Pre-U Pre-U 9794/2 (Pre-U Mathematics Paper 2) 2018 June

1 A geometric progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 32\) and \(u _ { n + 1 } = 0.75 u _ { n }\) for \(n \geqslant 1\).

1. Find \(u _ { 5 }\).
2. Find \(\sum _ { n = 1 } ^ { \infty } u _ { n }\).

2

1. Express \(2 x ^ { 2 } + 6 x + 5\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. State the equation of the line of symmetry of the curve \(y = 2 x ^ { 2 } + 6 x + 5\).
3. Find the value of the constant \(k\) for which the line \(y = k - 2 x\) is a tangent to the curve \(y = 2 x ^ { 2 } + 6 x + 5\).

3 Solve the equation \(6 ^ { 2 x - 1 } = 3 ^ { x + 2 }\), giving your answer in the form \(x = \frac { \ln a } { \ln b }\) where \(a\) and \(b\) are integers.

4 Solve the equation \(x + 2 \sqrt { x } - 6 = 0\), giving your answer in the form \(x = c + d \sqrt { 7 }\) where \(c\) and \(d\) are integers.

5 The complex numbers \(u\) and \(v\) are given by \(u = 3 + 2 \mathrm { i }\) and \(v = 1 + 4 \mathrm { i }\).

1. Given that \(a u ^ { 2 } + b v ^ { * } = 7 + 36 \mathrm { i }\) find the values of the real constants \(a\) and \(b\).
2. Show the points representing \(u\) and \(v\) on an Argand diagram and hence sketch the locus given by \(| z - u | = | z - v |\). Find the point of intersection of this locus with the imaginary axis.

6 \includegraphics[max width=\textwidth, alt={}, center]{f4b66aaa-16b9-4b15-b3f5-b9657fe98274-3_545_557_269_794} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius \(r\). Angle \(A O B\) is \(\theta\) radians. The point \(C\) lies on \(O B\), and \(A C\) is perpendicular to \(O B\). The area of the triangle \(A O C\) is equal to the area of the segment bounded by the chord \(A B\) and the \(\operatorname { arc } A B\).

1. Show that \(\theta = \sin \theta ( 1 + \cos \theta )\). The equation \(\theta = \sin \theta ( 1 + \cos \theta )\) has only one positive root.
2. Use an iterative process based on this equation to find the value of the root correct to 3 significant figures. Use a starting value of 1 and show the result of each iteration. Use a change of sign to verify that the value you have found is correct to 3 significant figures.

7 A curve is given parametrically by \(x = t ^ { 2 } + 1 , y = t ^ { 3 } - 2 t\) where \(t\) is any real number.

1. Show that the equation of the normal to the curve at the point where \(t = 2\) can be written in the form \(2 x + 5 y = 30\).
2. Show that this normal does not meet the curve again.

8

1. Use integration by parts twice to show that $$\int \mathrm { e } ^ { x } \sin x \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { x } ( \sin x - \cos x ) + c .$$
2. Hence find the equation of the curve which passes through the point \(( 0,2 )\) and for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { x } \sin x\).

9 In this question, \(x\) denotes an angle measured in degrees.

1. Express \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Give full details of the sequence of transformations which maps the graph of \(y = \cos x\) onto the graph of \(y = 4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\).
3. Find the smallest positive value of \(x\) that satisfies the equation \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x = 6\).

10

1. By using the substitution \(u = 3 - 2 x\), or otherwise, show that \(\int _ { 0 } ^ { 1 } \left( \frac { 4 x } { 3 - 2 x } \right) ^ { 2 } \mathrm {~d} x = 16 - 12 \ln 3\).
2. \includegraphics[max width=\textwidth, alt={}, center]{f4b66aaa-16b9-4b15-b3f5-b9657fe98274-4_595_588_927_817} The diagram shows the region \(R\), which is bounded by the curve \(y = \frac { 4 x } { 3 - 2 x }\), the \(y\)-axis and the line \(y = 4\). Find the exact volume generated when the region \(R\) is rotated completely around the \(x\)-axis. {www.cie.org.uk} after the live examination series. }