CAIE P1 (Pure Mathematics 1) 2021 June

1 A curve with equation \(y = \mathrm { f } ( x )\) is such that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } - \frac { 8 } { x ^ { 2 } }\). It is given that the curve passes through the point \(( 2,7 )\). Find \(\mathrm { f } ( x )\).

2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { 3 } ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } - 2 x\) for \(\frac { 1 } { 2 } < x < a\). It is given that f is a decreasing function. Find the maximum possible value of the constant \(a\).

3 A line with equation \(y = m x - 6\) is a tangent to the curve with equation \(y = x ^ { 2 } - 4 x + 3\).
Find the possible values of the constant \(m\), and the corresponding coordinates of the points at which the line touches the curve.

4

1. Show that the equation $$\frac { \tan x + \sin x } { \tan x - \sin x } = k$$ where \(k\) is a constant, may be expressed as $$\frac { 1 + \cos x } { 1 - \cos x } = k$$
2. Hence express \(\cos x\) in terms of \(k\).
3. Hence solve the equation \(\frac { \tan x + \sin x } { \tan x - \sin x } = 4\) for \(- \pi < x < \pi\).

5 The diagram shows a triangle \(A B C\), in which angle \(A B C = 90 ^ { \circ }\) and \(A B = 4 \mathrm {~cm}\). The sector \(A B D\) is part of a circle with centre \(A\). The area of the sector is \(10 \mathrm {~cm} ^ { 2 }\).

1. Find angle \(B A D\) in radians.
2. Find the perimeter of the shaded region.

6 Functions f and g are both defined for \(x \in \mathbb { R }\) and are given by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } - 2 x + 5
& \mathrm {~g} ( x ) = x ^ { 2 } + 4 x + 13 \end{aligned}$$

1. By first expressing each of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) in completed square form, express \(\mathrm { g } ( x )\) in the form \(\mathrm { f } ( x + p ) + q\), where \(p\) and \(q\) are constants.
2. Describe fully the transformation which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { g } ( x )\).

7

1. Write down the first four terms of the expansion, in ascending powers of \(x\), of \(( a - x ) ^ { 6 }\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + \frac { 2 } { a x } \right) ( a - x ) ^ { 6 }\) is - 20 , find in exact form the possible values of the constant \(a\).

8 Functions f and g are defined as follows: $$\begin{aligned} & \mathrm { f } : x \mapsto x ^ { 2 } - 1 \text { for } x < 0
& \mathrm {~g} : x \mapsto \frac { 1 } { 2 x + 1 } \text { for } x < - \frac { 1 } { 2 } \end{aligned}$$

1. Solve the equation \(\operatorname { fg } ( x ) = 3\).
2. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\).

9

1. A geometric progression is such that the second term is equal to \(24 \%\) of the sum to infinity. Find the possible values of the common ratio.
2. An arithmetic progression \(P\) has first term \(a\) and common difference \(d\). An arithmetic progression \(Q\) has first term 2( \(a + 1\) ) and common difference ( \(d + 1\) ). It is given that $$\frac { 5 \text { th term of } P } { 12 \text { th term of } Q } = \frac { 1 } { 3 } \quad \text { and } \quad \frac { \text { Sum of first } 5 \text { terms of } P } { \text { Sum of first } 5 \text { terms of } Q } = \frac { 2 } { 3 } .$$ Find the value of \(a\) and the value of \(d\).

10 Points \(A ( - 2,3 ) , B ( 3,0 )\) and \(C ( 6,5 )\) lie on the circumference of a circle with centre \(D\).

1. Show that angle \(A B C = 90 ^ { \circ }\).
2. Hence state the coordinates of \(D\).
3. Find an equation of the circle.
   The point \(E\) lies on the circumference of the circle such that \(B E\) is a diameter.
4. Find an equation of the tangent to the circle at \(E\).

11
\includegraphics[max width=\textwidth, alt={}, center]{aaba3158-b5be-464e-bea3-1a4c460f9637-16_622_1091_260_525} The diagram shows part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } + k ^ { 2 } x ^ { - \frac { 1 } { 2 } }\), where \(k\) is a positive constant.

1. Find the coordinates of the minimum point of the curve, giving your answer in terms of \(k\).
   The tangent at the point on the curve where \(x = 4 k ^ { 2 }\) intersects the \(y\)-axis at \(P\).
2. Find the \(y\)-coordinate of \(P\) in terms of \(k\).
   The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = \frac { 9 } { 4 } k ^ { 2 }\) and \(x = 4 k ^ { 2 }\).
3. Find the area of the shaded region in terms of \(k\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.