CAIE P1 (Pure Mathematics 1) 2015 November

1 In the expansion of \(\left( 1 - \frac { 2 x } { a } \right) ( a + x ) ^ { 5 }\), where \(a\) is a non-zero constant, show that the coefficient of \(x ^ { 2 }\) is zero.

2 The function f is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } - 7\) and \(\mathrm { f } ( 3 ) = 5\). Find \(\mathrm { f } ( x )\).

3 Solve the equation \(\sin ^ { - 1 } \left( 4 x ^ { 4 } + x ^ { 2 } \right) = \frac { 1 } { 6 } \pi\).

4

1. Show that the equation \(\frac { 4 \cos \theta } { \tan \theta } + 15 = 0\) can be expressed as $$4 \sin ^ { 2 } \theta - 15 \sin \theta - 4 = 0$$
2. Hence solve the equation \(\frac { 4 \cos \theta } { \tan \theta } + 15 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

5 A curve has equation \(y = \frac { 8 } { x } + 2 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary points and state, with a reason, the nature of each stationary point.

6 A curve has equation \(y = x ^ { 2 } - x + 3\) and a line has equation \(y = 3 x + a\), where \(a\) is a constant.

1. Show that the \(x\)-coordinates of the points of intersection of the line and the curve are given by the equation \(x ^ { 2 } - 4 x + ( 3 - a ) = 0\).
2. For the case where the line intersects the curve at two points, it is given that the \(x\)-coordinate of one of the points of intersection is - 1 . Find the \(x\)-coordinate of the other point of intersection.
3. For the case where the line is a tangent to the curve at a point \(P\), find the value of \(a\) and the coordinates of \(P\).

7
\includegraphics[max width=\textwidth, alt={}, center]{a9e04003-1e43-40c4-991a-36aa3a93654b-3_718_899_258_621} The diagram shows a circle with centre \(A\) and radius \(r\). Diameters \(C A D\) and \(B A E\) are perpendicular to each other. A larger circle has centre \(B\) and passes through \(C\) and \(D\).

1. Show that the radius of the larger circle is \(r \sqrt { } 2\).
2. Find the area of the shaded region in terms of \(r\).

8 The first term of a progression is \(4 x\) and the second term is \(x ^ { 2 }\).

1. For the case where the progression is arithmetic with a common difference of 12 , find the possible values of \(x\) and the corresponding values of the third term.
2. For the case where the progression is geometric with a sum to infinity of 8 , find the third term.

9

1. Express \(- x ^ { 2 } + 6 x - 5\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants. The function \(\mathrm { f } : x \mapsto - x ^ { 2 } + 6 x - 5\) is defined for \(x \geqslant m\), where \(m\) is a constant.
2. State the smallest value of \(m\) for which f is one-one.
3. For the case where \(m = 5\), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
   [0pt] [Questions 10 and 11 are printed on the next page.] \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. } \includegraphics[max width=\textwidth, alt={}, center]{a9e04003-1e43-40c4-991a-36aa3a93654b-4_773_641_260_753} The diagram shows a cuboid \(O A B C P Q R S\) with a horizontal base \(O A B C\) in which \(A B = 6 \mathrm {~cm}\) and \(O A = a \mathrm {~cm}\), where \(a\) is a constant. The height \(O P\) of the cuboid is 10 cm . The point \(T\) on \(B R\) is such that \(B T = 8 \mathrm {~cm}\), and \(M\) is the mid-point of \(A T\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O P\) respectively.

1. For the case where \(a = 2\), find the unit vector in the direction of \(\overrightarrow { P M }\).
2. For the case where angle \(A T P = \cos ^ { - 1 } \left( \frac { 2 } { 7 } \right)\), find the value of \(a\).

11
\includegraphics[max width=\textwidth, alt={}, center]{a9e04003-1e43-40c4-991a-36aa3a93654b-4_517_857_1594_644} The diagram shows part of the curve \(y = ( 1 + 4 x ) ^ { \frac { 1 } { 2 } }\) and a point \(P ( 6,5 )\) lying on the curve. The line \(P Q\) intersects the \(x\)-axis at \(Q ( 8,0 )\).

1. Show that \(P Q\) is a normal to the curve.
2. Find, showing all necessary working, the exact volume of revolution obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
   [0pt] [In part (ii) you may find it useful to apply the fact that the volume, \(V\), of a cone of base radius \(r\) and vertical height \(h\), is given by \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]