CAIE P1 (Pure Mathematics 1) 2016 November

1

1. Express \(x ^ { 2 } + 6 x + 2\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. Hence, or otherwise, find the set of values of \(x\) for which \(x ^ { 2 } + 6 x + 2 > 9\).

2 Find the term independent of \(x\) in the expansion of \(\left( 2 x + \frac { 1 } { 2 x ^ { 3 } } \right) ^ { 8 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-2_515_750_669_699} In the diagram \(O C A\) and \(O D B\) are radii of a circle with centre \(O\) and radius \(2 r \mathrm {~cm}\). Angle \(A O B = \alpha\) radians. \(C D\) and \(A B\) are arcs of circles with centre \(O\) and radii \(r \mathrm {~cm}\) and \(2 r \mathrm {~cm}\) respectively. The perimeter of the shaded region \(A B D C\) is \(4.4 r \mathrm {~cm}\).

1. Find the value of \(\alpha\).
2. It is given that the area of the shaded region is \(30 \mathrm {~cm} ^ { 2 }\). Find the value of \(r\). \(4 C\) is the mid-point of the line joining \(A ( 14 , - 7 )\) to \(B ( - 6,3 )\). The line through \(C\) perpendicular to \(A B\) crosses the \(y\)-axis at \(D\).

1. Find the equation of the line \(C D\), giving your answer in the form \(y = m x + c\).
2. Find the distance \(A D\).

5 The sum of the 1st and 2nd terms of a geometric progression is 50 and the sum of the 2nd and 3rd terms is 30 . Find the sum to infinity.

6

1. Show that \(\cos ^ { 4 } x \equiv 1 - 2 \sin ^ { 2 } x + \sin ^ { 4 } x\).
2. Hence, or otherwise, solve the equation \(8 \sin ^ { 4 } x + \cos ^ { 4 } x = 2 \cos ^ { 2 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-3_704_558_258_790} The diagram shows parts of the curves \(y = ( 2 x - 1 ) ^ { 2 }\) and \(y ^ { 2 } = 1 - 2 x\), intersecting at points \(A\) and \(B\).

1. State the coordinates of \(A\).
2. Find, showing all necessary working, the area of the shaded region.

8 The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) = \frac { 4 } { x } - 2 \quad \text { for } x > 0 \\ & \mathrm {~g} ( x ) = \frac { 4 } { 5 x + 2 } \quad \text { for } x \geqslant 0 \end{aligned}$$

1. Find and simplify an expression for \(\mathrm { fg } ( x )\) and state the range of fg.
2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { g } ^ { - 1 }\).

9 \includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-4_724_1488_257_330} The diagram shows a cuboid \(O A B C D E F G\) with a horizontal base \(O A B C\) in which \(O A = 4 \mathrm {~cm}\) and \(A B = 15 \mathrm {~cm}\). The height \(O D\) of the cuboid is 2 cm . The point \(X\) on \(A B\) is such that \(A X = 5 \mathrm {~cm}\) and the point \(P\) on \(D G\) is such that \(D P = p \mathrm {~cm}\), where \(p\) is a constant. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively.

1. Find the possible values of \(p\) such that angle \(O P X = 90 ^ { \circ }\).
2. For the case where \(p = 9\), find the unit vector in the direction of \(\overrightarrow { X P }\).
3. A point \(Q\) lies on the face \(C B F G\) and is such that \(X Q\) is parallel to \(A G\). Find \(\overrightarrow { X Q }\).

10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { \frac { 1 } { 2 } } - 2 x ^ { - \frac { 1 } { 2 } }\). The point \(A\) is the only point on the curve at which the gradient is - 1 .

1. Find the \(x\)-coordinate of \(A\).
2. Given that the curve also passes through the point \(( 4,10 )\), find the \(y\)-coordinate of \(A\), giving your answer as a fraction.

11 The point \(P ( 3,5 )\) lies on the curve \(y = \frac { 1 } { x - 1 } - \frac { 9 } { x - 5 }\).

1. Find the \(x\)-coordinate of the point where the normal to the curve at \(P\) intersects the \(x\)-axis.
2. Find the \(x\)-coordinate of each of the stationary points on the curve and determine the nature of each stationary point, justifying your answers. {www.cie.org.uk} after the live examination series. }