CAIE P1 (Pure Mathematics 1) 2019 June

1 Find the coefficient of \(x\) in the expansion of \(\left( \frac { 2 } { x } - 3 x \right) ^ { 5 }\).

2 Two points \(A\) and \(B\) have coordinates \(( 1,3 )\) and \(( 9 , - 1 )\) respectively. The perpendicular bisector of \(A B\) intersects the \(y\)-axis at the point \(C\). Find the coordinates of \(C\).

3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 3 } - \frac { 4 } { x ^ { 2 } }\). The point \(P ( 2,9 )\) lies on the curve.

1. A point moves on the curve in such a way that the \(x\)-coordinate is decreasing at a constant rate of 0.05 units per second. Find the rate of change of the \(y\)-coordinate when the point is at \(P\). [2]
2. Find the equation of the curve.

4 Angle \(x\) is such that \(\sin x = a + b\) and \(\cos x = a - b\), where \(a\) and \(b\) are constants.

1. Show that \(a ^ { 2 } + b ^ { 2 }\) has a constant value for all values of \(x\).
2. In the case where \(\tan x = 2\), express \(a\) in terms of \(b\).

5
\includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-06_355_634_255_753} The diagram shows a semicircle with diameter \(A B\), centre \(O\) and radius \(r\). The point \(C\) lies on the circumference and angle \(A O C = \theta\) radians. The perimeter of sector \(B O C\) is twice the perimeter of sector \(A O C\). Find the value of \(\theta\) correct to 2 significant figures.

6 The equation of a curve is \(y = 3 \cos 2 x\) and the equation of a line is \(2 y + \frac { 3 x } { \pi } = 5\).

1. State the smallest and largest values of \(y\) for both the curve and the line for \(0 \leqslant x \leqslant 2 \pi\).
2. Sketch, on the same diagram, the graphs of \(y = 3 \cos 2 x\) and \(2 y + \frac { 3 x } { \pi } = 5\) for \(0 \leqslant x \leqslant 2 \pi\).
3. State the number of solutions of the equation \(6 \cos 2 x = 5 - \frac { 3 x } { \pi }\) for \(0 \leqslant x \leqslant 2 \pi\).

7 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x - 2 , \quad x \in \mathbb { R } ,
& \mathrm {~g} : x \mapsto \frac { 2 x + 3 } { x - 1 } , \quad x \in \mathbb { R } , x \neq 1 \end{aligned}$$

1. Obtain expressions for \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\), stating the value of \(x\) for which \(\mathrm { g } ^ { - 1 } ( x )\) is not defined. [4]
2. Solve the equation \(\operatorname { fg } ( x ) = \frac { 7 } { 3 }\).

8 The position vectors of points \(A\) and \(B\), relative to an origin \(O\), are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 6
- 2
- 6 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 3
k
- 3 \end{array} \right)$$ where \(k\) is a constant.

1. Find the value of \(k\) for which angle \(A O B\) is \(90 ^ { \circ }\).
2. Find the values of \(k\) for which the lengths of \(O A\) and \(O B\) are equal.
   The point \(C\) is such that \(\overrightarrow { A C } = 2 \overrightarrow { C B }\).
3. In the case where \(k = 4\), find the unit vector in the direction of \(\overrightarrow { O C }\).

9 The curve \(C _ { 1 }\) has equation \(y = x ^ { 2 } - 4 x + 7\). The curve \(C _ { 2 }\) has equation \(y ^ { 2 } = 4 x + k\), where \(k\) is a constant. The tangent to \(C _ { 1 }\) at the point where \(x = 3\) is also the tangent to \(C _ { 2 }\) at the point \(P\). Find the value of \(k\) and the coordinates of \(P\).

10

1. In an arithmetic progression, the sum of the first ten terms is equal to the sum of the next five terms. The first term is \(a\).
   1. Show that the common difference of the progression is \(\frac { 1 } { 3 } a\).
   2. Given that the tenth term is 36 more than the fourth term, find the value of \(a\).
2. The sum to infinity of a geometric progression is 9 times the sum of the first four terms. Given that the first term is 12 , find the value of the fifth term.

11
\includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-16_723_942_260_598} The diagram shows part of the curve \(y = \sqrt { } ( 4 x + 1 ) + \frac { 9 } { \sqrt { } ( 4 x + 1 ) }\) and the minimum point \(M\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the coordinates of \(M\).
   The shaded region is bounded by the curve, the \(y\)-axis and the line through \(M\) parallel to the \(x\)-axis.
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.