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CAIE P1 2021 March Q3
3 Solve the equation \(\frac { \tan \theta + 2 \sin \theta } { \tan \theta - 2 \sin \theta } = 3\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2021 March Q4
4 A line has equation \(y = 3 x + k\) and a curve has equation \(y = x ^ { 2 } + k x + 6\), where \(k\) is a constant. Find the set of values of \(k\) for which the line and curve have two distinct points of intersection.
CAIE P1 2021 March Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-07_775_768_260_685} In the diagram, the graph of \(y = \mathrm { f } ( x )\) is shown with solid lines. The graph shown with broken lines is a transformation of \(y = \mathrm { f } ( x )\).
  1. Describe fully the two single transformations of \(y = \mathrm { f } ( x )\) that have been combined to give the resulting transformation.
  2. State in terms of \(y\), f and \(x\), the equation of the graph shown with broken lines.
CAIE P1 2021 March Q6
6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { ( 3 x - 2 ) ^ { 3 } }\) and \(A ( 1 , - 3 )\) lies on the curve. A point is moving along the curve and at \(A\) the \(y\)-coordinate of the point is increasing at 3 units per second.
  1. Find the rate of increase at \(A\) of the \(x\)-coordinate of the point.
  2. Find the equation of the curve.
CAIE P1 2021 March Q7
7 Functions f and g are defined as follows: $$\begin{aligned} & \mathrm { f } : x \mapsto x ^ { 2 } + 2 x + 3 \text { for } x \leqslant - 1 ,
& \mathrm {~g} : x \mapsto 2 x + 1 \text { for } x \geqslant - 1 . \end{aligned}$$
  1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\) and state the range of f .
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  3. Solve the equation \(\operatorname { gf } ( x ) = 13\).
CAIE P1 2021 March Q8
8 The points \(A ( 7,1 ) , B ( 7,9 )\) and \(C ( 1,9 )\) are on the circumference of a circle.
  1. Find an equation of the circle.
  2. Find an equation of the tangent to the circle at \(B\).
CAIE P1 2021 March Q9
9 The first term of a progression is \(\cos \theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\).
  1. For the case where the progression is geometric, the sum to infinity is \(\frac { 1 } { \cos \theta }\).
    1. Show that the second term is \(\cos \theta \sin ^ { 2 } \theta\).
    2. Find the sum of the first 12 terms when \(\theta = \frac { 1 } { 3 } \pi\), giving your answer correct to 4 significant figures.
  2. For the case where the progression is arithmetic, the first two terms are again \(\cos \theta\) and \(\cos \theta \sin ^ { 2 } \theta\) respectively. Find the 85 th term when \(\theta = \frac { 1 } { 3 } \pi\).
    \includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-16_547_421_264_863} The diagram shows a sector \(A B C\) which is part of a circle of radius \(a\). The points \(D\) and \(E\) lie on \(A B\) and \(A C\) respectively and are such that \(A D = A E = k a\), where \(k < 1\). The line \(D E\) divides the sector into two regions which are equal in area.
CAIE P1 2021 March Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-18_497_1049_264_548} The diagram shows the curve with equation \(y = 9 \left( x ^ { - \frac { 1 } { 2 } } - 4 x ^ { - \frac { 3 } { 2 } } \right)\). The curve crosses the \(x\)-axis at the point \(A\).
  1. Find the \(x\)-coordinate of \(A\).
  2. Find the equation of the tangent to the curve at \(A\).
  3. Find the \(x\)-coordinate of the maximum point of the curve.
  4. Find the area of the region bounded by the curve, the \(x\)-axis and the line \(x = 9\).
    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 March Q1
1 A curve with equation \(y = \mathrm { f } ( x )\) is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { - \frac { 1 } { 3 } } - x ^ { \frac { 1 } { 3 } }\). It is given that \(\mathrm { f } ( 8 ) = 5\).
Find \(\mathrm { f } ( x )\).
