CAIE P1 (Pure Mathematics 1) 2004 June

1 A geometric progression has first term 64 and sum to infinity 256. Find

1. the common ratio,
2. the sum of the first ten terms.

2 Evaluate \(\int _ { 0 } ^ { 1 } \sqrt { } ( 3 x + 1 ) \mathrm { d } x\).

3

1. Show that the equation \(\sin ^ { 2 } \theta + 3 \sin \theta \cos \theta = 4 \cos ^ { 2 } \theta\) can be written as a quadratic equation in \(\tan \theta\).
2. Hence, or otherwise, solve the equation in part (i) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

4 Find the coefficient of \(x ^ { 3 }\) in the expansion of

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

5
\includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-2_501_682_1302_735} In the diagram, \(O C D\) is an isosceles triangle with \(O C = O D = 10 \mathrm {~cm}\) and angle \(C O D = 0.8\) radians. The points \(A\) and \(B\), on \(O C\) and \(O D\) respectively, are joined by an arc of a circle with centre \(O\) and radius 6 cm . Find

1. the area of the shaded region,
2. the perimeter of the shaded region.

6 The curve \(y = 9 - \frac { 6 } { x }\) and the line \(y + x = 8\) intersect at two points. Find

1. the coordinates of the two points,
2. the equation of the perpendicular bisector of the line joining the two points.

7
\includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-3_646_841_593_651} The diagram shows part of the graph of \(y = \frac { 18 } { x }\) and the normal to the curve at \(P ( 6,3 )\). This normal meets the \(x\)-axis at \(R\). The point \(Q\) on the \(x\)-axis and the point \(S\) on the curve are such that \(P Q\) and \(S R\) are parallel to the \(y\)-axis.

1. Find the equation of the normal at \(P\) and show that \(R\) is the point ( \(4 \frac { 1 } { 2 } , 0\) ).
2. Show that the volume of the solid obtained when the shaded region \(P Q R S\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis is \(18 \pi\).

8
\includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-4_543_511_264_817} The diagram shows a glass window consisting of a rectangle of height \(h \mathrm {~m}\) and width \(2 r \mathrm {~m}\) and a semicircle of radius \(r \mathrm {~m}\). The perimeter of the window is 8 m .

1. Express \(h\) in terms of \(r\).
2. Show that the area of the window, \(A \mathrm {~m} ^ { 2 }\), is given by $$A = 8 r - 2 r ^ { 2 } - \frac { 1 } { 2 } \pi r ^ { 2 } .$$ Given that \(r\) can vary,
3. find the value of \(r\) for which \(A\) has a stationary value,
4. determine whether this stationary value is a maximum or a minimum.

9 Relative to an origin \(O\), the position vectors of the points \(A , B , C\) and \(D\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 1
3
- 1 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 3
- 1
3 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { l } 4
2
p \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } - 1
0
q \end{array} \right) ,$$ where \(p\) and \(q\) are constants. Find

1. the unit vector in the direction of \(\overrightarrow { A B }\),
2. the value of \(p\) for which angle \(A O C = 90 ^ { \circ }\),
3. the values of \(q\) for which the length of \(\overrightarrow { A D }\) is 7 units.

10 The functions \(f\) and \(g\) are defined as follows: $$\begin{array} { l l } \mathrm { f } : x \mapsto x ^ { 2 } - 2 x , & x \in \mathbb { R } ,
\mathrm {~g} : x \mapsto 2 x + 3 , & x \in \mathbb { R } . \end{array}$$

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > 15\).
2. Find the range of f and state, with a reason, whether f has an inverse.
3. Show that the equation \(\operatorname { gf } ( x ) = 0\) has no real solutions.
4. Sketch, in a single diagram, the graphs of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\), making clear the relationship between the graphs.