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CAIE P1 2021 November Q9
9 The line \(y = 2 x + 5\) intersects the circle with equation \(x ^ { 2 } + y ^ { 2 } = 20\) at \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\) in surd form and hence find the exact length of the chord \(A B\).
    A straight line through the point \(( 10,0 )\) with gradient \(m\) is a tangent to the circle.
  2. Find the two possible values of \(m\).
CAIE P1 2021 November Q10
10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that $$\mathrm { f } ^ { \prime } ( x ) = \left( \frac { 1 } { 2 } x + k \right) ^ { - 2 } - ( 1 + k ) ^ { - 2 }$$ where \(k\) is a constant. The curve has a minimum point at \(x = 2\).
  1. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) in terms of \(k\) and \(x\), and hence find the set of possible values of \(k\).
    It is now given that \(k = - 3\) and the minimum point is at \(\left( 2,3 \frac { 1 } { 2 } \right)\).
  2. Find \(\mathrm { f } ( x )\).
  3. Find the coordinates of the other stationary point and determine its nature.
    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 November Q1
1 Solve the equation \(3 x + 2 = \frac { 2 } { x - 1 }\).
CAIE P1 2022 November Q2
2 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 12 \left( \frac { 1 } { 2 } x - 1 \right) ^ { - 4 }\). It is given that the curve passes through the point \(P ( 6,4 )\).
  1. Find the equation of the tangent to the curve at \(P\).
  2. Find the equation of the curve.
CAIE P1 2022 November Q3
3 A curve has equation \(y = a x ^ { \frac { 1 } { 2 } } - 2 x\), where \(x > 0\) and \(a\) is a constant. The curve has a stationary point at the point \(P\), which has \(x\)-coordinate 9 . Find the \(y\)-coordinate of \(P\).
CAIE P1 2022 November Q4
4 The coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + \frac { 2 } { p } x \right) ^ { 5 } + ( 1 + p x ) ^ { 6 }\) is 70 .
Find the possible values of the constant \(p\).
CAIE P1 2022 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-06_494_542_260_799} The diagram shows a sector \(O A B\) of a circle with centre \(O\). The length of the \(\operatorname { arc } A B\) is 8 cm . It is given that the perimeter of the sector is 20 cm .
  1. Find the perimeter of the shaded segment.
  2. Find the area of the shaded segment.
CAIE P1 2022 November Q6
6
  1. Show that the equation $$\frac { 1 } { \sin \theta + \cos \theta } + \frac { 1 } { \sin \theta - \cos \theta } = 1$$ may be expressed in the form \(a \sin ^ { 2 } \theta + b \sin \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants to be found.
  2. Hence solve the equation \(\frac { 1 } { \sin \theta + \cos \theta } + \frac { 1 } { \sin \theta - \cos \theta } = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2022 November Q7
7 A tool for putting fence posts into the ground is called a 'post-rammer'. The distances in millimetres that the post sinks into the ground on each impact of the post-rammer follow a geometric progression. The first three impacts cause the post to sink into the ground by \(50 \mathrm {~mm} , 40 \mathrm {~mm}\) and 32 mm respectively.
  1. Verify that the 9th impact is the first in which the post sinks less than 10 mm into the ground.
  2. Find, to the nearest millimetre, the total depth of the post in the ground after 20 impacts.
  3. Find the greatest total depth in the ground which could theoretically be achieved.
CAIE P1 2022 November Q8
8 The function f is defined by \(\mathrm { f } ( x ) = 2 - \frac { 3 } { 4 x - p }\) for \(x > \frac { p } { 4 }\), where \(p\) is a constant.
  1. Find \(\mathrm { f } ^ { \prime } ( x )\) and hence determine whether f is an increasing function, a decreasing function or neither.
  2. Express \(\mathrm { f } ^ { - 1 } ( x )\) in the form \(\frac { p } { a } - \frac { b } { c x - d }\), where \(a , b , c\) and \(d\) are integers.
  3. Hence state the value of \(p\) for which \(\mathrm { f } ^ { - 1 } ( x ) \equiv \mathrm { f } ( x )\).
CAIE P1 2022 November Q9
9 Functions f and g are both defined for \(x \in \mathbb { R }\) and are given by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } - 4 x + 9
& \mathrm {~g} ( x ) = 2 x ^ { 2 } + 4 x + 12 \end{aligned}$$
  1. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\).
  2. Express \(\mathrm { g } ( x )\) in the form \(2 \left[ ( x + c ) ^ { 2 } + d \right]\).
  3. Express \(\mathrm { g } ( x )\) in the form \(k \mathrm { f } ( x + h )\), where \(k\) and \(h\) are integers.
  4. Describe fully the two transformations that have been combined to transform the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { g } ( x )\).
CAIE P1 2022 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-16_942_933_262_605} Curves with equations \(y = 2 x ^ { \frac { 1 } { 2 } } + 1\) and \(y = \frac { 1 } { 2 } x ^ { 2 } - x + 1\) intersect at \(A ( 0,1 )\) and \(B ( 4,5 )\), as shown in the diagram.
  1. Find the area of the region between the two curves.
    The acute angle between the two tangents at \(B\) is denoted by \(\alpha ^ { \circ }\), and the scales on the axes are the same.
  2. Find \(\alpha\).
    \includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-18_951_725_267_703} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } = 20\). Tangents touching the circle at points \(B\) and \(C\) pass through the point \(A ( 0,10 )\).
  3. By letting the equation of a tangent be \(y = m x + 10\), find the two possible values of \(m\).
  4. Find the coordinates of \(B\) and \(C\).
    The point \(D\) is where the circle crosses the positive \(x\)-axis.
