CAIE P1 (Pure Mathematics 1) 2006 June

1 A curve has equation \(y = \frac { k } { x }\). Given that the gradient of the curve is - 3 when \(x = 2\), find the value of the constant \(k\).

2 Solve the equation $$\sin 2 x + 3 \cos 2 x = 0$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

3 Each year a company gives a grant to a charity. The amount given each year increases by \(5 \%\) of its value in the preceding year. The grant in 2001 was \(
) 5000$. Find

1. the grant given in 2011,
2. the total amount of money given to the charity during the years 2001 to 2011 inclusive.

4 The first three terms in the expansion of \(( 2 + a x ) ^ { n }\), in ascending powers of \(x\), are \(32 - 40 x + b x ^ { 2 }\). Find the values of the constants \(n , a\) and \(b\).

5 The curve \(y ^ { 2 } = 12 x\) intersects the line \(3 y = 4 x + 6\) at two points. Find the distance between the two points.

6
\includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-2_389_995_1432_575} In the diagram, \(A B C\) is a triangle in which \(A B = 4 \mathrm {~cm} , B C = 6 \mathrm {~cm}\) and angle \(A B C = 150 ^ { \circ }\). The line \(C X\) is perpendicular to the line \(A B X\).

1. Find the exact length of \(B X\) and show that angle \(C A B = \tan ^ { - 1 } \left( \frac { 3 } { 4 + 3 \sqrt { } 3 } \right)\).
2. Show that the exact length of \(A C\) is \(\sqrt { } ( 52 + 24 \sqrt { } 3 ) \mathrm { cm }\).

7
\includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-3_545_759_269_694} The diagram shows a circle with centre \(O\) and radius 8 cm . Points \(A\) and \(B\) lie on the circle. The tangents at \(A\) and \(B\) meet at the point \(T\), and \(A T = B T = 15 \mathrm {~cm}\).

1. Show that angle \(A O B\) is 2.16 radians, correct to 3 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

8
\includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-3_517_1117_1362_514} The diagram shows the roof of a house. The base of the roof, \(O A B C\), is rectangular and horizontal with \(O A = C B = 14 \mathrm {~m}\) and \(O C = A B = 8 \mathrm {~m}\). The top of the roof \(D E\) is 5 m above the base and \(D E = 6 \mathrm {~m}\). The sloping edges \(O D , C D , A E\) and \(B E\) are all equal in length. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O C\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards.

1. Express the vector \(\overrightarrow { O D }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\), and find its magnitude.
2. Use a scalar product to find angle \(D O B\).

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { \sqrt { } ( 6 - 2 x ) }\), and \(P ( 1,8 )\) is a point on the curve.

1. The normal to the curve at the point \(P\) meets the coordinate axes at \(Q\) and at \(R\). Find the coordinates of the mid-point of \(Q R\).
2. Find the equation of the curve.

10
\includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-4_515_885_662_630} The diagram shows the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x + k\), where \(k\) is a constant. The curve has a minimum point on the \(x\)-axis.

1. Find the value of \(k\).
2. Find the coordinates of the maximum point of the curve.
3. State the set of values of \(x\) for which \(x ^ { 3 } - 3 x ^ { 2 } - 9 x + k\) is a decreasing function of \(x\).
4. Find the area of the shaded region.

11 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto k - x & \text { for } x \in \mathbb { R } , \text { where } k \text { is a constant, }
\mathrm { g } : x \mapsto \frac { 9 } { x + 2 } & \text { for } x \in \mathbb { R } , x \neq - 2 . \end{array}$$

1. Find the values of \(k\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has two equal roots and solve the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) in these cases.
2. Solve the equation \(\operatorname { fg } ( x ) = 5\) when \(k = 6\).
3. Express \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).