AQA C4 (Core Mathematics 4) 2009 January

1

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 7 x - 3\).
   1. Find \(\mathrm { f } ( - 1 )\).
   2. Use the Factor Theorem to show that \(2 x + 1\) is a factor of \(\mathrm { f } ( x )\).
   3. Simplify the algebraic fraction \(\frac { 4 x ^ { 3 } - 7 x - 3 } { 2 x ^ { 2 } + 3 x + 1 }\).
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 4 x ^ { 3 } - 7 x + d\). When \(\mathrm { g } ( x )\) is divided by \(2 x + 1\), the remainder is 2 . Find the value of \(d\).

2

1. Express \(\sin x - 3 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\) in radians to two decimal places.
2. Hence:
   1. write down the minimum value of \(\sin x - 3 \cos x\);
   2. find the value of \(x\) in the interval \(0 < x < 2 \pi\) at which this minimum value occurs, giving your value of \(x\) in radians to two decimal places.

3

1. 1. Express \(\frac { 2 x + 7 } { x + 2 }\) in the form \(A + \frac { B } { x + 2 }\), where \(A\) and \(B\) are integers. (2 marks)
   2. Hence find \(\int \frac { 2 x + 7 } { x + 2 } \mathrm {~d} x\).
2. 1. Express \(\frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } }\) in the form \(\frac { P } { 1 + 3 x } + \frac { Q } { 5 - x } + \frac { R } { ( 5 - x ) ^ { 2 } }\), where \(P , Q\) and \(R\) are constants.
   2. Hence find \(\int \frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } } \mathrm {~d} x\).

4

1. 1. Find the binomial expansion of \(( 1 - x ) ^ { \frac { 1 } { 2 } }\) up to and including the term in \(x ^ { 2 }\).
      (2 marks)
   2. Hence obtain the binomial expansion of \(\sqrt { 4 - x }\) up to and including the term in \(x ^ { 2 }\).
      (3 marks)
2. Use your answer to part (a)(ii) to find an approximate value for \(\sqrt { 3 }\). Give your answer to three decimal places.

5

1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
2. Solve the equation $$5 \sin 2 x + 3 \cos x = 0$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\) to the nearest \(0.1 ^ { \circ }\), where appropriate.
3. Given that \(\sin 2 x + \cos 2 x = 1 + \sin x\) and \(\sin x \neq 0\), show that \(2 ( \cos x - \sin x ) = 1\).

6 A curve is defined by the equation \(x ^ { 2 } y + y ^ { 3 } = 2 x + 1\).

1. Find the gradient of the curve at the point \(( 2,1 )\).
2. Show that the \(x\)-coordinate of any stationary point on this curve satisfies the equation $$\frac { 1 } { x ^ { 3 } } = x + 1$$ (4 marks)

7

1. A differential equation is given by \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - k t \mathrm { e } ^ { \frac { 1 } { 2 } x }\), where \(k\) is a positive constant.
   1. Solve the differential equation.
   2. Hence, given that \(x = 6\) when \(t = 0\), show that \(x = - 2 \ln \left( \frac { k t ^ { 2 } } { 4 } + \mathrm { e } ^ { - 3 } \right)\).
      (3 marks)
2. The population of a colony of insects is decreasing according to the model \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - k t \mathrm { e } ^ { \frac { 1 } { 2 } x }\), where \(x\) thousands is the number of insects in the colony after time \(t\) minutes. Initially, there were 6000 insects in the colony. Given that \(k = 0.004\), find:
   1. the population of the colony after 10 minutes, giving your answer to the nearest hundred;
   2. the time after which there will be no insects left in the colony, giving your answer to the nearest 0.1 of a minute.

8 The points \(A\) and \(B\) have coordinates \(( 2,1 , - 1 )\) and \(( 3,1 , - 2 )\) respectively. The angle \(O B A\) is \(\theta\), where \(O\) is the origin.

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Show that \(\cos \theta = \frac { 5 } { 2 \sqrt { 7 } }\).
2. The point \(C\) is such that \(\overrightarrow { O C } = 2 \overrightarrow { O B }\). The line \(l\) is parallel to \(\overrightarrow { A B }\) and passes through the point \(C\). Find a vector equation of \(l\).
3. The point \(D\) lies on \(l\) such that angle \(O D C = 90 ^ { \circ }\). Find the coordinates of \(D\).