CAIE P1 (Pure Mathematics 1) 2024 June

1 The coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 - 4 x ) ^ { 6 }\) is 12 times the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 + a x ) ^ { 5 }\). Find the value of the positive constant \(a\).

2 The curve \(y = x ^ { 2 }\) is transformed to the curve \(y = 4 ( x - 3 ) ^ { 2 } - 8\).
Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations have been applied.

3

1. Show that the equation \(\frac { 7 \tan \theta } { \cos \theta } + 12 = 0\) can be expressed as $$12 \sin ^ { 2 } \theta - 7 \sin \theta - 12 = 0$$
2. Hence solve the equation \(\frac { 7 \tan \theta } { \cos \theta } + 12 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4 The function f is defined as follows: $$\mathrm { f } ( x ) = \sqrt { x } - 1 \text { for } x > 1$$

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   \includegraphics[max width=\textwidth, alt={}, center]{84b3cd80-faf0-4522-8286-52bf7f86dc8a-06_362_1543_1023_251} The diagram shows the graph of \(\mathrm { y } = \mathrm { g } ( \mathrm { x } )\) where \(\mathrm { g } ( x ) = \frac { 1 } { x ^ { 2 } + 2 }\) for \(x \in \mathbb { R }\).
2. State the range of \(g\) and explain whether \(g ^ { - 1 }\) exists.
   The function h is defined by \(\mathrm { h } ( x ) = \frac { 1 } { x ^ { 2 } + 2 }\) for \(x \geqslant 0\).
3. Solve the equation \(\mathrm { hf } ( x ) = \mathrm { f } \left( \frac { 25 } { 16 } \right)\). Give your answer in the form \(\mathrm { a } + \mathrm { b } \sqrt { \mathrm { c } }\), where \(a , b\) and \(c\) are
   integers. integers.

5 The first and second terms of an arithmetic progression are \(\tan \theta\) and \(\sin \theta\) respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).

1. Given that \(\theta = \frac { 1 } { 4 } \pi\), find the exact sum of the first 40 terms of the progression.
   The first and second terms of a geometric progression are \(\tan \theta\) and \(\sin \theta\) respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).
2. 1. Find the sum to infinity of the progression in terms of \(\theta\).
   2. Given that \(\theta = \frac { 1 } { 3 } \pi\), find the sum of the first 10 terms of the progression. Give your answer correct to 3 significant figures.

6 The curve with equation \(y = 2 x - 8 x ^ { \frac { 1 } { 2 } }\) has a minimum point at \(A\) and intersects the positive \(x\)-axis at \(B\).

1. Find the coordinates of \(A\) and \(B\).
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{84b3cd80-faf0-4522-8286-52bf7f86dc8a-11_522_1561_296_244} The diagram shows the curve with equation \(\mathrm { y } = 2 \mathrm { x } - 8 \mathrm { x } ^ { \frac { 1 } { 2 } }\) and the line \(A B\). It is given that the equation of \(A B\) is \(y = \frac { 2 x - 32 } { 3 }\). Find the area of the shaded region between the curve and the line.

7 The equation of a circle is \(( x - 6 ) ^ { 2 } + ( y + a ) ^ { 2 } = 18\). The line with equation \(y = 2 a - x\) is a tangent to the circle.

1. Find the two possible values of the constant \(a\).
2. For the greater value of \(a\), find the equation of the diameter which is perpendicular to the given tangent.
   \includegraphics[max width=\textwidth, alt={}, center]{84b3cd80-faf0-4522-8286-52bf7f86dc8a-14_280_1358_317_349} The diagram shows a symmetrical plate \(A B C D E F\). The line \(A B C D\) is straight and the length of \(B C\) is 2 cm . Each of the two sectors \(A B F\) and \(D C E\) is of radius \(r \mathrm {~cm}\) and each of the angles \(A B F\) and \(D C E\) is equal to \(\frac { 1 } { 3 } \pi\) radians.
3. It is given that \(r = 0.4 \mathrm {~cm}\).
   1. Show that the length \(\mathrm { EF } = 2.4 \mathrm {~cm}\).
   2. Find the area of the plate. Give your answer correct to 3 significant figures.
4. It is given instead that the perimeter of the plate is 6 cm . Find the value of \(r\). Give your answer correct to 3 significant figures.

9 A function f is such that \(\mathrm { f } ^ { \prime } ( x ) = 6 ( 2 x - 3 ) ^ { 2 } - 6 x\) for \(x \in \mathbb { R }\).

1. Determine the set of values of \(x\) for which \(\mathrm { f } ( x )\) is decreasing.
2. Given that \(\mathrm { f } ( 1 ) = - 1\), find \(\mathrm { f } ( x )\).

10 The equation of a curve is \(\mathrm { y } = ( 5 - 2 \mathrm { x } ) ^ { \frac { 3 } { 2 } } + 5\) for \(x < \frac { 5 } { 2 }\).

1. A point \(P\) is moving along the curve in such a way that the \(y\)-coordinate of point \(P\) is decreasing at 5 units per second. Find the rate at which the \(x\)-coordinate of point \(P\) is increasing when \(y = 32\).
2. Point \(A\) on the curve has \(y\)-coordinate 32. Point \(B\) on the curve is such that the gradient of the curve at \(B\) is - 3 . Find the equation of the perpendicular bisector of \(A B\). Give your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are integers.
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