CAIE P1 (Pure Mathematics 1) 2019 June

1 The term independent of \(x\) in the expansion of \(\left( 2 x + \frac { k } { x } \right) ^ { 6 }\), where \(k\) is a constant, is 540.

1. Find the value of \(k\).
2. For this value of \(k\), find the coefficient of \(x ^ { 2 }\) in the expansion.

2 The line \(4 y = x + c\), where \(c\) is a constant, is a tangent to the curve \(y ^ { 2 } = x + 3\) at the point \(P\) on the curve.

1. Find the value of \(c\).
2. Find the coordinates of \(P\).

3 A sector of a circle of radius \(r \mathrm {~cm}\) has an area of \(A \mathrm {~cm} ^ { 2 }\). Express the perimeter of the sector in terms of \(r\) and \(A\).

4
\includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-06_625_750_260_699} The diagram shows a trapezium \(A B C D\) in which the coordinates of \(A , B\) and \(C\) are (4, 0), (0, 2) and \(( h , 3 h )\) respectively. The lines \(B C\) and \(A D\) are parallel, angle \(A B C = 90 ^ { \circ }\) and \(C D\) is parallel to the \(x\)-axis.

1. Find, by calculation, the value of \(h\).
2. Hence find the coordinates of \(D\).

5 The function f is defined by \(\mathrm { f } ( x ) = - 2 x ^ { 2 } + 12 x - 3\) for \(x \in \mathbb { R }\).

1. Express \(- 2 x ^ { 2 } + 12 x - 3\) in the form \(- 2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. State the greatest value of \(\mathrm { f } ( x )\).
   The function g is defined by \(\mathrm { g } ( x ) = 2 x + 5\) for \(x \in \mathbb { R }\).
3. Find the values of \(x\) for which \(\operatorname { gf } ( x ) + 1 = 0\).

6

1. Prove the identity \(\left( \frac { 1 } { \cos x } - \tan x \right) ^ { 2 } \equiv \frac { 1 - \sin x } { 1 + \sin x }\).
2. Hence solve the equation \(\left( \frac { 1 } { \cos 2 x } - \tan 2 x \right) ^ { 2 } = \frac { 1 } { 3 }\) for \(0 \leqslant x \leqslant \pi\).

7
\includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-12_775_823_260_662} The diagram shows a three-dimensional shape in which the base \(O A B C\) and the upper surface \(D E F G\) are identical horizontal squares. The parallelograms \(O A E D\) and \(C B F G\) both lie in vertical planes. The point \(M\) is the mid-point of \(A F\). Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O C\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards. The position vectors of \(A\) and \(D\) are given by \(\overrightarrow { O A } = 8 \mathbf { i }\) and \(\overrightarrow { O D } = 3 \mathbf { i } + 10 \mathbf { k }\).

1. Express each of the vectors \(\overrightarrow { A M }\) and \(\overrightarrow { G M }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to find angle \(G M A\) correct to the nearest degree.

8

1. The third and fourth terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression.
2. Two schemes are proposed for increasing the amount of household waste that is recycled each week. Scheme \(A\) is to increase the amount of waste recycled each month by 0.16 tonnes.
   Scheme \(B\) is to increase the amount of waste recycled each month by \(6 \%\) of the amount recycled in the previous month.
   The proposal is to operate the scheme for a period of 24 months. The amount recycled in the first month is 2.5 tonnes. For each scheme, find the total amount of waste that would be recycled over the 24 -month period. Scheme \(A\)
   Scheme \(B\) \(\_\_\_\_\)

9 The function f is defined by \(\mathrm { f } ( x ) = 2 - 3 \cos x\) for \(0 \leqslant x \leqslant 2 \pi\).

1. State the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\). The function g is defined by \(\mathrm { g } ( x ) = 2 - 3 \cos x\) for \(0 \leqslant x \leqslant p\), where \(p\) is a constant.
3. State the largest value of \(p\) for which g has an inverse.
4. For this value of \(p\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

10 A curve for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 5\) has a stationary point at \(( 3,6 )\).

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the other stationary point on the curve.
3. Determine the nature of each of the stationary points.
   \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-20_700_616_262_762} The diagram shows part of the curve \(y = \frac { 3 } { \sqrt { ( 1 + 4 x ) } }\) and a point \(P ( 2,1 )\) lying on the curve. The normal to the curve at \(P\) intersects the \(x\)-axis at \(Q\).
4. Show that the \(x\)-coordinate of \(Q\) is \(\frac { 16 } { 9 }\).
5. 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.