CAIE P1 (Pure Mathematics 1) 2012 November

1 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 2 - \frac { 1 } { 2 } x \right) ^ { 7 }\).

2 It is given that \(\mathrm { f } ( x ) = \frac { 1 } { x ^ { 3 } } - x ^ { 3 }\), for \(x > 0\). Show that f is a decreasing function.

3 Solve the equation \(7 \cos x + 5 = 2 \sin ^ { 2 } x\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

4
\includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-2_478_828_708_660} In the diagram, \(D\) lies on the side \(A B\) of triangle \(A B C\) and \(C D\) is an arc of a circle with centre \(A\) and radius 2 cm . The line \(B C\) is of length \(2 \sqrt { } 3 \mathrm {~cm}\) and is perpendicular to \(A C\). Find the area of the shaded region \(B D C\), giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).

5 The first term of a geometric progression is \(5 \frac { 1 } { 3 }\) and the fourth term is \(2 \frac { 1 } { 4 }\). Find

1. the common ratio,
2. the sum to infinity.

6 The functions f and g are defined for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\) by $$\begin{aligned} & f ( x ) = \frac { 1 } { 2 } x + \frac { 1 } { 6 } \pi
& g ( x ) = \cos x \end{aligned}$$ Solve the following equations for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. \(\operatorname { gf } ( x ) = 1\), giving your answer in terms of \(\pi\).
2. \(\operatorname { fg } ( x ) = 1\), giving your answers correct to 2 decimal places.

7
\includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-3_821_688_255_731}

1. The diagram shows part of the curve \(y = 11 - x ^ { 2 }\) and part of the straight line \(y = 5 - x\) meeting at the point \(A ( p , q )\), where \(p\) and \(q\) are positive constants. Find the values of \(p\) and \(q\).
2. The function f is defined for the domain \(x \geqslant 0\) by $$f ( x ) = \begin{cases} 11 - x ^ { 2 } & \text { for } 0 \leqslant x \leqslant p
   5 - x & \text { for } x > p \end{cases}$$ Express \(\mathrm { f } ^ { - 1 } ( x )\) in a similar way.

8 A curve is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( 3 x + 4 ) ^ { \frac { 3 } { 2 } } - 6 x - 8 .$$

1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Verify that the curve has a stationary point when \(x = - 1\) and determine its nature.
3. It is now given that the stationary point on the curve has coordinates \(( - 1,5 )\). Find the equation of the curve.

9 The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { c } p
1
1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { l } 4
2
p \end{array} \right)$$ where \(p\) is a constant.

1. In the case where \(O A B\) is a straight line, state the value of \(p\) and find the unit vector in the direction of \(\overrightarrow { O A }\).
2. In the case where \(O A\) is perpendicular to \(A B\), find the possible values of \(p\).
3. In the case where \(p = 3\), the point \(C\) is such that \(O A B C\) is a parallelogram. Find the position vector of \(C\).

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

1. Show that the \(x\)-coordinates of any points of intersection of the line and curve are given by the equation \(2 x ^ { 2 } - ( 6 + k ) x + ( 2 + 3 k ) = 0\).
2. Find the two values of \(k\) for which the line is a tangent to the curve. The two tangents, given by the values of \(k\) found in part (ii), touch the curve at points \(A\) and \(B\).
3. Find the coordinates of \(A\) and \(B\) and the equation of the line \(A B\).

11
\includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-4_526_974_822_587} The diagram shows the curve with equation \(y = x ( x - 2 ) ^ { 2 }\). The minimum point on the curve has coordinates \(( a , 0 )\) and the \(x\)-coordinate of the maximum point is \(b\), where \(a\) and \(b\) are constants.

1. State the value of \(a\).
2. Find the value of \(b\).
3. Find the area of the shaded region.
4. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of the curve has a minimum value \(m\). Find the value of \(m\).