CAIE P1 2022 March Q2
2 A curve has equation \(y = x ^ { 2 } + 2 c x + 4\) and a straight line has equation \(y = 4 x + c\), where \(c\) is a constant. Find the set of values of \(c\) for which the curve and line intersect at two distinct points.
CAIE P1 2022 March Q3
3 Find the term independent of \(x\) in each of the following expansions.
  1. \(\left( 3 x + \frac { 2 } { x ^ { 2 } } \right) ^ { 6 }\)
  2. \(\left( 3 x + \frac { 2 } { x ^ { 2 } } \right) ^ { 6 } \left( 1 - x ^ { 3 } \right)\)
CAIE P1 2022 March Q4
4 The first term of a geometric progression and the first term of an arithmetic progression are both equal to \(a\). The third term of the geometric progression is equal to the second term of the arithmetic progression.
The fifth term of the geometric progression is equal to the sixth term of the arithmetic progression.
Given that the terms are all positive and not all equal, find the sum of the first twenty terms of the arithmetic progression in terms of \(a\).
CAIE P1 2022 March Q5
5
  1. Express \(2 x ^ { 2 } - 8 x + 14\) in the form \(2 \left[ ( x - a ) ^ { 2 } + b \right]\).
    The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } \quad \text { for } x \in \mathbb { R }
    & \mathrm {~g} ( x ) = 2 x ^ { 2 } - 8 x + 14 \quad \text { for } x \in \mathbb { R } \end{aligned}$$
  2. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), making clear the order in which the transformations are applied.
    \includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-08_679_1043_260_552} The circle with equation \(( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 85\) and the straight line with equation \(y = 3 x - 20\) are shown in the diagram. The line intersects the circle at \(A\) and \(B\), and the centre of the circle is at \(C\).
CAIE P1 2022 March Q7
7
  1. Show that \(\frac { \sin \theta + 2 \cos \theta } { \cos \theta - 2 \sin \theta } - \frac { \sin \theta - 2 \cos \theta } { \cos \theta + 2 \sin \theta } \equiv \frac { 4 } { 5 \cos ^ { 2 } \theta - 4 }\).
  2. Hence solve the equation \(\frac { \sin \theta + 2 \cos \theta } { \cos \theta - 2 \sin \theta } - \frac { \sin \theta - 2 \cos \theta } { \cos \theta + 2 \sin \theta } = 5\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2022 March Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-12_771_839_262_651} The diagram shows the circle with equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 8\). The chord \(A B\) of the circle intersects the positive \(y\)-axis at \(A\) and is parallel to the \(x\)-axis.
  1. Find, by calculation, the coordinates of \(A\) and \(B\).
  2. Find the volume of revolution when the shaded segment, bounded by the circle and the chord \(A B\), is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2022 March Q9
9 Functions f, g and h are defined as follows: $$\begin{aligned} & \mathrm { f } : x \mapsto x - 4 x ^ { \frac { 1 } { 2 } } + 1 \quad \text { for } x \geqslant 0
& \mathrm {~g} : x \mapsto m x ^ { 2 } + n \quad \text { for } x \geqslant - 2 , \text { where } m \text { and } n \text { are constants, }
& \mathrm { h } : x \mapsto x ^ { \frac { 1 } { 2 } } - 2 \quad \text { for } x \geqslant 0 . \end{aligned}$$
  1. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your solutions in the form \(x = a + b \sqrt { c }\), where \(a , b\) and \(c\) are integers.
  2. Given that \(\mathrm { f } ( x ) \equiv \mathrm { gh } ( x )\), find the values of \(m\) and \(n\).
    \includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-16_652_1045_255_550} The diagram shows a circle with centre \(A\) of radius 5 cm and a circle with centre \(B\) of radius 8 cm . The circles touch at the point \(C\) so that \(A C B\) is a straight line. The tangent at the point \(D\) on the smaller circle intersects the larger circle at \(E\) and passes through \(B\).
