AQA C4 (Core Mathematics 4) 2011 January

1

1. Express \(2 \sin x + 5 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
2. 1. Write down the maximum value of \(2 \sin x + 5 \cos x\).
   2. Find the value of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\) at which this maximum occurs, giving the value of \(x\) to the nearest \(0.1 ^ { \circ }\).

2

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } - x - 2\).
   1. Use the Factor Theorem to show that \(3 x + 1\) is a factor of \(\mathrm { f } ( x )\).
   2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
   3. Simplify \(\frac { 9 x ^ { 3 } + 21 x ^ { 2 } + 6 x } { \mathrm { f } ( x ) }\).
2. When the polynomial \(9 x ^ { 3 } + p x ^ { 2 } - x - 2\) is divided by \(3 x - 2\), the remainder is - 4 . Find the value of the constant \(p\).

3

1. Express \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) in the form \(\frac { A } { 1 + x } + \frac { B } { 3 + 5 x }\), where \(A\) and \(B\) are integers.
2. Hence, or otherwise, find the binomial expansion of \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) up to and including the term in \(x ^ { 2 }\).
3. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) is valid.
   (2 marks)

4 A curve is defined by the parametric equations $$x = 3 \mathrm { e } ^ { t } , \quad y = \mathrm { e } ^ { 2 t } - \mathrm { e } ^ { - 2 t }$$

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

5 A model for the radioactive decay of a form of iodine is given by $$m = m _ { 0 } 2 ^ { - \frac { 1 } { 8 } t }$$ The mass of the iodine after \(t\) days is \(m\) grams. Its initial mass is \(m _ { 0 }\) grams.

1. Use the given model to find the mass that remains after 10 grams of this form of iodine have decayed for 14 days, giving your answer to the nearest gram.
2. A mass of \(m _ { 0 }\) grams of this form of iodine decays to \(\frac { m _ { 0 } } { 16 }\) grams in \(d\) days. Find the value of \(d\).
3. After \(n\) days, a mass of this form of iodine has decayed to less than \(1 \%\) of its initial mass. Find the minimum integer value of \(n\).

6

1. 1. Given that \(\tan 2 x + \tan x = 0\), show that \(\tan x = 0\) or \(\tan ^ { 2 } x = 3\).
   2. Hence find all solutions of \(\tan 2 x + \tan x = 0\) in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
      (l mark)
2. 1. Given that \(\cos x \neq 0\), show that the equation $$\sin 2 x = \cos x \cos 2 x$$ can be written in the form $$2 \sin ^ { 2 } x + 2 \sin x - 1 = 0$$
   2. Show that all solutions of the equation \(2 \sin ^ { 2 } x + 2 \sin x - 1 = 0\) are given by \(\sin x = \frac { \sqrt { 3 } - 1 } { p }\), where \(p\) is an integer.

7

1. 1. Solve the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \sqrt { x } \sin \left( \frac { t } { 2 } \right)\) to find \(x\) in terms of \(t\).
   2. Given that \(x = 1\) when \(t = 0\), show that the solution can be written as $$x = ( a - \cos b t ) ^ { 2 }$$ where \(a\) and \(b\) are constants to be found.
2. The height, \(x\) metres, above the ground of a car in a fairground ride at time \(t\) seconds is modelled by the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \sqrt { x } \sin \left( \frac { t } { 2 } \right)\). The car is 1 metre above the ground when \(t = 0\).
   1. Find the greatest height above the ground reached by the car during the ride.
   2. Find the value of \(t\) when the car is first 5 metres above the ground, giving your answer to one decimal place.

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

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Calculate the acute angle between \(\overrightarrow { A B }\) and the line \(l _ { 1 }\), giving your answer to the nearest \(0.1 ^ { \circ }\).
2. The point \(D\) lies on \(l _ { 1 }\) where \(\lambda = 2\). The line \(l _ { 2 }\) passes through \(D\) and is parallel to \(A B\).
   1. Find a vector equation of line \(l _ { 2 }\) with parameter \(\mu\).
   2. The diagram shows a symmetrical trapezium \(A B C D\), with angle \(D A B\) equal to angle \(A B C\).
      \includegraphics[max width=\textwidth, alt={}, center]{5fe2527a-33da-4076-b3fa-4cab545336ec-9_620_675_1197_726} The point \(C\) lies on line \(l _ { 2 }\). The length of \(A D\) is equal to the length of \(B C\). Find the coordinates of \(C\).