CAIE P1 (Pure Mathematics 1) 2016 March

1

1. Find the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) in the expansion of \(( 1 - 2 x ) ^ { 5 }\).
2. It is given that, when \(( 1 + p x ) ( 1 - 2 x ) ^ { 5 }\) is expanded, there is no term in \(x ^ { 5 }\). Find the value of the constant \(p\).

2 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - \frac { 2 } { x ^ { 3 } }\) passes through \(( - 1,3 )\). Find the equation of the curve.

3 The 12th term of an arithmetic progression is 17 and the sum of the first 31 terms is 1023. Find the 31st term.

4

1. Solve the equation \(\sin ^ { - 1 } ( 3 x ) = - \frac { 1 } { 3 } \pi\), giving the solution in an exact form.
2. Solve, by factorising, the equation \(2 \cos \theta \sin \theta - 2 \cos \theta - \sin \theta + 1 = 0\) for \(0 \leqslant \theta \leqslant \pi\).

5 Two points have coordinates \(A ( 5,7 )\) and \(B ( 9 , - 1 )\).

1. Find the equation of the perpendicular bisector of \(A B\). The line through \(C ( 1,2 )\) parallel to \(A B\) meets the perpendicular bisector of \(A B\) at the point \(X\).
2. Find, by calculation, the distance \(B X\).

6 A vacuum flask (for keeping drinks hot) is modelled as a closed cylinder in which the internal radius is \(r \mathrm {~cm}\) and the internal height is \(h \mathrm {~cm}\). The volume of the flask is \(1000 \mathrm {~cm} ^ { 3 }\). A flask is most efficient when the total internal surface area, \(A \mathrm {~cm} ^ { 2 }\), is a minimum.

1. Show that \(A = 2 \pi r ^ { 2 } + \frac { 2000 } { r }\).
2. Given that \(r\) can vary, find the value of \(r\), correct to 1 decimal place, for which \(A\) has a stationary value and verify that the flask is most efficient when \(r\) takes this value.

7
\includegraphics[max width=\textwidth, alt={}, center]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-3_529_698_260_721} The diagram shows a pyramid \(O A B C\) with a horizontal triangular base \(O A B\) and vertical height \(O C\). Angles \(A O B , B O C\) and \(A O C\) are each right angles. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O C\) respectively, with \(O A = 4\) units, \(O B = 2.4\) units and \(O C = 3\) units. The point \(P\) on \(C A\) is such that \(C P = 3\) units.

1. Show that \(\overrightarrow { C P } = 2.4 \mathbf { i } - 1.8 \mathbf { k }\).
2. Express \(\overrightarrow { O P }\) and \(\overrightarrow { B P }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Use a scalar product to find angle \(B P C\).

8 The function f is such that \(\mathrm { f } ( x ) = a ^ { 2 } x ^ { 2 } - a x + 3 b\) for \(x \leqslant \frac { 1 } { 2 a }\), where \(a\) and \(b\) are constants.

1. For the case where \(\mathrm { f } ( - 2 ) = 4 a ^ { 2 } - b + 8\) and \(\mathrm { f } ( - 3 ) = 7 a ^ { 2 } - b + 14\), find the possible values of \(a\) and \(b\).
2. For the case where \(a = 1\) and \(b = - 1\), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and give the domain of \(\mathrm { f } ^ { - 1 }\).

9

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_433_476_264_872} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} In Fig. 1, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). \(A X\) is the tangent at \(A\) to the arc \(A B\) and angle \(B A X = \alpha\).
   1. Show that angle \(A O B = 2 \alpha\).
   2. Find the area of the shaded segment in terms of \(r\) and \(\alpha\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_451_503_1162_861} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} In Fig. 2, \(A B C\) is an equilateral triangle of side 4 cm . The lines \(A X , B X\) and \(C X\) are tangents to the equal circular \(\operatorname { arcs } A B , B C\) and \(C A\). Use the results in part (a) to find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
   [0pt] [6]

10
\includegraphics[max width=\textwidth, alt={}, center]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-5_650_1038_260_550} The diagram shows part of the curve \(y = \frac { 1 } { 16 } ( 3 x - 1 ) ^ { 2 }\), which touches the \(x\)-axis at the point \(P\). The point \(Q ( 3,4 )\) lies on the curve and the tangent to the curve at \(Q\) crosses the \(x\)-axis at \(R\).

1. State the \(x\)-coordinate of \(P\). Showing all necessary working, find by calculation
2. the \(x\)-coordinate of \(R\),
3. the area of the shaded region \(P Q R\).