CAIE P1 (Pure Mathematics 1) 2008 June

1 In the triangle \(A B C , A B = 12 \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A C B = 45 ^ { \circ }\). Find the exact length of \(B C\).

2

1. Show that the equation \(2 \tan ^ { 2 } \theta \cos \theta = 3\) can be written in the form \(2 \cos ^ { 2 } \theta + 3 \cos \theta - 2 = 0\).
2. Hence solve the equation \(2 \tan ^ { 2 } \theta \cos \theta = 3\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

3

1. Find the first 3 terms in the expansion, in ascending powers of \(x\), of \(\left( 2 + x ^ { 2 } \right) ^ { 5 }\).
2. Hence find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) ^ { 2 } \left( 2 + x ^ { 2 } \right) ^ { 5 }\).

4 The equation of a curve \(C\) is \(y = 2 x ^ { 2 } - 8 x + 9\) and the equation of a line \(L\) is \(x + y = 3\).

1. Find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\).
2. Show that one of these points is also the stationary point of \(C\).

5
\includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-2_543_883_1274_630} The diagram shows a circle with centre \(O\) and radius 5 cm . The point \(P\) lies on the circle, \(P T\) is a tangent to the circle and \(P T = 12 \mathrm {~cm}\). The line \(O T\) cuts the circle at the point \(Q\).

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

6 The function f is such that \(\mathrm { f } ( x ) = ( 3 x + 2 ) ^ { 3 } - 5\) for \(x \geqslant 0\).

1. Obtain an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why f is an increasing function.
2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).

7 The first term of a geometric progression is 81 and the fourth term is 24 . Find

1. the common ratio of the progression,
2. the sum to infinity of the progression. The second and third terms of this geometric progression are the first and fourth terms respectively of an arithmetic progression.
3. Find the sum of the first ten terms of the arithmetic progression.

8 Functions f and g are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 4 x - 2 k & \text { for } x \in \mathbb { R } , \text { where } k \text { is a constant, }
\mathrm { g } : x \mapsto \frac { 9 } { 2 - x } & \text { for } x \in \mathbb { R } , x \neq 2 . \end{array}$$

1. Find the values of \(k\) for which the equation \(\mathrm { fg } ( x ) = x\) has two equal roots.
2. Determine the roots of the equation \(\operatorname { fg } ( x ) = x\) for the values of \(k\) found in part (i).

9
\includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-3_791_885_1281_630} The diagram shows a curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { k } { x ^ { 3 } }\), where \(k\) is a constant. The curve passes through the points \(( 1,18 )\) and \(( 4,3 )\).

1. Show, by integration, that the equation of the curve is \(y = \frac { 16 } { x ^ { 2 } } + 2\). The point \(P\) lies on the curve and has \(x\)-coordinate 1.6.
2. Find the area of the shaded region.

10 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and \(3 \mathbf { i } - 2 \mathbf { j } + p \mathbf { k }\) respectively.

1. Find the value of \(p\) for which \(O A\) and \(O B\) are perpendicular.
2. In the case where \(p = 6\), use a scalar product to find angle \(A O B\), correct to the nearest degree.
3. Express the vector \(\overrightarrow { A B }\) is terms of \(p\) and hence find the values of \(p\) for which the length of \(A B\) is 3.5 units.

11
\includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-4_563_965_813_591} In the diagram, the points \(A\) and \(C\) lie on the \(x\) - and \(y\)-axes respectively and the equation of \(A C\) is \(2 y + x = 16\). The point \(B\) has coordinates ( 2,2 ). The perpendicular from \(B\) to \(A C\) meets \(A C\) at the point \(X\).

1. Find the coordinates of \(X\). The point \(D\) is such that the quadrilateral \(A B C D\) has \(A C\) as a line of symmetry.
2. Find the coordinates of \(D\).
3. Find, correct to 1 decimal place, the perimeter of \(A B C D\). \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. }