CAIE P1 (Pure Mathematics 1) 2013 November

1 Given that \(\cos x = p\), where \(x\) is an acute angle in degrees, find, in terms of \(p\),

1. \(\sin x\),
2. \(\tan x\),
3. \(\tan \left( 90 ^ { \circ } - x \right)\).

2 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Fig. 1 shows a hollow cone with no base, made of paper. The radius of the cone is 6 cm and the height is 8 cm . The paper is cut from \(A\) to \(O\) and opened out to form the sector shown in Fig. 2. The circular bottom edge of the cone in Fig. 1 becomes the arc of the sector in Fig. 2. The angle of the sector is \(\theta\) radians. Calculate

1. the value of \(\theta\),
2. the area of paper needed to make the cone.

3 The equation of a curve is \(y = \frac { 2 } { \sqrt { } ( 5 x - 6 ) }\).

1. Find the gradient of the curve at the point where \(x = 2\).
2. Find \(\int \frac { 2 } { \sqrt { } ( 5 x - 6 ) } \mathrm { d } x\) and hence evaluate \(\int _ { 2 } ^ { 3 } \frac { 2 } { \sqrt { } ( 5 x - 6 ) } \mathrm { d } x\).

4 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } \quad \text { and } \quad \overrightarrow { O B } = 4 \mathbf { i } + p \mathbf { k } .$$

1. In the case where \(p = 6\), find the unit vector in the direction of \(\overrightarrow { A B }\).
2. Find the values of \(p\) for which angle \(A O B = \cos ^ { - 1 } \left( \frac { 1 } { 5 } \right)\).

5
\includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-3_636_811_255_667} The diagram shows a rectangle \(A B C D\) in which point \(A\) is ( 0,8 ) and point \(B\) is ( 4,0 ). The diagonal \(A C\) has equation \(8 y + x = 64\). Find, by calculation, the coordinates of \(C\) and \(D\).

6
\includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-3_465_663_1160_740} In the diagram, \(S\) is the point ( 0,12 ) and \(T\) is the point ( 16,0 ). The point \(Q\) lies on \(S T\), between \(S\) and \(T\), and has coordinates \(( x , y )\). The points \(P\) and \(R\) lie on the \(x\)-axis and \(y\)-axis respectively and \(O P Q R\) is a rectangle.

1. Show that the area, \(A\), of the rectangle \(O P Q R\) is given by \(A = 12 x - \frac { 3 } { 4 } x ^ { 2 }\).
2. Given that \(x\) can vary, find the stationary value of \(A\) and determine its nature.

7

1. An athlete runs the first mile of a marathon in 5 minutes. His speed reduces in such a way that each mile takes 12 seconds longer than the preceding mile.
   1. Given that the \(n\)th mile takes 9 minutes, find the value of \(n\).
   2. Assuming that the length of the marathon is 26 miles, find the total time, in hours and minutes, to complete the marathon.
2. The second and third terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression.

8 A function f is defined by \(\mathrm { f } : x \mapsto 3 \cos x - 2\) for \(0 \leqslant x \leqslant 2 \pi\).

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Find the range of f .
3. Sketch the graph of \(y = \mathrm { f } ( x )\). A function g is defined by \(\mathrm { g } : x \mapsto 3 \cos x - 2\) for \(0 \leqslant x \leqslant k\).
4. State the maximum value of \(k\) for which g has an inverse.
5. Obtain an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

9
\includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-4_584_670_881_740} The diagram shows part of the curve \(y = \frac { 8 } { x } + 2 x\) and three points \(A , B\) and \(C\) on the curve with \(x\)-coordinates 1, 2 and 5 respectively.

1. A point \(P\) moves along the curve in such a way that its \(x\)-coordinate increases at a constant rate of 0.04 units per second. Find the rate at which the \(y\)-coordinate of \(P\) is changing as \(P\) passes through \(A\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

10 A curve has equation \(y = 2 x ^ { 2 } - 3 x\).

1. Find the set of values of \(x\) for which \(y > 9\).
2. Express \(2 x ^ { 2 } - 3 x\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants, and state the coordinates of the vertex of the curve. The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 2 x ^ { 2 } - 3 x \quad \text { and } \quad \mathrm { g } ( x ) = 3 x + k$$ where \(k\) is a constant.
3. Find the value of \(k\) for which the equation \(\mathrm { gf } ( x ) = 0\) has equal roots.