CAIE P1 (Pure Mathematics 1) 2014 June

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\includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-2_750_1287_258_427} The diagram shows part of the graph of \(y = a + b \sin x\). State the values of the constants \(a\) and \(b\). [2

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1. Express \(4 x ^ { 2 } - 12 x\) in the form \(( 2 x + a ) ^ { 2 } + b\).
2. Hence, or otherwise, find the set of values of \(x\) satisfying \(4 x ^ { 2 } - 12 x > 7\).

3 Find the term independent of \(x\) in the expansion of \(\left( 4 x ^ { 3 } + \frac { 1 } { 2 x } \right) ^ { 8 }\).

4 A curve has equation \(y = \frac { 4 } { ( 3 x + 1 ) ^ { 2 } }\). Find the equation of the tangent to the curve at the point where the line \(x = - 1\) intersects the curve.

5 An arithmetic progression has first term \(a\) and common difference \(d\). It is given that the sum of the first 200 terms is 4 times the sum of the first 100 terms.

1. Find \(d\) in terms of \(a\).
2. Find the 100th term in terms of \(a\).

6
\includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-3_625_897_260_623} The diagram shows triangle \(A B C\) in which \(A B\) is perpendicular to \(B C\). The length of \(A B\) is 4 cm and angle \(C A B\) is \(\alpha\) radians. The arc \(D E\) with centre \(A\) and radius 2 cm meets \(A C\) at \(D\) and \(A B\) at \(E\). Find, in terms of \(\alpha\),

1. the area of the shaded region,
2. the perimeter of the shaded region.

7 The coordinates of points \(A\) and \(B\) are \(( a , 2 )\) and \(( 3 , b )\) respectively, where \(a\) and \(b\) are constants. The distance \(A B\) is \(\sqrt { } ( 125 )\) units and the gradient of the line \(A B\) is 2 . Find the possible values of \(a\) and of \(b\).

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

1. Find the values of \(p\) for which angle \(A O B\) is \(90 ^ { \circ }\).
2. For the case where \(p = 3\), find the unit vector in the direction of \(\overrightarrow { B A }\).

9

1. Prove the identity \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } \equiv \frac { 1 } { \tan \theta }\).
2. Hence solve the equation \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = 4 \tan \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

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\includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-4_819_812_255_662} The diagram shows the function f defined for \(- 1 \leqslant x \leqslant 4\), where $$f ( x ) = \begin{cases} 3 x - 2 & \text { for } - 1 \leqslant x \leqslant 1
\frac { 4 } { 5 - x } & \text { for } 1 < x \leqslant 4 \end{cases}$$

1. State the range of f .
2. Copy the diagram and on your copy sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\).
3. Obtain expressions to define the function \(\mathrm { f } ^ { - 1 }\), giving also the set of values for which each expression is valid.

11 A line has equation \(y = 2 x + c\) and a curve has equation \(y = 8 - 2 x - x ^ { 2 }\).

1. For the case where the line is a tangent to the curve, find the value of the constant \(c\).
2. For the case where \(c = 11\), find the \(x\)-coordinates of the points of intersection of the line and the curve. Find also, by integration, the area of the region between the line and the curve.

12 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { \frac { 1 } { 2 } } - x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(\left( 4 , \frac { 2 } { 3 } \right)\).

1. Find the equation of the curve.
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
3. Find the coordinates of the stationary point and determine its nature.