CAIE P1 (Pure Mathematics 1) 2018 November

1 Find the coefficient of \(\frac { 1 } { x ^ { 2 } }\) in the expansion of \(\left( 3 x + \frac { 2 } { 3 x ^ { 2 } } \right) ^ { 7 }\).

2 Showing all necessary working, find \(\int _ { 1 } ^ { 4 } \left( \sqrt { } x + \frac { 2 } { \sqrt { } x } \right) \mathrm { d } x\).

3
\includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-04_540_554_260_792} The diagram shows part of the curve \(y = x \left( 9 - x ^ { 2 } \right)\) and the line \(y = 5 x\), intersecting at the origin \(O\) and the point \(R\). Point \(P\) lies on the line \(y = 5 x\) between \(O\) and \(R\) and the \(x\)-coordinate of \(P\) is \(t\). Point \(Q\) lies on the curve and \(P Q\) is parallel to the \(y\)-axis.

1. Express the length of \(P Q\) in terms of \(t\), simplifying your answer.
2. Given that \(t\) can vary, find the maximum value of the length of \(P Q\).

4 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 - 3 \cos x \quad \text { for } 0 \leqslant x \leqslant 2 \pi
& \mathrm {~g} : x \mapsto \frac { 1 } { 2 } x \quad \text { for } 0 \leqslant x \leqslant 2 \pi \end{aligned}$$

1. Solve the equation \(\operatorname { fg } ( x ) = 1\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\).

5 The first three terms of an arithmetic progression are \(4 , x\) and \(y\) respectively. The first three terms of a geometric progression are \(x , y\) and 18 respectively. It is given that both \(x\) and \(y\) are positive.

1. Find the value of \(x\) and the value of \(y\).
2. Find the fourth term of each progression.
   \includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-08_389_716_260_712} The diagram shows a triangle \(A B C\) in which \(B C = 20 \mathrm {~cm}\) and angle \(A B C = 90 ^ { \circ }\). The perpendicular from \(B\) to \(A C\) meets \(A C\) at \(D\) and \(A D = 9 \mathrm {~cm}\). Angle \(B C A = \theta ^ { \circ }\).
3. By expressing the length of \(B D\) in terms of \(\theta\) in each of the triangles \(A B D\) and \(D B C\), show that \(20 \sin ^ { 2 } \theta = 9 \cos \theta\).
4. Hence, showing all necessary working, calculate \(\theta\).

7
\includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-10_819_497_262_826} The diagram shows a solid cylinder standing on a horizontal circular base with centre \(O\) and radius 4 units. Points \(A , B\) and \(C\) lie on the circumference of the base such that \(A B\) is a diameter and angle \(B O C = 90 ^ { \circ }\). Points \(P , Q\) and \(R\) lie on the upper surface of the cylinder vertically above \(A , B\) and \(C\) respectively. The height of the cylinder is 12 units. The mid-point of \(C R\) is \(M\) and \(N\) lies on \(B Q\) with \(B N = 4\) units. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O B\) and \(O C\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards.
Evaluate \(\overrightarrow { P N } \cdot \overrightarrow { P M }\) and hence find angle \(M P N\).
\includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-12_483_574_262_788} The diagram shows an isosceles triangle \(A C B\) in which \(A B = B C = 8 \mathrm {~cm}\) and \(A C = 12 \mathrm {~cm}\). The arc \(X C\) is part of a circle with centre \(A\) and radius 12 cm , and the arc \(Y C\) is part of circle with centre \(B\) and radius 8 cm . The points \(A , B , X\) and \(Y\) lie on a straight line.

1. Show that angle \(C B Y = 1.445\) radians, correct to 4 significant figures.
2. Find the perimeter of the shaded region.

9 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 12 x + 7\) for \(x \in \mathbb { R }\).

1. Express \(2 x ^ { 2 } - 12 x + 7\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. State the range of f .
   The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 12 x + 7\) for \(x \leqslant k\).
3. State the largest value of \(k\) for which g has an inverse.
4. Given that g has an inverse, find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

10 The equation of a curve is \(y = 2 x + \frac { 12 } { x }\) and the equation of a line is \(y + x = k\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the line does not meet the curve.
   In the case where \(k = 15\), the curve intersects the line at points \(A\) and \(B\).
2. Find the coordinates of \(A\) and \(B\).
3. Find the equation of the perpendicular bisector of the line joining \(A\) and \(B\).

11
\includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-18_661_698_260_717} The diagram shows part of the curve \(y = 3 \sqrt { } ( 4 x + 1 ) - 2 x\). The curve crosses the \(y\)-axis at \(A\) and the stationary point on the curve is \(M\).

1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the coordinates of \(M\).
3. Find, showing all necessary working, the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.