  5. Find angle \(B D C\) in degrees.
    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 November Q1
1 Points \(A\) and \(B\) have coordinates \(( 5,2 )\) and \(( 10 , - 1 )\) respectively.
  1. Find the equation of the perpendicular bisector of \(A B\).
  2. Find the equation of the circle with centre \(A\) which passes through \(B\).
CAIE P1 2022 November Q2
2 The first, second and third terms of an arithmetic progression are \(a , 2 a\) and \(a ^ { 2 }\) respectively, where \(a\) is a positive constant. Find the sum of the first 50 terms of the progression.
CAIE P1 2022 November Q3
3
  1. Find the set of values of \(k\) for which the equation \(8 x ^ { 2 } + k x + 2 = 0\) has no real roots.
  2. Solve the equation \(8 \cos ^ { 2 } \theta - 10 \cos \theta + 2 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2022 November Q4
4 A geometric progression is such that the third term is 1764 and the sum of the second and third terms is 3444 . Find the 50th term.
CAIE P1 2022 November Q5
5 The graph with equation \(y = \mathrm { f } ( x )\) is transformed to the graph with equation \(y = \mathrm { g } ( x )\) by a stretch in the \(x\)-direction with factor 0.5 , followed by a translation of \(\binom { 0 } { 1 }\).
  1. The diagram below shows the graph of \(y = \mathrm { f } ( x )\). On the diagram sketch the graph of \(y = \mathrm { g } ( x )\).
    \includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-07_613_1527_623_342}
  2. Find an expression for \(\mathrm { g } ( x )\) in terms of \(\mathrm { f } ( x )\).
CAIE P1 2022 November Q6
6 The equation of a curve is \(y = 4 x ^ { 2 } + 20 x + 6\).
  1. Express the equation in the form \(y = a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are constants.
  2. Hence solve the equation \(4 x ^ { 2 } + 20 x + 6 = 45\).
  3. Sketch the graph of \(y = 4 x ^ { 2 } + 20 x + 6\) showing the coordinates of the stationary point. You are not required to indicate where the curve crosses the \(x\) - and \(y\)-axes.
CAIE P1 2022 November Q7
7
  1. Prove the identity \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } \equiv \frac { \tan ^ { 2 } \theta + 1 } { \tan ^ { 2 } \theta - 1 }\).
  2. Hence find the exact solutions of the equation \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } = 2\) for \(0 \leqslant \theta \leqslant \pi\).
CAIE P1 2022 November Q8
8 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 3 x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(( 3,5 )\).
  1. Find the equation of the curve.
  2. Find the \(x\)-coordinate of the stationary point.
  3. State the set of values of \(x\) for which \(y\) increases as \(x\) increases.
CAIE P1 2022 November Q9
9 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = x + \frac { 1 } { x } \quad \text { for } x > 0
& \mathrm {~g} ( x ) = a x + 1 \quad \text { for } x \in \mathbb { R } \end{aligned}$$ where \(a\) is a constant.
  1. Find an expression for \(\operatorname { gf } ( x )\).
  2. Given that \(\operatorname { gf } ( 2 ) = 11\), find the value of \(a\).
  3. Given that the graph of \(y = \mathrm { f } ( x )\) has a minimum point when \(x = 1\), explain whether or not f has an inverse.
    It is given instead that \(a = 5\).
  4. Find and simplify an expression for \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x )\).
  5. Explain why the composite function fg cannot be formed.
    \includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-16_1143_1008_267_566} The diagram shows a cross-section RASB of the body of an aircraft. The cross-section consists of a sector \(O A R B\) of a circle of radius 2.5 m , with centre \(O\), a sector \(P A S B\) of another circle of radius 2.24 m with centre \(P\) and a quadrilateral \(O A P B\). Angle \(A O B = \frac { 2 } { 3 } \pi\) and angle \(A P B = \frac { 5 } { 6 } \pi\).
CAIE P1 2022 November Q11
11
  1. Find the coordinates of the minimum point of the curve \(y = \frac { 9 } { 4 } x ^ { 2 } - 12 x + 18\).
    \includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-18_675_901_1270_612} The diagram shows the curves with equations \(y = \frac { 9 } { 4 } x ^ { 2 } - 12 x + 18\) and \(y = 18 - \frac { 3 } { 8 } x ^ { \frac { 5 } { 2 } }\). The curves intersect at the points \(( 0,18 )\) and \(( 4,6 )\).
  2. Find the area of the shaded region.
  3. A point \(P\) is moving along the curve \(y = 18 - \frac { 3 } { 8 } x ^ { \frac { 5 } { 2 } }\) in such a way that the \(x\)-coordinate of \(P\) is increasing at a constant rate of 2 units per second. Find the rate at which the \(y\)-coordinate of \(P\) is changing when \(x = 4\).
    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 November Q1
1 Solve the equation \(8 \sin ^ { 2 } \theta + 6 \cos \theta + 1 = 0\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2022 November Q2
2 The function f is defined by \(\mathrm { f } ( x ) = - 2 x ^ { 2 } - 8 x - 13\) for \(x < - 3\).
  1. Express \(\mathrm { f } ( x )\) in the form \(- 2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers.
  2. Find the range of f.
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
CAIE P1 2022 November Q3
3
  1. Find the first three terms in ascending powers of \(x\) of the expansion of \(( 1 + 2 x ) ^ { 5 }\).
  2. Find the first three terms in ascending powers of \(x\) of the expansion of \(( 1 - 3 x ) ^ { 4 }\).
  3. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ^ { 5 } ( 1 - 3 x ) ^ { 4 }\).