AQA C4 (Core Mathematics 4) 2005 June

1

1. Express \(2 \sin x + \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R\) is a positive constant and \(\alpha\) is an acute angle. Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
2. Solve the equation \(2 \sin x + \cos x = 1\) for \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).

2

1. Express \(\frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) }\) in the form \(\frac { A } { x + 3 } + \frac { B } { 2 x - 1 }\).
2. Hence find \(\int \frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) } \mathrm { d } x\).

3

1. Find the remainder when \(2 x ^ { 3 } - x ^ { 2 } + 2 x - 2\) is divided by \(2 x - 1\).
2. Given that \(\frac { 2 x ^ { 3 } - x ^ { 2 } + 2 x - 2 } { 2 x - 1 } = x ^ { 2 } + a + \frac { b } { 2 x - 1 }\), find the values of \(a\) and \(b\).

4

1. Find the binomial expansion of \(( 1 + x ) ^ { - \frac { 1 } { 2 } }\) up to the term in \(x ^ { 2 }\).
2. Hence, or otherwise, obtain the binomial expansion of \(\frac { 1 } { \sqrt { 1 + 2 x } }\) up to the term in \(x ^ { 2 }\), in simplified form.
3. Use your answer to part (b) with \(x = - 0.1\) to show that \(\sqrt { 5 } \approx 2.23\).

5 A curve is defined by the parametric equations $$x = 2 t + \frac { 1 } { t } , \quad y = \frac { 1 } { t } , \quad t \neq 0$$

1. Find the coordinates of the point on the curve where \(t = \frac { 1 } { 2 }\).
2. Show that the cartesian equation of the curve can be written as $$x y - y ^ { 2 } = 2$$
3. Show that the gradient of the curve at the point \(( 3,2 )\) is 2 .

6

1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
2. Using the identity \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\) :
   1. express \(\cos 2 x\) in terms of \(\sin x\) and \(\cos x\);
   2. show, by writing \(3 x\) as \(( 2 x + x )\), that $$\cos 3 x = 4 \cos ^ { 3 } x - 3 \cos x$$
3. Show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 3 } x \mathrm {~d} x = \frac { 2 } { 3 }\).

7 The points \(A\) and \(B\) have coordinates \(( 1,4,2 )\) and \(( 2 , - 1,3 )\) respectively.
The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 2
- 1
3 \end{array} \right] + \lambda \left[ \begin{array} { r } 1
- 1
1 \end{array} \right]\).

1. Show that the distance between the points \(A\) and \(B\) is \(3 \sqrt { 3 }\).
2. The line \(A B\) makes an acute angle \(\theta\) with \(l\). Show that \(\cos \theta = \frac { 7 } { 9 }\).
3. The point \(P\) on the line \(l\) is where \(\lambda = p\).
   1. Show that $$\overrightarrow { A P } \cdot \left[ \begin{array} { r } 1
      - 1
      1 \end{array} \right] = 7 + 3 p$$
   2. Hence find the coordinates of the foot of the perpendicular from the point \(A\) to the line \(l\).

8

1. A cup of coffee is cooling down in a room. At time \(t\) minutes after the coffee is made, its temperature is \(x ^ { \circ } \mathrm { C }\), where $$x = 15 + 70 \mathrm { e } ^ { - \frac { t } { 40 } }$$
   1. Find the temperature of the coffee when it is made.
   2. Find the temperature of the coffee 30 minutes after it is made.
   3. Find how long it will take for the coffee to cool down to \(60 ^ { \circ } \mathrm { C }\).
2. 1. Use integration to solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { 1 } { 40 } ( x - 15 ) , \quad x > 15$$ given that \(x = 85\) when \(t = 0\), expressing \(t\) in terms of \(x\).
   2. Hence show that \(x = 15 + 70 \mathrm { e } ^ { - \frac { t } { 40 } }\).