CAIE P1 (Pure Mathematics 1) 2009 November

1 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { \sqrt { x } } - x\). Given that the curve passes through the point (4,6), find the equation of the curve.

2

1. Find, in terms of the non-zero constant \(k\), the first 4 terms in the expansion of \(( k + x ) ^ { 8 }\) in ascending powers of \(x\).
2. Given that the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in this expansion are equal, find the value of \(k\).

3 A progression has a second term of 96 and a fourth term of 54. Find the first term of the progression in each of the following cases:

1. the progression is arithmetic,
2. the progression is geometric with a positive common ratio.

4 The function f is defined by f : \(x \mapsto 5 - 3 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\).

1. Find the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. State, with a reason, whether f has an inverse.

5

1. Prove the identity \(( \sin x + \cos x ) ( 1 - \sin x \cos x ) \equiv \sin ^ { 3 } x + \cos ^ { 3 } x\).
2. Solve the equation \(( \sin x + \cos x ) ( 1 - \sin x \cos x ) = 9 \sin ^ { 3 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-2_590_666_1720_737} In the diagram, \(O A B C D E F G\) is a cube in which each side has length 6 . Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively. The point \(P\) is such that \(\overrightarrow { A P } = \frac { 1 } { 3 } \overrightarrow { A B }\) and the point \(Q\) is the mid-point of \(D F\).

1. Express each of the vectors \(\overrightarrow { O Q }\) and \(\overrightarrow { P Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Find the angle \(O Q P\).

7
\includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-3_301_485_264_829} A piece of wire of length 50 cm is bent to form the perimeter of a sector \(P O Q\) of a circle. The radius of the circle is \(r \mathrm {~cm}\) and the angle \(P O Q\) is \(\theta\) radians (see diagram).

1. Express \(\theta\) in terms of \(r\) and show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the sector is given by $$A = 25 r - r ^ { 2 } .$$
2. Given that \(r\) can vary, find the stationary value of \(A\) and determine its nature.

8 The function f is such that \(\mathrm { f } ( x ) = \frac { 3 } { 2 x + 5 }\) for \(x \in \mathbb { R } , x \neq - 2.5\).

1. Obtain an expression for \(\mathrm { f } ^ { \prime } ( x )\) and explain why f is a decreasing function.
2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. A curve has the equation \(y = \mathrm { f } ( x )\). Find the volume obtained when the region bounded by the curve, the coordinate axes and the line \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

9
\includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-3_554_723_1557_712} The diagram shows a rectangle \(A B C D\). The point \(A\) is \(( 0 , - 2 )\) and \(C\) is \(( 12,14 )\). The diagonal \(B D\) is parallel to the \(x\)-axis.

1. Explain why the \(y\)-coordinate of \(D\) is 6 . The \(x\)-coordinate of \(D\) is \(h\).
2. Express the gradients of \(A D\) and \(C D\) in terms of \(h\).
3. Calculate the \(x\)-coordinates of \(D\) and \(B\).
4. Calculate the area of the rectangle \(A B C D\).

10
\includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-4_702_625_260_758}

1. The diagram shows the line \(2 y = x + 5\) and the curve \(y = x ^ { 2 } - 4 x + 7\), which intersect at the points \(A\) and \(B\). Find
   (a) the \(x\)-coordinates of \(A\) and \(B\),
   (b) the equation of the tangent to the curve at \(B\),
   (c) the acute angle, in degrees correct to 1 decimal place, between this tangent and the line \(2 y = x + 5\).
2. Determine the set of values of \(k\) for which the line \(2 y = x + k\) does not intersect the curve \(y = x ^ { 2 } - 4 x + 7\). \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. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }