CAIE P1 (Pure Mathematics 1) 2014 November

1 In the expansion of \(( 2 + a x ) ^ { 7 }\), the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\). Find the value of the non-zero constant \(a\).

2 Find the value of \(x\) satisfying the equation \(\sin ^ { - 1 } ( x - 1 ) = \tan ^ { - 1 } ( 3 )\).

3 Solve the equation \(\frac { 13 \sin ^ { 2 } \theta } { 2 + \cos \theta } + \cos \theta = 2\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

4 The line \(4 x + k y = 20\) passes through the points \(A ( 8 , - 4 )\) and \(B ( b , 2 b )\), where \(k\) and \(b\) are constants.

1. Find the values of \(k\) and \(b\).
2. Find the coordinates of the mid-point of \(A B\).

5 Find the set of values of \(k\) for which the line \(y = 2 x - k\) meets the curve \(y = x ^ { 2 } + k x - 2\) at two distinct points.

6 Relative to an origin \(O\), the position vector of \(A\) is \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and the position vector of \(B\) is \(7 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\).

1. Show that angle \(O A B\) is a right angle.
2. Find the area of triangle \(O A B\).

7

1. A geometric progression has first term \(a ( a \neq 0 )\), common ratio \(r\) and sum to infinity \(S\). A second geometric progression has first term \(a\), common ratio \(2 r\) and sum to infinity \(3 S\). Find the value of \(r\).
2. An arithmetic progression has first term 7. The \(n\)th term is 84 and the ( \(3 n\) )th term is 245 . Find the value of \(n\).

8 \includegraphics[max width=\textwidth, alt={}, center]{77543862-ed95-42bf-b788-a9a43f039a89-3_408_686_264_731} In the diagram, \(A B\) is an arc of a circle with centre \(O\) and radius 4 cm . Angle \(A O B\) is \(\alpha\) radians. The point \(D\) on \(O B\) is such that \(A D\) is perpendicular to \(O B\). The arc \(D C\), with centre \(O\), meets \(O A\) at \(C\).

1. Find an expression in terms of \(\alpha\) for the perimeter of the shaded region \(A B D C\).
2. For the case where \(\alpha = \frac { 1 } { 6 } \pi\), find the area of the shaded region \(A B D C\), giving your answer in the form \(k \pi\), where \(k\) is a constant to be determined.

9 The function f is defined for \(x > 0\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - \frac { 2 } { x ^ { 2 } }\). The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( 2,6 )\).

1. Find the equation of the normal to the curve at \(P\).
2. Find the equation of the curve.
3. Find the \(x\)-coordinate of the stationary point and state with a reason whether this point is a maximum or a minimum.

10

1. Express \(x ^ { 2 } - 2 x - 15\) in the form \(( x + a ) ^ { 2 } + b\). The function f is defined for \(p \leqslant x \leqslant q\), where \(p\) and \(q\) are positive constants, by $$f : x \mapsto x ^ { 2 } - 2 x - 15$$ The range of f is given by \(c \leqslant \mathrm { f } ( x ) \leqslant d\), where \(c\) and \(d\) are constants.
2. State the smallest possible value of \(c\). For the case where \(c = 9\) and \(d = 65\),
3. find \(p\) and \(q\),
4. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

11 \includegraphics[max width=\textwidth, alt={}, center]{77543862-ed95-42bf-b788-a9a43f039a89-4_995_905_260_621} The diagram shows parts of the curves \(y = ( 4 x + 1 ) ^ { \frac { 1 } { 2 } }\) and \(y = \frac { 1 } { 2 } x ^ { 2 } + 1\) intersecting at points \(P ( 0,1 )\) and \(Q ( 2,3 )\). The angle between the tangents to the two curves at \(Q\) is \(\alpha\).

1. Find \(\alpha\), giving your answer in degrees correct to 3 significant figures.
2. Find by integration the area of the shaded region.