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

1 Write down the coordinates of the centre and the radius of the circle with equation $$( x + 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 36 .$$

2

1. Show that \(x = 2\) is a root of the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
2. Hence solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).

3

1. In an arithmetic progression, the first term is 7 and the sum of the first 40 terms is 4960. Find the common difference.
2. A geometric progression has first term 14 and common ratio 0.3. Find the sum to infinity.

4 A sector \(A O B\) of a circle has radius \(r \mathrm {~cm}\) and the angle \(A O B\) is \(\theta\) radians. The perimeter of the sector is 40 cm and its area is \(100 \mathrm {~cm} ^ { 2 }\).

1. Write down equations for the perimeter and area of the sector in terms of \(r\) and \(\theta\).
2. Use your equations to show that \(r ^ { 2 } - 20 r + 100 = 0\) and hence find the value of \(r\) and of \(\theta\).

5

1. Find \(\int \left( \frac { 1 } { x - 2 } - \frac { 2 } { 2 x + 3 } \right) \mathrm { d } x\) giving your answer in its simplest form.
2. Use integration by parts to find \(\int x ^ { 2 } \ln x \mathrm {~d} x\).

6

1. Find and simplify the first four terms in the expansion of \(( 1 - 2 x ) ^ { 9 }\) in ascending powers of \(x\).
2. In the expansion of $$( 2 + a x ) ( 1 - 2 x ) ^ { 9 }$$ the coefficient of \(x ^ { 2 }\) is 66 . Find the value of \(a\).

7 Given that the equation \(x = 2 - \frac { 1 } { ( x + 1 ) ^ { 2 } }\) has a root between \(x = 1\) and \(x = 2\), use the Newton-Raphson formula with \(x _ { 0 } = 2\) to find this root correct to 3 decimal places.

8 A curve has equation \(y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 1\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the \(x\)-coordinates of the stationary points of the curve.
3. By using the second derivative, determine whether each of the stationary points is a maximum or a minimum.

9

1. On the same axes, sketch the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).
2. Find the exact area of the region contained between the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).

10 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 5 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\).

1. Find the length of \(A B\).
2. Use a scalar product to find angle \(O A B\).

11 Solve the differential equation \(x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \sec y\) given that \(y = \frac { \pi } { 6 }\) when \(x = 4\) giving your answer in the form \(y = \mathrm { f } ( x )\).

12 Calculate the maximum and minimum values of \(\frac { 1 } { 2 + \cos \theta + \sqrt { 2 } \sin \theta }\) and the smallest positive values of \(\theta\) for which they occur.