CAIE P1 (Pure Mathematics 1) 2016 November

1 Find the set of values of \(k\) for which the curve \(y = k x ^ { 2 } - 3 x\) and the line \(y = x - k\) do not meet.

2 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - 3 x ) ^ { 6 } + ( 1 + a x ) ^ { 5 }\) is 100 . Find the value of the constant \(a\).

3 Showing all necessary working, solve the equation \(6 \sin ^ { 2 } x - 5 \cos ^ { 2 } x = 2 \sin ^ { 2 } x + \cos ^ { 2 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

4 The function f is such that \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 2\) for \(x > n\), where \(n\) is an integer. It is given that f is an increasing function. Find the least possible value of \(n\).

5
\includegraphics[max width=\textwidth, alt={}, center]{5fed65b9-a848-4343-858c-3cbac0608b24-2_609_533_938_804} The diagram shows a major arc \(A B\) of a circle with centre \(O\) and radius 6 cm . Points \(C\) and \(D\) on \(O A\) and \(O B\) respectively are such that the line \(A B\) is a tangent at \(E\) to the arc \(C E D\) of a smaller circle also with centre \(O\). Angle \(C O D = 1.8\) radians.

1. Show that the radius of the \(\operatorname { arc } C E D\) is 3.73 cm , correct to 3 significant figures.
2. Find the area of the shaded region.

6 Three points, \(A , B\) and \(C\), are such that \(B\) is the mid-point of \(A C\). The coordinates of \(A\) are ( \(2 , m\) ) and the coordinates of \(B\) are \(( n , - 6 )\), where \(m\) and \(n\) are constants.

1. Find the coordinates of \(C\) in terms of \(m\) and \(n\). The line \(y = x + 1\) passes through \(C\) and is perpendicular to \(A B\).
2. Find the values of \(m\) and \(n\).

7
\includegraphics[max width=\textwidth, alt={}, center]{5fed65b9-a848-4343-858c-3cbac0608b24-3_736_399_260_872} The diagram shows a triangular pyramid \(A B C D\). It is given that $$\overrightarrow { A B } = 3 \mathbf { i } + \mathbf { j } + \mathbf { k } , \quad \overrightarrow { A C } = \mathbf { i } - 2 \mathbf { j } - \mathbf { k } \quad \text { and } \quad \overrightarrow { A D } = \mathbf { i } + 4 \mathbf { j } - 7 \mathbf { k }$$

1. Verify, showing all necessary working, that each of the angles \(D A B , D A C\) and \(C A B\) is \(90 ^ { \circ }\).
2. Find the exact value of the area of the triangle \(A B C\), and hence find the exact value of the volume of the pyramid.
   [0pt] [The volume \(V\) of a pyramid of base area \(A\) and vertical height \(h\) is given by \(V = \frac { 1 } { 3 } A h\).]

8

1. Express \(4 x ^ { 2 } + 12 x + 10\) in the form \(( a x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
2. Functions f and g are both defined for \(x > 0\). It is given that \(\mathrm { f } ( x ) = x ^ { 2 } + 1\) and \(\mathrm { fg } ( x ) = 4 x ^ { 2 } + 12 x + 10\). Find \(\mathrm { g } ( x )\).
3. Find \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and give the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
   (a) Two convergent geometric progressions, \(P\) and \(Q\), have the same sum to infinity. The first and second terms of \(P\) are 6 and \(6 r\) respectively. The first and second terms of \(Q\) are 12 and \(- 12 r\) respectively. Find the value of the common sum to infinity.
   (b) The first term of an arithmetic progression is \(\cos \theta\) and the second term is \(\cos \theta + \sin ^ { 2 } \theta\), where \(0 \leqslant \theta \leqslant \pi\). The sum of the first 13 terms is 52 . Find the possible values of \(\theta\).

10 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { a } x ^ { - \frac { 1 } { 2 } } + a x ^ { - \frac { 3 } { 2 } }\), where \(a\) is a positive constant. The point \(A \left( a ^ { 2 } , 3 \right)\) lies on the curve. Find, in terms of \(a\),

1. the equation of the tangent to the curve at \(A\), simplifying your answer,
2. the equation of the curve. It is now given that \(B ( 16,8 )\) also lies on the curve.
3. Find the value of \(a\) and, using this value, find the distance \(A B\).

11 A curve has equation \(y = ( k x - 3 ) ^ { - 1 } + ( k x - 3 )\), where \(k\) is a non-zero constant.

1. Find the \(x\)-coordinates of the stationary points in terms of \(k\), and determine the nature of each stationary point, justifying your answers.
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{5fed65b9-a848-4343-858c-3cbac0608b24-4_556_855_1032_685} The diagram shows part of the curve for the case when \(k = 1\). Showing all necessary working, find the volume obtained when the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), shown shaded in the diagram, is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. \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. }