CAIE P1 (Pure Mathematics 1) 2024 March

1 Find the exact value of \(\int _ { 3 } ^ { \infty } \frac { 2 } { x ^ { 2 } } d x\).

2
\includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-03_451_597_255_735} The diagram shows part of the curve with equation \(\mathrm { y } = \mathrm { ksin } \frac { 1 } { 2 } \mathrm { x }\), where \(k\) is a positive constant and \(x\) is measured in radians. The curve has a minimum point \(A\).

1. State the coordinates of \(A\).
2. A sequence of transformations is applied to the curve in the following order. Translation of 2 units in the negative \(y\)-direction
   Reflection in the \(x\)-axis
   Find the equation of the new curve and determine the coordinates of the point on the new curve corresponding to \(A\).

3 A curve is such that \(\frac { \mathrm { dy } } { \mathrm { dx } } = 3 ( 4 \mathrm { x } + 5 ) ^ { \frac { 1 } { 2 } }\). It is given that the points \(( 1,9 )\) and \(( 5 , a )\) lie on the curve. Find the value of \(a\).

4

1. Prove that \(\frac { ( \sin \theta + \cos \theta ) ^ { 2 } - 1 } { \cos ^ { 2 } \theta } \equiv 2 \tan \theta\).
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2. Hence solve the equation \(\frac { ( \sin \theta + \cos \theta ) ^ { 2 } - 1 } { \cos ^ { 2 } \theta } = 5 \tan ^ { 3 } \theta\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).

5 A curve has the equation \(\mathrm { y } = \frac { 3 } { 2 \mathrm { x } ^ { 2 } - 5 }\).
Find the equation of the normal to the curve at the point \(( 2,1 )\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are integers.

6 It is given that the coefficient of \(x ^ { 3 }\) in the expansion of $$( 2 + a x ) ^ { 4 } ( 5 - a x )$$ is 432 .
Find the value of the constant \(a\).

7 The straight line \(\mathrm { y } = \mathrm { x } + 5\) meets the curve \(2 \mathrm { x } ^ { 2 } + 3 \mathrm { y } ^ { 2 } = \mathrm { k }\) at a single point \(P\).

1. Find the value of the constant \(k\).
2. Find the coordinates of \(P\).

8

1. An arithmetic progression is such that its first term is 6 and its tenth term is 19.5 .
   Find the sum of the first 100 terms of this arithmetic progression.
2. A geometric progression \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that \(a _ { 1 } = 24\) and the common ratio is \(\frac { 1 } { 2 }\). The sum to infinity of this geometric progression is denoted by \(S\). The sum to infinity of the even-numbered terms (i.e. \(a _ { 2 } , a _ { 4 } , a _ { 6 } , \ldots\) ) is denoted by \(S _ { E }\). Find the values of \(S\) and \(S _ { E }\).

9 The functions f and g are defined for all real values of \(x\) by $$f ( x ) = ( 3 x - 2 ) ^ { 2 } + k \quad \text { and } \quad g ( x ) = 5 x - 1$$ where \(k\) is a constant.

1. Given that the range of the function gf is \(\mathrm { gf } ( x ) \geqslant 39\), find the value of \(k\).
2. For this value of \(k\), determine the range of the function fg .
3. The function h is defined for all real values of \(x\) and is such that \(\mathrm { gh } ( x ) = 35 x + 19\). Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and hence, or otherwise, find an expression for \(\mathrm { h } ( x )\).
   \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-12_739_625_260_721} The diagram shows the circle with centre \(C ( - 4,5 )\) and radius \(\sqrt { 20 }\) units. The circle intersects the \(y\)-axis at the points \(A\) and \(B\). The size of angle \(A C B\) is \(\theta\) radians.

1. Find the equation of the tangent to the circle at the point \(( - 6,9 )\).
2. Find the equation of the circle in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
3. Find the value of \(\theta\) correct to 4 significant figures.
4. Find the perimeter and area of the segment shaded in the diagram.

11
\includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-14_467_757_262_653} The diagram shows the curve with equation \(\mathrm { y } = 2 \mathrm { x } ^ { - \frac { 2 } { 3 } } - 3 \mathrm { x } ^ { - \frac { 1 } { 3 } } + 1\) for \(x > 0\). The curve crosses the \(x\)-axis at points \(A\) and \(B\) and has a minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the area of the region bounded by the curve and the line segment \(A B\).
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