Edexcel C4 (Core Mathematics 4) 2010 June

1. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \sqrt { } \left( 0.75 + \cos ^ { 2 } x \right)\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(y\)-axis, the \(x\)-axis and the line with equation \(x = \frac { \pi } { 3 }\).

1. Complete the table with values of \(y\) corresponding to \(x = \frac { \pi } { 6 }\) and \(x = \frac { \pi } { 4 }\).

   \(x\)	0	\(\frac { \pi } { 12 }\)	\(\frac { \pi } { 6 }\)	\(\frac { \pi } { 4 }\)	\(\frac { \pi } { 3 }\)
   \(y\)	1.3229	1.2973			1

2. Use the trapezium rule
   1. with the values of \(y\) at \(x = 0 , x = \frac { \pi } { 6 }\) and \(x = \frac { \pi } { 3 }\) to find an estimate of the area of \(R\). Give your answer to 3 decimal places.
   2. with the values of \(y\) at \(x = 0 , x = \frac { \pi } { 12 } , x = \frac { \pi } { 6 } , x = \frac { \pi } { 4 }\) and \(x = \frac { \pi } { 3 }\) to find a further estimate of the area of \(R\). Give your answer to 3 decimal places.
      (6) \section*{LU}

2. Using the substitution \(u = \cos x + 1\), or otherwise, show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { \cos x + 1 } \sin x \mathrm {~d} x = \mathrm { e } ( \mathrm { e } - 1 )$$ (6)

3. A curve \(C\) has equation $$2 ^ { x } + y ^ { 2 } = 2 x y$$ Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point on \(C\) with coordinates \(( 3,2 )\).

4. A curve \(C\) has parametric equations $$x = \sin ^ { 2 } t , \quad y = 2 \tan t , \quad 0 \leqslant t < \frac { \pi } { 2 }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The tangent to \(C\) at the point where \(t = \frac { \pi } { 3 }\) cuts the \(x\)-axis at the point \(P\).
2. Find the \(x\)-coordinate of \(P\).
   \section*{LU}

5. $$\frac { 2 x ^ { 2 } + 5 x - 10 } { ( x - 1 ) ( x + 2 ) } \equiv A + \frac { B } { x - 1 } + \frac { C } { x + 2 }$$

1. Find the values of the constants \(A , B\) and \(C\).
2. Hence, or otherwise, expand \(\frac { 2 x ^ { 2 } + 5 x - 10 } { ( x - 1 ) ( x + 2 ) }\) in ascending powers of \(x\), as far as the term in \(x ^ { 2 }\). Give each coefficient as a simplified fraction.

6. $$f ( \theta ) = 4 \cos ^ { 2 } \theta - 3 \sin ^ { 2 } \theta$$

1. Show that \(f ( \theta ) = \frac { 1 } { 2 } + \frac { 7 } { 2 } \cos 2 \theta\).
2. Hence, using calculus, find the exact value of \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \theta \mathrm { f } ( \theta ) \mathrm { d } \theta\).

7. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2
3
- 4 \end{array} \right) + \lambda \left( \begin{array} { l } 1
2
1 \end{array} \right)\), where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 0
9
- 3 \end{array} \right) + \mu \left( \begin{array} { l } 5
0
2 \end{array} \right)\), where \(\mu\) is a scalar parameter.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(C\), find

1. the coordinates of \(C\). The point \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 0\) and the point \(B\) is the point on \(l _ { 2 }\) where \(\mu = - 1\).
2. Find the size of the angle \(A C B\). Give your answer in degrees to 2 decimal places.
3. Hence, or otherwise, find the area of the triangle \(A B C\).

8. \begin{figure}[h] \end{figure} Figure 2 shows a cylindrical water tank. The diameter of a circular cross-section of the tank is 6 m . Water is flowing into the tank at a constant rate of \(0.48 \pi \mathrm {~m} ^ { 3 } \mathrm {~min} ^ { - 1 }\). At time \(t\) minutes, the depth of the water in the tank is \(h\) metres. There is a tap at a point \(T\) at the bottom of the tank. When the tap is open, water leaves the tank at a rate of \(0.6 \pi h \mathrm {~m} ^ { 3 } \mathrm {~min} ^ { - 1 }\).

1. Show that \(t\) minutes after the tap has been opened $$75 \frac { \mathrm {~d} h } { \mathrm {~d} t } = ( 4 - 5 h )$$ When \(t = 0 , h = 0.2\)
2. Find the value of \(t\) when \(h = 0.5\)