CAIE P1 (Pure Mathematics 1) 2015 June

1 Express \(2 x ^ { 2 } - 12 x + 7\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.

2 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x + 1 ) ^ { \frac { 1 } { 2 } }\) and the point (4,7) lies on the curve. Find the equation of the curve.

3

1. Write down the first 4 terms, in ascending powers of \(x\), of the expansion of \(( a - x ) ^ { 5 }\).
2. The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - a x ) ( a - x ) ^ { 5 }\) is - 200 . Find the possible values of the constant \(a\).

4

1. Express the equation \(3 \sin \theta = \cos \theta\) in the form \(\tan \theta = k\) and solve the equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
2. Solve the equation \(3 \sin ^ { 2 } 2 x = \cos ^ { 2 } 2 x\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

5 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ 2 \\ - 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 5 \\ - 1 \\ - 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 6 \\ 1 \\ 2 \end{array} \right)$$

1. Show that angle \(A B C\) is \(90 ^ { \circ }\).
2. Find the area of triangle \(A B C\), giving your answer correct to 1 decimal place.

6 \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-2_519_670_1640_735} The diagram shows the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\), where \(\mathrm { f } ^ { - 1 }\) is defined by \(\mathrm { f } ^ { - 1 } ( x ) = \frac { 1 - 5 x } { 2 x }\) for \(0 < x \leqslant 2\).

1. Find an expression for \(\mathrm { f } ( x )\) and state the domain of f .
2. The function g is defined by \(\mathrm { g } ( x ) = \frac { 1 } { x }\) for \(x \geqslant 1\). Find an expression for \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\), giving your answer in the form \(a x + b\), where \(a\) and \(b\) are constants to be found.

7 The point \(A\) has coordinates \(( p , 1 )\) and the point \(B\) has coordinates \(( 9,3 p + 1 )\), where \(p\) is a constant.

1. For the case where the distance \(A B\) is 13 units, find the possible values of \(p\).
2. For the case in which the line with equation \(2 x + 3 y = 9\) is perpendicular to \(A B\), find the value of \(p\).

8 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }\) for \(x > - 1\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. State, with a reason, whether f is an increasing function, a decreasing function or neither. The function g is defined by \(\mathrm { g } ( x ) = \frac { 1 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }\) for \(x < - 1\).
3. Find the coordinates of the stationary point on the curve \(y = \mathrm { g } ( x )\).

9

1. The first term of an arithmetic progression is - 2222 and the common difference is 17 . Find the value of the first positive term.
2. The first term of a geometric progression is \(\sqrt { } 3\) and the second term is \(2 \cos \theta\), where \(0 < \theta < \pi\). Find the set of values of \(\theta\) for which the progression is convergent.

10 \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-3_682_1319_1525_413} Points \(A ( 2,9 )\) and \(B ( 3,0 )\) lie on the curve \(y = 9 + 6 x - 3 x ^ { 2 }\), as shown in the diagram. The tangent at \(A\) intersects the \(x\)-axis at \(C\). Showing all necessary working,

1. find the equation of the tangent \(A C\) and hence find the \(x\)-coordinate of \(C\),
2. find the area of the shaded region \(A B C\).
   [0pt] [Question 11 is printed on the next page.]

11 \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-4_519_560_260_797} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). The point \(C\) on \(O B\) is such that angle \(A C O\) is a right angle. Angle \(A O B\) is \(\alpha\) radians and is such that \(A C\) divides the sector into two regions of equal area.

1. Show that \(\sin \alpha \cos \alpha = \frac { 1 } { 2 } \alpha\). It is given that the solution of the equation in part (i) is \(\alpha = 0.9477\), correct to 4 decimal places.
2. Find the ratio perimeter of region \(O A C\) : perimeter of region \(A C B\), giving your answer in the form \(k : 1\), where \(k\) is given correct to 1 decimal place.
3. Find angle \(A O B\) in degrees. {www.cie.org.uk} after the live examination series. }