Pre-U Pre-U 9794/1 (Pre-U Mathematics Paper 1) 2012 June

1 The first term of a geometric progression is 16 and the common ratio is 0.8 .

1. Calculate the sum of the first 12 terms.
2. Find the sum to infinity.

2 Let \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15\).

1. Show that \(\mathrm { f } ( 1 ) = 0\) and hence factorise \(x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15\) completely.
2. Hence solve the equation \(x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15 = 0\).

3 The equation of a curve is \(y = x ^ { 3 } + x ^ { 2 } - x + 3\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the stationary points on the curve.

4

1. Show that the equation \(x ^ { 3 } - 6 x + 2 = 0\) has a root between \(x = 0\) and \(x = 1\).
2. Use the iterative formula \(x _ { n + 1 } = \frac { 2 + x _ { n } ^ { 3 } } { 6 }\) with \(x _ { 0 } = 0.5\) to find this root correct to 4 decimal places, showing the result of each iteration.

5 Let \(\mathrm { f } ( x ) = x ^ { 2 }\) and \(\mathrm { g } ( x ) = 7 x - 2\) for all real values of \(x\).

1. Give a reason why f has no inverse function.
2. Write down an expression for \(\mathrm { gf } ( x )\).
3. Find \(\mathrm { g } ^ { - 1 } ( x )\).

6 The roots of the equation \(z ^ { 2 } - 6 z + 10 = 0\) are \(z _ { 1 }\) and \(z _ { 2 }\), where \(z _ { 1 } = 3 + \mathrm { i }\).

1. Write down the value of \(z _ { 2 }\).
2. Show \(z _ { 1 }\) and \(z _ { 2 }\) on an Argand diagram.
3. Show that \(z _ { 1 } ^ { 2 } = 8 + 6 \mathrm { i }\).

7

1. Show that the first three terms in the expansion of \(( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) are \(1 - x - \frac { 1 } { 2 } x ^ { 2 }\) and find the next term.
2. State the range of values of \(x\) for which this expansion is valid.
3. Hence show that the first four terms in the expansion of \(( 2 + x ) ( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) are \(2 - x + a x ^ { 2 } + b x ^ { 3 }\) and state the values of \(a\) and \(b\).

8

1. Given that \(\frac { 2 x + 11 } { ( 2 x + 1 ) ( x + 3 ) } \equiv \frac { A } { 2 x + 1 } + \frac { B } { x + 3 }\), find the values of the constants \(A\) and \(B\).
2. Hence show that \(\int _ { 0 } ^ { 2 } \frac { 2 x + 11 } { ( 2 x + 1 ) ( x + 3 ) } \mathrm { d } x = \ln 15\).

9 Three points \(A , B\) and \(C\) have coordinates \(( 1,0,7 ) , ( 13,9,1 )\) and \(( 2 , - 1 , - 7 )\) respectively.

1. Use a scalar product to find angle \(A C B\).
2. Hence find the area of triangle \(A C B\).
3. Show that a vector equation of the line \(A B\) is given by \(\mathbf { r } = \mathbf { i } + 7 \mathbf { k } + \lambda ( 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )\), where \(\lambda\) is a scalar parameter.

10

1. Prove that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta \right)$$ and hence show that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \cos 2 \theta$$
2. Hence solve the equation \(\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 2\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).

11

1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
2. Find
   1. \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
   2. \(\quad \int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).