AQA C4 (Core Mathematics 4) 2008 January

1

1. Given that \(\frac { 3 } { 9 - x ^ { 2 } }\) can be expressed in the form \(k \left( \frac { 1 } { 3 + x } + \frac { 1 } { 3 - x } \right)\), find the value of the rational number \(k\).
2. Show that \(\int _ { 1 } ^ { 2 } \frac { 3 } { 9 - x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.

2

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 18 x + 8\).
   1. Use the Factor Theorem to show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
   2. Write \(\mathrm { f } ( x )\) in the form \(( 2 x - 1 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
   3. Simplify the algebraic fraction \(\frac { 4 x ^ { 2 } + 16 x } { 2 x ^ { 3 } + 3 x ^ { 2 } - 18 x + 8 }\).
2. Express the algebraic fraction \(\frac { 2 x ^ { 2 } } { ( x + 5 ) ( x - 3 ) }\) in the form \(A + \frac { B + C x } { ( x + 5 ) ( x - 3 ) }\), where \(A , B\) and \(C\) are integers.

3

1. Obtain the binomial expansion of \(( 1 + x ) ^ { \frac { 1 } { 2 } }\) up to and including the term in \(x ^ { 2 }\).
2. Hence obtain the binomial expansion of \(\sqrt { 1 + \frac { 3 } { 2 } x }\) up to and including the term in \(x ^ { 2 }\).
3. Hence show that \(\sqrt { \frac { 2 + 3 x } { 8 } } \approx a + b x + c x ^ { 2 }\) for small values of \(x\), where \(a , b\) and \(c\) are constants to be found.

4 David is researching changes in the selling price of houses. One particular house was sold on 1 January 1885 for \(\pounds 20\). Sixty years later, on 1 January 1945, it was sold for \(\pounds 2000\). David proposes a model $$P = A k ^ { t }$$ for the selling price, \(\pounds P\), of this house, where \(t\) is the time in years after 1 January 1885 and \(A\) and \(k\) are constants.

1. 1. Write down the value of \(A\).
   2. Show that, to six decimal places, \(k = 1.079775\).
   3. Use the model, with this value of \(k\), to estimate the selling price of this house on 1 January 2008. Give your answer to the nearest \(\pounds 1000\).
2. For another house, which was sold for \(\pounds 15\) on 1 January 1885, David proposes the model $$Q = 15 \times 1.082709 ^ { t }$$ for the selling price, \(\pounds Q\), of this house \(t\) years after 1 January 1885. Calculate the year in which, according to these models, these two houses would have had the same selling price.

5 A curve is defined by the parametric equations \(x = 2 t + \frac { 1 } { t ^ { 2 } } , \quad y = 2 t - \frac { 1 } { t ^ { 2 } }\).

1. At the point \(P\) on the curve, \(t = \frac { 1 } { 2 }\).
   1. Find the coordinates of \(P\).
   2. Find an equation of the tangent to the curve at \(P\).
2. Show that the cartesian equation of the curve can be written as $$( x - y ) ( x + y ) ^ { 2 } = k$$ where \(k\) is an integer.

6 A curve has equation \(3 x y - 2 y ^ { 2 } = 4\).
Find the gradient of the curve at the point \(( 2,1 )\).

7

1. 1. Express \(6 \sin \theta + 8 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give your value for \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation \(6 \sin 2 x + 8 \cos 2 x = 7\), giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
2. 1. Prove the identity \(\frac { \sin 2 x } { 1 - \cos 2 x } = \frac { 1 } { \tan x }\).
   2. Hence solve the equation $$\frac { \sin 2 x } { 1 - \cos 2 x } = \tan x$$ giving all solutions in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

8 Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 \cos 3 x } { y }$$ given that \(y = 2\) when \(x = \frac { \pi } { 2 }\). Give your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

9 The points \(A\) and \(B\) lie on the line \(l _ { 1 }\) and have coordinates \(( 2,5,1 )\) and \(( 4,1 , - 2 )\) respectively.

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Find a vector equation of the line \(l _ { 1 }\), with parameter \(\lambda\).
2. The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 1
   - 3
   - 1 \end{array} \right] + \mu \left[ \begin{array} { r } 1
   0
   - 2 \end{array} \right]\).
   1. Show that the point \(P ( - 2 , - 3,5 )\) lies on \(l _ { 2 }\).
   2. The point \(Q\) lies on \(l _ { 1 }\) and is such that \(P Q\) is perpendicular to \(l _ { 2 }\). Find the coordinates of \(Q\).