CAIE P1 (Pure Mathematics 1) 2016 June

1 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 10 - 3 x , \quad x \in \mathbb { R } ,
& \mathrm {~g} : x \mapsto \frac { 10 } { 3 - 2 x } , \quad x \in \mathbb { R } , x \neq \frac { 3 } { 2 } \end{aligned}$$ Solve the equation \(\mathrm { ff } ( x ) = \mathrm { gf } ( 2 )\).

2 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 } { ( 5 - 2 x ) ^ { 2 } }\). Given that the curve passes through ( 2,7 ), find the equation of the curve.

3 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } - 5 \mathbf { j } - 2 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 4 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }$$ The point \(C\) is such that \(\overrightarrow { A B } = \overrightarrow { B C }\). Find the unit vector in the direction of \(\overrightarrow { O C }\).

4 Find the term that is independent of \(x\) in the expansion of

1. \(\left( x - \frac { 2 } { x } \right) ^ { 6 }\),
2. \(\left( 2 + \frac { 3 } { x ^ { 2 } } \right) \left( x - \frac { 2 } { x } \right) ^ { 6 }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-2_663_446_1562_847} In the diagram, triangle \(A B C\) is right-angled at \(C\) and \(M\) is the mid-point of \(B C\). It is given that angle \(A B C = \frac { 1 } { 3 } \pi\) radians and angle \(B A M = \theta\) radians. Denoting the lengths of \(B M\) and \(M C\) by \(x\),

1. find \(A M\) in terms of \(x\),
2. show that \(\theta = \frac { 1 } { 6 } \pi - \tan ^ { - 1 } \left( \frac { 1 } { 2 \sqrt { 3 } } \right)\).

6
\includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-3_552_734_255_703} The diagram shows a circle with radius \(r \mathrm {~cm}\) and centre \(O\). The line \(P T\) is the tangent to the circle at \(P\) and angle \(P O T = \alpha\) radians. The line \(O T\) meets the circle at \(Q\).

1. Express the perimeter of the shaded region \(P Q T\) in terms of \(r\) and \(\alpha\).
2. In the case where \(\alpha = \frac { 1 } { 3 } \pi\) and \(r = 10\), find the area of the shaded region correct to 2 significant figures.

7

1. Prove the identity \(\frac { 1 + \cos \theta } { 1 - \cos \theta } - \frac { 1 - \cos \theta } { 1 + \cos \theta } \equiv \frac { 4 } { \sin \theta \tan \theta }\).
2. Hence solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation $$\sin \theta \left( \frac { 1 + \cos \theta } { 1 - \cos \theta } - \frac { 1 - \cos \theta } { 1 + \cos \theta } \right) = 3 .$$

8 Three points have coordinates \(A ( 0,7 ) , B ( 8,3 )\) and \(C ( 3 k , k )\). Find the value of the constant \(k\) for which

1. \(C\) lies on the line that passes through \(A\) and \(B\),
2. \(C\) lies on the perpendicular bisector of \(A B\).

9 A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost.

1. On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day.
   (a) How many litres will be lost on the 30th day after filling?
   (b) The tank becomes empty during the \(n\)th day after filling. Find the value of \(n\).
2. Assume instead that 10 litres of water are lost on the first day and that the amount of water lost increases by \(10 \%\) on each succeeding day. Find what percentage of the original 2000 litres is left in the tank at the end of the 30th day after filling.
   [0pt] [Questions 10 and 11 are printed on the next page.]

10
\includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-4_687_488_262_826} The diagram shows the part of the curve \(y = \frac { 8 } { x } + 2 x\) for \(x > 0\), and the minimum point \(M\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\int y ^ { 2 } \mathrm {~d} x\).
2. Find the coordinates of \(M\) and determine the coordinates and nature of the stationary point on the part of the curve for which \(x < 0\).
3. Find 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.

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

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 3\).
2. Given that the line \(y = m x + c\) is a tangent to the curve \(y = \mathrm { f } ( x )\), show that \(4 c = m ^ { 2 } - 12 m + 16\). The function g is defined by \(\mathrm { g } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \geqslant k\), where \(k\) is a constant.
3. Express \(6 x - x ^ { 2 } - 5\) in the form \(a - ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
4. State the smallest value of \(k\) for which g has an inverse.
5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
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