CAIE P1 (Pure Mathematics 1) 2013 June

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

2 Find the coefficient of \(x ^ { 2 }\) in the expansion of

1. \(\left( 2 x - \frac { 1 } { 2 x } \right) ^ { 6 }\),
2. \(\left( 1 + x ^ { 2 } \right) \left( 2 x - \frac { 1 } { 2 x } \right) ^ { 6 }\).

3 The straight line \(y = m x + 14\) is a tangent to the curve \(y = \frac { 12 } { x } + 2\) at the point \(P\). Find the value of the constant \(m\) and the coordinates of \(P\).

4
\includegraphics[max width=\textwidth, alt={}, center]{fe4c3555-5736-48c4-b61a-9f6b9a1ee46e-2_645_652_1023_744} The diagram shows a square \(A B C D\) of side 10 cm . The mid-point of \(A D\) is \(O\) and \(B X C\) is an arc of a circle with centre \(O\).

1. Show that angle \(B O C\) is 0.9273 radians, correct to 4 decimal places.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

5 It is given that \(a = \sin \theta - 3 \cos \theta\) and \(b = 3 \sin \theta + \cos \theta\), where \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

1. Show that \(a ^ { 2 } + b ^ { 2 }\) has a constant value for all values of \(\theta\).
2. Find the values of \(\theta\) for which \(2 a = b\).

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

1. State the values of \(p\) and \(q\) for which \(\overrightarrow { O A }\) is parallel to \(\overrightarrow { O B }\).
2. In the case where \(q = 2 p\), find the value of \(p\) for which angle \(B O A\) is \(90 ^ { \circ }\).
3. In the case where \(p = 1\) and \(q = 8\), find the unit vector in the direction of \(\overrightarrow { A B }\).

7 The point \(R\) is the reflection of the point \(( - 1,3 )\) in the line \(3 y + 2 x = 33\). Find by calculation the coordinates of \(R\).

8 The volume of a solid circular cylinder of radius \(r \mathrm {~cm}\) is \(250 \pi \mathrm {~cm} ^ { 3 }\).

1. Show that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the cylinder is given by $$S = 2 \pi r ^ { 2 } + \frac { 500 \pi } { r }$$
2. Given that \(r\) can vary, find the stationary value of \(S\).
3. Determine the nature of this stationary value.

9 A function f is defined by \(\mathrm { f } ( x ) = \frac { 5 } { 1 - 3 x }\), for \(x \geqslant 1\).

1. Find an expression for \(\mathrm { f } ^ { \prime } ( x )\).
2. Determine, with a reason, whether \(f\) is an increasing function, a decreasing function or neither.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain and range of \(\mathrm { f } ^ { - 1 }\).

10

1. The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression.
2. The third term of a geometric progression is four times the first term. The sum of the first six terms is \(k\) times the first term. Find the possible values of \(k\).

11
\includegraphics[max width=\textwidth, alt={}, center]{fe4c3555-5736-48c4-b61a-9f6b9a1ee46e-4_598_789_255_678} The diagram shows the curve \(y = \sqrt { } ( 1 + 4 x )\), which intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The normal to the curve at \(B\) meets the \(x\)-axis at \(C\). Find

1. the equation of \(B C\),
2. the area of the shaded region.