CAIE P1 (Pure Mathematics 1) 2016 June

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

2 Solve the equation \(3 \sin ^ { 2 } \theta = 4 \cos \theta - 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-2_497_1106_554_515} The diagram shows part of the curve \(x = \frac { 12 } { y ^ { 2 } } - 2\). The shaded region is bounded by the curve, the \(y\)-axis and the lines \(y = 1\) and \(y = 2\). Showing all necessary working, find the volume, in terms of \(\pi\), when this shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis.

4 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 8 ( 3 x + 4 ) ^ { - \frac { 1 } { 2 } }\).

1. A point \(P\) moves along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of change of the \(y\)-coordinate as \(P\) crosses the \(y\)-axis. The curve intersects the \(y\)-axis where \(y = \frac { 4 } { 3 }\).
2. Find the equation of the curve.

5
\includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-2_364_625_1873_762} A farmer divides a rectangular piece of land into 8 equal-sized rectangular sheep pens as shown in the diagram. Each sheep pen measures \(x \mathrm {~m}\) by \(y \mathrm {~m}\) and is fully enclosed by metal fencing. The farmer uses 480 m of fencing.

1. Show that the total area of land used for the sheep pens, \(A \mathrm {~m} ^ { 2 }\), is given by $$A = 384 x - 9.6 x ^ { 2 }$$
2. Given that \(x\) and \(y\) can vary, find the dimensions of each sheep pen for which the value of \(A\) is a maximum. (There is no need to verify that the value of \(A\) is a maximum.)

6

1. Find the values of the constant \(m\) for which the line \(y = m x\) is a tangent to the curve \(y = 2 x ^ { 2 } - 4 x + 8\).
2. The function f is defined for \(x \in \mathbb { R }\) by \(\mathrm { f } ( x ) = x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants. The solutions of the equation \(\mathrm { f } ( x ) = 0\) are \(x = 1\) and \(x = 9\). Find
   1. the values of \(a\) and \(b\),
   2. the coordinates of the vertex of the curve \(y = \mathrm { f } ( x )\).

7
\includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-3_408_451_721_845} In the diagram, \(A O B\) is a quarter circle with centre \(O\) and radius \(r\). The point \(C\) lies on the arc \(A B\) and the point \(D\) lies on \(O B\). The line \(C D\) is parallel to \(A O\) and angle \(A O C = \theta\) radians.

1. Express the perimeter of the shaded region in terms of \(r , \theta\) and \(\pi\).
2. For the case where \(r = 5 \mathrm {~cm}\) and \(\theta = 0.6\), find the area of the shaded region.

8 A curve has equation \(y = 3 x - \frac { 4 } { x }\) and passes through the points \(A ( 1 , - 1 )\) and \(B ( 4,11 )\). At each of the points \(C\) and \(D\) on the curve, the tangent is parallel to \(A B\). Find the equation of the perpendicular bisector of \(C D\).

9

1. The first term of a geometric progression in which all the terms are positive is 50 . The third term is 32 . Find the sum to infinity of the progression.
2. The first three terms of an arithmetic progression are \(2 \sin x , 3 \cos x\) and ( \(\sin x + 2 \cos x\) ) respectively, where \(x\) is an acute angle.
   1. Show that \(\tan x = \frac { 4 } { 3 }\).
   2. Find the sum of the first twenty terms of the progression.

10 Relative to an origin \(O\), the position vectors of points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2
1
- 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 5
- 1
k \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2
6
- 3 \end{array} \right)$$ respectively, where \(k\) is a constant.

1. Find the value of \(k\) in the case where angle \(A O B = 90 ^ { \circ }\).
2. Find the possible values of \(k\) for which the lengths of \(A B\) and \(O C\) are equal. The point \(D\) is such that \(\overrightarrow { O D }\) is in the same direction as \(\overrightarrow { O A }\) and has magnitude 9 units. The point \(E\) is such that \(\overrightarrow { O E }\) is in the same direction as \(\overrightarrow { O C }\) and has magnitude 14 units.
3. Find the magnitude of \(\overrightarrow { D E }\) in the form \(\sqrt { } n\) where \(n\) is an integer.

11 The function f is defined by \(\mathrm { f } : x \mapsto 4 \sin x - 1\) for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. State the range of f .
2. Find the coordinates of the points at which the curve \(y = \mathrm { f } ( x )\) intersects the coordinate axes.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).
4. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\), stating both the domain and range of \(\mathrm { f } ^ { - 1 }\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
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