CAIE P1 (Pure Mathematics 1) 2022 March

1 A curve with equation \(y = \mathrm { f } ( x )\) is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { - \frac { 1 } { 3 } } - x ^ { \frac { 1 } { 3 } }\). It is given that \(\mathrm { f } ( 8 ) = 5\).
Find \(\mathrm { f } ( x )\).

2 A curve has equation \(y = x ^ { 2 } + 2 c x + 4\) and a straight line has equation \(y = 4 x + c\), where \(c\) is a constant. Find the set of values of \(c\) for which the curve and line intersect at two distinct points.

3 Find the term independent of \(x\) in each of the following expansions.

1. \(\left( 3 x + \frac { 2 } { x ^ { 2 } } \right) ^ { 6 }\)
2. \(\left( 3 x + \frac { 2 } { x ^ { 2 } } \right) ^ { 6 } \left( 1 - x ^ { 3 } \right)\)

4 The first term of a geometric progression and the first term of an arithmetic progression are both equal to \(a\). The third term of the geometric progression is equal to the second term of the arithmetic progression.
The fifth term of the geometric progression is equal to the sixth term of the arithmetic progression.
Given that the terms are all positive and not all equal, find the sum of the first twenty terms of the arithmetic progression in terms of \(a\).

5

1. Express \(2 x ^ { 2 } - 8 x + 14\) in the form \(2 \left[ ( x - a ) ^ { 2 } + b \right]\).
   The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } \quad \text { for } x \in \mathbb { R }
   & \mathrm {~g} ( x ) = 2 x ^ { 2 } - 8 x + 14 \quad \text { for } x \in \mathbb { R } \end{aligned}$$
2. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), making clear the order in which the transformations are applied.
   \includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-08_679_1043_260_552} The circle with equation \(( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 85\) and the straight line with equation \(y = 3 x - 20\) are shown in the diagram. The line intersects the circle at \(A\) and \(B\), and the centre of the circle is at \(C\).

1. Find, by calculation, the coordinates of \(A\) and \(B\).
2. Find an equation of the circle which has its centre at \(C\) and for which the line with equation \(y = 3 x - 20\) is a tangent to the circle.

7

1. Show that \(\frac { \sin \theta + 2 \cos \theta } { \cos \theta - 2 \sin \theta } - \frac { \sin \theta - 2 \cos \theta } { \cos \theta + 2 \sin \theta } \equiv \frac { 4 } { 5 \cos ^ { 2 } \theta - 4 }\).
2. Hence solve the equation \(\frac { \sin \theta + 2 \cos \theta } { \cos \theta - 2 \sin \theta } - \frac { \sin \theta - 2 \cos \theta } { \cos \theta + 2 \sin \theta } = 5\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

8
\includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-12_771_839_262_651} The diagram shows the circle with equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 8\). The chord \(A B\) of the circle intersects the positive \(y\)-axis at \(A\) and is parallel to the \(x\)-axis.

1. Find, by calculation, the coordinates of \(A\) and \(B\).
2. Find the volume of revolution when the shaded segment, bounded by the circle and the chord \(A B\), is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

9 Functions f, g and h are defined as follows: $$\begin{aligned} & \mathrm { f } : x \mapsto x - 4 x ^ { \frac { 1 } { 2 } } + 1 \quad \text { for } x \geqslant 0
& \mathrm {~g} : x \mapsto m x ^ { 2 } + n \quad \text { for } x \geqslant - 2 , \text { where } m \text { and } n \text { are constants, }
& \mathrm { h } : x \mapsto x ^ { \frac { 1 } { 2 } } - 2 \quad \text { for } x \geqslant 0 . \end{aligned}$$

1. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your solutions in the form \(x = a + b \sqrt { c }\), where \(a , b\) and \(c\) are integers.
2. Given that \(\mathrm { f } ( x ) \equiv \mathrm { gh } ( x )\), find the values of \(m\) and \(n\).
   \includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-16_652_1045_255_550} The diagram shows a circle with centre \(A\) of radius 5 cm and a circle with centre \(B\) of radius 8 cm . The circles touch at the point \(C\) so that \(A C B\) is a straight line. The tangent at the point \(D\) on the smaller circle intersects the larger circle at \(E\) and passes through \(B\).

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

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

1. Find, in terms of \(k\), the values of \(x\) at which there is a stationary point.
   The function f has a stationary value at \(x = a\) and is defined by $$f ( x ) = 4 ( 3 x - 4 ) ^ { - 1 } + 3 x \quad \text { for } x \geqslant \frac { 3 } { 2 }$$
2. Find the value of \(a\) and determine the nature of the stationary value.
3. The function g is defined by \(\mathrm { g } ( x ) = - ( 3 x + 1 ) ^ { - 1 } + 3 x\) for \(x \geqslant 0\). Determine, making your reasoning clear, whether \(g\) is an increasing function, a decreasing function or neither.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.