CAIE P1 (Pure Mathematics 1) 2014 June

1 Find the coordinates of the point at which the perpendicular bisector of the line joining (2, 7) to \(( 10,3 )\) meets the \(x\)-axis.

2 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) \left( \frac { x } { 2 } - \frac { 4 } { x } \right) ^ { 6 }\).

3 The reflex angle \(\theta\) is such that \(\cos \theta = k\), where \(0 < k < 1\).

1. Find an expression, in terms of \(k\), for
   (a) \(\sin \theta\),
   (b) \(\tan \theta\).
2. Explain why \(\sin 2 \theta\) is negative for \(0 < k < 1\).

4
\includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-2_358_618_1082_762} The diagram shows a sector of a circle with radius \(r \mathrm {~cm}\) and centre \(O\). The chord \(A B\) divides the sector into a triangle \(A O B\) and a segment \(A X B\). Angle \(A O B\) is \(\theta\) radians.

1. In the case where the areas of the triangle \(A O B\) and the segment \(A X B\) are equal, find the value of the constant \(p\) for which \(\theta = p \sin \theta\).
2. In the case where \(r = 8\) and \(\theta = 2.4\), find the perimeter of the segment \(A X B\).

5

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

6 The 1st, 2nd and 3rd terms of a geometric progression are the 1st, 9th and 21st terms respectively of an arithmetic progression. The 1st term of each progression is 8 and the common ratio of the geometric progression is \(r\), where \(r \neq 1\). Find

1. the value of \(r\),
2. the 4th term of each progression.

7
\includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-3_394_750_260_699} The diagram shows a trapezium \(A B C D\) in which \(B A\) is parallel to \(C D\). The position vectors of \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 3
4
0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 1
3
2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 4
5
6 \end{array} \right)$$

1. Use a scalar product to show that \(A B\) is perpendicular to \(B C\).
2. Given that the length of \(C D\) is 12 units, find the position vector of \(D\).

8 The equation of a curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 1\). Given that the curve has a minimum point at \(( 3 , - 10 )\), find the coordinates of the maximum point.

9
\includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-3_849_565_1466_790} The diagram shows part of the curve \(y = 8 - \sqrt { } ( 4 - x )\) and the tangent to the curve at \(P ( 3,7 )\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the equation of the tangent to the curve at \(P\) in the form \(y = m x + c\).
3. Find, showing all necessary working, the area of the shaded region.

10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x - 3 , \quad x \in \mathbb { R } ,
& \mathrm {~g} : x \mapsto x ^ { 2 } + 4 x , \quad x \in \mathbb { R } . \end{aligned}$$

1. Solve the equation \(\mathrm { ff } ( x ) = 11\).
2. Find the range of g .
3. Find the set of values of \(x\) for which \(\mathrm { g } ( x ) > 12\).
4. Find the value of the constant \(p\) for which the equation \(\mathrm { gf } ( x ) = p\) has two equal roots. Function h is defined by \(\mathrm { h } : x \mapsto x ^ { 2 } + 4 x\) for \(x \geqslant k\), and it is given that h has an inverse.
5. State the smallest possible value of \(k\).
6. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).