CAIE P1 (Pure Mathematics 1) 2003 November

1 Find the coordinates of the points of intersection of the line \(y + 2 x = 11\) and the curve \(x y = 12\).

2

1. Show that the equation \(4 \sin ^ { 4 } \theta + 5 = 7 \cos ^ { 2 } \theta\) may be written in the form \(4 x ^ { 2 } + 7 x - 2 = 0\), where \(x = \sin ^ { 2 } \theta\).
2. Hence solve the equation \(4 \sin ^ { 4 } \theta + 5 = 7 \cos ^ { 2 } \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

3

1. A debt of \(
   ) 3726\( is repaid by weekly payments which are in arithmetic progression. The first payment is \)\\( 60\) and the debt is fully repaid after 48 weeks. Find the third payment.
2. Find the sum to infinity of the geometric progression whose first term is 6 and whose second term is 4 .

4 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 4 x + 1\). The curve passes through the point \(( 1,5 )\).

1. Find the equation of the curve.
2. Find the set of values of \(x\) for which the gradient of the curve is positive.

5
\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-2_594_778_1360_682} The diagram shows a trapezium \(A B C D\) in which \(B C\) is parallel to \(A D\) and angle \(B C D = 90 ^ { \circ }\). The coordinates of \(A , B\) and \(D\) are \(( 2,0 ) , ( 4,6 )\) and \(( 12,5 )\) respectively.

1. Find the equations of \(B C\) and \(C D\).
2. Calculate the coordinates of \(C\).

6
\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-3_293_502_269_826} The diagram shows the sector \(O P Q\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The angle \(P O Q\) is \(\theta\) radians and the perimeter of the sector is 20 cm .

1. Show that \(\theta = \frac { 20 } { r } - 2\).
2. Hence express the area of the sector in terms of \(r\).
3. In the case where \(r = 8\), find the length of the chord \(P Q\).

7
\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-3_636_1047_1153_550} The diagram shows a triangular prism with a horizontal rectangular base \(A D F C\), where \(C F = 12\) units and \(D F = 6\) units. The vertical ends \(A B C\) and \(D E F\) are isosceles triangles with \(A B = B C = 5\) units. The mid-points of \(B E\) and \(D F\) are \(M\) and \(N\) respectively. The origin \(O\) is at the mid-point of \(A C\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O C , O N\) and \(O B\) respectively.

1. Find the length of \(O B\).
2. Express each of the vectors \(\overrightarrow { M C }\) and \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Evaluate \(\overrightarrow { M C } \cdot \overrightarrow { M N }\) and hence find angle \(C M N\), giving your answer correct to the nearest degree.

8 A solid rectangular block has a base which measures \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\). The height of the block is \(y \mathrm {~cm}\) and the volume of the block is \(72 \mathrm {~cm} ^ { 3 }\).

1. Express \(y\) in terms of \(x\) and show that the total surface area, \(A \mathrm {~cm} ^ { 2 }\), of the block is given by $$A = 4 x ^ { 2 } + \frac { 216 } { x }$$ Given that \(x\) can vary,
2. find the value of \(x\) for which \(A\) has a stationary value,
3. find this stationary value and determine whether it is a maximum or a minimum.

9
\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-4_563_679_938_733} The diagram shows points \(A ( 0,4 )\) and \(B ( 2,1 )\) on the curve \(y = \frac { 8 } { 3 x + 2 }\). The tangent to the curve at \(B\) crosses the \(x\)-axis at \(C\). The point \(D\) has coordinates \(( 2,0 )\).

1. Find the equation of the tangent to the curve at \(B\) and hence show that the area of triangle \(B D C\) is \(\frac { 4 } { 3 }\).
2. Show that the volume of the solid formed when the shaded region \(O D B A\) is rotated completely about the \(x\)-axis is \(8 \pi\).

10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x - 5 , \quad x \in \mathbb { R } ,
& \mathrm {~g} : x \mapsto \frac { 4 } { 2 - x } , \quad x \in \mathbb { R } , \quad x \neq 2 . \end{aligned}$$

1. Find the value of \(x\) for which \(\mathrm { fg } ( x ) = 7\).
2. Express each of \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
3. Show that the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has no real roots.
4. Sketch, on a single diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between these two graphs.