CAIE P1 2022 March Q11
11 It is given that a curve has equation \(y = k ( 3 x - k ) ^ { - 1 } + 3 x\), where \(k\) is a constant.
  1. Find, in terms of \(k\), the values of \(x\) at which there is a stationary point.
    The function f has a stationary value at \(x = a\) and is defined by $$f ( x ) = 4 ( 3 x - 4 ) ^ { - 1 } + 3 x \quad \text { for } x \geqslant \frac { 3 } { 2 }$$
  2. Find the value of \(a\) and determine the nature of the stationary value.
  3. The function g is defined by \(\mathrm { g } ( x ) = - ( 3 x + 1 ) ^ { - 1 } + 3 x\) for \(x \geqslant 0\). Determine, making your reasoning clear, whether \(g\) is an increasing function, a decreasing function or neither.
    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 2023 March Q1
1 A line has equation \(y = 3 x - 2 k\) and a curve has equation \(y = x ^ { 2 } - k x + 2\), where \(k\) is a constant. Show that the line and the curve meet for all values of \(k\).
CAIE P1 2023 March Q2
2 A function f is defined by \(\mathrm { f } ( x ) = x ^ { 2 } - 2 x + 5\) for \(x \in \mathbb { R }\). A sequence of transformations is applied in the following order to the graph of \(y = \mathrm { f } ( x )\) to give the graph of \(y = \mathrm { g } ( x )\). Stretch parallel to the \(x\)-axis with scale factor \(\frac { 1 } { 2 }\)
Reflection in the \(y\)-axis
Stretch parallel to the \(y\)-axis with scale factor 3
Find \(\mathrm { g } ( x )\), giving your answer in the form \(a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants.
CAIE P1 2023 March Q3
3 A curve has equation \(y = \frac { 1 } { 60 } ( 3 x + 1 ) ^ { 2 }\) and a point is moving along the curve.
Find the \(x\)-coordinate of the point on the curve at which the \(x\) - and \(y\)-coordinates are increasing at the same rate.
CAIE P1 2023 March Q4
4 The circumference round the trunk of a large tree is measured and found to be 5.00 m . After one year the circumference is measured again and found to be 5.02 m .
  1. Given that the circumferences at yearly intervals form an arithmetic progression, find the circumference 20 years after the first measurement.
  2. Given instead that the circumferences at yearly intervals form a geometric progression, find the circumference 20 years after the first measurement.
CAIE P1 2023 March Q5
5 Points \(A ( 7,12 )\) and \(B\) lie on a circle with centre \(( - 2,5 )\). The line \(A B\) has equation \(y = - 2 x + 26\).
Find the coordinates of \(B\).
CAIE P1 2023 March Q6
6 In the expansion of \(\left( \frac { x } { a } + \frac { a } { x ^ { 2 } } \right) ^ { 7 }\), it is given that $$\frac { \text { the coefficient of } x ^ { 4 } } { \text { the coefficient of } x } = 3 \text {. }$$ Find the possible values of the constant \(a\).
CAIE P1 2023 March Q7
7
  1. By first obtaining a quadratic equation in \(\cos \theta\), solve the equation $$\tan \theta \sin \theta = 1$$ for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
  2. Show that \(\frac { \tan \theta } { \sin \theta } - \frac { \sin \theta } { \tan \theta } \equiv \tan \theta \sin \theta\).
CAIE P1 2023 March Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{3bad1d9f-5b9e-4895-aa4e-3e6d9f6c072e-10_454_744_255_703} The diagram shows triangle \(A B C\) in which angle \(B\) is a right angle. The length of \(A B\) is 8 cm and the length of \(B C\) is 4 cm . The point \(D\) on \(A B\) is such that \(A D = 5 \mathrm {~cm}\). The sector \(D A C\) is part of a circle with centre \(D\).
  1. Find the perimeter of the shaded region.
  2. Find the area of the shaded region.