CAIE P1 (Pure Mathematics 1) 2019 November

1 Find the term independent of \(x\) in the expansion of \(\left( 2 x + \frac { 1 } { 4 x ^ { 2 } } \right) ^ { 6 }\).

2 An increasing function, f , is defined for \(x > n\), where \(n\) is an integer. It is given that \(\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } - 6 x + 8\). Find the least possible value of \(n\).

3 The line \(y = a x + b\) is a tangent to the curve \(y = 2 x ^ { 3 } - 5 x ^ { 2 } - 3 x + c\) at the point \(( 2,6 )\). Find the values of the constants \(a , b\) and \(c\).

4 A runner who is training for a long-distance race plans to run increasing distances each day for 21 days. She will run \(x \mathrm {~km}\) on day 1 , and on each subsequent day she will increase the distance by \(10 \%\) of the previous day's distance. On day 21 she will run 20 km .

1. Find the distance she must run on day 1 in order to achieve this. Give your answer in km correct to 1 decimal place.
2. Find the total distance she runs over the 21 days.

5

1. Given that \(4 \tan x + 3 \cos x + \frac { 1 } { \cos x } = 0\), show, without using a calculator, that \(\sin x = - \frac { 2 } { 3 }\).
2. Hence, showing all necessary working, solve the equation $$4 \tan \left( 2 x - 20 ^ { \circ } \right) + 3 \cos \left( 2 x - 20 ^ { \circ } \right) + \frac { 1 } { \cos \left( 2 x - 20 ^ { \circ } \right) } = 0$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

6 A straight line has gradient \(m\) and passes through the point ( \(0 , - 2\) ). Find the two values of \(m\) for which the line is a tangent to the curve \(y = x ^ { 2 } - 2 x + 7\) and, for each value of \(m\), find the coordinates of the point where the line touches the curve.

7 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto \frac { 3 } { 2 x + 1 } \quad \text { for } x > 0
& \mathrm {~g} : x \mapsto \frac { 1 } { x } + 2 \quad \text { for } x > 0 \end{aligned}$$

1. Find the range of f and the range of g .
2. Find an expression for \(\mathrm { fg } ( x )\), giving your answer in the form \(\frac { a x } { b x + c }\), where \(a , b\) and \(c\) are integers.
3. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\), giving your answer in the same form as for part (ii).

8
\includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-12_560_574_258_781} The diagram shows a sector \(O A C\) of a circle with centre \(O\). Tangents \(A B\) and \(C B\) to the circle meet at \(B\). The arc \(A C\) is of length 6 cm and angle \(A O C = \frac { 3 } { 8 } \pi\) radians.

1. Find the length of \(O A\) correct to 4 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

9 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 5 x - 1 ) ^ { \frac { 1 } { 2 } } - 2\) passes through the point ( 2,3 ).

1. Find the equation of the curve.
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
3. Find the coordinates of the stationary point on the curve and, showing all necessary working, determine the nature of this stationary point.

10
\includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-16_318_1006_260_568} Relative to an origin \(O\), the position vectors of the points \(A , B , C\) and \(D\), shown in the diagram, are given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 1
3
- 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2
- 3
5 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r } 4
- 2
5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } 2
2
- 1 \end{array} \right) .$$

1. Show that \(A B\) is perpendicular to \(B C\).
2. Show that \(A B C D\) is a trapezium.
3. Find the area of \(A B C D\), giving your answer correct to 2 decimal places.

11
\includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-18_650_611_260_762} The diagram shows a shaded region bounded by the \(y\)-axis, the line \(y = - 1\) and the part of the curve \(y = x ^ { 2 } + 4 x + 3\) for which \(x \geqslant - 2\).

1. Express \(y = x ^ { 2 } + 4 x + 3\) in the form \(y = ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants. Hence, for \(x \geqslant - 2\), express \(x\) in terms of \(y\).
2. Hence, showing all necessary working, find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.