CAIE P1 (Pure Mathematics 1) 2010 June

1

1. Show that the equation $$3 ( 2 \sin x - \cos x ) = 2 ( \sin x - 3 \cos x )$$ can be written in the form \(\tan x = - \frac { 3 } { 4 }\).
2. Solve the equation \(3 ( 2 \sin x - \cos x ) = 2 ( \sin x - 3 \cos x )\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

2
\includegraphics[max width=\textwidth, alt={}, center]{dcc8cfc5-7ed3-4e4d-9856-b12e38ac69ef-2_486_727_625_708} The diagram shows part of the curve \(y = \frac { a } { x }\), where \(a\) is a positive constant. Given that the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis is \(24 \pi\), find the value of \(a\).

3 The functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 4 x - 2 x ^ { 2 }
& \mathrm {~g} : x \mapsto 5 x + 3 \end{aligned}$$

1. Find the range of f .
2. Find the value of the constant \(k\) for which the equation \(\mathrm { gf } ( x ) = k\) has equal roots.
   \includegraphics[max width=\textwidth, alt={}, center]{dcc8cfc5-7ed3-4e4d-9856-b12e38ac69ef-2_607_780_1909_685} In the diagram, \(A\) is the point \(( - 1,3 )\) and \(B\) is the point \(( 3,1 )\). The line \(L _ { 1 }\) passes through \(A\) and is parallel to \(O B\). The line \(L _ { 2 }\) passes through \(B\) and is perpendicular to \(A B\). The lines \(L _ { 1 }\) and \(L _ { 2 }\) meet at \(C\). Find the coordinates of \(C\).

5 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 2
3
1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { l } 4
1
p \end{array} \right)$$

1. Find the value of \(p\) for which \(\overrightarrow { O A }\) is perpendicular to \(\overrightarrow { O B }\).
2. Find the values of \(p\) for which the magnitude of \(\overrightarrow { A B }\) is 7 .

6

1. Find the first 3 terms in the expansion of \(( 1 + a x ) ^ { 5 }\) in ascending powers of \(x\).
2. Given that there is no term in \(x\) in the expansion of \(( 1 - 2 x ) ( 1 + a x ) ^ { 5 }\), find the value of the constant \(a\).
3. For this value of \(a\), find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 - 2 x ) ( 1 + a x ) ^ { 5 }\).

7

1. Find the sum of all the multiples of 5 between 100 and 300 inclusive.
2. A geometric progression has a common ratio of \(- \frac { 2 } { 3 }\) and the sum of the first 3 terms is 35 . Find
   1. the first term of the progression,
   2. the sum to infinity.

8 A solid rectangular block has a square base of side \(x \mathrm {~cm}\). The height of the block is \(h \mathrm {~cm}\) and the total surface area of the block is \(96 \mathrm {~cm} ^ { 2 }\).

1. Express \(h\) in terms of \(x\) and show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the block is given by $$V = 24 x - \frac { 1 } { 2 } x ^ { 3 }$$ Given that \(x\) can vary,
2. find the stationary value of \(V\),
3. determine whether this stationary value is a maximum or a minimum.

9
\includegraphics[max width=\textwidth, alt={}, center]{dcc8cfc5-7ed3-4e4d-9856-b12e38ac69ef-4_723_919_248_612} The diagram shows the curve \(y = ( x - 2 ) ^ { 2 }\) and the line \(y + 2 x = 7\), which intersect at points \(A\) and \(B\). Find the area of the shaded region.

10 The equation of a curve is \(y = \frac { 1 } { 6 } ( 2 x - 3 ) ^ { 3 } - 4 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the equation of the tangent to the curve at the point where the curve intersects the \(y\)-axis.
3. Find the set of values of \(x\) for which \(\frac { 1 } { 6 } ( 2 x - 3 ) ^ { 3 } - 4 x\) is an increasing function of \(x\).

11 The function f : \(x \mapsto 4 - 3 \sin x\) is defined for the domain \(0 \leqslant x \leqslant 2 \pi\).

1. Solve the equation \(\mathrm { f } ( x ) = 2\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has no solution. The function \(\mathrm { g } : x \mapsto 4 - 3 \sin x\) is defined for the domain \(\frac { 1 } { 2 } \pi \leqslant x \leqslant A\).
4. State the largest value of \(A\) for which g has an inverse.
5. For this value of \(A\), find the value of \(\mathrm { g } ^ { - 1 } ( 3 )\). \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. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }