CAIE P1 (Pure Mathematics 1) 2011 November

1 Find the term independent of \(x\) in the expansion of \(\left( 2 x + \frac { 1 } { x ^ { 2 } } \right) ^ { 6 }\).

2 A curve has equation \(y = 3 x ^ { 3 } - 6 x ^ { 2 } + 4 x + 2\). Show that the gradient of the curve is never negative.

3

1. Sketch, on a single diagram, the graphs of \(y = \cos 2 \theta\) and \(y = \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).
2. Write down the number of roots of the equation \(2 \cos 2 \theta - 1 = 0\) in the interval \(0 \leqslant \theta \leqslant 2 \pi\).
3. Deduce the number of roots of the equation \(2 \cos 2 \theta - 1 = 0\) in the interval \(10 \pi \leqslant \theta \leqslant 20 \pi\).

4 A function f is defined for \(x \in \mathbb { R }\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - 6\). The range of the function is given by \(\mathrm { f } ( x ) \geqslant - 4\).

1. State the value of \(x\) for which \(\mathrm { f } ( x )\) has a stationary value.
2. Find an expression for \(\mathrm { f } ( x )\) in terms of \(x\).

5
\includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-2_512_903_1302_621} The diagram represents a metal plate \(O A B C\), consisting of a sector \(O A B\) of a circle with centre \(O\) and radius \(r\), together with a triangle \(O C B\) which is right-angled at \(C\). Angle \(A O B = \theta\) radians and \(O C\) is perpendicular to \(O A\).

1. Find an expression in terms of \(r\) and \(\theta\) for the perimeter of the plate.
2. For the case where \(r = 10\) and \(\theta = \frac { 1 } { 5 } \pi\), find the area of the plate.

6

1. The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200 . Find the seventh term.
2. A geometric progression has first term 1 and common ratio \(r\). A second geometric progression has first term 4 and common ratio \(\frac { 1 } { 4 } r\). The two progressions have the same sum to infinity, \(S\). Find the values of \(r\) and \(S\).

7
\includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-3_534_895_255_625} The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the garden is 48 m .

1. Find an expression for \(y\) in terms of \(x\).
2. Given that the area of the garden is \(A \mathrm {~m} ^ { 2 }\), show that \(A = 48 x - 8 x ^ { 2 }\).
3. Given that \(x\) can vary, find the maximum area of the garden, showing that this is a maximum value rather than a minimum value.

8 Relative to an origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } + 7 \mathbf { j } - p \mathbf { k }\) and the point \(B\) has position vector \(8 \mathbf { i } - \mathbf { j } - p \mathbf { k }\), where \(p\) is a constant.

1. Find \(\overrightarrow { O A } \cdot \overrightarrow { O B }\).
2. Hence show that there are no real values of \(p\) for which \(O A\) and \(O B\) are perpendicular to each other.
3. Find the values of \(p\) for which angle \(A O B = 60 ^ { \circ }\).

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

1. For the case where \(k = 2\), the line and the curve intersect at points \(A\) and \(B\). Find the distance \(A B\) and the coordinates of the mid-point of \(A B\).
2. Find the two values of \(k\) for which the line is a tangent to the curve.

10
\includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-4_799_1390_255_376} The diagram shows the curve \(y = \sqrt { } ( 1 + 2 x )\) meeting the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The \(y\)-coordinate of the point \(C\) on the curve is 3 .

1. Find the coordinates of \(B\) and \(C\).
2. Find the equation of the normal to the curve at \(C\).
3. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.

11 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 10 & \text { for } 0 \leqslant x \leqslant 2
\mathrm {~g} : x \mapsto x & \text { for } 0 \leqslant x \leqslant 10 \end{array}$$

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. State the domain of \(\mathrm { f } ^ { - 1 }\).
4. Sketch on the same diagram the graphs of \(y = \mathrm { f } ( x ) , y = \mathrm { g } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.
5. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).