CAIE P1 (Pure Mathematics 1) 2004 November

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

2 Find

1. the sum of the first ten terms of the geometric progression \(81,54,36 , \ldots\),
2. the sum of all the terms in the arithmetic progression \(180,175,170 , \ldots , 25\).

3 In the diagram, \(A C\) is an arc of a circle, centre \(O\) and radius 6 cm . The line \(B C\) is perpendicular to \(O C\) and \(O A B\) is a straight line. Angle \(A O C = \frac { 1 } { 3 } \pi\) radians. Find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).

4

1. Sketch and label, on the same diagram, the graphs of \(y = 2 \sin x\) and \(y = \cos 2 x\), for the interval \(0 \leqslant x \leqslant \pi\).
2. Hence state the number of solutions of the equation \(2 \sin x = \cos 2 x\) in the interval \(0 \leqslant x \leqslant \pi\).

5 The equation of a curve is \(y = x ^ { 2 } - 4 x + 7\) and the equation of a line is \(y + 3 x = 9\). The curve and the line intersect at the points \(A\) and \(B\).

1. The mid-point of \(A B\) is \(M\). Show that the coordinates of \(M\) are \(\left( \frac { 1 } { 2 } , 7 \frac { 1 } { 2 } \right)\).
2. Find the coordinates of the point \(Q\) on the curve at which the tangent is parallel to the line \(y + 3 x = 9\).
3. Find the distance \(M Q\).

6 The function \(\mathrm { f } : x \mapsto 5 \sin ^ { 2 } x + 3 \cos ^ { 2 } x\) is defined for the domain \(0 \leqslant x \leqslant \pi\).

1. Express \(\mathrm { f } ( x )\) in the form \(a + b \sin ^ { 2 } x\), stating the values of \(a\) and \(b\).
2. Hence find the values of \(x\) for which \(\mathrm { f } ( x ) = 7 \sin x\).
3. State the range of f .

7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { \sqrt { } ( 4 x - 3 ) }\) and \(P ( 3,3 )\) is a point on the curve.

1. Find the equation of the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\).
2. Find the equation of the curve.

8 The points \(A\) and \(B\) have position vectors \(\mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k }\) and \(- 5 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\) respectively, relative to an origin \(O\).

1. Use a scalar product to calculate angle \(A O B\), giving your answer in radians correct to 3 significant figures.
2. The point \(C\) is such that \(\overrightarrow { A B } = 2 \overrightarrow { B C }\). Find the unit vector in the direction of \(\overrightarrow { O C }\).

9 The function f : \(x \mapsto 2 x - a\), where \(a\) is a constant, is defined for all real \(x\).

1. In the case where \(a = 3\), solve the equation \(\mathrm { ff } ( x ) = 11\). The function \(\mathrm { g } : x \mapsto x ^ { 2 } - 6 x\) is defined for all real \(x\).
2. Find the value of \(a\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has exactly one real solution. The function \(\mathrm { h } : x \mapsto x ^ { 2 } - 6 x\) is defined for the domain \(x \geqslant 3\).
3. Express \(x ^ { 2 } - 6 x\) in the form \(( x - p ) ^ { 2 } - q\), where \(p\) and \(q\) are constants.
4. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { h } ^ { - 1 }\).

10 A curve has equation \(y = x ^ { 2 } + \frac { 2 } { x }\).

1. Write down expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point on the curve and determine its nature.
3. Find the volume of the solid formed when the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated completely about the \(x\)-axis.