CAIE P1 (Pure Mathematics 1) 2011 June

1 Find \(\int \left( x ^ { 3 } + \frac { 1 } { x ^ { 3 } } \right) \mathrm { d } x\).

2

1. Find the terms in \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(\left( 1 - \frac { 3 } { 2 } x \right) ^ { 6 }\).
2. Given that there is no term in \(x ^ { 3 }\) in the expansion of \(( k + 2 x ) \left( 1 - \frac { 3 } { 2 } x \right) ^ { 6 }\), find the value of the constant \(k\).

3 The equation \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are constants, has roots - 3 and 5 .

1. Find the values of \(p\) and \(q\).
2. Using these values of \(p\) and \(q\), find the value of the constant \(r\) for which the equation \(x ^ { 2 } + p x + q + r = 0\) has equal roots.

4 A curve has equation \(y = \frac { 4 } { 3 x - 4 }\) and \(P ( 2,2 )\) is a point on the curve.

1. Find the equation of the tangent to the curve at \(P\).
2. Find the angle that this tangent makes with the \(x\)-axis.

5

1. Prove the identity \(\frac { \cos \theta } { \tan \theta ( 1 - \sin \theta ) } \equiv 1 + \frac { 1 } { \sin \theta }\).
2. Hence solve the equation \(\frac { \cos \theta } { \tan \theta ( 1 - \sin \theta ) } = 4\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

6 The function f is defined by \(\mathrm { f } : x \mapsto \frac { x + 3 } { 2 x - 1 } , x \in \mathbb { R } , x \neq \frac { 1 } { 2 }\).

1. Show that \(\operatorname { ff } ( x ) = x\).
2. Hence, or otherwise, obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

7 The line \(L _ { 1 }\) passes through the points \(A ( 2,5 )\) and \(B ( 10,9 )\). The line \(L _ { 2 }\) is parallel to \(L _ { 1 }\) and passes through the origin. The point \(C\) lies on \(L _ { 2 }\) such that \(A C\) is perpendicular to \(L _ { 2 }\). Find

1. the coordinates of \(C\),
2. the distance \(A C\).

8 Relative to the origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2
3
5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 4
2
3 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r }

10
0
6 \end{array} \right)$$

1. Find angle \(A B C\). The point \(D\) is such that \(A B C D\) is a parallelogram.
2. Find the position vector of \(D\). 9 The function f is such that \(\mathrm { f } ( x ) = 3 - 4 \cos ^ { k } x\), for \(0 \leqslant x \leqslant \pi\), where \(k\) is a constant.
3. In the case where \(k = 2\),
   (a) find the range of f,
   (b) find the exact solutions of the equation \(\mathrm { f } ( x ) = 1\).
4. In the case where \(k = 1\),
   (a) sketch the graph of \(y = \mathrm { f } ( x )\),
   (b) state, with a reason, whether f has an inverse. 10 (a) A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that the radius of the circle is 5 cm , find the perimeter of the smallest sector.
   (b) The first, second and third terms of a geometric progression are \(2 k + 3 , k + 6\) and \(k\), respectively. Given that all the terms of the geometric progression are positive, calculate
5. the value of the constant \(k\),
6. the sum to infinity of the progression.

11
\includegraphics[max width=\textwidth, alt={}, center]{5e2a6be8-b260-4b03-90c5-7485adbc39cc-3_513_1023_1838_561} The diagram shows part of the curve \(y = 4 \sqrt { } x - x\). The curve has a maximum point at \(M\) and meets the \(x\)-axis at \(O\) and \(A\).

1. Find the coordinates of \(A\) and \(M\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in terms of \(\pi\).