CAIE P1 (Pure Mathematics 1) 2022 June

1 The coefficient of \(x ^ { 3 }\) in the expansion of \(\left( p + \frac { 1 } { p } x \right) ^ { 4 }\) is 144 .
Find the possible values of the constant \(p\).

2
\includegraphics[max width=\textwidth, alt={}, center]{89a18f20-a4d6-4a42-8b00-849f4fb89692-04_657_1253_269_431} The diagram shows part of the curve with equation \(y = p \sin ( q \theta ) + r\), where \(p , q\) and \(r\) are constants.

1. State the value of \(p\).
2. State the value of \(q\).
3. State the value of \(r\).

3 An arithmetic progression has first term 4 and common difference \(d\). The sum of the first \(n\) terms of the progression is 5863.

1. Show that \(( n - 1 ) d = \frac { 11726 } { n } - 8\).
2. Given that the \(n\)th term is 139 , find the values of \(n\) and \(d\), giving the value of \(d\) as a fraction.

4

1. The curve with equation \(y = x ^ { 2 } + 2 x - 5\) is translated by \(\binom { - 1 } { 3 }\).
   Find the equation of the translated curve, giving your answer in the form \(y = a x ^ { 2 } + b x + c\).
2. The curve with equation \(y = x ^ { 2 } + 2 x - 5\) is transformed to a curve with equation \(y = 4 x ^ { 2 } + 4 x - 5\). Describe fully the single transformation that has been applied.

5

1. Solve the equation \(6 \sqrt { y } + \frac { 2 } { \sqrt { y } } - 7 = 0\).
2. Hence solve the equation \(6 \sqrt { \tan x } + \frac { 2 } { \sqrt { \tan x } } - 7 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

6 The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } - 16 x + 23\) for \(x < 3\).

1. Express \(\mathrm { f } ( x )\) in the form \(2 ( x + a ) ^ { 2 } + b\).
2. Find the range of f.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The function g is defined by \(\mathrm { g } ( x ) = 2 x + 4\) for \(x < - 1\).
4. Find and simplify an expression for \(\mathrm { fg } ( x )\).

7
\includegraphics[max width=\textwidth, alt={}, center]{89a18f20-a4d6-4a42-8b00-849f4fb89692-10_887_1003_258_571} The diagram shows the circle with equation \(( x - 2 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 20\) and with centre \(C\). The point \(B\) has coordinates \(( 0,2 )\) and the line segment \(B C\) intersects the circle at \(P\).

1. Find the equation of \(B C\).
2. Hence find the coordinates of \(P\), giving your answer in exact form.

8
\includegraphics[max width=\textwidth, alt={}, center]{89a18f20-a4d6-4a42-8b00-849f4fb89692-12_577_1088_260_523} The diagram shows the curve with equation \(y = x ^ { \frac { 1 } { 2 } } + 4 x ^ { - \frac { 1 } { 2 } }\). The line \(y = 5\) intersects the curve at the points \(A ( 1,5 )\) and \(B ( 16,5 )\).

1. Find the equation of the tangent to the curve at the point \(A\).
2. Calculate the area of the shaded region.

9 The diagram shows triangle \(A B C\) with \(A B = B C = 6 \mathrm {~cm}\) and angle \(A B C = 1.8\) radians. The arc \(C D\) is part of a circle with centre \(A\) and \(A B D\) is a straight line.

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

10 The function f is defined by \(\mathrm { f } ( x ) = ( 4 x + 2 ) ^ { - 2 }\) for \(x > - \frac { 1 } { 2 }\).

1. Find \(\int _ { 1 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x\).
   A point is moving along the curve \(y = \mathrm { f } ( x )\) in such a way that, as it passes through the point \(A\), its \(y\)-coordinate is decreasing at the rate of \(k\) units per second and its \(x\)-coordinate is increasing at the rate of \(k\) units per second.
2. Find the coordinates of \(A\).

11 The point \(P\) lies on the line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. A curve has equation \(y = - \frac { m } { x }\). There is a single point \(P\) on the curve such that the straight line is a tangent to the curve at \(P\).

1. Find the coordinates of \(P\), giving the \(y\)-coordinate in terms of \(m\).
   The normal to the curve at \(P\) intersects the curve again at the point \(Q\).
2. Find the coordinates of \(Q\) in terms of \(m\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.