CAIE P1 (Pure Mathematics 1) 2015 June

1 Express \(2 x ^ { 2 } - 12 x + 7\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.

2 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x + 1 ) ^ { \frac { 1 } { 2 } }\) and the point (4,7) lies on the curve. Find the equation of the curve.

3

1. Write down the first 4 terms, in ascending powers of \(x\), of the expansion of \(( a - x ) ^ { 5 }\).
2. The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - a x ) ( a - x ) ^ { 5 }\) is - 200 . Find the possible values of the constant \(a\).

4

1. Express the equation \(3 \sin \theta = \cos \theta\) in the form \(\tan \theta = k\) and solve the equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
2. Solve the equation \(3 \sin ^ { 2 } 2 x = \cos ^ { 2 } 2 x\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

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

6
1
2 \end{array} \right)$$

1. Show that angle \(A B C\) is \(90 ^ { \circ }\).
2. Find the area of triangle \(A B C\), giving your answer correct to 1 decimal place. 6
   \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-2_519_670_1640_735} The diagram shows the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\), where \(\mathrm { f } ^ { - 1 }\) is defined by \(\mathrm { f } ^ { - 1 } ( x ) = \frac { 1 - 5 x } { 2 x }\) for \(0 < x \leqslant 2\).
3. Find an expression for \(\mathrm { f } ( x )\) and state the domain of f .
4. The function g is defined by \(\mathrm { g } ( x ) = \frac { 1 } { x }\) for \(x \geqslant 1\). Find an expression for \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\), giving your answer in the form \(a x + b\), where \(a\) and \(b\) are constants to be found.

7 The point \(A\) has coordinates \(( p , 1 )\) and the point \(B\) has coordinates \(( 9,3 p + 1 )\), where \(p\) is a constant.

1. For the case where the distance \(A B\) is 13 units, find the possible values of \(p\).
2. For the case in which the line with equation \(2 x + 3 y = 9\) is perpendicular to \(A B\), find the value of \(p\).

8 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }\) for \(x > - 1\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. State, with a reason, whether f is an increasing function, a decreasing function or neither. The function g is defined by \(\mathrm { g } ( x ) = \frac { 1 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }\) for \(x < - 1\).
3. Find the coordinates of the stationary point on the curve \(y = \mathrm { g } ( x )\).

9

1. The first term of an arithmetic progression is - 2222 and the common difference is 17 . Find the value of the first positive term.
2. The first term of a geometric progression is \(\sqrt { } 3\) and the second term is \(2 \cos \theta\), where \(0 < \theta < \pi\). Find the set of values of \(\theta\) for which the progression is convergent.

10
\includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-3_682_1319_1525_413} Points \(A ( 2,9 )\) and \(B ( 3,0 )\) lie on the curve \(y = 9 + 6 x - 3 x ^ { 2 }\), as shown in the diagram. The tangent at \(A\) intersects the \(x\)-axis at \(C\). Showing all necessary working,

1. find the equation of the tangent \(A C\) and hence find the \(x\)-coordinate of \(C\),
2. find the area of the shaded region \(A B C\).
   [0pt] [Question 11 is printed on the next page.]

11
\includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-4_519_560_260_797} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). The point \(C\) on \(O B\) is such that angle \(A C O\) is a right angle. Angle \(A O B\) is \(\alpha\) radians and is such that \(A C\) divides the sector into two regions of equal area.

1. Show that \(\sin \alpha \cos \alpha = \frac { 1 } { 2 } \alpha\). It is given that the solution of the equation in part (i) is \(\alpha = 0.9477\), correct to 4 decimal places.
2. Find the ratio perimeter of region \(O A C\) : perimeter of region \(A C B\), giving your answer in the form \(k : 1\), where \(k\) is given correct to 1 decimal place.
3. Find angle \(A O B\) in degrees. \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.
   To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. 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. }