CAIE P1 (Pure Mathematics 1) 2023 November

1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 + 2 a x ) ^ { 5 }\) is six times the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 + a x ) ^ { 6 }\). Find the value of the constant \(a\).

2 Find the exact solution of the equation $$\frac { 1 } { 6 } \pi + \tan ^ { - 1 } ( 4 x ) = - \cos ^ { - 1 } \left( \frac { 1 } { 2 } \sqrt { 3 } \right)$$

3 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 } x + \frac { 72 } { x ^ { 4 } }\). The curve passes through the point \(P ( 2,8 )\).

1. Find the equation of the normal to the curve at \(P\).
2. Find the equation of the curve.
   \includegraphics[max width=\textwidth, alt={}, center]{e48188bc-3332-4248-971d-ebbbbbfb1280-05_456_488_264_826} The diagram shows the shape of a coin. The three \(\operatorname { arcs } A B , B C\) and \(C A\) are parts of circles with centres \(C , A\) and \(B\) respectively. \(A B C\) is an equilateral triangle with sides of length 2 cm .
3. Find the perimeter of the coin.
4. Find the area of the face \(A B C\) of the coin, giving the answer in terms of \(\pi\) and \(\sqrt { 3 }\).

5 The first, second and third terms of a geometric progression are \(\sin \theta , \cos \theta\) and \(2 - \sin \theta\) respectively, where \(\theta\) radians is an acute angle.

1. Find the value of \(\theta\).
2. Using this value of \(\theta\), find the sum of the first 10 terms of the progression. Give the answer in the form \(\frac { b } { \sqrt { c } - 1 }\), where \(b\) and \(c\) are integers to be found.

6 The equation of a curve is \(y = x ^ { 2 } - 8 x + 5\).

1. Find the coordinates of the minimum point of the curve.
   The curve is stretched by a factor of 2 parallel to the \(y\)-axis and then translated by \(\binom { 4 } { 1 }\).
2. Find the coordinates of the minimum point of the transformed curve.
3. Find the equation of the transformed curve. Give the answer in the form \(y = a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are integers to be found.

7

1. Verify the identity \(( 2 x - 1 ) \left( 4 x ^ { 2 } + 2 x - 1 \right) \equiv 8 x ^ { 3 } - 4 x + 1\).
2. Prove the identity \(\frac { \tan ^ { 2 } \theta + 1 } { \tan ^ { 2 } \theta - 1 } \equiv \frac { 1 } { 1 - 2 \cos ^ { 2 } \theta }\).
3. Using the results of (a) and (b), solve the equation $$\frac { \tan ^ { 2 } \theta + 1 } { \tan ^ { 2 } \theta - 1 } = 4 \cos \theta$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

8 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = ( x + a ) ^ { 2 } - a \text { for } x \leqslant - a ,
& \mathrm {~g} ( x ) = 2 x - 1 \text { for } x \in \mathbb { R } , \end{aligned}$$ where \(a\) is a positive constant.

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
2. 1. State the domain of the function \(f ^ { - 1 }\).
   2. State the range of the function \(f ^ { - 1 }\).
3. Given that \(a = \frac { 7 } { 2 }\), solve the equation \(\mathrm { gf } ( x ) = 0\).

9
\includegraphics[max width=\textwidth, alt={}, center]{e48188bc-3332-4248-971d-ebbbbbfb1280-14_807_1278_274_427} The diagram shows curves with equations \(y = 2 x ^ { \frac { 1 } { 2 } } + 13 x ^ { - \frac { 1 } { 2 } }\) and \(y = 3 x ^ { - \frac { 1 } { 2 } } + 12\). The curves intersect at points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\).
2. Hence find the area of the shaded region.

10 The equation of a curve is \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = ( 4 x - 3 ) ^ { \frac { 5 } { 3 } } - \frac { 20 } { 3 } x\).

1. Find the \(x\)-coordinates of the stationary points of the curve and determine their nature.
2. State the set of values for which the function f is increasing.

11 The coordinates of points \(A , B\) and \(C\) are (6, 4), ( \(p , 7\) ) and (14, 18) respectively, where \(p\) is a constant. The line \(A B\) is perpendicular to the line \(B C\).

1. Given that \(p < 10\), find the value of \(p\).
   A circle passes through the points \(A , B\) and \(C\).
2. Find the equation of the circle.
3. Find the equation of the tangent to the circle at \(C\), giving the answer in the form \(d x + e y + f = 0\), where \(d , e\) and \(f\) are integers.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.