CAIE P1 (Pure Mathematics 1) 2021 November

1 The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = 3 - \mathrm { f } ( x )\).
Describe fully, in the correct order, the two transformations that have been combined.

2

1. Find the first three terms, in ascending powers of \(x\), in the expansion of \(( 1 + a x ) ^ { 6 }\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 - 3 x ) ( 1 + a x ) ^ { 6 }\) is - 3 , find the possible values of the constant \(a\).

3

1. Express \(5 y ^ { 2 } - 30 y + 50\) in the form \(5 ( y + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. The function f is defined by \(\mathrm { f } ( x ) = x ^ { 5 } - 10 x ^ { 3 } + 50 x\) for \(x \in \mathbb { R }\). Determine whether \(f\) is an increasing function, a decreasing function or neither.

4 The first term of an arithmetic progression is 84 and the common difference is - 3 .

1. Find the smallest value of \(n\) for which the \(n\)th term is negative.
   It is given that the sum of the first \(2 k\) terms of this progression is equal to the sum of the first \(k\) terms.
2. Find the value of \(k\).

5
\includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-06_720_686_260_733} In the diagram, \(X\) and \(Y\) are points on the line \(A B\) such that \(B X = 9 \mathrm {~cm}\) and \(A Y = 11 \mathrm {~cm}\). Arc \(B C\) is part of a circle with centre \(X\) and radius 9 cm , where \(C X\) is perpendicular to \(A B\). Arc \(A C\) is part of a circle with centre \(Y\) and radius 11 cm .

1. Show that angle \(X Y C = 0.9582\) radians, correct to 4 significant figures.
2. Find the perimeter of \(A B C\).

6
\includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-08_608_597_258_772} The diagram shows the graph of \(y = \mathrm { f } ( x )\).

1. On this diagram sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\). It is now given that \(\mathrm { f } ( x ) = - \frac { x } { \sqrt { 4 - x ^ { 2 } } }\) where \(- 2 < x < 2\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The function g is defined by \(\mathrm { g } ( x ) = 2 x\) for \(- a < x < a\), where \(a\) is a constant.
3. State the maximum possible value of \(a\) for which fg can be formed.
4. Assuming that fg can be formed, find and simplify an expression for \(\mathrm { fg } ( x )\).

7

1. Show that the equation \(\frac { \tan x + \cos x } { \tan x - \cos x } = k\), where \(k\) is a constant, can be expressed as $$( k + 1 ) \sin ^ { 2 } x + ( k - 1 ) \sin x - ( k + 1 ) = 0$$
2. Hence solve the equation \(\frac { \tan x + \cos x } { \tan x - \cos x } = 4\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

8
\includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-12_570_961_260_591} The diagram shows the curves with equations \(y = x ^ { - \frac { 1 } { 2 } }\) and \(y = \frac { 5 } { 2 } - x ^ { \frac { 1 } { 2 } }\). The curves intersect at the points \(A \left( \frac { 1 } { 4 } , 2 \right)\) and \(B \left( 4 , \frac { 1 } { 2 } \right)\).

1. Find the area of the region between the two curves.
2. The normal to the curve \(y = x ^ { - \frac { 1 } { 2 } }\) at the point \(( 1,1 )\) intersects the \(y\)-axis at the point \(( 0 , p )\). Find the value of \(p\).

9 The line \(y = 2 x + 5\) intersects the circle with equation \(x ^ { 2 } + y ^ { 2 } = 20\) at \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\) in surd form and hence find the exact length of the chord \(A B\).
   A straight line through the point \(( 10,0 )\) with gradient \(m\) is a tangent to the circle.
2. Find the two possible values of \(m\).

10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that $$\mathrm { f } ^ { \prime } ( x ) = \left( \frac { 1 } { 2 } x + k \right) ^ { - 2 } - ( 1 + k ) ^ { - 2 }$$ where \(k\) is a constant. The curve has a minimum point at \(x = 2\).

1. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) in terms of \(k\) and \(x\), and hence find the set of possible values of \(k\).
   It is now given that \(k = - 3\) and the minimum point is at \(\left( 2,3 \frac { 1 } { 2 } \right)\).
2. Find \(\mathrm { f } ( x )\).
3. Find the coordinates of the other stationary point and determine its nature.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.