \sec ^ { 2 } y = 537\)

1. find the two possible values of \(m\). Given that the angle \(x\) and the angle \(y\) are acute, find the exact value of
2. \(\sin x\)
3. \(\cot y\)

6. Relative to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have coordinates ( \(2,1,9 ) , ( 5,2,7 )\) and \(( 4 , - 3,3 )\) respectively. The line \(l\) passes through the points \(A\) and \(B\).

1. Find a vector equation for the line \(l\).
2. Find, in degrees, the acute angle between the line \(I\) and the line \(A C\). The point \(D\) lies on the line \(l\) such that angle \(A C D\) is \(90 ^ { \circ }\)
3. Find the coordinates of \(D\).
4. Find the exact area of triangle \(A D C\), giving your answer as a fully simplified surd.

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = \frac { x + 7 } { \sqrt { 2 x - 3 } } \quad x > \frac { 3 } { 2 }$$ The region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 6\)

1. Use the trapezium rule with 4 strips of equal width to find an estimate for the area of \(R\), giving your answer to 2 decimal places.
2. Using the substitution \(u = 2 x - 3\), or otherwise, use calculus to find the exact area of \(R\), giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are constants to be found.

8. A curve has parametric equations $$x = t ^ { 2 } - t , \quad y = \frac { 4 t } { 1 - t } \quad t \neq 1$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer as a simplified fraction.
2. Find an equation for the tangent to the curve at the point \(P\) where \(t = - 1\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. The tangent to the curve at \(P\) cuts the curve at the point \(Q\).
3. Use algebra to find the coordinates of \(Q\).

9. \begin{figure}[h] \end{figure} (c) Find the exact value for the volume of this solid, giving your answer as a single, simplified fraction. \section*{Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x + \sin 2 x\).
The region \(R\), shown shaded in Figure 2, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\)
The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x + \sin 2 x\). The region \(R\), shown shaded in Figure 2, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\) The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.} \(\_\_\_\_\) simplified fraction.

10. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 3 shows a container in the shape of an inverted right circular cone which contains some water. The cone has an internal radius of 3 m and a vertical height of 5 m as shown in Figure 3. At time \(t\) seconds,the height of the water is \(h\) metres,the volume of the water is \(V \mathrm {~m} ^ { 3 }\) and water is leaking from a hole in the bottom of the container at a constant rate of \(0.02 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
[The volume of a cone of radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) .]

1. Show that,while the water is leaking, $$h ^ { 2 } \frac { \mathrm {~d} h } { \mathrm {~d} t } = - \frac { 1 } { \mathrm { k } \pi }$$ where \(k\) is a constant to be found. Given that the container is initially full of water,
2. express \(h\) in terms of \(t\) .
3. Find the time taken for the container to empty,giving your answer to the nearest minute.

11. (a) Given that \(0 \leqslant \mathrm { f } ( x ) \leqslant \pi\), sketch the graph of \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \arccos ( x - 1 ) , \quad 0 \leqslant x \leqslant 2$$ The equation \(\arccos ( x - 1 ) - \tan x = 0\) has a single root \(\alpha\).
(b) Show that \(0.9 < \alpha < 1.1\) The iteration formula $$x _ { n + 1 } = \arctan \left( \arccos \left( x _ { n } - 1 \right) \right)$$ can be used to find an approximation for \(\alpha\).
(c) Taking \(x _ { 0 } = 1.1\) find, to 3 decimal places, the values of \(x _ { 1 }\) and \(x _ { 2 }\)

12. Given that \(k\) is a positive constant,

1. sketch the graph with equation $$y = 2 | x | - k$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and the \(y\)-axis.
2. Find, in terms of \(k\), the values of \(x\) for which $$2 | x | - k = \frac { 1 } { 2 } x + \frac { 1 } { 4 } k$$

13. A scientist is studying a population of insects. The number of insects, \(N\), in the population, \(t\) days after the start of the study is modelled by the equation $$N = \frac { 240 } { 1 + k \mathrm { e } ^ { - \frac { t } { 16 } } }$$ where \(k\) is a constant.
Given that there were 50 insects at the start of the study,

1. find the value of \(k\)
2. use the model to find the value of \(t\) when \(N = 100\)
3. Show that $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { 1 } { p } N - \frac { 1 } { q } N ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
   

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