CAIE P1 (Pure Mathematics 1) 2021 November

1

1. Expand \(\left( 1 - \frac { 1 } { 2 x } \right) ^ { 2 }\).
2. Find the first four terms in the expansion, in ascending powers of \(x\), of \(( 1 + 2 x ) ^ { 6 }\).
3. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 - \frac { 1 } { 2 x } \right) ^ { 2 } ( 1 + 2 x ) ^ { 6 }\).

2 A curve has equation \(y = k x ^ { 2 } + 2 x - k\) and a line has equation \(y = k x - 2\), where \(k\) is a constant. Find the set of values of \(k\) for which the curve and line do not intersect.

3 Solve, by factorising, the equation $$6 \cos \theta \tan \theta - 3 \cos \theta + 4 \tan \theta - 2 = 0$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

4 The first term of an arithmetic progression is \(a\) and the common difference is - 4 . The first term of a geometric progression is \(5 a\) and the common ratio is \(- \frac { 1 } { 4 }\). The sum to infinity of the geometric progression is equal to the sum of the first eight terms of the arithmetic progression.

1. Find the value of \(a\).
   The \(k\) th term of the arithmetic progression is zero.
2. Find the value of \(k\).

5
\includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-07_778_878_255_630} The diagram shows part of the graph of \(y = a \cos ( b x ) + c\).

1. Find the values of the positive integers \(a , b\) and \(c\).
2. For these values of \(a\), \(b\) and \(c\), use the given diagram to determine the number of solutions in the interval \(0 \leqslant x \leqslant 2 \pi\) for each of the following equations.
   1. \(a \cos ( b x ) + c = \frac { 6 } { \pi } x\)
   2. \(a \cos ( b x ) + c = 6 - \frac { 6 } { \pi } x\)
      The diagram shows a metal plate \(A B C\) in which the sides are the straight line \(A B\) and the arcs \(A C\) and \(B C\). The line \(A B\) has length 6 cm . The arc \(A C\) is part of a circle with centre \(B\) and radius 6 cm , and the arc \(B C\) is part of a circle with centre \(A\) and radius 6 cm .

1. Find the perimeter of the plate, giving your answer in terms of \(\pi\).
2. Find the area of the plate, giving your answer in terms of \(\pi\) and \(\sqrt { 3 }\).

7 A circle with centre \(( 5,2 )\) passes through the point \(( 7,5 )\).

1. Find an equation of the circle.
   The line \(y = 5 x - 10\) intersects the circle at \(A\) and \(B\).
2. Find the exact length of the chord \(A B\).

8

1. Express \(- 3 x ^ { 2 } + 12 x + 2\) in the form \(- 3 ( x - a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
   The one-one function f is defined by \(\mathrm { f } : x \mapsto - 3 x ^ { 2 } + 12 x + 2\) for \(x \leqslant k\).
2. State the largest possible value of the constant \(k\).
   It is now given that \(k = - 1\).
3. State the range of f.
4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The result of translating the graph of \(y = \mathrm { f } ( x )\) by \(\binom { - 3 } { 1 }\) is the graph of \(y = \mathrm { g } ( x )\).
5. Express \(\mathrm { g } ( x )\) in the form \(p x ^ { 2 } + q x + r\), where \(p , q\) and \(r\) are constants.

9 A curve has equation \(y = \mathrm { f } ( x )\), and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { 2 } - 7 - \frac { 4 } { x ^ { 2 } }\).

1. Given that \(\mathrm { f } ( 1 ) = - \frac { 1 } { 3 }\), find \(\mathrm { f } ( x )\).
2. Find the coordinates of the stationary points on the curve.
3. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
4. Hence, or otherwise, determine the nature of each of the stationary points.

10

1. Find \(\int _ { 1 } ^ { \infty } \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\).
   \includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-16_499_689_1322_726} The diagram shows the curve with equation \(y = \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } }\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\). The shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. Find the volume of revolution.
   The normal to the curve at the point \(( 1,1 )\) crosses the \(y\)-axis at the point \(A\).
3. Find the \(y\)-coordinate of \(A\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.