Pre-U Pre-U 9794/2 (Pre-U Mathematics Paper 2) 2017 June

1 Find the equation of the line which passes through the points \(( 2,5 )\) and \(( 8 , - 1 )\). Show that this line also passes through the point \(( - 2,9 )\).

2

1. 1. Find the value of the discriminant of \(x ^ { 2 } + 3 x + 5\).
   2. Use your value from part (i) to determine the number of real roots of the equation \(x ^ { 2 } + 3 x + 5 = 0\).
2. Find the non-zero value of \(k\) for which the equation \(k x ^ { 2 } + 3 x + 5 = 0\) has only one distinct real root.

3 Solve the equation \(\tan \left( \theta + 10 ^ { \circ } \right) = 0.1\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4 A sequence of complex numbers is defined by $$u _ { 1 } = 1 + \mathrm { i } \quad \text { and } \quad u _ { n + 1 } = \mathrm { i } u _ { n } ( n = 1,2,3 , \ldots )$$

1. Find \(u _ { 2 } , u _ { 3 } , u _ { 4 } , u _ { 5 }\) and \(u _ { 6 }\).
2. Describe the behaviour of the sequence.
3. Hence evaluate \(\sum _ { n = 1 } ^ { 73 } u _ { n }\).

5

1. Differentiate \(\frac { x } { \sqrt { 1 + x ^ { 2 } } }\) with respect to \(x\).
2. Hence show that \(\frac { x } { \sqrt { 1 + x ^ { 2 } } }\) is increasing for all \(x\).

6 Find the solution of the differential equation $$x y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x + 1$$ given that \(y = 3\) when \(x = 1\). Give your answer in the form \(y = \mathrm { f } ( x )\).

7 A curve, \(C\), is given parametrically by \(x = 2 \cos \theta , y = 3 \sin \theta , 0 < \theta < \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 } { 2 } \cot \theta\). A tangent to \(C\) intersects the \(x\)-axis and \(y\)-axis at \(P\) and \(Q\) respectively.
2. Show that the midpoint of \(P Q\) has coordinates \(\left( \sec \theta , \frac { 3 } { 2 } \operatorname { cosec } \theta \right)\).
3. Hence show that the midpoint of \(P Q\) lies on the curve \(\frac { 4 } { x ^ { 2 } } + \frac { 9 } { y ^ { 2 } } = 4\).

8

1. Express \(\frac { 7 x ^ { 2 } - 12 x + 1 } { \left( x ^ { 2 } + 1 \right) ( x - 2 ) }\) in the form \(\frac { A x + B } { x ^ { 2 } + 1 } + \frac { C } { x - 2 }\) where \(A , B\) and \(C\) are constants to be found.
2. Hence find the exact value of \(\int _ { 0 } ^ { 1 } \frac { 7 x ^ { 2 } - 12 x + 1 } { \left( x ^ { 2 } + 1 \right) ( x - 2 ) } \mathrm { d } x\).

9

1. Show that \(\int x ( x - 2 ) ^ { \frac { 3 } { 2 } } \mathrm {~d} x = \frac { 2 } { 35 } ( 5 x + 4 ) ( x - 2 ) ^ { \frac { 5 } { 2 } } + c\).
2. Hence find the coordinates of the stationary points of the curve $$y = \frac { 2 } { 35 } ( 5 x + 4 ) ( x - 2 ) ^ { \frac { 5 } { 2 } } + x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 }$$

10 An arithmetic sequence and a geometric sequence have \(n\)th terms \(a _ { n }\) and \(g _ { n }\) respectively, where \(n = 1,2,3 , \ldots\). It is given that \(a _ { 1 } = g _ { 1 } , a _ { 2 } = g _ { 2 } , a _ { 5 } = g _ { 3 } , a _ { 1 } \neq a _ { 2 }\) and \(a _ { 1 } \neq 0\).

1. Show that the common ratio of the geometric sequence is 3 .
2. Find the common difference of the arithmetic sequence in terms of \(a _ { 1 }\).
3. Let \(a _ { 1 } = g _ { 1 } = 5\).
   1. Find the first three terms of both sequences.
   2. Show that every term of the geometric sequence is also a term of the arithmetic sequence.