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CAIE P1 2022 June Q6
6 The function \(f\) is defined as follows: $$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 4 } { x ^ { 2 } + 4 } \quad \text { for } x > 2$$
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  2. Show that \(1 - \frac { 8 } { x ^ { 2 } + 4 }\) can be expressed as \(\frac { x ^ { 2 } - 4 } { x ^ { 2 } + 4 }\) and hence state the range of f .
  3. Explain why the composite function ff cannot be formed.
CAIE P1 2022 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{574b96b2-62f2-41b3-a178-8e68e16429ff-12_631_1031_267_534} The diagram shows the curve with equation \(y = ( 3 x - 2 ) ^ { \frac { 1 } { 2 } }\) and the line \(y = \frac { 1 } { 2 } x + 1\). The curve and the line intersect at points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\).
  2. Hence find the area of the region enclosed between the curve and the line.
CAIE P1 2022 June Q8
8
  1. The curve \(y = \sin x\) is transformed to the curve \(y = 4 \sin \left( \frac { 1 } { 2 } x - 30 ^ { \circ } \right)\).
    Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations are applied.
  2. Find the exact solutions of the equation \(4 \sin \left( \frac { 1 } { 2 } x - 30 ^ { \circ } \right) = 2 \sqrt { 2 }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2022 June Q9
9 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 26 = 0\).
  1. Find the coordinates of the centre of the circle and the radius. Hence find the coordinates of the lowest point on the circle.
  2. Find the set of values of the constant \(k\) for which the line with equation \(y = k x - 5\) intersects the circle at two distinct points.
CAIE P1 2022 June Q10
10 The equation of a curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 x ^ { 2 } - \frac { 4 } { x ^ { 3 } }\). The curve has a stationary point at \(\left( - 1 , \frac { 9 } { 2 } \right)\).
  1. Determine the nature of the stationary point at \(\left( - 1 , \frac { 9 } { 2 } \right)\).
  2. Find the equation of the curve.
  3. Show that the curve has no other stationary points.
  4. A point \(A\) is moving along the curve and the \(y\)-coordinate of \(A\) is increasing at a rate of 5 units per second. Find the rate of increase of the \(x\)-coordinate of \(A\) at the point where \(x = 1\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P1 2022 June Q1
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 \(a\).
CAIE P1 2022 June Q2
2 The second and third terms of a geometric progression are 10 and 8 respectively.
Find the sum to infinity.
CAIE P1 2022 June Q3
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.
CAIE P1 2022 June Q4
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.
CAIE P1 2022 June Q5
5 marks
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]
CAIE P1 2022 June Q6
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.
CAIE P1 2022 June Q7
4 marks
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]
CAIE P1 2022 June Q8
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.
CAIE P1 2022 June Q9
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.
CAIE P1 2022 June Q10
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.
CAIE P1 2022 June Q11
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.
CAIE P1 2022 June Q1
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 \(p\).
CAIE P1 2022 June Q2
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 \(r\).
CAIE P1 2022 June Q3
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.
CAIE P1 2022 June Q4
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.
CAIE P1 2022 June Q5
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 \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2022 June Q6
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 \(\mathrm { fg } ( x )\).
CAIE P1 2022 June Q7
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.
CAIE P1 2022 June Q8
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.
CAIE P1 2022 June Q9
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.