CAIE P1 (Pure Mathematics 1) 2010 June

1 The acute angle \(x\) radians is such that \(\tan x = k\), where \(k\) is a positive constant. Express, in terms of \(k\),

1. \(\tan ( \pi - x )\),
2. \(\tan \left( \frac { 1 } { 2 } \pi - x \right)\),
3. \(\sin x\).

2

1. Find the first 3 terms in the expansion of \(\left( 2 x - \frac { 3 } { x } \right) ^ { 5 }\) in descending powers of \(x\).
2. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + \frac { 2 } { x ^ { 2 } } \right) \left( 2 x - \frac { 3 } { x } \right) ^ { 5 }\).

3 The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is 49 .

1. Find the first term of the progression and the common difference. The \(n\)th term of the progression is 46 .
2. Find the value of \(n\).

4 \includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-2_428_550_1343_794} The diagram shows the curve \(y = 6 x - x ^ { 2 }\) and the line \(y = 5\). Find the area of the shaded region.

5 The function f is such that \(\mathrm { f } ( x ) = 2 \sin ^ { 2 } x - 3 \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \pi\).

1. Express \(\mathrm { f } ( x )\) in the form \(a + b \cos ^ { 2 } x\), stating the values of \(a\) and \(b\).
2. State the greatest and least values of \(\mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) + 1 = 0\).

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 6\) and the point \(( 9,2 )\) lies on the curve.

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the stationary point on the curve and determine the nature of the stationary point.

7 \includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-3_766_589_251_778} The diagram shows part of the curve \(y = 2 - \frac { 18 } { 2 x + 3 }\), which crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The normal to the curve at \(A\) crosses the \(y\)-axis at \(C\).

1. Show that the equation of the line \(A C\) is \(9 x + 4 y = 27\).
2. Find the length of \(B C\).

8 \includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-3_625_547_1489_797} The diagram shows a triangle \(A B C\) in which \(A\) is \(( 3 , - 2 )\) and \(B\) is \(( 15,22 )\). The gradients of \(A B , A C\) and \(B C\) are \(2 m , - 2 m\) and \(m\) respectively, where \(m\) is a positive constant.

1. Find the gradient of \(A B\) and deduce the value of \(m\).
2. Find the coordinates of \(C\). The perpendicular bisector of \(A B\) meets \(B C\) at \(D\).
3. Find the coordinates of \(D\).

9 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 12 x + 7\) for \(x \in \mathbb { R }\).

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x - b ) ^ { 2 } - c\).
2. State the range of f .
3. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) < 21\). The function g is defined by \(\mathrm { g } : x \mapsto 2 x + k\) for \(x \in \mathbb { R }\).
4. Find the value of the constant \(k\) for which the equation \(\operatorname { gf } ( x ) = 0\) has two equal roots.

10 \includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-4_552_629_842_758} The diagram shows the parallelogram \(O A B C\). Given that \(\overrightarrow { O A } = \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { O C } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k }\), find

1. the unit vector in the direction of \(\overrightarrow { O B }\),
2. the acute angle between the diagonals of the parallelogram,
3. the perimeter of the parallelogram, correct to 1 decimal place.