CAIE P1 (Pure Mathematics 1) 2023 March

1 A line has equation \(y = 3 x - 2 k\) and a curve has equation \(y = x ^ { 2 } - k x + 2\), where \(k\) is a constant. Show that the line and the curve meet for all values of \(k\).

2 A function f is defined by \(\mathrm { f } ( x ) = x ^ { 2 } - 2 x + 5\) for \(x \in \mathbb { R }\). A sequence of transformations is applied in the following order to the graph of \(y = \mathrm { f } ( x )\) to give the graph of \(y = \mathrm { g } ( x )\). Stretch parallel to the \(x\)-axis with scale factor \(\frac { 1 } { 2 }\)
Reflection in the \(y\)-axis
Stretch parallel to the \(y\)-axis with scale factor 3
Find \(\mathrm { g } ( x )\), giving your answer in the form \(a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants.

3 A curve has equation \(y = \frac { 1 } { 60 } ( 3 x + 1 ) ^ { 2 }\) and a point is moving along the curve.
Find the \(x\)-coordinate of the point on the curve at which the \(x\) - and \(y\)-coordinates are increasing at the same rate.

4 The circumference round the trunk of a large tree is measured and found to be 5.00 m . After one year the circumference is measured again and found to be 5.02 m .

1. Given that the circumferences at yearly intervals form an arithmetic progression, find the circumference 20 years after the first measurement.
2. Given instead that the circumferences at yearly intervals form a geometric progression, find the circumference 20 years after the first measurement.

5 Points \(A ( 7,12 )\) and \(B\) lie on a circle with centre \(( - 2,5 )\). The line \(A B\) has equation \(y = - 2 x + 26\).
Find the coordinates of \(B\).

6 In the expansion of \(\left( \frac { x } { a } + \frac { a } { x ^ { 2 } } \right) ^ { 7 }\), it is given that $$\frac { \text { the coefficient of } x ^ { 4 } } { \text { the coefficient of } x } = 3 \text {. }$$ Find the possible values of the constant \(a\).

7

1. By first obtaining a quadratic equation in \(\cos \theta\), solve the equation $$\tan \theta \sin \theta = 1$$ for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
2. Show that \(\frac { \tan \theta } { \sin \theta } - \frac { \sin \theta } { \tan \theta } \equiv \tan \theta \sin \theta\).

8
\includegraphics[max width=\textwidth, alt={}, center]{3bad1d9f-5b9e-4895-aa4e-3e6d9f6c072e-10_454_744_255_703} The diagram shows triangle \(A B C\) in which angle \(B\) is a right angle. The length of \(A B\) is 8 cm and the length of \(B C\) is 4 cm . The point \(D\) on \(A B\) is such that \(A D = 5 \mathrm {~cm}\). The sector \(D A C\) is part of a circle with centre \(D\).

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

9 The function f is defined by \(\mathrm { f } ( x ) = - 3 x ^ { 2 } + 2\) for \(x \leqslant - 1\).

1. State the range of f.
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The function g is defined by \(\mathrm { g } ( x ) = - x ^ { 2 } - 1\) for \(x \leqslant - 1\).
3. Solve the equation \(\mathrm { fg } ( x ) - \mathrm { gf } ( x ) + 8 = 0\).

10 At the point \(( 4 , - 1 )\) on a curve, the gradient of the curve is \(- \frac { 3 } { 2 }\). It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { - \frac { 1 } { 2 } } + k\), where \(k\) is a constant.

1. Show that \(k = - 2\).
2. Find the equation of the curve.
3. Find the coordinates of the stationary point.
4. Determine the nature of the stationary point.

11
\includegraphics[max width=\textwidth, alt={}, center]{3bad1d9f-5b9e-4895-aa4e-3e6d9f6c072e-16_599_780_274_671} The diagram shows the curve with equation \(x = y ^ { 2 } + 1\). The points \(A ( 5,2 )\) and \(B ( 2 , - 1 )\) lie on the curve.

1. Find an equation of the line \(A B\).
2. Find the volume of revolution when the region between the curve and the line \(A B\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.