CAIE P1 (Pure Mathematics 1) 2024 June

1 Find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 2 - 5 x ) ( 1 + 3 x ) ^ { 10 }$$

2

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-04_582_922_335_575} The diagram shows the curve \(y = k \cos \left( x - \frac { 1 } { 6 } \pi \right)\) where \(k\) is a positive constant and \(x\) is measured in radians. The curve crosses the \(x\)-axis at point \(A\) and \(B\) is a minimum point. Find the coordinates of \(A\) and \(B\).
2. Find the exact value of \(t\) that satisfies the equation $$3 \sin ^ { - 1 } ( 3 t ) + 2 \cos ^ { - 1 } \left( \frac { 1 } { 2 } \sqrt { 2 } \right) = \pi .$$ \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-04_2718_33_141_2013}

3
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-05_483_561_287_753} The diagram shows a sector of a circle with centre \(C\). The radii \(C A\) and \(C B\) each have length \(r \mathrm {~cm}\) and the size of the reflex angle \(A C B\) is \(\theta\) radians. The sector, shaded in the diagram, has a perimeter of 65 cm and an area of \(225 \mathrm {~cm} ^ { 2 }\).

1. Find the values of \(r\) and \(\theta\).
2. Find the area of triangle \(A C B\).

4

1. Show that the equation \(\cos \theta ( 7 \tan \theta - 5 \cos \theta ) = 1\) can be written in the form \(a \sin ^ { 2 } \theta + b \sin \theta + c = 0\), where \(a , b\) and \(c\) are integers to be found.
2. Hence solve the equation \(\cos 2 x ( 7 \tan 2 x - 5 \cos 2 x ) = 1\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
   \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-06_2718_35_141_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-07_2714_33_144_22}

5 The equation of a curve is \(y = 2 x ^ { 2 } - \frac { 1 } { 2 x } + 3\).

1. Find the coordinates of the stationary point.
2. Determine the nature of the stationary point.
3. For positive values of \(x\), determine whether the curve shows a function that is increasing, decreasing or neither. Give a reason for your answer.

6 A curve passes through the point \(\left( \frac { 4 } { 5 } , - 3 \right)\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 20 } { ( 5 x - 3 ) ^ { 2 } }\).

1. Find the equation of the curve.
2. The curve is transformed by a stretch in the \(x\)-direction with scale factor \(\frac { 1 } { 2 }\) followed by a translation of \(\binom { 2 } { 10 }\). Find the equation of the new curve.
   \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-08_2716_38_143_2009}

7 The first term of an arithmetic progression is 1.5 and the sum of the first ten terms is 127.5 .

1. Find the common difference.
2. Find the sum of all the terms of the arithmetic progression whose values are between 25 and 100 .

8 A circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 2 y - 15 = 0\) meets the \(y\)-axis at the points \(A\) and \(B\). The tangents to the circle at \(A\) and \(B\) meet at the point \(P\). Find the coordinates of \(P\).
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-10_71_1659_466_244}
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-10_2723_37_136_2010}

9
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-12_764_967_292_555} The diagram shows the curve with equation \(y = \sqrt { 2 x ^ { 3 } + 10 }\).

1. Find the equation of the tangent to the curve at the point where \(x = 3\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
   \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-12_2716_35_141_2013}
2. The region shaded in the diagram is enclosed by the curve and the straight lines \(x = 1 , x = 3\) and \(y = 0\). Find the volume of the solid obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

10 The geometric progression \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) has first term 2 and common ratio \(r\) where \(r > 0\). It is given that \(\frac { 9 } { 2 } a _ { 5 } + 7 a _ { 3 } = 8\).

1. Find the value of \(r\).
2. Find the sum of the first 20 terms of the geometric progression. Give your answer correct to 4 significant figures.
   \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-14_2725_42_134_2008}
3. Find the sum to infinity of the progression \(a _ { 2 } , a _ { 5 } , a _ { 8 } , \ldots\).

11 The function f is defined by \(\mathrm { f } ( x ) = 10 + 6 x - x ^ { 2 }\) for \(x \in \mathbb { R }\).

1. By completing the square, find the range of f .
   \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-16_2715_37_143_2010} The function g is defined by \(\mathrm { g } ( x ) = 4 x + k\) for \(x \in \mathbb { R }\) where \(k\) is a constant.
2. It is given that the graph of \(y = \mathrm { g } ^ { - 1 } \mathrm { f } ( x )\) meets the graph of \(y = \mathrm { g } ( x )\) at a single point \(P\). Determine the coordinates of \(P\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.
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