CAIE P1 (Pure Mathematics 1) 2023 November

1

1. Expand \(( 1 + 3 x ) ^ { 6 }\) in ascending powers of \(x\) up to, and including, the term in \(x ^ { 2 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 - 7 x + x ^ { 2 } \right) ( 1 + 3 x ) ^ { 6 }\).

2 A line has equation \(y = 2 c x + 3\) and a curve has equation \(y = c x ^ { 2 } + 3 x - c\), where \(c\) is a constant.
Showing all necessary working, determine which of the following statements is correct.
A The line and curve intersect only for a particular set of values of \(c\).
B The line and curve intersect for all values of \(c\).
C The line and curve do not intersect for any values of \(c\).

3
\includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-05_424_529_248_815} The diagram shows a cubical closed container made of a thin elastic material which is filled with water and frozen. During the freezing process the length, \(x \mathrm {~cm}\), of each edge of the container increases at the constant rate of 0.01 cm per minute. The volume of the container at time \(t\) minutes is \(V \mathrm {~cm} ^ { 3 }\). Find the rate of increase of \(V\) when \(x = 20\).

4 The transformation R denotes a reflection in the \(x\)-axis and the transformation T denotes a translation of \(\binom { 3 } { - 1 }\).

1. Find the equation, \(y = \mathrm { g } ( x )\), of the curve with equation \(y = x ^ { 2 }\) after it has been transformed by the sequence of transformations R followed by T .
2. Find the equation, \(y = \mathrm { h } ( x )\), of the curve with equation \(y = x ^ { 2 }\) after it has been transformed by the sequence of transformations T followed by R .
3. State fully the transformation that maps the curve \(y = \mathrm { g } ( x )\) onto the curve \(y = \mathrm { h } ( x )\).

5

1. Show that the equation $$4 \sin x + \frac { 5 } { \tan x } + \frac { 2 } { \sin x } = 0$$ may be expressed in the form \(a \cos ^ { 2 } x + b \cos x + c = 0\), where \(a , b\) and \(c\) are integers to be found.
2. Hence solve the equation \(4 \sin x + \frac { 5 } { \tan x } + \frac { 2 } { \sin x } = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-08_534_506_255_815} The diagram shows a motif formed by the major arc \(A B\) of a circle with radius \(r\) and centre \(O\), and the minor arc \(A O B\) of a circle, also with radius \(r\) but with centre \(C\). The point \(C\) lies on the circle with centre \(O\).

1. Given that angle \(A C B = k \pi\) radians, state the value of the fraction \(k\).
2. State the perimeter of the shaded motif in terms of \(\pi\) and \(r\).
3. Find the area of the shaded motif, giving your answer in terms of \(\pi , r\) and \(\sqrt { 3 }\).

7 The sum of the first two terms of a geometric progression is 15 and the sum to infinity is \(\frac { 125 } { 7 }\). The common ratio of the progression is negative. Find the third term of the progression.

8
\includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-12_684_776_274_680} The diagram shows the curves with equations \(y = 2 ( 2 x - 3 ) ^ { 4 }\) and \(y = ( 2 x - 3 ) ^ { 2 } + 1\) meeting at points \(A\) and \(B\).

1. By using the substitution \(u = 2 x - 3\) find, by calculation, the coordinates of \(A\) and \(B\).
2. Find the exact area of the shaded region.

9

1. Express \(4 x ^ { 2 } - 12 x + 13\) in the form \(( 2 x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
   The function f is defined by \(\mathrm { f } ( x ) = 4 x ^ { 2 } - 12 x + 13\) for \(p < x < q\), where \(p\) and \(q\) are constants. The function g is defined by \(\mathrm { g } ( x ) = 3 x + 1\) for \(x < 8\).
2. Given that it is possible to form the composite function gf , find the least possible value of \(p\) and the greatest possible value of \(q\).
3. Find an expression for \(\operatorname { gf } ( x )\).
   The function h is defined by \(\mathrm { h } ( x ) = 4 x ^ { 2 } - 12 x + 13\) for \(x < 0\).
4. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

10 A curve has a stationary point at \(( 2 , - 10 )\) and is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the equation of the curve.
3. Find the coordinates of the other stationary point and determine its nature.
4. Find the equation of the tangent to the curve at the point where the curve crosses the \(y\)-axis.
   \includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-18_689_828_276_646} The diagram shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 40\). Parallel tangents, each with gradient 1 , touch the circle at points \(A\) and \(B\).
5. Find the equation of the line \(A B\), giving the answer in the form \(y = m x + c\).
6. Find the coordinates of \(A\), giving each coordinate in surd form.
7. Find the equation of the tangent at \(A\), giving the answer in the form \(y = m x + c\), where \(c\) is in surd form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.