CAIE P1 (Pure Mathematics 1) 2024 November

1 In the expansion of \(\left( k x + \frac { 2 } { x } \right) ^ { 4 }\) ,where \(k\) is a positive constant,the term independent of \(x\) is equal to 150 .
Find the value of \(k\) and hence determine the coefficient of \(x ^ { 2 }\) in the expansion.
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2 The curve \(y = x ^ { 2 } - \frac { a } { x }\) has a stationary point at \(( - 3 , b )\).
Find the values of the constants \(a\) and \(b\).
\includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-04_403_517_251_776} The diagram shows a sector of a circle, centre \(O\), where \(O B = O C = 15 \mathrm {~cm}\). The size of angle \(B O C\) is \(\frac { 2 } { 5 } \pi\) radians. Points \(A\) and \(D\) on the lines \(O B\) and \(O C\) respectively are joined by an arc \(A D\) of circle with centre \(O\). The shaded region is bounded by the \(\operatorname { arcs } A D\) and \(B C\) and by the straight lines \(A B\) and \(D C\). It is given that the area of the shaded region is \(\frac { 209 } { 5 } \pi \mathrm {~cm} ^ { 2 }\). Find the perimeter of the shaded region. Give your answer in terms of \(\pi\).
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\includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-05_2723_35_101_20}

4 Show that the curve with equation \(x ^ { 2 } - 3 x y - 40 = 0\) and the line with equation \(3 x + y + k = 0\) meet for all values of the constant \(k\).

5 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x - 3 \sqrt { x } + 1\).

1. Find the \(x\)-coordinate of the point on the curve at which the gradient is \(\frac { 11 } { 2 }\).
2. Given that the curve passes through the point \(( 4,11 )\), find the equation of the curve.
   \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-06_2715_31_106_2016}

6 Circles \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$x ^ { 2 } + y ^ { 2 } + 6 x - 10 y + 18 = 0 \text { and } ( x - 9 ) ^ { 2 } + ( y + 4 ) ^ { 2 } - 64 = 0$$ respectively.

1. Find the distance between the centres of the circles.
   \(P\) and \(Q\) are points on \(C _ { 1 }\) and \(C _ { 2 }\) respectively. The distance between \(P\) and \(Q\) is denoted by \(d\).
2. Find the greatest and least possible values of \(d\).

7
\includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-08_534_462_248_804} The diagram shows part of the curve with equation \(y = \frac { 12 } { \sqrt [ 3 ] { 2 x + 1 } }\). The point \(A\) on the curve has coordinates \(\left( \frac { 7 } { 2 } , 6 \right)\).

1. Find the equation of the tangent to the curve at \(A\). Give your answer in the form \(y = m x + c\). [4]
   \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-08_2716_38_109_2012}
2. Find the area of the region bounded by the curve and the lines \(x = 0 , x = \frac { 7 } { 2 }\) and \(y = 0\).

8

1. It is given that \(\beta\) is an angle between \(90 ^ { \circ }\) and \(180 ^ { \circ }\) such that \(\sin \beta = a\).
   Express \(\tan ^ { 2 } \beta - 3 \sin \beta \cos \beta\) in terms of \(a\).
   \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-11_2726_35_97_20}
2. Solve the equation \(\sin ^ { 2 } \theta + 2 \cos ^ { 2 } \theta = 4 \sin \theta + 3\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

9 The equation of a curve is \(y = 4 + 5 x + 6 x ^ { 2 } - 3 x ^ { 3 }\).

1. Find the set of values of \(x\) for which \(y\) decreases as \(x\) increases.
2. It is given that \(y = 9 x + k\) is a tangent to the curve. Find the value of the constant \(k\).

10 An arithmetic progression has first term 5 and common difference \(d\), where \(d > 0\). The second, fifth and eleventh terms of the arithmetic progression, in that order, are the first three terms of a geometric progression.

1. Find the value of \(d\).
2. The sum of the first 77 terms of the arithmetic progression is denoted by \(S _ { 77 }\). The sum of the first 10 terms of the geometric progression is denoted by \(G _ { 10 }\). Find the value of \(S _ { 77 } - G _ { 10 }\).

11 The function f is defined by \(\mathrm { f } ( x ) = 3 + 6 x - 2 x ^ { 2 }\) for \(x \in \mathbb { R }\).

1. Express \(\mathrm { f } ( x )\) in the form \(a - b ( x - c ) ^ { 2 }\), where \(a , b\) and \(c\) are constants, and state the range of f .
2. The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = \mathrm { h } ( x )\) by a reflection in one of the axes followed by a translation. It is given that the graph of \(y = \mathrm { h } ( x )\) has a minimum point at the origin. Give details of the reflection and translation involved.
   \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-16_2715_38_109_2009} The function g is defined by \(\mathrm { g } ( x ) = 3 + 6 x - 2 x ^ { 2 }\) for \(x \leqslant 0\).
3. Sketch the graph of \(y = \mathrm { g } ( x )\) and explain why g is a one-one function. You are not required to find the coordinates of any intersections with the axes.
4. Sketch the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\) on your diagram in (c), and find an expression for \(\mathrm { g } ^ { - 1 } ( x )\). You should label the two graphs in your diagram appropriately and show any relevant mirror line.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
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