CAIE P1 (Pure Mathematics 1) 2013 November

1

1. Find the first three terms when \(( 2 + 3 x ) ^ { 6 }\) is expanded in ascending powers of \(x\).
2. In the expansion of \(( 1 + a x ) ( 2 + 3 x ) ^ { 6 }\), the coefficient of \(x ^ { 2 }\) is zero. Find the value of \(a\).

2 A curve has equation \(y = f ( x )\). It is given that \(f ^ { \prime } ( x ) = \frac { 1 } { \sqrt { } ( x + 6 ) } + \frac { 6 } { x ^ { 2 } }\) and that \(f ( 3 ) = 1\). Find \(f ( x )\).

3
\includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-2_397_949_657_596} The diagram shows a pyramid \(O A B C D\) in which the vertical edge \(O D\) is 3 units in length. The point \(E\) is the centre of the horizontal rectangular base \(O A B C\). The sides \(O A\) and \(A B\) have lengths of 6 units and 4 units respectively. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively.

1. Express each of the vectors \(\overrightarrow { D B }\) and \(\overrightarrow { D E }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to find angle \(B D E\).

4

1. Solve the equation \(4 \sin ^ { 2 } x + 8 \cos x - 7 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Hence find the solution of the equation \(4 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) + 8 \cos \left( \frac { 1 } { 2 } \theta \right) - 7 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

5 The function f is defined by $$\mathrm { f } : x \mapsto x ^ { 2 } + 1 \text { for } x \geqslant 0$$

1. Define in a similar way the inverse function \(\mathrm { f } ^ { - 1 }\).
2. Solve the equation \(\operatorname { ff } ( x ) = \frac { 185 } { 16 }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-3_412_629_258_758} The diagram shows a metal plate made by fixing together two pieces, \(O A B C D\) (shaded) and \(O A E D\) (unshaded). The piece \(O A B C D\) is a minor sector of a circle with centre \(O\) and radius \(2 r\). The piece \(O A E D\) is a major sector of a circle with centre \(O\) and radius \(r\). Angle \(A O D\) is \(\alpha\) radians. Simplifying your answers where possible, find, in terms of \(\alpha , \pi\) and \(r\),

1. the perimeter of the metal plate,
2. the area of the metal plate. It is now given that the shaded and unshaded pieces are equal in area.
3. Find \(\alpha\) in terms of \(\pi\).

7 The point \(A\) has coordinates ( \(- 1,6\) ) and the point \(B\) has coordinates (7,2).

1. Find the equation of the perpendicular bisector of \(A B\), giving your answer in the form \(y = m x + c\).
2. A point \(C\) on the perpendicular bisector has coordinates \(( p , q )\). The distance \(O C\) is 2 units, where \(O\) is the origin. Write down two equations involving \(p\) and \(q\) and hence find the coordinates of the possible positions of \(C\).

8
\includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-3_365_663_1813_740} The inside lane of a school running track consists of two straight sections each of length \(x\) metres, and two semicircular sections each of radius \(r\) metres, as shown in the diagram. The straight sections are perpendicular to the diameters of the semicircular sections. The perimeter of the inside lane is 400 metres.

1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the region enclosed by the inside lane is given by \(A = 400 r - \pi r ^ { 2 }\).
2. Given that \(x\) and \(r\) can vary, show that, when \(A\) has a stationary value, there are no straight sections in the track. Determine whether the stationary value is a maximum or a minimum. [5]

9

1. In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten terms is 1000 . Find the common difference and the first term.
2. A geometric progression has first term \(a\), common ratio \(r\) and sum to infinity 6. A second geometric progression has first term \(2 a\), common ratio \(r ^ { 2 }\) and sum to infinity 7 . Find the values of \(a\) and \(r\).

10
\includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-4_654_974_614_587} The diagram shows the curve \(y = ( 3 - 2 x ) ^ { 3 }\) and the tangent to the curve at the point \(\left( \frac { 1 } { 2 } , 8 \right)\).

1. Find the equation of this tangent, giving your answer in the form \(y = m x + c\).
2. Find the area of the shaded region.