CAIE P1 (Pure Mathematics 1) 2022 June

1 The coefficient of \(x ^ { 4 }\) in the expansion of \(( 3 + x ) ^ { 5 }\) is equal to the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 x + \frac { a } { x } \right) ^ { 6 }\). Find the value of the positive constant \(a\).

2 The second and third terms of a geometric progression are 10 and 8 respectively.
Find the sum to infinity.

3 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 ( 4 x - 7 ) ^ { \frac { 1 } { 2 } } - 4 x ^ { - \frac { 1 } { 2 } }\). It is given that the curve passes through the point \(\left( 4 , \frac { 5 } { 2 } \right)\). Find the equation of the curve.

4 The first, second and third terms of an arithmetic progression are \(k , 6 k\) and \(k + 6\) respectively.

1. Find the value of the constant \(k\).
2. Find the sum of the first 30 terms of the progression.

5 The equation of a curve is \(y = 4 x ^ { 2 } - k x + \frac { 1 } { 2 } k ^ { 2 }\) and the equation of a line is \(y = x - a\), where \(k\) and \(a\) are constants.

1. Given that the curve and the line intersect at the points with \(x\)-coordinates 0 and \(\frac { 3 } { 4 }\), find the values of \(k\) and \(a\).
2. Given instead that \(a = - \frac { 7 } { 2 }\), find the values of \(k\) for which the line is a tangent to the curve. [5]

6
\includegraphics[max width=\textwidth, alt={}, center]{bb7595c9-93ae-49e8-9cc5-9ecc802e6060-08_613_865_262_632} The diagram shows the curve with equation \(y = 5 x ^ { \frac { 1 } { 2 } }\) and the line with equation \(y = 2 x + 2\).
Find the exact area of the shaded region which is bounded by the line and the curve.

7
\includegraphics[max width=\textwidth, alt={}, center]{bb7595c9-93ae-49e8-9cc5-9ecc802e6060-10_593_841_260_651} The diagram shows a sector \(O B A C\) of a circle with centre \(O\) and radius 10 cm . The point \(P\) lies on \(O C\) and \(B P\) is perpendicular to \(O C\). Angle \(A O C = \frac { 1 } { 6 } \pi\) and the length of the \(\operatorname { arc } A B\) is 2 cm .

1. Find the angle \(B O C\).
2. Hence find the area of the shaded region \(B P C\) giving your answer correct to 3 significant figures. [4]

8 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + a x + b y - 12 = 0\). The points \(A ( 1,1 )\) and \(B ( 2 , - 6 )\) lie on the circle.

1. Find the values of \(a\) and \(b\) and hence find the coordinates of the centre of the circle.
2. Find the equation of the tangent to the circle at the point \(A\), giving your answer in the form \(p x + q y = k\), where \(p , q\) and \(k\) are integers.

9 The equation of a curve is \(y = 3 x + 1 - 4 ( 3 x + 1 ) ^ { \frac { 1 } { 2 } }\) for \(x > - \frac { 1 } { 3 }\).

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 of the curve and determine its nature.

10 Functions \(f\) and \(g\) are defined as follows: $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 2 x + 1 } { 2 x - 1 } & \text { for } x \neq \frac { 1 } { 2 }
\mathrm {~g} ( x ) = x ^ { 2 } + 4 & \text { for } x \in \mathbb { R } \end{array}$$

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{bb7595c9-93ae-49e8-9cc5-9ecc802e6060-16_773_1182_555_511} The diagram shows part of the graph of \(y = \mathrm { f } ( x )\).
   State the domain of \(\mathrm { f } ^ { - 1 }\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. Find \(\mathrm { gf } ^ { - 1 } ( 3 )\).
4. Explain why \(\mathrm { g } ^ { - 1 } ( x )\) cannot be found.
5. Show that \(1 + \frac { 2 } { 2 x - 1 }\) can be expressed as \(\frac { 2 x + 1 } { 2 x - 1 }\). Hence find the area of the triangle enclosed by the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = 1\) and the \(x\) - and \(y\)-axes.

11 The function f is given by \(\mathrm { f } ( x ) = 4 \cos ^ { 4 } x + \cos ^ { 2 } x - k\) for \(0 \leqslant x \leqslant 2 \pi\), where \(k\) is a constant.

1. Given that \(k = 3\), find the exact solutions of the equation \(\mathrm { f } ( x ) = 0\).
2. Use the quadratic formula to show that, when \(k > 5\), the equation \(\mathrm { f } ( x ) = 0\) has no solutions.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.