Edexcel C4 (Core Mathematics 4) 2018 June

1. (a) Find the binomial series expansion of

$$\sqrt { 4 - 9 x } , | x | < \frac { 4 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) Give each coefficient in its simplest form.
(b) Use the expansion from part (a), with a suitable value of \(x\), to find an approximate value for \(\sqrt { 310 }\)
Show all your working and give your answer to 3 decimal places.

1. The curve \(C\) has equation

$$x ^ { 2 } + x y + y ^ { 2 } - 4 x - 5 y + 1 = 0$$

1. Use implicit differentiation to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find the \(x\) coordinates of the two points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) Give exact answers in their simplest form.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

3. (i) Given that $$\frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \equiv \frac { A } { ( 2 x + 1 ) } + \frac { B } { ( 2 x + 1 ) ^ { 2 } } + \frac { C } { ( x + 3 ) }$$

1. find the values of the constants \(A , B\) and \(C\).
2. Hence find $$\int \frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \mathrm { d } x , \quad x > - \frac { 1 } { 2 }$$ (ii) Find $$\int \left( \mathrm { e } ^ { x } + 1 \right) ^ { 3 } \mathrm {~d} x$$ (iii) Using the substitution \(u ^ { 3 } = x\), or otherwise, find $$\int \frac { 1 } { 4 x + 5 x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x , \quad x > 0$$

4. \begin{figure}[h] \end{figure} A water container is made in the shape of a hollow inverted right circular cone with semi-vertical angle of \(30 ^ { \circ }\), as shown in Figure 1. The height of the container is 50 cm . When the depth of the water in the container is \(h \mathrm {~cm}\), the surface of the water has radius \(r \mathrm {~cm}\) and the volume of water is \(V \mathrm {~cm} ^ { 3 }\).

1. Show that \(V = \frac { 1 } { 9 } \pi h ^ { 3 }\)
   [0pt] [You may assume the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.] Given that the volume of water in the container increases at a constant rate of \(200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\),
2. find the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 15\) Give your answer in its simplest form in terms of \(\pi\).

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 1 + t - 5 \sin t , \quad y = 2 - 4 \cos t , \quad - \pi \leqslant t \leqslant \pi$$ The point \(A\) lies on the curve \(C\). Given that the coordinates of \(A\) are ( \(k , 2\) ), where \(k > 0\)

1. find the exact value of \(k\), giving your answer in a fully simplified form.
2. Find the equation of the tangent to \(C\) at the point \(A\). Give your answer in the form \(y = p x + q\), where \(p\) and \(q\) are exact real values.

1. Given that \(y = 2\) when \(x = - \frac { \pi } { 8 }\), solve the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } } { 3 \cos ^ { 2 } 2 x } \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).

7. The point \(A\) with coordinates ( \(- 3,7,2\) ) lies on a line \(l _ { 1 }\) The point \(B\) also lies on the line \(l _ { 1 }\) Given that \(\quad \overrightarrow { A B } = \left( \begin{array} { r } 4
- 6
2 \end{array} \right)\),

1. find the coordinates of point \(B\). The point \(P\) has coordinates ( \(9,1,8\) )
2. Find the cosine of the angle \(P A B\), giving your answer as a simplified surd.
3. Find the exact area of triangle \(P A B\), giving your answer in its simplest form. The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
4. Find a vector equation for the line \(l _ { 2 }\) The point \(Q\) lies on the line \(l _ { 2 }\) Given that the line segment \(A P\) is perpendicular to the line segment \(B Q\),
5. find the coordinates of the point \(Q\).

8. \begin{figure}[h] \end{figure}

1. Find \(\int x \cos 4 x d x\) Figure 3 shows part of the curve with equation \(y = \sqrt { x } \sin 2 x , \quad x \geqslant 0\)
   The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \frac { \pi } { 4 }\) The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
2. Find the exact value of the volume of this solid of revolution, giving your answer in its simplest form.
   (Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-32_2630_1828_121_121}