CAIE P1 (Pure Mathematics 1) 2020 Specimen

1 The following points $$A ( 0,1 ) , \quad B ( 1,6 ) , \quad C ( 1.5,7.75 ) , \quad D ( 1.9,8.79 ) \quad \text { and } \quad E ( 2,9 )$$ lie on the curve \(y = \mathrm { f } ( x )\). The table below shows the gradients of the chords \(A E\) and \(B E\).

1. Complete the table to show the gradients of \(C E\) and \(D E\).
2. State what the values in the table indicate about the value of \(\mathrm { f } ^ { \prime } ( 2 )\).

2 Functions \(f\) and \(g\) are defined by $$\begin{aligned} \mathrm { f } : x & \mapsto 3 x + 2 , \quad x \in \mathbb { R } , \\ \mathrm {~g} : x & \mapsto 4 x - 12 , \quad x \in \mathbb { R } . \end{aligned}$$ Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { gf } ( x )\).

3 An arithmetic progression has first term 7. The \(n\)th term is 84 and the ( \(3 n\) )th term is 245 .
Find the value of \(n\).

4 A curve has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 } { \sqrt { x + 6 } } + \frac { 6 } { x ^ { 2 } }\) and that \(\mathrm { f } ( 3 ) = 1\). Find \(\mathrm { f } ( x )\).

5

1. The curve \(y = x ^ { 2 } + 3 x + 4\) is translated by \(\binom { 2 } { 0 }\).
   Find and simplify the equation of the translated curve.
2. The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = 3 \mathrm { f } ( - x )\). Describe fully the two single transformations which have been combined to give the resulting transformation.

6

1. Find the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(( 2 - x ) ^ { 6 }\).
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 x + 1 ) ( 2 - x ) ^ { 6 }\).

7

1. Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as $$6 \cos ^ { 2 } x - \cos x - 1 = 0$$
2. Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

8 A curve has equation \(y = \frac { 12 } { 3 - 2 x }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   A point moves along this curve. As the point passes through \(A\), the \(x\)-coordinate is increasing at a rate of 0.15 units per second and the \(y\)-coordinate is increasing at a rate of 0.4 units per second.
2. Find the possible \(x\)-coordinates of \(A\).

9 \includegraphics[max width=\textwidth, alt={}, center]{9803d51b-215e-4d03-884f-a67fb7ed6442-14_713_912_258_573} The diagram shows a circle with centre \(A\) and radius \(r\). Diameters CAD and BAE are perpendicular to each other. A larger circle has centre \(B\) and passes through \(C\) and \(D\).

1. Show that the radius of the larger circle is \(r \sqrt { 2 }\).
2. Find the area of the shaded region in terms of \(r\).

10 The circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - 20 = 0\) has centre \(C\) and passes through points \(A\) and \(B\).

1. State the coordinates of \(C\).
   It is given that the midpoint, \(D\), of \(A B\) has coordinates \(\left( 1 \frac { 1 } { 2 } , 1 \frac { 1 } { 2 } \right)\).
2. Find the equation of \(A B\), giving your answer in the form \(y = m x + c\).
3. Find, by calculation, the \(x\)-coordinates of \(A\) and \(B\).

11 The function f is defined, for \(x \in \mathbb { R }\), by \(\mathrm { f } : x \mapsto x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.

1. It is given that \(a = 6\) and \(b = - 8\). Find the range of f .
2. It is given instead that \(a = 5\) and that the roots of the equation \(\mathrm { f } ( x ) = 0\) are \(k\) and \(- 2 k\), where \(k\) is a constant. Find the values of \(b\) and \(k\).
3. Show that if the equation \(\mathrm { f } ( x + a ) = a\) has no real roots then \(a ^ { 2 } < 4 ( b - a )\).

12 \includegraphics[max width=\textwidth, alt={}, center]{9803d51b-215e-4d03-884f-a67fb7ed6442-20_524_972_274_548} The diagram shows the curve with equation \(y = x ( x - 2 ) ^ { 2 }\). The minimum point on the curve has coordinates \(( a , 0 )\) and the \(x\)-coordinate of the maximum point is \(b\), where \(a\) and \(b\) are constants.

1. State the value of \(a\).
2. Calculate the value of \(b\).
3. Find the area of the shaded region.
4. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of the curve has a minimum value \(m\). Calculate the value of \(m\).