CAIE P1 (Pure Mathematics 1) 2024 November

1 An arithmetic progression has fourth term 15 and eighth term 25.
Find the 30th term of the progression.

2 Find the exact solution of the equation $$\cos \frac { 1 } { 6 } \pi + \tan 2 x + \frac { \sqrt { 3 } } { 2 } = 0 \text { for } - \frac { 1 } { 4 } \pi < x < \frac { 1 } { 4 } \pi .$$

3

1. Find the coefficients of \(x ^ { 3 }\) and \(x ^ { 4 }\) in the expansion of \(( 3 - a x ) ^ { 5 }\), where \(a\) is a constant. Give your answers in terms of \(a\).
2. Given that the coefficient of \(x ^ { 4 }\) in the expansion of \(( a x + 7 ) ( 3 - a x ) ^ { 5 }\) is 240 , find the positive value of \(a\).
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-05_2723_33_99_21}

4 Solve the equation \(4 \sin ^ { 4 } \theta + 12 \sin ^ { 2 } \theta - 7 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-06_631_1500_260_285} In the diagram, the graph with equation \(y = \mathrm { f } ( x )\) is shown with solid lines and the graph with equation \(y = \mathrm { g } ( x )\) is shown with broken lines.

1. Describe fully a sequence of three transformations which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { g } ( x )\).
2. Find an expression for \(\mathrm { g } ( x )\) in the form \(a \mathrm { f } ( b x + c )\), where \(a , b\) and \(c\) are integers.
   □
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-06_2716_31_106_2016}
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-07_2723_35_101_20}

6 The first term of a convergent geometric progression is 10 . The sum of the first 4 terms of the progression is \(p\) and the sum of the first 8 terms of the progression is \(q\). It is given that \(\frac { q } { p } = \frac { 17 } { 16 }\). Find the two possible values of the sum to infinity.
\includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-08_801_730_255_669} The diagram shows a metal plate \(A B C D E F\) consisting of five parts. The parts \(B C D\) and \(D E F\) are semicircles. The part \(B A F O\) is a sector of a circle with centre \(O\) and radius 20 cm , and \(D\) lies on this circle. The parts \(O B D\) and \(O D F\) are triangles. Angles \(B O D\) and \(D O F\) are both \(\theta\) radians.

1. Given that \(\theta = 1.2\), find the area of the metal plate. Give your answer correct to 3 significant figures.
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-08_2715_42_110_2006}
2. Given instead that the area of each semicircle is \(50 \pi \mathrm {~cm} ^ { 2 }\), find the exact perimeter of the metal plate.

8

1. Express \(3 x ^ { 2 } - 12 x + 14\) in the form \(3 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants to be found.
   The function \(\mathrm { f } ( x ) = 3 x ^ { 2 } - 12 x + 14\) is defined for \(x \geqslant k\), where \(k\) is a constant.
2. Find the least value of \(k\) for which the function \(\mathrm { f } ^ { - 1 }\) exists.
   For the rest of this question, you should assume that \(k\) has the value found in part (b).
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-11_2726_35_97_20}
4. Hence or otherwise solve the equation \(\mathrm { ff } ( x ) = 29\).

9
\includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-12_883_1703_267_182} The diagram shows the curves with equations \(y = x ^ { 3 } - 3 x + 3\) and \(y = 2 x ^ { 3 } - 4 x ^ { 2 } + 3\).

1. Find the \(x\)-coordinates of the points of intersection of the curves.
2. Find the area of the shaded region.

10 Points \(A\) and \(B\) have coordinates \(( 4,3 )\) and \(( 8 , - 5 )\) respectively. A circle with radius 10 passes through the points \(A\) and \(B\).

1. Show that the centre of the circle lies on the line \(y = \frac { 1 } { 2 } x - 4\).
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-14_2715_35_109_2010}
2. Find the two possible equations of the circle.

11 The equation of a curve is \(y = k x ^ { \frac { 1 } { 2 } } - 4 x ^ { 2 } + 2\), where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(k\).
2. It is given that \(k = 2\). Find the coordinates of the stationary point and determine its nature.
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-16_2717_38_110_2010}
3. Points \(A\) and \(B\) on the curve have \(x\)-coordinates 0.25 and 1 respectively. For a different value of \(k\), the tangents to the curve at the points \(A\) and \(B\) meet at a point with \(x\)-coordinate 0.6 . Find this value of \(k\).
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-17_58_1569_429_328}
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
   …………………………………………………………………………………………………………....................
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-18_2716_35_109_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-19_2717_35_106_20}