CAIE P1 (Pure Mathematics 1) 2020 November

1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 + k x ) ( 1 - 2 x ) ^ { 5 }\) is 20 .
Find the value of the constant \(k\).

2 The first, second and third terms of a geometric progression are \(2 p + 6 , - 2 p\) and \(p + 2\) respectively, where \(p\) is positive. Find the sum to infinity of the progression.

3 The equation of a curve is \(y = 2 x ^ { 2 } + m ( 2 x + 1 )\), where \(m\) is a constant, and the equation of a line is \(y = 6 x + 4\). Show that, for all values of \(m\), the line intersects the curve at two distinct points.

4 The sum, \(S _ { n }\), of the first \(n\) terms of an arithmetic progression is given by $$S _ { n } = n ^ { 2 } + 4 n$$ The \(k\) th term in the progression is greater than 200.
Find the smallest possible value of \(k\).

5 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = 4 x - 2 , \quad \text { for } x \in \mathbb { R } , \\ & \mathrm {~g} ( x ) = \frac { 4 } { x + 1 } , \quad \text { for } x \in \mathbb { R } , x \neq - 1 \end{aligned}$$

1. Find the value of fg (7).
2. Find the values of \(x\) for which \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\).

6

1. Prove the identity \(\left( \frac { 1 } { \cos x } - \tan x \right) \left( \frac { 1 } { \sin x } + 1 \right) \equiv \frac { 1 } { \tan x }\).
2. Hence solve the equation \(\left( \frac { 1 } { \cos x } - \tan x \right) \left( \frac { 1 } { \sin x } + 1 \right) = 2 \tan ^ { 2 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

7 The point \(( 4,7 )\) lies on the curve \(y = \mathrm { f } ( x )\) and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { - \frac { 1 } { 2 } } - 4 x ^ { - \frac { 3 } { 2 } }\).

1. A point moves along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.12 units per second. Find the rate of increase of the \(y\)-coordinate when \(x = 4\).
2. Find the equation of the curve.

8 \includegraphics[max width=\textwidth, alt={}, center]{6bcc553c-4938-46ef-bba4-97391b4d58d4-10_348_700_262_721} In the diagram, \(A B C\) is an isosceles triangle with \(A B = B C = r \mathrm {~cm}\) and angle \(B A C = \theta\) radians. The point \(D\) lies on \(A C\) and \(A B D\) is a sector of a circle with centre \(A\).

1. Express the area of the shaded region in terms of \(r\) and \(\theta\).
2. In the case where \(r = 10\) and \(\theta = 0.6\), find the perimeter of the shaded region.

9 A circle has centre at the point \(B ( 5,1 )\). The point \(A ( - 1 , - 2 )\) lies on the circle.

1. Find the equation of the circle.
   Point \(C\) is such that \(A C\) is a diameter of the circle. Point \(D\) has coordinates (5, 16).
2. Show that \(D C\) is a tangent to the circle.
   The other tangent from \(D\) to the circle touches the circle at \(E\).
3. Find the coordinates of \(E\).

10 \includegraphics[max width=\textwidth, alt={}, center]{6bcc553c-4938-46ef-bba4-97391b4d58d4-14_378_666_264_737} The diagram shows part of the curve \(y = \frac { 2 } { ( 3 - 2 x ) ^ { 2 } } - x\) and its minimum point \(M\), which lies on the \(x\)-axis.

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\int y \mathrm {~d} x\).
2. Find, by calculation, the \(x\)-coordinate of \(M\).
3. Find the area of the shaded region bounded by the curve and the coordinate axes.

11 A curve has equation \(y = 3 \cos 2 x + 2\) for \(0 \leqslant x \leqslant \pi\).

1. State the greatest and least values of \(y\).
2. Sketch the graph of \(y = 3 \cos 2 x + 2\) for \(0 \leqslant x \leqslant \pi\).
3. By considering the straight line \(y = k x\), where \(k\) is a constant, state the number of solutions of the equation \(3 \cos 2 x + 2 = k x\) for \(0 \leqslant x \leqslant \pi\) in each of the following cases.
   1. \(k = - 3\)
   2. \(k = 1\)
   3. \(k = 3\) Functions \(\mathrm { f } , \mathrm { g }\) and h are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } ( x ) = 3 \cos 2 x + 2 \\ & \mathrm {~g} ( x ) = \mathrm { f } ( 2 x ) + 4 \\ & \mathrm {~h} ( x ) = 2 \mathrm { f } \left( x + \frac { 1 } { 2 } \pi \right) \end{aligned}$$
4. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) on to \(y = \mathrm { g } ( x )\).
5. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) on to \(y = \mathrm { h } ( x )\). [2]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.