CAIE P1 (Pure Mathematics 1) 2020 June

1 Find the set of values of \(m\) for which the line with equation \(y = m x + 1\) and the curve with equation \(y = 3 x ^ { 2 } + 2 x + 4\) intersect at two distinct points.

2 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 3 x ^ { - \frac { 1 } { 2 } }\). It is given that the point (4,7) lies on the curve. Find the equation of the curve.

3 In each of parts (a), (b) and (c), the graph shown with solid lines has equation \(y = \mathrm { f } ( x )\). The graph shown with broken lines is a transformation of \(y = \mathrm { f } ( x )\).

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-04_412_645_367_788} State, in terms of f , the equation of the graph shown with broken lines.
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-04_650_423_1046_900} State, in terms of f , the equation of the graph shown with broken lines.
3. 
   \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-04_550_631_1975_804} State, in terms of f , the equation of the graph shown with broken lines.

4

1. Expand \(( 1 + a ) ^ { 5 }\) in ascending powers of \(a\) up to and including the term in \(a ^ { 3 }\).
2. Hence expand \(\left[ 1 + \left( x + x ^ { 2 } \right) \right] ^ { 5 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying your answer.

5
\includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-06_761_460_258_840} The diagram shows a cord going around a pulley and a pin. The pulley is modelled as a circle with centre \(O\) and radius 5 cm . The thickness of the cord and the size of the pin \(P\) can be neglected. The pin is situated 13 cm vertically below \(O\). Points \(A\) and \(B\) are on the circumference of the circle such that \(A P\) and \(B P\) are tangents to the circle. The cord passes over the major arc \(A B\) of the circle and under the pin such that the cord is taut. Calculate the length of the cord.

6 A point \(P\) is moving along a curve in such a way that the \(x\)-coordinate of \(P\) is increasing at a constant rate of 2 units per minute. The equation of the curve is \(y = ( 5 x - 1 ) ^ { \frac { 1 } { 2 } }\).

1. Find the rate at which the \(y\)-coordinate is increasing when \(x = 1\).
2. Find the value of \(x\) when the \(y\)-coordinate is increasing at \(\frac { 5 } { 8 }\) units per minute.

7

1. Show that \(\frac { \tan \theta } { 1 + \cos \theta } + \frac { \tan \theta } { 1 - \cos \theta } \equiv \frac { 2 } { \sin \theta \cos \theta }\).
2. Hence solve the equation \(\frac { \tan \theta } { 1 + \cos \theta } + \frac { \tan \theta } { 1 - \cos \theta } = \frac { 6 } { \tan \theta }\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

8 The first term of a progression is \(\sin ^ { 2 } \theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\). The second term of the progression is \(\sin ^ { 2 } \theta \cos ^ { 2 } \theta\).

1. Given that the progression is geometric, find the sum to infinity.
   It is now given instead that the progression is arithmetic.
2. 1. Find the common difference of the progression in terms of \(\sin \theta\).
   2. Find the sum of the first 16 terms when \(\theta = \frac { 1 } { 3 } \pi\).

9 The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } - 4 x + 3 \text { for } x > c , \text { where } c \text { is a constant, }
& \mathrm { g } ( x ) = \frac { 1 } { x + 1 } \quad \text { for } x > - 1 \end{aligned}$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\).
   It is given that f is a one-one function.
2. State the smallest possible value of \(c\).
   It is now given that \(c = 5\).
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
4. Find an expression for \(\mathrm { gf } ( x )\) and state the range of gf .

10

1. The coordinates of two points \(A\) and \(B\) are \(( - 7,3 )\) and \(( 5,11 )\) respectively.
   Show that the equation of the perpendicular bisector of \(A B\) is \(3 x + 2 y = 11\).
2. A circle passes through \(A\) and \(B\) and its centre lies on the line \(12 x - 5 y = 70\). Find an equation of the circle.

11
\includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-18_387_920_260_609} The diagram shows part of the curve with equation \(y = x ^ { 3 } - 2 b x ^ { 2 } + b ^ { 2 } x\) and the line \(O A\), where \(A\) is the maximum point on the curve. The \(x\)-coordinate of \(A\) is \(a\) and the curve has a minimum point at ( \(b , 0\) ), where \(a\) and \(b\) are positive constants.

1. Show that \(b = 3 a\).
2. Show that the area of the shaded region between the line and the curve is \(k a ^ { 4 }\), where \(k\) is a fraction to be found.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.