CAIE P1 (Pure Mathematics 1) 2013 November

1 Solve the inequality \(x ^ { 2 } - x - 2 > 0\).

2 A curve has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = x ^ { - \frac { 3 } { 2 } } + 1\) and that \(\mathrm { f } ( 4 ) = 5\). Find \(\mathrm { f } ( x )\).

3 The point \(A\) has coordinates \(( 3,1 )\) and the point \(B\) has coordinates \(( - 21,11 )\). The point \(C\) is the mid-point of \(A B\).

1. Find the equation of the line through \(A\) that is perpendicular to \(y = 2 x - 7\).
2. Find the distance \(A C\).

4 \includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-2_711_643_900_753} The diagram shows a pyramid \(O A B C\) in which the edge \(O C\) is vertical. The horizontal base \(O A B\) is a triangle, right-angled at \(O\), and \(D\) is the mid-point of \(A B\). The edges \(O A , O B\) and \(O C\) have lengths of 8 units, 6 units and 10 units respectively. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O B }\) and \(\overrightarrow { O C }\) respectively.

1. Express each of the vectors \(\overrightarrow { O D }\) and \(\overrightarrow { C D }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to find angle ODC.

5

1. In a geometric progression, the sum to infinity is equal to eight times the first term. Find the common ratio.
2. In an arithmetic progression, the fifth term is 197 and the sum of the first ten terms is 2040. Find the common difference.

6 \includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-3_463_621_255_762} The diagram shows sector \(O A B\) with centre \(O\) and radius 11 cm . Angle \(A O B = \alpha\) radians. Points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively. Arc \(C D\) has centre \(O\) and radius 5 cm .

1. The area of the shaded region \(A B D C\) is equal to \(k\) times the area of the unshaded region \(O C D\). Find \(k\).
2. The perimeter of the shaded region \(A B D C\) is equal to twice the perimeter of the unshaded region \(O C D\). Find the exact value of \(\alpha\).

7

1. Find the possible values of \(x\) for which \(\sin ^ { - 1 } \left( x ^ { 2 } - 1 \right) = \frac { 1 } { 3 } \pi\), giving your answers correct to 3 decimal places.
2. Solve the equation \(\sin \left( 2 \theta + \frac { 1 } { 3 } \pi \right) = \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant \pi\), giving \(\theta\) in terms of \(\pi\) in your answers.

8

1. Find the coefficient of \(x ^ { 8 }\) in the expansion of \(\left( x + 3 x ^ { 2 } \right) ^ { 4 }\).
2. Find the coefficient of \(x ^ { 8 }\) in the expansion of \(\left( x + 3 x ^ { 2 } \right) ^ { 5 }\).
3. Hence find the coefficient of \(x ^ { 8 }\) in the expansion of \(\left[ 1 + \left( x + 3 x ^ { 2 } \right) \right] ^ { 5 }\).

9 A curve has equation \(y = \frac { k ^ { 2 } } { x + 2 } + x\), where \(k\) is a positive constant. Find, in terms of \(k\), the values of \(x\) for which the curve has stationary points and determine the nature of each stationary point.

10 The function f is defined by \(\mathrm { f } : x \mapsto x ^ { 2 } + 4 x\) for \(x \geqslant c\), where \(c\) is a constant. It is given that f is a one-one function.

1. State the range of f in terms of \(c\) and find the smallest possible value of \(c\). The function g is defined by \(\mathrm { g } : x \mapsto a x + b\) for \(x \geqslant 0\), where \(a\) and \(b\) are positive constants. It is given that, when \(c = 0 , \operatorname { gf } ( 1 ) = 11\) and \(\operatorname { fg } ( 1 ) = 21\).
2. Write down two equations in \(a\) and \(b\) and solve them to find the values of \(a\) and \(b\).

11 \includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-4_547_1057_255_543} The diagram shows the curve \(y = \sqrt { } \left( x ^ { 4 } + 4 x + 4 \right)\).

1. Find the equation of the tangent to the curve at the point ( 0,2 ).
2. Show that the \(x\)-coordinates of the points of intersection of the line \(y = x + 2\) and the curve are given by the equation \(( x + 2 ) ^ { 2 } = x ^ { 4 } + 4 x + 4\). Hence find these \(x\)-coordinates.
3. The region shaded in the diagram is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of revolution.