CAIE P1 (Pure Mathematics 1) 2015 June

1 The function f is such that \(\mathrm { f } ^ { \prime } ( x ) = 5 - 2 x ^ { 2 }\) and ( 3,5 ) is a point on the curve \(y = \mathrm { f } ( x )\). Find \(\mathrm { f } ( x )\).

2
\includegraphics[max width=\textwidth, alt={}, center]{5eb35648-db3f-4f29-b835-1aff4ef927b9-2_624_588_395_776} In the diagram, \(A Y B\) is a semicircle with \(A B\) as diameter and \(O A X B\) is a sector of a circle with centre \(O\) and radius \(r\). Angle \(A O B = 2 \theta\) radians. Find an expression, in terms of \(r\) and \(\theta\), for the area of the shaded region.

3

1. Find the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(( 2 - x ) ^ { 6 }\).
2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 x + 1 ) ( 2 - x ) ^ { 6 }\).

4 Variables \(u , x\) and \(y\) are such that \(u = 2 x ( y - x )\) and \(x + 3 y = 12\). Express \(u\) in terms of \(x\) and hence find the stationary value of \(u\).

1. Prove the identity \(\frac { \sin \theta - \cos \theta } { \sin \theta + \cos \theta } \equiv \frac { \tan \theta - 1 } { \tan \theta + 1 }\).
2. Hence solve the equation \(\frac { \sin \theta - \cos \theta } { \sin \theta + \cos \theta } = \frac { \tan \theta } { 6 }\), for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

6 A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel turns in such a way that the height, \(h \mathrm {~m}\), of a passenger above the ground is given by the formula \(h = 60 ( 1 - \cos k t )\). In this formula, \(k\) is a constant, \(t\) is the time in minutes that has elapsed since the passenger started the ride at ground level and \(k t\) is measured in radians.

1. Find the greatest height of the passenger above the ground. One complete revolution of the wheel takes 30 minutes.
2. Show that \(k = \frac { 1 } { 15 } \pi\).
3. Find the time for which the passenger is above a height of 90 m .

7 The point \(C\) lies on the perpendicular bisector of the line joining the points \(A ( 4,6 )\) and \(B ( 10,2 )\). \(C\) also lies on the line parallel to \(A B\) through \(( 3,11 )\).

1. Find the equation of the perpendicular bisector of \(A B\).
2. Calculate the coordinates of \(C\).

8

1. The first, second and last terms in an arithmetic progression are 56, 53 and -22 respectively. Find the sum of all the terms in the progression.
2. The first, second and third terms of a geometric progression are \(2 k + 6,2 k\) and \(k + 2\) respectively, where \(k\) is a positive constant.
   1. Find the value of \(k\).
   2. Find the sum to infinity of the progression.

9 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }$$

1. Use a vector method to find angle \(A O B\). The point \(C\) is such that \(\overrightarrow { A B } = \overrightarrow { B C }\).
2. Find the unit vector in the direction of \(\overrightarrow { O C }\).
3. Show that triangle \(O A C\) is isosceles.

10 The equation of a curve is \(y = \frac { 4 } { 2 x - 1 }\).

1. Find, showing all necessary working, the volume obtained when the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. Given that the line \(2 y = x + c\) is a normal to the curve, find the possible values of the constant \(c\).

11 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 6 x + 5\) for \(x \in \mathbb { R }\).

1. Find the set of values of \(p\) for which the equation \(\mathrm { f } ( x ) = p\) has no real roots. The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 6 x + 5\) for \(0 \leqslant x \leqslant 4\).
2. Express \(\mathrm { g } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
3. Find the range of g . The function h is defined by \(\mathrm { h } : x \mapsto 2 x ^ { 2 } - 6 x + 5\) for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.
4. State the smallest value of \(k\) for which h has an inverse.
5. For this value of \(k\), find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).