CAIE P1 (Pure Mathematics 1) 2015 June

1 Given that \(\theta\) is an obtuse angle measured in radians and that \(\sin \theta = k\), find, in terms of \(k\), an expression for

1. \(\cos \theta\),
2. \(\tan \theta\),
3. \(\sin ( \theta + \pi )\).

2
\includegraphics[max width=\textwidth, alt={}, center]{c8925c7a-cb3b-43b8-9d09-8adc800c6887-2_627_828_641_657} The diagram shows the curve \(y = 2 x ^ { 2 }\) and the points \(X ( - 2,0 )\) and \(P ( p , 0 )\). The point \(Q\) lies on the curve and \(P Q\) is parallel to the \(y\)-axis.

1. Express the area, \(A\), of triangle \(X P Q\) in terms of \(p\). The point \(P\) moves along the \(x\)-axis at a constant rate of 0.02 units per second and \(Q\) moves along the curve so that \(P Q\) remains parallel to the \(y\)-axis.
2. Find the rate at which \(A\) is increasing when \(p = 2\).

3

1. Find the first three terms, in ascending powers of \(x\), in the expansion of
   (a) \(\quad ( 1 - x ) ^ { 6 }\),
   (b) \(( 1 + 2 x ) ^ { 6 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \([ ( 1 - x ) ( 1 + 2 x ) ] ^ { 6 }\).

4 Relative to the origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3
0
- 4 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r }

6
- 3
2 \end{array} \right)$$

1. Find the cosine of angle \(A O B\). The position vector of \(C\) is given by \(\overrightarrow { O C } = \left( \begin{array} { c } k
   - 2 k
   2 k - 3 \end{array} \right)\).
2. Given that \(A B\) and \(O C\) have the same length, find the possible values of \(k\). 5 A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius \(r \mathrm {~cm}\).
3. Show that the area of the sector, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 12 r - r ^ { 2 }\).
4. Express \(A\) in the form \(a - ( r - b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
5. Given that \(r\) can vary, state the greatest value of \(A\) and find the corresponding angle of the sector. [2] 6 The line with gradient - 2 passing through the point \(P ( 3 t , 2 t )\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
6. Find the area of triangle \(A O B\) in terms of \(t\). The line through \(P\) perpendicular to \(A B\) intersects the \(x\)-axis at \(C\).
7. Show that the mid-point of \(P C\) lies on the line \(y = x\).

7

1. The third and fourth terms of a geometric progression are \(\frac { 1 } { 3 }\) and \(\frac { 2 } { 9 }\) respectively. Find the sum to infinity of the progression.
2. A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic progression. Given that the angle of the largest sector is 4 times the angle of the smallest sector, find the angle of the largest sector.

8 The function f : \(x \mapsto 5 + 3 \cos \left( \frac { 1 } { 2 } x \right)\) is defined for \(0 \leqslant x \leqslant 2 \pi\).

1. Solve the equation \(\mathrm { f } ( x ) = 7\), giving your answer correct to 2 decimal places.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Explain why f has an inverse.
4. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

9 The equation of a curve is \(y = x ^ { 3 } + p x ^ { 2 }\), where \(p\) is a positive constant.

1. Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of \(p\).
2. Find the nature of each of the stationary points. Another curve has equation \(y = x ^ { 3 } + p x ^ { 2 } + p x\).
3. Find the set of values of \(p\) for which this curve has no stationary points.
   [0pt] [Question 10 is printed on the next page.]

10
\includegraphics[max width=\textwidth, alt={}, center]{c8925c7a-cb3b-43b8-9d09-8adc800c6887-4_798_805_258_669} The diagram shows part of the curve \(y = \frac { 8 } { \sqrt { } ( 3 x + 4 ) }\). The curve intersects the \(y\)-axis at \(A ( 0,4 )\). The normal to the curve at \(A\) intersects the line \(x = 4\) at the point \(B\).

1. Find the coordinates of \(B\).
2. Show, with all necessary working, that the areas of the regions marked \(P\) and \(Q\) are equal.