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

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
   1. the \(x\)-coordinates of \(A\) and \(B\),
   2. the equation of the tangent to the curve at \(B\),
   3. the acute angle, in degrees correct to 1 decimal place, between this tangent and the line \(2 y = x + 5\).
   4. Determine the set of values of \(k\) for which the line \(2 y = x + k\) does not intersect the curve \(y = x ^ { 2 } - 4 x + 7\).