CAIE P1 (Pure Mathematics 1) 2011 June

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

2 The volume of a spherical balloon is increasing at a constant rate of \(50 \mathrm {~cm} ^ { 3 }\) per second. Find the rate of increase of the radius when the radius is 10 cm . [Volume of a sphere \(= \frac { 4 } { 3 } \pi r ^ { 3 }\).]

3

1. Sketch the curve \(y = ( x - 2 ) ^ { 2 }\).
2. The region enclosed by the curve, the \(x\)-axis and the \(y\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume obtained, giving your answer in terms of \(\pi\).

4
\includegraphics[max width=\textwidth, alt={}, center]{53839c8c-07ea-4545-9c00-a6884aa2afc3-2_750_855_902_646} The diagram shows a prism \(A B C D P Q R S\) with a horizontal square base \(A P S D\) with sides of length 6 cm . The cross-section \(A B C D\) is a trapezium and is such that the vertical edges \(A B\) and \(D C\) are of lengths 5 cm and 2 cm respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(A D , A P\) and \(A B\) respectively.

1. Express each of the vectors \(\overrightarrow { C P }\) and \(\overrightarrow { C Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to calculate angle \(P C Q\).

5

1. Show that the equation \(2 \tan ^ { 2 } \theta \sin ^ { 2 } \theta = 1\) can be written in the form $$2 \sin ^ { 4 } \theta + \sin ^ { 2 } \theta - 1 = 0 .$$
2. Hence solve the equation \(2 \tan ^ { 2 } \theta \sin ^ { 2 } \theta = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

6 The variables \(x , y\) and \(z\) can take only positive values and are such that $$z = 3 x + 2 y \quad \text { and } \quad x y = 600 .$$

1. Show that \(z = 3 x + \frac { 1200 } { x }\).
2. Find the stationary value of \(z\) and determine its nature.

7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { ( 1 + 2 x ) ^ { 2 } }\) and the point \(\left( 1 , \frac { 1 } { 2 } \right)\) lies on the curve.

1. Find the equation of the curve.
2. Find the set of values of \(x\) for which the gradient of the curve is less than \(\frac { 1 } { 3 }\).

8 A television quiz show takes place every day. On day 1 the prize money is \(
) 1000$. If this is not won the prize money is increased for day 2 . The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money. Model 1: Increase the prize money by \(
) 1000$ each day.
Model 2: Increase the prize money by \(10 \%\) each day.
On each day that the prize money is not won the television company makes a donation to charity. The amount donated is \(5 \%\) of the value of the prize on that day. After 40 days the prize money has still not been won. Calculate the total amount donated to charity

1. if Model 1 is used,
2. if Model 2 is used.

9
\includegraphics[max width=\textwidth, alt={}, center]{53839c8c-07ea-4545-9c00-a6884aa2afc3-3_387_1175_1781_486} In the diagram, \(O A B\) is an isosceles triangle with \(O A = O B\) and angle \(A O B = 2 \theta\) radians. Arc \(P S T\) has centre \(O\) and radius \(r\), and the line \(A S B\) is a tangent to the \(\operatorname { arc } P S T\) at \(S\).

1. Find the total area of the shaded regions in terms of \(r\) and \(\theta\).
2. In the case where \(\theta = \frac { 1 } { 3 } \pi\) and \(r = 6\), find the total perimeter of the shaded regions, leaving your answer in terms of \(\sqrt { } 3\) and \(\pi\).

10

1. Express \(2 x ^ { 2 } - 4 x + 1\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point, \(A\), on the curve \(y = 2 x ^ { 2 } - 4 x + 1\). The line \(x - y + 4 = 0\) intersects the curve \(y = 2 x ^ { 2 } - 4 x + 1\) at points \(P\) and \(Q\). It is given that the coordinates of \(P\) are \(( 3,7 )\).
2. Find the coordinates of \(Q\).
3. Find the equation of the line joining \(Q\) to the mid-point of \(A P\).

11 Functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x + 1
& \mathrm {~g} : x \mapsto x ^ { 2 } - 2 \end{aligned}$$

1. Find and simplify expressions for \(\mathrm { fg } ( x )\) and \(\mathrm { gf } ( x )\).
2. Hence find the value of \(a\) for which \(\mathrm { fg } ( a ) = \mathrm { gf } ( a )\).
3. Find the value of \(b ( b \neq a )\) for which \(\mathrm { g } ( b ) = b\).
4. Find and simplify an expression for \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\). The function h is defined by $$\mathrm { h } : x \mapsto x ^ { 2 } - 2 , \quad \text { for } x \leqslant 0$$
5. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).