CAIE P1 (Pure Mathematics 1) 2017 November

1 An arithmetic progression has first term - 12 and common difference 6 . The sum of the first \(n\) terms exceeds 3000 . Calculate the least possible value of \(n\).

2 Find the set of values of \(a\) for which the curve \(y = - \frac { 2 } { x }\) and the straight line \(y = a x + 3 a\) meet at two distinct points.

3

1. Find the term independent of \(x\) in the expansion of \(\left( \frac { 2 } { x } - 3 x \right) ^ { 6 }\).
2. Find the value of \(a\) for which there is no term independent of \(x\) in the expansion of $$\left( 1 + a x ^ { 2 } \right) \left( \frac { 2 } { x } - 3 x \right) ^ { 6 }$$

4 The function f is such that \(\mathrm { f } ( x ) = ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } - 6 x\) for \(\frac { 1 } { 2 } < x < k\), where \(k\) is a constant. Find the largest value of \(k\) for which f is a decreasing function.

5

1. Show that the equation \(\frac { \cos \theta + 4 } { \sin \theta + 1 } + 5 \sin \theta - 5 = 0\) may be expressed as \(5 \cos ^ { 2 } \theta - \cos \theta - 4 = 0\).
   [0pt] [3]
2. Hence solve the equation \(\frac { \cos \theta + 4 } { \sin \theta + 1 } + 5 \sin \theta - 5 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

6 The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = \frac { 2 } { x ^ { 2 } - 1 } \text { for } x < - 1
& \mathrm {~g} ( x ) = x ^ { 2 } + 1 \text { for } x > 0 \end{aligned}$$

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
2. Solve the equation \(\operatorname { gf } ( x ) = 5\).

7
\includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-10_401_561_260_790} The diagram shows a rectangle \(A B C D\) in which \(A B = 5\) units and \(B C = 3\) units. Point \(P\) lies on \(D C\) and \(A P\) is an arc of a circle with centre \(B\). Point \(Q\) lies on \(D C\) and \(A Q\) is an arc of a circle with centre \(D\).

1. Show that angle \(A B P = 0.6435\) radians, correct to 4 decimal places.
2. Calculate the areas of the sectors \(B A P\) and \(D A Q\).
3. Calculate the area of the shaded region.

8
\includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-12_485_570_262_790} The diagram shows parts of the graphs of \(y = 3 - 2 x\) and \(y = 4 - 3 \sqrt { } x\) intersecting at points \(A\) and \(B\).

1. Find by calculation the \(x\)-coordinates of \(A\) and \(B\).
2. Find, showing all necessary working, the area of the shaded region.

9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 8
- 6
5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 10
3
- 13 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2
- 3
- 1 \end{array} \right)$$ A fourth point, \(D\), is such that the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\) are the first, second and third terms respectively of a geometric progression.

1. Find the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\).
2. Given that \(D\) is a point lying on the line through \(B\) and \(C\), find the two possible position vectors of the point \(D\).

10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that \(\mathrm { f } ^ { \prime } ( x ) = a x ^ { 2 } + b x\), where \(a\) and \(b\) are positive constants.

1. Find, in terms of \(a\) and \(b\), the non-zero value of \(x\) for which the curve has a stationary point and determine, showing all necessary working, the nature of the stationary point.
2. It is now given that the curve has a stationary point at \(( - 2 , - 3 )\) and that the gradient of the curve at \(x = 1\) is 9 . Find \(\mathrm { f } ( x )\).

11
\includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-18_428_857_260_644} The diagram shows the curve \(y = ( x - 1 ) ^ { \frac { 1 } { 2 } }\) and points \(A ( 1,0 )\) and \(B ( 5,2 )\) lying on the curve.

1. Find the equation of the line \(A B\), giving your answer in the form \(y = m x + c\).
2. Find, showing all necessary working, the equation of the tangent to the curve which is parallel to \(A B\).
3. Find the perpendicular distance between the line \(A B\) and the tangent parallel to \(A B\). Give your answer correct to 2 decimal places.