CAIE P1 (Pure Mathematics 1) 2005 June

1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 2 } - 5\). Given that the point \(( 3,8 )\) lies on the curve, find the equation of the curve.

2 Find the gradient of the curve \(y = \frac { 12 } { x ^ { 2 } - 4 x }\) at the point where \(x = 3\).

3

1. Show that the equation \(\sin \theta + \cos \theta = 2 ( \sin \theta - \cos \theta )\) can be expressed as \(\tan \theta = 3\).
2. Hence solve the equation \(\sin \theta + \cos \theta = 2 ( \sin \theta - \cos \theta )\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4

1. Find the first 3 terms in the expansion of \(( 2 - x ) ^ { 6 }\) in ascending powers of \(x\).
2. Find the value of \(k\) for which there is no term in \(x ^ { 2 }\) in the expansion of \(( 1 + k x ) ( 2 - x ) ^ { 6 }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{e439eea6-76f0-41eb-aa91-bd0f3e4e1a07-2_591_1061_1098_541} The diagram shows a rhombus \(A B C D\). The points \(B\) and \(D\) have coordinates \(( 2,10 )\) and \(( 6,2 )\) respectively, and \(A\) lies on the \(x\)-axis. The mid-point of \(B D\) is \(M\). Find, by calculation, the coordinates of each of \(M , A\) and \(C\).

6 A geometric progression has 6 terms. The first term is 192 and the common ratio is 1.5. An arithmetic progression has 21 terms and common difference 1.5. Given that the sum of all the terms in the geometric progression is equal to the sum of all the terms in the arithmetic progression, find the first term and the last term of the arithmetic progression.

7 A function f is defined by f : \(x \mapsto 3 - 2 \sin x\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1. Find the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\). A function g is defined by \(\mathrm { g } : x \mapsto 3 - 2 \sin x\), for \(0 ^ { \circ } \leqslant x \leqslant A ^ { \circ }\), where \(A\) is a constant.
3. State the largest value of \(A\) for which g has an inverse.
4. When \(A\) has this value, obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).

8 \includegraphics[max width=\textwidth, alt={}, center]{e439eea6-76f0-41eb-aa91-bd0f3e4e1a07-3_438_805_849_669} In the diagram, \(A B C\) is a semicircle, centre \(O\) and radius 9 cm . The line \(B D\) is perpendicular to the diameter \(A C\) and angle \(A O B = 2.4\) radians.

1. Show that \(B D = 6.08 \mathrm {~cm}\), correct to 3 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

9 A curve has equation \(y = \frac { 4 } { \sqrt { } x }\).

1. The normal to the curve at the point \(( 4,2 )\) meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Find the length of \(P Q\), correct to 3 significant figures.
2. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).

10 The equation of a curve is \(y = x ^ { 2 } - 3 x + 4\).

1. Show that the whole of the curve lies above the \(x\)-axis.
2. Find the set of values of \(x\) for which \(x ^ { 2 } - 3 x + 4\) is a decreasing function of \(x\). The equation of a line is \(y + 2 x = k\), where \(k\) is a constant.
3. In the case where \(k = 6\), find the coordinates of the points of intersection of the line and the curve.
4. Find the value of \(k\) for which the line is a tangent to the curve.

11 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$$

1. Use a scalar product to find angle \(A O B\), correct to the nearest degree.
2. Find the unit vector in the direction of \(\overrightarrow { A B }\).
3. The point \(C\) is such that \(\overrightarrow { O C } = 6 \mathbf { j } + p \mathbf { k }\), where \(p\) is a constant. Given that the lengths of \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\) are equal, find the possible values of \(p\).