CAIE P1 (Pure Mathematics 1) 2020 November

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1. Express \(x ^ { 2 } + 6 x + 5\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. The curve with equation \(y = x ^ { 2 }\) is transformed to the curve with equation \(y = x ^ { 2 } + 6 x + 5\). Describe fully the transformation(s) involved.

2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 } { ( x + 2 ) ^ { 2 } }\) for \(x > - 2\).

1. Find \(\int _ { 1 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x\).
2. The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x )\). It is given that the point \(( - 1 , - 1 )\) lies on the curve. Find the equation of the curve.

3 Solve the equation \(3 \tan ^ { 2 } \theta + 1 = \frac { 2 } { \tan ^ { 2 } \theta }\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

4 A curve has equation \(y = 3 x ^ { 2 } - 4 x + 4\) and a straight line has equation \(y = m x + m - 1\), where \(m\) is a constant. Find the set of values of \(m\) for which the curve and the line have two distinct points of intersection.

5 In the expansion of \(( a + b x ) ^ { 7 }\), where \(a\) and \(b\) are non-zero constants, the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 4 }\) are the first, second and third terms respectively of a geometric progression. Find the value of \(\frac { a } { b }\).

6 The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 x } { 3 x - 1 }\) for \(x > \frac { 1 } { 3 }\).

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
2. Show that \(\frac { 2 } { 3 } + \frac { 2 } { 3 ( 3 x - 1 ) }\) can be expressed as \(\frac { 2 x } { 3 x - 1 }\).
3. State the range of f.

7 The first and second terms of an arithmetic progression are \(\frac { 1 } { \cos ^ { 2 } \theta }\) and \(- \frac { \tan ^ { 2 } \theta } { \cos ^ { 2 } \theta }\), respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).

1. Show that the common difference is \(- \frac { 1 } { \cos ^ { 4 } \theta }\).
2. Find the exact value of the 13th term when \(\theta = \frac { 1 } { 6 } \pi\).

8 The equation of a curve is \(y = 2 x + 1 + \frac { 1 } { 2 x + 1 }\) for \(x > - \frac { 1 } { 2 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point and determine the nature of the stationary point.

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\includegraphics[max width=\textwidth, alt={}, center]{3b44e558-f91d-4175-acda-eceb70dad82c-12_497_652_260_744} In the diagram, arc \(A B\) is part of a circle with centre \(O\) and radius 8 cm . Arc \(B C\) is part of a circle with centre \(A\) and radius 12 cm , where \(A O C\) is a straight line.

1. Find angle \(B A O\) in radians.
2. Find the area of the shaded region.
3. Find the perimeter of the shaded region.

10 A curve has equation \(y = \frac { 1 } { k } x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } + \frac { 1 } { k ^ { 2 } }\) where \(x > 0\) and \(k\) is a positive constant.

1. It is given that when \(x = \frac { 1 } { 4 }\), the gradient of the curve is 3 . Find the value of \(k\).
2. It is given instead that \(\int _ { \frac { 1 } { 4 } k ^ { 2 } } ^ { k ^ { 2 } } \left( \frac { 1 } { k } x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } + \frac { 1 } { k ^ { 2 } } \right) \mathrm { d } x = \frac { 13 } { 12 }\). Find the value of \(k\).

11 A circle with centre \(C\) has equation \(( x - 8 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 100\).

1. Show that the point \(T ( - 6,6 )\) is outside the circle.
   Two tangents from \(T\) to the circle are drawn.
2. Show that the angle between one of the tangents and \(C T\) is exactly \(45 ^ { \circ }\).
   The two tangents touch the circle at \(A\) and \(B\).
3. Find the equation of the line \(A B\), giving your answer in the form \(y = m x + c\).
4. Find the \(x\)-coordinates of \(A\) and \(B\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.