CAIE P1 (Pure Mathematics 1) 2012 June

1
\includegraphics[max width=\textwidth, alt={}, center]{fa90db86-a73a-40db-b416-3c9f470fa207-2_618_533_246_808} The diagram shows the region enclosed by the curve \(y = \frac { 6 } { 2 x - 3 }\), the \(x\)-axis and the lines \(x = 2\) and \(x = 3\). Find, in terms of \(\pi\), the volume obtained when this region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

2 The equation of a curve is \(y = 4 \sqrt { } x + \frac { 2 } { \sqrt { } x }\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.12 units per second. Find the rate of change of the \(y\)-coordinate when \(x = 4\).

3 The coefficient of \(x ^ { 3 }\) in the expansion of \(( a + x ) ^ { 5 } + ( 2 - x ) ^ { 6 }\) is 90 . Find the value of the positive constant \(a\).

4 The point \(A\) has coordinates \(( - 1 , - 5 )\) and the point \(B\) has coordinates \(( 7,1 )\). The perpendicular bisector of \(A B\) meets the \(x\)-axis at \(C\) and the \(y\)-axis at \(D\). Calculate the length of \(C D\).

1. Prove the identity \(\tan x + \frac { 1 } { \tan x } \equiv \frac { 1 } { \sin x \cos x }\).
2. Solve the equation \(\frac { 2 } { \sin x \cos x } = 1 + 3 \tan x\), for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{fa90db86-a73a-40db-b416-3c9f470fa207-3_446_645_258_751} The diagram shows a metal plate made by removing a segment from a circle with centre \(O\) and radius 8 cm . The line \(A B\) is a chord of the circle and angle \(A O B = 2.4\) radians. Find

1. the length of \(A B\),
2. the perimeter of the plate,
3. the area of the plate.

7

1. In an arithmetic progression, the sum of the first \(n\) terms, denoted by \(S _ { n }\), is given by $$S _ { n } = n ^ { 2 } + 8 n .$$ Find the first term and the common difference.
2. In a geometric progression, the second term is 9 less than the first term. The sum of the second and third terms is 30 . Given that all the terms of the progression are positive, find the first term.

8

1. Find the angle between the vectors \(3 \mathbf { i } - 4 \mathbf { k }\) and \(2 \mathbf { i } + 3 \mathbf { j } - 6 \mathbf { k }\). The vector \(\overrightarrow { O A }\) has a magnitude of 15 units and is in the same direction as the vector \(3 \mathbf { i } - 4 \mathbf { k }\). The vector \(\overrightarrow { O B }\) has a magnitude of 14 units and is in the same direction as the vector \(2 \mathbf { i } + 3 \mathbf { j } - 6 \mathbf { k }\).
2. Express \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Find the unit vector in the direction of \(\overrightarrow { A B }\).

9
\includegraphics[max width=\textwidth, alt={}, center]{fa90db86-a73a-40db-b416-3c9f470fa207-4_762_848_255_646} The diagram shows part of the curve \(y = - x ^ { 2 } + 8 x - 10\) which passes through the points \(A\) and \(B\). The curve has a maximum point at \(A\) and the gradient of the line \(B A\) is 2 .

1. Find the coordinates of \(A\) and \(B\).
2. Find \(\int y \mathrm {~d} x\) and hence evaluate the area of the shaded region.

10 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x + 5 & \text { for } x \in \mathbb { R } ,
\mathrm {~g} : x \mapsto \frac { 8 } { x - 3 } & \text { for } x \in \mathbb { R } , x \neq 3 \end{array}$$

1. Obtain expressions, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\), stating the value of \(x\) for which \(\mathrm { g } ^ { - 1 } ( x )\) is not defined.
2. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram, making clear the relationship between the two graphs.
3. Given that the equation \(\operatorname { fg } ( x ) = 5 - k x\), where \(k\) is a constant, has no solutions, find the set of possible values of \(k\).