CAIE P1 (Pure Mathematics 1) 2023 November

1 A curve is such that its gradient at a point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x - 3 x ^ { - \frac { 1 } { 2 } }\). It is given that the curve passes through the point \(( 4,1 )\). Find the equation of the curve.

2 The circle with equation \(( x - 3 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 40\) intersects the \(y\)-axis at points \(A\) and \(B\).

1. Find the \(y\)-coordinates of \(A\) and \(B\), expressing your answers in terms of surds.
2. Find the equation of the circle which has \(A B\) as its diameter.

3

1. Show that the equation $$5 \cos \theta - \sin \theta \tan \theta + 1 = 0$$ may be expressed in the form \(a \cos ^ { 2 } \theta + b \cos \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants to be found.
2. Hence solve the equation \(5 \cos \theta - \sin \theta \tan \theta + 1 = 0\) for \(0 < \theta < 2 \pi\).

4

1. Expand the following in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
   1. \(( 1 + 2 x ) ^ { 5 }\).
   2. \(( 1 - a x ) ^ { 6 }\), where \(a\) is a constant.
      In the expansion of \(( 1 + 2 x ) ^ { 5 } ( 1 - a x ) ^ { 6 }\), the coefficient of \(x ^ { 2 }\) is - 5 .
2. Find the possible values of \(a\).

5 The first, second and third terms of a geometric progression are \(2 p + 6,5 p\) and \(8 p + 2\) respectively.

1. Find the possible values of the constant \(p\).
2. One of the values of \(p\) found in (a) is a negative fraction. Use this value of \(p\) to find the sum to infinity of this progression.

6 A line has equation \(y = 6 x - c\) and a curve has equation \(y = c x ^ { 2 } + 2 x - 3\), where \(c\) is a constant. The line is a tangent to the curve at point \(P\). Find the possible values of \(c\) and the corresponding coordinates of \(P\).

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

1. State the range of f.
2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
   The function g is defined by \(\mathrm { g } ( x ) = 2 x - 2\) for \(x > 0\).
3. Obtain a simplified expression for \(\mathrm { gf } ( x )\).

8
\includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-10_515_936_274_577} The diagram shows part of the graph of \(y = \sin ( a ( x + b ) )\), where \(a\) and \(b\) are positive constants.

1. State the value of \(a\) and one possible value of \(b\).
   Another curve, with equation \(y = \mathrm { f } ( x )\), has a single stationary point at the point \(( p , q )\), where \(p\) and \(q\) are constants. This curve is transformed to a curve with equation $$y = - 3 f \left( \frac { 1 } { 4 } ( x + 8 ) \right) .$$
2. For the transformed curve, find the coordinates of the stationary point, giving your answer in terms of \(p\) and \(q\).

9 A curve has equation \(y = 2 x ^ { \frac { 1 } { 2 } } - 1\).

1. Find the equation of the normal to the curve at the point \(A ( 4,3 )\), giving your answer in the form \(y = m x + c\).
   A point is moving along the curve \(y = 2 x ^ { \frac { 1 } { 2 } } - 1\) in such a way that at \(A\) the rate of increase of the \(x\)-coordinate is \(3 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
2. Find the rate of increase of the \(y\)-coordinate at \(A\).
   At \(A\) the moving point suddenly changes direction and speed, and moves down the normal in such a way that the rate of decrease of the \(y\)-coordinate is constant at \(5 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
3. As the point moves down the normal, find the rate of change of its \(x\)-coordinate.

10
\includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-12_552_582_255_778} The diagram shows points \(A , B\) and \(C\) lying on a circle with centre \(O\) and radius \(r\). Angle \(A O B\) is 2.8 radians. The shaded region is bounded by two arcs. The upper arc is part of the circle with centre \(O\) and radius \(r\). The lower arc is part of a circle with centre \(C\) and radius \(R\).

1. State the size of angle \(A C O\) in radians.
2. Find \(R\) in terms of \(r\).
3. Find the area of the shaded region in terms of \(r\).

11
\includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-14_693_782_267_669} The diagram shows part of the curve with equation \(y = x + \frac { 2 } { ( 2 x - 1 ) ^ { 2 } }\). The lines \(x = 1\) and \(x = 2\) intersect the curve at \(P\) and \(Q\) respectively and \(R\) is the stationary point on the curve.

1. Verify that the \(x\)-coordinate of \(R\) is \(\frac { 3 } { 2 }\) and find the \(y\)-coordinate of \(R\).
2. Find the exact value of the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.