CAIE P1 (Pure Mathematics 1) 2022 June

1

1. Express \(x ^ { 2 } - 8 x + 11\) in the form \(( x + p ) ^ { 2 } + q\) where \(p\) and \(q\) are constants.
2. Hence find the exact solutions of the equation \(x ^ { 2 } - 8 x + 11 = 1\).

2 The thirteenth term of an arithmetic progression is 12 and the sum of the first 30 terms is - 15 .
Find the sum of the first 50 terms of the progression.

3 The coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 2 x ^ { 2 } + \frac { k ^ { 2 } } { x } \right) ^ { 5 }\) is \(a\). The coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 k x - 1 ) ^ { 4 }\) is \(b\).

1. Find \(a\) and \(b\) in terms of the constant \(k\).
2. Given that \(a + b = 216\), find the possible values of \(k\).

4

1. Prove the identity \(\frac { \sin ^ { 3 } \theta } { \sin \theta - 1 } - \frac { \sin ^ { 2 } \theta } { 1 + \sin \theta } \equiv - \tan ^ { 2 } \theta \left( 1 + \sin ^ { 2 } \theta \right)\).
2. Hence solve the equation $$\frac { \sin ^ { 3 } \theta } { \sin \theta - 1 } - \frac { \sin ^ { 2 } \theta } { 1 + \sin \theta } = \tan ^ { 2 } \theta \left( 1 - \sin ^ { 2 } \theta \right)$$ for \(0 < \theta < 2 \pi\).

5
\includegraphics[max width=\textwidth, alt={}, center]{574b96b2-62f2-41b3-a178-8e68e16429ff-08_509_654_264_751} The diagram shows a sector \(A B C\) of a circle with centre \(A\) and radius \(r\). The line \(B D\) is perpendicular to \(A C\). Angle \(C A B\) is \(\theta\) radians.

1. Given that \(\theta = \frac { 1 } { 6 } \pi\), find the exact area of \(B C D\) in terms of \(r\).
2. Given instead that the length of \(B D\) is \(\frac { \sqrt { 3 } } { 2 } r\), find the exact perimeter of \(B C D\) in terms of \(r\). [4]

6 The function \(f\) is defined as follows: $$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 4 } { x ^ { 2 } + 4 } \quad \text { for } x > 2$$

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
2. Show that \(1 - \frac { 8 } { x ^ { 2 } + 4 }\) can be expressed as \(\frac { x ^ { 2 } - 4 } { x ^ { 2 } + 4 }\) and hence state the range of f .
3. Explain why the composite function ff cannot be formed.

7
\includegraphics[max width=\textwidth, alt={}, center]{574b96b2-62f2-41b3-a178-8e68e16429ff-12_631_1031_267_534} The diagram shows the curve with equation \(y = ( 3 x - 2 ) ^ { \frac { 1 } { 2 } }\) and the line \(y = \frac { 1 } { 2 } x + 1\). The curve and the line intersect at points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\).
2. Hence find the area of the region enclosed between the curve and the line.

8

1. The curve \(y = \sin x\) is transformed to the curve \(y = 4 \sin \left( \frac { 1 } { 2 } x - 30 ^ { \circ } \right)\).
   Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations are applied.
2. Find the exact solutions of the equation \(4 \sin \left( \frac { 1 } { 2 } x - 30 ^ { \circ } \right) = 2 \sqrt { 2 }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

9 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 26 = 0\).

1. Find the coordinates of the centre of the circle and the radius. Hence find the coordinates of the lowest point on the circle.
2. Find the set of values of the constant \(k\) for which the line with equation \(y = k x - 5\) intersects the circle at two distinct points.

10 The equation of a curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 x ^ { 2 } - \frac { 4 } { x ^ { 3 } }\). The curve has a stationary point at \(\left( - 1 , \frac { 9 } { 2 } \right)\).

1. Determine the nature of the stationary point at \(\left( - 1 , \frac { 9 } { 2 } \right)\).
2. Find the equation of the curve.
3. Show that the curve has no other stationary points.
4. A point \(A\) is moving along the curve and the \(y\)-coordinate of \(A\) is increasing at a rate of 5 units per second. Find the rate of increase of the \(x\)-coordinate of \(A\) at the point where \(x = 1\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.