CAIE P1 (Pure Mathematics 1) 2021 March

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1. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 + x ) ^ { 5 }\).
2. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 6 }\).
3. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + x ) ^ { 5 } ( 1 - 2 x ) ^ { 6 }\).

2 By using a suitable substitution, solve the equation $$( 2 x - 3 ) ^ { 2 } - \frac { 4 } { ( 2 x - 3 ) ^ { 2 } } - 3 = 0$$

3 Solve the equation \(\frac { \tan \theta + 2 \sin \theta } { \tan \theta - 2 \sin \theta } = 3\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

4 A line has equation \(y = 3 x + k\) and a curve has equation \(y = x ^ { 2 } + k x + 6\), where \(k\) is a constant. Find the set of values of \(k\) for which the line and curve have two distinct points of intersection.

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\includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-07_775_768_260_685} In the diagram, the graph of \(y = \mathrm { f } ( x )\) is shown with solid lines. The graph shown with broken lines is a transformation of \(y = \mathrm { f } ( x )\).

1. Describe fully the two single transformations of \(y = \mathrm { f } ( x )\) that have been combined to give the resulting transformation.
2. State in terms of \(y\), f and \(x\), the equation of the graph shown with broken lines.

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { ( 3 x - 2 ) ^ { 3 } }\) and \(A ( 1 , - 3 )\) lies on the curve. A point is moving along the curve and at \(A\) the \(y\)-coordinate of the point is increasing at 3 units per second.

1. Find the rate of increase at \(A\) of the \(x\)-coordinate of the point.
2. Find the equation of the curve.

7 Functions f and g are defined as follows: $$\begin{aligned} & \mathrm { f } : x \mapsto x ^ { 2 } + 2 x + 3 \text { for } x \leqslant - 1 ,
& \mathrm {~g} : x \mapsto 2 x + 1 \text { for } x \geqslant - 1 . \end{aligned}$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\) and state the range of f .
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. Solve the equation \(\operatorname { gf } ( x ) = 13\).

8 The points \(A ( 7,1 ) , B ( 7,9 )\) and \(C ( 1,9 )\) are on the circumference of a circle.

1. Find an equation of the circle.
2. Find an equation of the tangent to the circle at \(B\).

9 The first term of a progression is \(\cos \theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\).

1. For the case where the progression is geometric, the sum to infinity is \(\frac { 1 } { \cos \theta }\).
   1. Show that the second term is \(\cos \theta \sin ^ { 2 } \theta\).
   2. Find the sum of the first 12 terms when \(\theta = \frac { 1 } { 3 } \pi\), giving your answer correct to 4 significant figures.
2. For the case where the progression is arithmetic, the first two terms are again \(\cos \theta\) and \(\cos \theta \sin ^ { 2 } \theta\) respectively. Find the 85 th term when \(\theta = \frac { 1 } { 3 } \pi\).
   \includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-16_547_421_264_863} The diagram shows a sector \(A B C\) which is part of a circle of radius \(a\). The points \(D\) and \(E\) lie on \(A B\) and \(A C\) respectively and are such that \(A D = A E = k a\), where \(k < 1\). The line \(D E\) divides the sector into two regions which are equal in area.

1. For the case where angle \(B A C = \frac { 1 } { 6 } \pi\) radians, find \(k\) correct to 4 significant figures.
2. For the general case in which angle \(B A C = \theta\) radians, where \(0 < \theta < \frac { 1 } { 2 } \pi\), it is given that \(\frac { \theta } { \sin \theta } > 1\). Find the set of possible values of \(k\).

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\includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-18_497_1049_264_548} The diagram shows the curve with equation \(y = 9 \left( x ^ { - \frac { 1 } { 2 } } - 4 x ^ { - \frac { 3 } { 2 } } \right)\). The curve crosses the \(x\)-axis at the point \(A\).

1. Find the \(x\)-coordinate of \(A\).
2. Find the equation of the tangent to the curve at \(A\).
3. Find the \(x\)-coordinate of the maximum point of the curve.
4. Find the area of the region bounded by the curve, the \(x\)-axis and the line \(x = 9\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.