CAIE P1 (Pure Mathematics 1) 2019 November

1

1. Expand \(( 1 + y ) ^ { 6 }\) in ascending powers of \(y\) as far as the term in \(y ^ { 2 }\).
2. In the expansion of \(\left( 1 + \left( p x - 2 x ^ { 2 } \right) \right) ^ { 6 }\) the coefficient of \(x ^ { 2 }\) is 48 . Find the value of the positive constant \(p\).

2 The function g is defined by \(\mathrm { g } ( x ) = x ^ { 2 } - 6 x + 7\) for \(x > 4\). By first completing the square, find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { g } ^ { - 1 }\).

3 The equation of a curve is \(y = x ^ { 3 } + x ^ { 2 } - 8 x + 7\). The curve has no stationary points in the interval \(a < x < b\). Find the least possible value of \(a\) and the greatest possible value of \(b\).

4
\includegraphics[max width=\textwidth, alt={}, center]{17e813c6-890f-4198-b20a-557b133e8c34-05_360_639_255_753} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). Arc \(O C\) is part of a circle with centre \(A\).

1. Express angle \(C A O\) in radians in terms of \(\pi\).
2. Find the area of the shaded region in terms of \(r , \pi\) and \(\sqrt { } 3\), simplifying your answer.

5
\includegraphics[max width=\textwidth, alt={}, center]{17e813c6-890f-4198-b20a-557b133e8c34-06_462_878_258_635} The dimensions of a cuboid are \(x \mathrm {~cm} , 2 x \mathrm {~cm}\) and \(4 x \mathrm {~cm}\), as shown in the diagram.

1. Show that the surface area \(S \mathrm {~cm} ^ { 2 }\) and the volume \(V \mathrm {~cm} ^ { 3 }\) are connected by the relation $$S = 7 V ^ { \frac { 2 } { 3 } }$$
2. When the volume of the cuboid is \(1000 \mathrm {~cm} ^ { 3 }\) the surface area is increasing at \(2 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of the volume at this instant.

6 A line has equation \(y = 3 k x - 2 k\) and a curve has equation \(y = x ^ { 2 } - k x + 2\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the line and curve meet at two distinct points.
2. For each of two particular values of \(k\), the line is a tangent to the curve. Show that these two tangents meet on the \(x\)-axis.

7

1. Show that the equation \(3 \cos ^ { 4 } \theta + 4 \sin ^ { 2 } \theta - 3 = 0\) can be expressed as \(3 x ^ { 2 } - 4 x + 1 = 0\), where \(x = \cos ^ { 2 } \theta\).
2. Hence solve the equation \(3 \cos ^ { 4 } \theta + 4 \sin ^ { 2 } \theta - 3 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

8 A function f is defined for \(x > \frac { 1 } { 2 }\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 ( 2 x - 1 ) ^ { \frac { 1 } { 2 } } - 6\).

1. Find the set of values of \(x\) for which f is decreasing.
2. It is now given that \(\mathrm { f } ( 1 ) = - 3\). Find \(\mathrm { f } ( x )\).

9 The first, second and third terms of a geometric progression are \(3 k , 5 k - 6\) and \(6 k - 4\), respectively.

1. Show that \(k\) satisfies the equation \(7 k ^ { 2 } - 48 k + 36 = 0\).
2. Find, showing all necessary working, the exact values of the common ratio corresponding to each of the possible values of \(k\).
3. One of these ratios gives a progression which is convergent. Find the sum to infinity.

10 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(X\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 8
- 4
2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 10
2

11 \end{array} \right) \quad \text { and } \quad \overrightarrow { O X } = \left( \begin{array} { r } - 2
- 2
5 \end{array} \right)$$

1. Find \(\overrightarrow { A X }\) and show that \(A X B\) is a straight line.
   The position vector of a point \(C\) is given by \(\overrightarrow { O C } = \left( \begin{array} { r } 1
   - 8
   3 \end{array} \right)\).
2. Show that \(C X\) is perpendicular to \(A X\).
3. Find the area of triangle \(A B C\).
   \includegraphics[max width=\textwidth, alt={}, center]{17e813c6-890f-4198-b20a-557b133e8c34-18_949_1087_260_529} The diagram shows part of the curve \(y = ( x - 1 ) ^ { - 2 } + 2\), and the lines \(x = 1\) and \(x = 3\). The point \(A\) on the curve has coordinates \(( 2,3 )\). The normal to the curve at \(A\) crosses the line \(x = 1\) at \(B\).
4. Show that the normal \(A B\) has equation \(y = \frac { 1 } { 2 } x + 2\).
5. Find, showing all necessary working, the volume of revolution obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.