AQA C4 (Core Mathematics 4) 2009 June

1

1. Use the Remainder Theorem to find the remainder when \(3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5\) is divided by \(3 x - 1\).
2. Express \(\frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5 } { 3 x - 1 }\) in the form \(a x ^ { 2 } + b x + \frac { c } { 3 x - 1 }\), where \(a , b\) and \(c\) are integers.

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find an equation of the normal to the curve at the point where \(t = 1\).
3. Show that the cartesian equation of the curve can be written in the form $$x ^ { 2 } - 2 x y + k = 0$$ where \(k\) is an integer.

3

1. Find the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
2. 1. Express \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) in the form \(\frac { A } { 1 - x } + \frac { B } { 2 - 3 x }\), where \(A\) and \(B\) are integers.
   2. Find the binomial expansion of \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) up to and including the term in \(x ^ { 2 }\).
3. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) is valid.

4 A car depreciates in value according to the model $$V = A k ^ { t }$$ where \(\pounds V\) is the value of the car \(t\) months from when it was new, and \(A\) and \(k\) are constants. Its value when new was \(\pounds 12499\) and 36 months later its value was \(\pounds 7000\).

1. 1. Write down the value of \(A\).
   2. Show that the value of \(k\) is 0.984025 , correct to six decimal places.
2. The value of this car first dropped below \(\pounds 5000\) during the \(n\)th month from new. Find the value of \(n\).

5 A curve is defined by the equation \(4 x ^ { 2 } + y ^ { 2 } = 4 + 3 x y\).
Find the gradient at the point ( 1,3 ) on this curve.

6

1. 1. Show that the equation \(3 \cos 2 x + 7 \cos x + 5 = 0\) can be written in the form \(a \cos ^ { 2 } x + b \cos x + c = 0\), where \(a , b\) and \(c\) are integers.
   2. Hence find the possible values of \(\cos x\).
2. 1. Express \(7 \sin \theta + 3 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(\alpha\) is an acute angle. Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation \(7 \sin \theta + 3 \cos \theta = 4\) for all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), giving \(\theta\) to the nearest \(0.1 ^ { \circ }\).
3. 1. Given that \(\beta\) is an acute angle and that \(\tan \beta = 2 \sqrt { 2 }\), show that \(\cos \beta = \frac { 1 } { 3 }\).
   2. Hence show that \(\sin 2 \beta = p \sqrt { 2 }\), where \(p\) is a rational number.

7 The points \(A\) and \(B\) have coordinates ( \(3 , - 2,5\) ) and ( \(4,0,1\) ) respectively. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 6 \\ - 1 \\ 5 \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ - 1 \\ 4 \end{array} \right]\).

1. Find the distance between the points \(A\) and \(B\).
2. Verify that \(B\) lies on \(l _ { 1 }\).
   (2 marks)
3. The line \(l _ { 2 }\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { r } - 1 \\ 3 \\ - 8 \end{array} \right]\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(C\). Show that the points \(A , B\) and \(C\) form an isosceles triangle.
   (6 marks)

8

1. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 150 \cos 2 t } { x }$$ given that \(x = 20\) when \(t = \frac { \pi } { 4 }\), giving your solution in the form \(x ^ { 2 } = \mathrm { f } ( t )\). (6 marks)
2. The oscillations of a 'baby bouncy cradle' are modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 150 \cos 2 t } { x }$$ where \(x \mathrm {~cm}\) is the height of the cradle above its base \(t\) seconds after the cradle begins to oscillate. Given that the cradle is 20 cm above its base at time \(t = \frac { \pi } { 4 }\) seconds, find:
   1. the height of the cradle above its base 13 seconds after it starts oscillating, giving your answer to the nearest millimetre;
   2. the time at which the cradle will first be 11 cm above its base, giving your answer to the nearest tenth of a second.
      (2 marks)