Edexcel C4 (Core Mathematics 4) 2011 June

1. $$\frac { 9 x ^ { 2 } } { ( x - 1 ) ^ { 2 } ( 2 x + 1 ) } = \frac { A } { ( x - 1 ) } + \frac { B } { ( x - 1 ) ^ { 2 } } + \frac { C } { ( 2 x + 1 ) }$$ Find the values of the constants \(A , B\) and \(C\).

2. $$f ( x ) = \frac { 1 } { \sqrt { } \left( 9 + 4 x ^ { 2 } \right) } , \quad | x | < \frac { 3 } { 2 }$$ Find the first three non-zero terms of the binomial expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\). Give each coefficient as a simplified fraction.

3. \begin{figure}[h] \end{figure} A hollow hemispherical bowl is shown in Figure 1. Water is flowing into the bowl. When the depth of the water is \(h \mathrm {~m}\), the volume \(V \mathrm {~m} ^ { 3 }\) is given by $$V = \frac { 1 } { 12 } \pi \cdot h ^ { 2 } ( 3 - 4 h ) , \quad 0 \leqslant h \leqslant 0.25$$

1. Find, in terms of \(\pi , \frac { \mathrm { d } V } { \mathrm {~d} h }\) when \(h = 0.1\) Water flows into the bowl at a rate of \(\frac { \pi } { 800 } \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
2. Find the rate of change of \(h\), in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), when \(h = 0.1\)

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = x ^ { 3 } \ln \left( x ^ { 2 } + 2 \right) , x \geqslant 0\).
The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the line \(x = \sqrt { } 2\). The table below shows corresponding values of \(x\) and \(y\) for \(y = x ^ { 3 } \ln \left( x ^ { 2 } + 2 \right)\).

1. Complete the table above giving the missing values of \(y\) to 4 decimal places.
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places.
3. Use the substitution \(u = x ^ { 2 } + 2\) to show that the area of \(R\) is $$\frac { 1 } { 2 } \int _ { 2 } ^ { 4 } ( u - 2 ) \ln u \mathrm {~d} u$$
4. Hence, or otherwise, find the exact area of \(R\).

1. Find the gradient of the curve with equation

$$\ln y = 2 x \ln x , \quad x > 0 , y > 0$$ at the point on the curve where \(x = 2\). Give your answer as an exact value.

6. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 6
- 3
- 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
3 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 5

7. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = \sin \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( \sqrt { } 3 , \frac { 1 } { 2 } \sqrt { } 3 \right)\).

1. Find the value of \(\theta\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
2. Show that \(Q\) has coordinates \(( k \sqrt { } 3,0 )\), giving the value of the constant \(k\). The finite shaded region \(S\) shown in Figure 3 is bounded by the curve \(C\), the line \(x = \sqrt { } 3\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
3. Find the volume of the solid of revolution, giving your answer in the form \(p \pi \sqrt { } 3 + q \pi ^ { 2 }\), where \(p\) and \(q\) are constants.

15
3 \end{array} \right) + \mu \left( \begin{array} { r } 2
- 3
1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection \(A\).
2. Find, to the nearest \(0.1 ^ { \circ }\), the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\). The point \(B\) has position vector \(\left( \begin{array} { r } 5
   - 1
   1 \end{array} \right)\).
3. Show that \(B\) lies on \(l _ { 1 }\).
4. Find the shortest distance from \(B\) to the line \(l _ { 2 }\), giving your answer to 3 significant figures.
   7. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9d513d77-b8f9-4223-832f-f566c5f50457-10_643_999_276_475} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = \sin \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( \sqrt { } 3 , \frac { 1 } { 2 } \sqrt { } 3 \right)\).
5. Find the value of \(\theta\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
6. Show that \(Q\) has coordinates \(( k \sqrt { } 3,0 )\), giving the value of the constant \(k\). The finite shaded region \(S\) shown in Figure 3 is bounded by the curve \(C\), the line \(x = \sqrt { } 3\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
7. Find the volume of the solid of revolution, giving your answer in the form \(p \pi \sqrt { } 3 + q \pi ^ { 2 }\), where \(p\) and \(q\) are constants. 8. (a) Find \(\int ( 4 y + 3 ) ^ { - \frac { 1 } { 2 } } \mathrm {~d} y\)
8. Given that \(y = 1.5\) at \(x = - 2\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { } ( 4 y + 3 ) } { x ^ { 2 } }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).