CAIE P1 (Pure Mathematics 1) 2022 November

1 Points \(A\) and \(B\) have coordinates \(( 5,2 )\) and \(( 10 , - 1 )\) respectively.

1. Find the equation of the perpendicular bisector of \(A B\).
2. Find the equation of the circle with centre \(A\) which passes through \(B\).

2 The first, second and third terms of an arithmetic progression are \(a , 2 a\) and \(a ^ { 2 }\) respectively, where \(a\) is a positive constant. Find the sum of the first 50 terms of the progression.

3

1. Find the set of values of \(k\) for which the equation \(8 x ^ { 2 } + k x + 2 = 0\) has no real roots.
2. Solve the equation \(8 \cos ^ { 2 } \theta - 10 \cos \theta + 2 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

4 A geometric progression is such that the third term is 1764 and the sum of the second and third terms is 3444 . Find the 50th term.

5 The graph with equation \(y = \mathrm { f } ( x )\) is transformed to the graph with equation \(y = \mathrm { g } ( x )\) by a stretch in the \(x\)-direction with factor 0.5 , followed by a translation of \(\binom { 0 } { 1 }\).

1. The diagram below shows the graph of \(y = \mathrm { f } ( x )\). On the diagram sketch the graph of \(y = \mathrm { g } ( x )\).
   \includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-07_613_1527_623_342}
2. Find an expression for \(\mathrm { g } ( x )\) in terms of \(\mathrm { f } ( x )\).

6 The equation of a curve is \(y = 4 x ^ { 2 } + 20 x + 6\).

1. Express the equation in the form \(y = a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are constants.
2. Hence solve the equation \(4 x ^ { 2 } + 20 x + 6 = 45\).
3. Sketch the graph of \(y = 4 x ^ { 2 } + 20 x + 6\) showing the coordinates of the stationary point. You are not required to indicate where the curve crosses the \(x\) - and \(y\)-axes.

7

1. Prove the identity \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } \equiv \frac { \tan ^ { 2 } \theta + 1 } { \tan ^ { 2 } \theta - 1 }\).
2. Hence find the exact solutions of the equation \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } = 2\) for \(0 \leqslant \theta \leqslant \pi\).

8 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 3 x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(( 3,5 )\).

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the stationary point.
3. State the set of values of \(x\) for which \(y\) increases as \(x\) increases.

9 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = x + \frac { 1 } { x } \quad \text { for } x > 0
& \mathrm {~g} ( x ) = a x + 1 \quad \text { for } x \in \mathbb { R } \end{aligned}$$ where \(a\) is a constant.

1. Find an expression for \(\operatorname { gf } ( x )\).
2. Given that \(\operatorname { gf } ( 2 ) = 11\), find the value of \(a\).
3. Given that the graph of \(y = \mathrm { f } ( x )\) has a minimum point when \(x = 1\), explain whether or not f has an inverse.
   It is given instead that \(a = 5\).
4. Find and simplify an expression for \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x )\).
5. Explain why the composite function fg cannot be formed.
   \includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-16_1143_1008_267_566} The diagram shows a cross-section RASB of the body of an aircraft. The cross-section consists of a sector \(O A R B\) of a circle of radius 2.5 m , with centre \(O\), a sector \(P A S B\) of another circle of radius 2.24 m with centre \(P\) and a quadrilateral \(O A P B\). Angle \(A O B = \frac { 2 } { 3 } \pi\) and angle \(A P B = \frac { 5 } { 6 } \pi\).

1. Find the perimeter of the cross-section RASB, giving your answer correct to 2 decimal places.
2. Find the difference in area of the two triangles \(A O B\) and \(A P B\), giving your answer correct to 2 decimal places.
3. Find the area of the cross-section RASB, giving your answer correct to 1 decimal place.

11

1. Find the coordinates of the minimum point of the curve \(y = \frac { 9 } { 4 } x ^ { 2 } - 12 x + 18\).
   \includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-18_675_901_1270_612} The diagram shows the curves with equations \(y = \frac { 9 } { 4 } x ^ { 2 } - 12 x + 18\) and \(y = 18 - \frac { 3 } { 8 } x ^ { \frac { 5 } { 2 } }\). The curves intersect at the points \(( 0,18 )\) and \(( 4,6 )\).
2. Find the area of the shaded region.
3. A point \(P\) is moving along the curve \(y = 18 - \frac { 3 } { 8 } x ^ { \frac { 5 } { 2 } }\) in such a way that the \(x\)-coordinate of \(P\) is increasing at a constant rate of 2 units per second. Find the rate at which the \(y\)-coordinate of \(P\) is changing when \(x = 4\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.