CAIE P1 (Pure Mathematics 1) 2012 June

1 Solve the equation \(\sin 2 x = 2 \cos 2 x\), for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

2 Find the coefficient of \(x ^ { 6 }\) in the expansion of \(\left( 2 x ^ { 3 } - \frac { 1 } { x ^ { 2 } } \right) ^ { 7 }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-2_584_659_575_742} In the diagram, \(A B C\) is an equilateral triangle of side 2 cm . The mid-point of \(B C\) is \(Q\). An arc of a circle with centre \(A\) touches \(B C\) at \(Q\), and meets \(A B\) at \(P\) and \(A C\) at \(R\). Find the total area of the shaded regions, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).

4 A watermelon is assumed to be spherical in shape while it is growing. Its mass, \(M \mathrm {~kg}\), and radius, \(r \mathrm {~cm}\), are related by the formula \(M = k r ^ { 3 }\), where \(k\) is a constant. It is also assumed that the radius is increasing at a constant rate of 0.1 centimetres per day. On a particular day the radius is 10 cm and the mass is 3.2 kg . Find the value of \(k\) and the rate at which the mass is increasing on this day.

5
\includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-2_636_947_1738_598} The diagram shows the curve \(y = 7 \sqrt { } x\) and the line \(y = 6 x + k\), where \(k\) is a constant. The curve and the line intersect at the points \(A\) and \(B\).

1. For the case where \(k = 2\), find the \(x\)-coordinates of \(A\) and \(B\).
2. Find the value of \(k\) for which \(y = 6 x + k\) is a tangent to the curve \(y = 7 \sqrt { } x\).

6 Two vectors \(\mathbf { u }\) and \(\mathbf { v }\) are such that \(\mathbf { u } = \left( \begin{array} { c } p ^ { 2 }
- 2
6 \end{array} \right)\) and \(\mathbf { v } = \left( \begin{array} { c } 2
p - 1
2 p + 1 \end{array} \right)\), where \(p\) is a constant.

1. Find the values of \(p\) for which \(\mathbf { u }\) is perpendicular to \(\mathbf { v }\).
2. For the case where \(p = 1\), find the angle between the directions of \(\mathbf { u }\) and \(\mathbf { v }\).

7

1. The first two terms of an arithmetic progression are 1 and \(\cos ^ { 2 } x\) respectively. Show that the sum of the first ten terms can be expressed in the form \(a - b \sin ^ { 2 } x\), where \(a\) and \(b\) are constants to be found.
2. The first two terms of a geometric progression are 1 and \(\frac { 1 } { 3 } \tan ^ { 2 } \theta\) respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).
   1. Find the set of values of \(\theta\) for which the progression is convergent.
   2. Find the exact value of the sum to infinity when \(\theta = \frac { 1 } { 6 } \pi\).

8 The function \(\mathrm { f } : x \mapsto x ^ { 2 } - 4 x + k\) is defined for the domain \(x \geqslant p\), where \(k\) and \(p\) are constants.

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b + k\), where \(a\) and \(b\) are constants.
2. State the range of f in terms of \(k\).
3. State the smallest value of \(p\) for which f is one-one.
4. For the value of \(p\) found in part (iii), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain \(\mathrm { f } ^ { - 1 }\), giving your answers in terms of \(k\).

9 The coordinates of \(A\) are \(( - 3,2 )\) and the coordinates of \(C\) are (5,6). The mid-point of \(A C\) is \(M\) and the perpendicular bisector of \(A C\) cuts the \(x\)-axis at \(B\).

1. Find the equation of \(M B\) and the coordinates of \(B\).
2. Show that \(A B\) is perpendicular to \(B C\).
3. Given that \(A B C D\) is a square, find the coordinates of \(D\) and the length of \(A D\).

10 It is given that a curve has equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + x\).

1. Find the set of values of \(x\) for which the gradient of the curve is less than 5 .
2. Find the values of \(\mathrm { f } ( x )\) at the two stationary points on the curve and determine the nature of each stationary point.

11
\includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-4_636_951_255_596} The diagram shows the line \(y = 1\) and part of the curve \(y = \frac { 2 } { \sqrt { } ( x + 1 ) }\).

1. Show that the equation \(y = \frac { 2 } { \sqrt { } ( x + 1 ) }\) can be written in the form \(x = \frac { 4 } { y ^ { 2 } } - 1\).
2. Find \(\int \left( \frac { 4 } { y ^ { 2 } } - 1 \right) \mathrm { d } y\). Hence find the area of the shaded region.
3. The shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis. Find the exact value of the volume of revolution obtained.