CAIE P1 (Pure Mathematics 1) 2007 November

1 Determine the set of values of the constant \(k\) for which the line \(y = 4 x + k\) does not intersect the curve \(y = x ^ { 2 }\).

2 Find the area of the region enclosed by the curve \(y = 2 \sqrt { } x\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).

3

1. Find the first three terms in the expansion of \(( 2 + u ) ^ { 5 }\) in ascending powers of \(u\).
2. Use the substitution \(u = x + x ^ { 2 }\) in your answer to part (i) to find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 + x + x ^ { 2 } \right) ^ { 5 }\).

4 The 1st term of an arithmetic progression is \(a\) and the common difference is \(d\), where \(d \neq 0\).

1. Write down expressions, in terms of \(a\) and \(d\), for the 5th term and the 15th term. The 1st term, the 5th term and the 15th term of the arithmetic progression are the first three terms of a geometric progression.
2. Show that \(3 a = 8 d\).
3. Find the common ratio of the geometric progression.

5

1. Show that the equation \(3 \sin x \tan x = 8\) can be written as \(3 \cos ^ { 2 } x + 8 \cos x - 3 = 0\).
2. Hence solve the equation \(3 \sin x \tan x = 8\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-2_627_748_1685_699} The three points \(A ( 3,8 ) , B ( 6,2 )\) and \(C ( 10,2 )\) are shown in the diagram. The point \(D\) is such that the line \(D A\) is perpendicular to \(A B\) and \(D C\) is parallel to \(A B\). Calculate the coordinates of \(D\).

7
\includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-3_579_659_269_744} In the diagram, \(A B\) is an arc of a circle, centre \(O\) and radius \(r \mathrm {~cm}\), and angle \(A O B = \theta\) radians. The point \(X\) lies on \(O B\) and \(A X\) is perpendicular to \(O B\).

1. Show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the shaded region \(A X B\) is given by $$A = \frac { 1 } { 2 } r ^ { 2 } ( \theta - \sin \theta \cos \theta ) .$$
2. In the case where \(r = 12\) and \(\theta = \frac { 1 } { 6 } \pi\), find the perimeter of the shaded region \(A X B\), leaving your answer in terms of \(\sqrt { } 3\) and \(\pi\).

8 The equation of a curve is \(y = ( 2 x - 3 ) ^ { 3 } - 6 x\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(x\).
2. Find the \(x\)-coordinates of the two stationary points and determine the nature of each stationary point.

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 - x\) and the point \(P ( 2,9 )\) lies on the curve. The normal to the curve at \(P\) meets the curve again at \(Q\). Find

1. the equation of the curve,
2. the equation of the normal to the curve at \(P\),
3. the coordinates of \(Q\).

10
\includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-4_597_693_274_726} The diagram shows a cube \(O A B C D E F G\) in which the length of each side is 4 units. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively. The mid-points of \(O A\) and \(D G\) are \(P\) and \(Q\) respectively and \(R\) is the centre of the square face \(A B F E\).

1. Express each of the vectors \(\overrightarrow { P R }\) and \(\overrightarrow { P Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to find angle \(Q P R\).
3. Find the perimeter of triangle \(P Q R\), giving your answer correct to 1 decimal place.

11 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 11\) for \(x \in \mathbb { R }\).

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are constants.
2. State the range of f .
3. Explain why f does not have an inverse. The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 8 x + 11\) for \(x \leqslant A\), where \(A\) is a constant.
4. State the largest value of \(A\) for which g has an inverse.
5. When \(A\) has this value, obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\) and state the range of \(\mathrm { g } ^ { - 1 }\).