CAIE P1 (Pure Mathematics 1) 2017 November

1 Find the term independent of \(x\) in the expansion of \(\left( 2 x - \frac { 1 } { 4 x ^ { 2 } } \right) ^ { 9 }\).

2 A function f is defined by \(\mathrm { f } : x \mapsto 4 - 5 x\) for \(x \in \mathbb { R }\).

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and find the point of intersection of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\).
2. Sketch, on the same diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.

3

1. Each year, the value of a certain rare stamp increases by \(5 \%\) of its value at the beginning of the year. A collector bought the stamp for \(
   ) 10000\( at the beginning of 2005. Find its value at the beginning of 2015 correct to the nearest \)\\( 100\).
2. The sum of the first \(n\) terms of an arithmetic progression is \(\frac { 1 } { 2 } n ( 3 n + 7 )\). Find the 1 st term and the common difference of the progression.

4
\includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-06_401_698_255_721} The diagram shows a semicircle with centre \(O\) and radius 6 cm . The radius \(O C\) is perpendicular to the diameter \(A B\). The point \(D\) lies on \(A B\), and \(D C\) is an arc of a circle with centre \(B\).

1. Calculate the length of the \(\operatorname { arc } D C\).
2. Find the value of
   \(\frac { \text { area of region } P } { \text { area of region } Q }\),
   giving your answer correct to 3 significant figures.

5

1. Show that the equation \(\cos 2 x \left( \tan ^ { 2 } 2 x + 3 \right) + 3 = 0\) can be expressed as $$2 \cos ^ { 2 } 2 x + 3 \cos 2 x + 1 = 0$$
2. Hence solve the equation \(\cos 2 x \left( \tan ^ { 2 } 2 x + 3 \right) + 3 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

6

1. The function f , defined by \(\mathrm { f } : x \mapsto a + b \sin x\) for \(x \in \mathbb { R }\), is such that \(\mathrm { f } \left( \frac { 1 } { 6 } \pi \right) = 4\) and \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 3\).
   1. Find the values of the constants \(a\) and \(b\).
   2. Evaluate \(\mathrm { ff } ( 0 )\).
2. The function g is defined by \(\mathrm { g } : x \mapsto c + d \sin x\) for \(x \in \mathbb { R }\). The range of g is given by \(- 4 \leqslant \mathrm {~g} ( x ) \leqslant 10\). Find the values of the constants \(c\) and \(d\).

7 Points \(A\) and \(B\) lie on the curve \(y = x ^ { 2 } - 4 x + 7\). Point \(A\) has coordinates \(( 4,7 )\) and \(B\) is the stationary point of the curve. The equation of a line \(L\) is \(y = m x - 2\), where \(m\) is a constant.

1. In the case where \(L\) passes through the mid-point of \(A B\), find the value of \(m\).
2. Find the set of values of \(m\) for which \(L\) does not meet the curve.

8 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 2 } + 5 x - 4\).

1. Find the \(x\)-coordinate of each of the stationary points of the curve.
2. Obtain an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence or otherwise find the nature of each of the stationary points.
3. Given that the curve passes through the point \(( 6,2 )\), find the equation of the curve.

9
\includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-16_533_601_258_772} The diagram shows a trapezium \(O A B C\) in which \(O A\) is parallel to \(C B\). The position vectors of \(A\) and \(B\) relative to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2
- 2
- 1 \end{array} \right)\) and \(\overrightarrow { O B } = \left( \begin{array} { l } 6
1
1 \end{array} \right)\).

1. Show that angle \(O A B\) is \(90 ^ { \circ }\).
   The magnitude of \(\overrightarrow { C B }\) is three times the magnitude of \(\overrightarrow { O A }\).
2. Find the position vector of \(C\).
3. Find the exact area of the trapezium \(O A B C\), giving your answer in the form \(a \sqrt { } b\), where \(a\) and \(b\) are integers.

10
\includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-18_551_689_260_726} The diagram shows part of the curve \(y = \sqrt { } ( 5 x - 1 )\) and the normal to the curve at the point \(P ( 2,3 )\). This normal meets the \(x\)-axis at \(Q\).

1. Find the equation of the normal at \(P\).
2. Find, showing all necessary working, the area of the shaded region.