CAIE P1 (Pure Mathematics 1) 2009 November

1 Solve the equation \(3 \tan \left( 2 x + 15 ^ { \circ } \right) = 4\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

2 The equation of a curve is \(y = 3 \cos 2 x\). The equation of a line is \(x + 2 y = \pi\). On the same diagram, sketch the curve and the line for \(0 \leqslant x \leqslant \pi\).

3

1. Find the first 3 terms in the expansion of \(( 2 - x ) ^ { 6 }\) in ascending powers of \(x\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + 2 x + a x ^ { 2 } \right) ( 2 - x ) ^ { 6 }\) is 48 , find the value of the constant \(a\).

4 The equation of a curve is \(y = x ^ { 4 } + 4 x + 9\).

1. Find the coordinates of the stationary point on the curve and determine its nature.
2. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 1\).

5
\includegraphics[max width=\textwidth, alt={}, center]{7f66531c-2de6-44b7-9c48-7944acfea4d9-2_439_787_1174_678} The diagram shows a semicircle \(A B C\) with centre \(O\) and radius 6 cm . The point \(B\) is such that angle \(B O A\) is \(90 ^ { \circ }\) and \(B D\) is an arc of a circle with centre \(A\). Find

1. the length of the \(\operatorname { arc } B D\),
2. the area of the shaded region.

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k - 2 x\), where \(k\) is a constant.

1. Given that the tangents to the curve at the points where \(x = 2\) and \(x = 3\) are perpendicular, find the value of \(k\).
2. Given also that the curve passes through the point \(( 4,9 )\), find the equation of the curve.

7 The equation of a curve is \(y = \frac { 12 } { x ^ { 2 } + 3 }\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the equation of the normal to the curve at the point \(P ( 1,3 )\).
3. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.012 units per second. Find the rate of change of the \(y\)-coordinate as the point passes through \(P\).

8 The first term of an arithmetic progression is 8 and the common difference is \(d\), where \(d \neq 0\). The first term, the fifth term and the eighth term of this arithmetic progression are the first term, the second term and the third term, respectively, of a geometric progression whose common ratio is \(r\).

1. Write down two equations connecting \(d\) and \(r\). Hence show that \(r = \frac { 3 } { 4 }\) and find the value of \(d\).
2. Find the sum to infinity of the geometric progression.
3. Find the sum of the first 8 terms of the arithmetic progression.

9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2
3
- 6 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 0
- 6
8 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } - 2
5
- 2 \end{array} \right)$$

1. Find angle \(A O B\).
2. Find the vector which is in the same direction as \(\overrightarrow { A C }\) and has magnitude 30 .
3. Find the value of the constant \(p\) for which \(\overrightarrow { O A } + p \overrightarrow { O B }\) is perpendicular to \(\overrightarrow { O C }\).

10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x + 1 , \quad x \in \mathbb { R } , \quad x > 0
& \mathrm {~g} : x \mapsto \frac { 2 x - 1 } { x + 3 } , \quad x \in \mathbb { R } , \quad x \neq - 3 \end{aligned}$$

1. Solve the equation \(\operatorname { gf } ( x ) = x\).
2. Express \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
3. Show that the equation \(\mathrm { g } ^ { - 1 } ( x ) = x\) has no solutions.
4. Sketch in a single diagram the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.