CAIE P1 (Pure Mathematics 1) 2013 June

1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { } ( 2 x + 5 )\) and \(( 2,5 )\) is a point on the curve. Find the equation of the curve.

2 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-2_501_641_461_753} The diagram shows a circle \(C\) with centre \(O\) and radius 3 cm . The radii \(O P\) and \(O Q\) are extended to \(S\) and \(R\) respectively so that \(O R S\) is a sector of a circle with centre \(O\). Given that \(P S = 6 \mathrm {~cm}\) and that the area of the shaded region is equal to the area of circle \(C\),

1. show that angle \(P O Q = \frac { 1 } { 4 } \pi\) radians,
2. find the perimeter of the shaded region.

3

1. Express the equation \(2 \cos ^ { 2 } \theta = \tan ^ { 2 } \theta\) as a quadratic equation in \(\cos ^ { 2 } \theta\).
2. Solve the equation \(2 \cos ^ { 2 } \theta = \tan ^ { 2 } \theta\) for \(0 \leqslant \theta \leqslant \pi\), giving solutions in terms of \(\pi\).

4

1. Find the first three terms in the expansion of \(( 2 + a x ) ^ { 5 }\) in ascending powers of \(x\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ( 2 + a x ) ^ { 5 }\) is 240 , find the possible values of \(a\).

5

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

6 The non-zero variables \(x , y\) and \(u\) are such that \(u = x ^ { 2 } y\). Given that \(y + 3 x = 9\), find the stationary value of \(u\) and determine whether this is a maximum or a minimum value.

7 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-3_465_554_255_794} The diagram shows three points \(A ( 2,14 ) , B ( 14,6 )\) and \(C ( 7,2 )\). The point \(X\) lies on \(A B\), and \(C X\) is perpendicular to \(A B\). Find, by calculation,

1. the coordinates of \(X\),
2. the ratio \(A X : X B\).

8 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-3_716_437_1137_854} The diagram shows a parallelogram \(O A B C\) in which $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ 3 \\ - 4 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right)$$

1. Use a scalar product to find angle \(B O C\).
2. Find a vector which has magnitude 35 and is parallel to the vector \(\overrightarrow { O C }\).

9

1. In an arithmetic progression, the sum, \(S _ { n }\), of the first \(n\) terms is given by \(S _ { n } = 2 n ^ { 2 } + 8 n\). Find the first term and the common difference of the progression.
2. The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the 1st term, the 9th term and the \(n\)th term respectively of an arithmetic progression. Find the value of \(n\).

10 The function f is defined by \(\mathrm { f } : x \mapsto 2 x + k , x \in \mathbb { R }\), where \(k\) is a constant.

1. In the case where \(k = 3\), solve the equation \(\mathrm { ff } ( x ) = 25\). The function g is defined by \(\mathrm { g } : x \mapsto x ^ { 2 } - 6 x + 8 , x \in \mathbb { R }\).
2. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has no real solutions. The function \(h\) is defined by \(h : x \mapsto x ^ { 2 } - 6 x + 8 , x > 3\).
3. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

11 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-4_643_570_849_790} The diagram shows part of the curve \(y = \frac { 8 } { \sqrt { } x } - x\) and points \(A ( 1,7 )\) and \(B ( 4,0 )\) which lie on the curve. The tangent to the curve at \(B\) intersects the line \(x = 1\) at the point \(C\).

1. Find the coordinates of \(C\).
2. Find the area of the shaded region.