CAIE P1 (Pure Mathematics 1) 2020 November

1 Find the set of values of \(m\) for which the line with equation \(y = m x - 3\) and the curve with equation \(y = 2 x ^ { 2 } + 5\) do not meet.

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

3 Air is being pumped into a balloon in the shape of a sphere so that its volume is increasing at a constant rate of \(50 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate at which the radius of the balloon is increasing when the radius is 10 cm .

4
\includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-05_615_1169_260_488} In the diagram, the lower curve has equation \(y = \cos \theta\). The upper curve shows the result of applying a combination of transformations to \(y = \cos \theta\). Find, in terms of a cosine function, the equation of the upper curve.

5 In the expansion of \(\left( 2 x ^ { 2 } + \frac { a } { x } \right) ^ { 6 }\), the coefficients of \(x ^ { 6 }\) and \(x ^ { 3 }\) are equal.

1. Find the value of the non-zero constant \(a\).
2. Find the coefficient of \(x ^ { 6 }\) in the expansion of \(\left( 1 - x ^ { 3 } \right) \left( 2 x ^ { 2 } + \frac { a } { x } \right) ^ { 6 }\).

6 The equation of a curve is \(y = 2 + \sqrt { 25 - x ^ { 2 } }\).
Find the coordinates of the point on the curve at which the gradient is \(\frac { 4 } { 3 }\).

7

1. Show that \(\frac { \sin \theta } { 1 - \sin \theta } - \frac { \sin \theta } { 1 + \sin \theta } \equiv 2 \tan ^ { 2 } \theta\).
2. Hence solve the equation \(\frac { \sin \theta } { 1 - \sin \theta } - \frac { \sin \theta } { 1 + \sin \theta } = 8\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

8 A geometric progression has first term \(a\), common ratio \(r\) and sum to infinity \(S\). A second geometric progression has first term \(a\), common ratio \(R\) and sum to infinity \(2 S\).

1. Show that \(r = 2 R - 1\).
   It is now given that the 3rd term of the first progression is equal to the 2nd term of the second progression.
2. Express \(S\) in terms of \(a\).

9
\includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-12_583_661_260_740} The diagram shows a circle with centre \(A\) passing through the point \(B\). A second circle has centre \(B\) and passes through \(A\). The tangent at \(B\) to the first circle intersects the second circle at \(C\) and \(D\). The coordinates of \(A\) are ( \(- 1,4\) ) and the coordinates of \(B\) are ( 3,2 ).

1. Find the equation of the tangent CBD.
2. Find an equation of the circle with centre \(B\).
3. Find, by calculation, the \(x\)-coordinates of \(C\) and \(D\).
   \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-14_746_652_262_744} The diagram shows a sector \(C A B\) which is part of a circle with centre \(C\). A circle with centre \(O\) and radius \(r\) lies within the sector and touches it at \(D , E\) and \(F\), where \(C O D\) is a straight line and angle \(A C D\) is \(\theta\) radians.

1. Find \(C D\) in terms of \(r\) and \(\sin \theta\).
   It is now given that \(r = 4\) and \(\theta = \frac { 1 } { 6 } \pi\).
2. Find the perimeter of sector \(C A B\) in terms of \(\pi\).
3. Find the area of the shaded region in terms of \(\pi\) and \(\sqrt { 3 }\).

11 The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } f ( x ) = x ^ { 2 } + 3 & \text { for } x > 0
g ( x ) = 2 x + 1 & \text { for } x > - \frac { 1 } { 2 } \end{array}$$

1. Find an expression for \(\mathrm { fg } ( x )\).
2. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and state the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
3. Solve the equation \(\mathrm { fg } ( x ) - 3 = \mathrm { gf } ( x )\).

12
\includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-18_557_677_264_733} The diagram shows a curve with equation \(y = 4 x ^ { \frac { 1 } { 2 } } - 2 x\) for \(x \geqslant 0\), and a straight line with equation \(y = 3 - x\). The curve crosses the \(x\)-axis at \(A ( 4,0 )\) and crosses the straight line at \(B\) and \(C\).

1. Find, by calculation, the \(x\)-coordinates of \(B\) and \(C\).
2. Show that \(B\) is a stationary point on the curve.
3. Find the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.