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CAIE P2 2023 June Q2
2 A curve has equation \(y = \frac { 2 + 3 \ln x } { 1 + 2 x }\).
Find the equation of the tangent to the curve at the point \(\left( 1 , \frac { 2 } { 3 } \right)\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
CAIE P2 2023 June Q3
3 marks
3 It is given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { 2 x } - 1 \right) \mathrm { d } x = 12\), where \(a\) is a positive constant.
  1. Show that \(a = \frac { 1 } { 2 } \ln \left( 9 + \frac { 2 } { 3 } a \right)\).
  2. Use an iterative formula, based on the equation in (a), to find the value of \(a\) correct to 4 significant figures. Use an initial value of 1 and give the result of each iteration to 6 significant figures. [3]
CAIE P2 2023 June Q4
4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + k x - 30$$ where \(k\) is a constant. It is given that \(( x - 3 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(k\).
  2. Hence find the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x - 3\) ) and factorise \(\mathrm { p } ( x )\) completely.
  3. It is given that \(a\) is one of the roots of the equation \(\mathrm { p } ( x ) = 0\). Given also that the equation \(| 4 y - 5 | = a\) is satisfied by two real values of \(y\), find these two values of \(y\).
CAIE P2 2023 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{3966e088-0a2f-434a-94fc-40765cd157a7-06_376_848_269_644} The diagram shows the curve with parametric equations $$x = 4 \mathrm { e } ^ { 2 t } , \quad y = 5 \mathrm { e } ^ { - t } \cos 2 t$$ for \(- \frac { 1 } { 4 } \pi \leqslant t \leqslant \frac { 1 } { 4 } \pi\). The curve has a maximum point \(M\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the coordinates of \(M\), giving each coordinate correct to 3 significant figures.
CAIE P2 2023 June Q6
6 Show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \left( 4 \cos ^ { 2 } 2 x + \frac { 1 } { \cos ^ { 2 } x } \right) \mathrm { d } x = \frac { 3 } { 4 } \sqrt { 3 } + \frac { 1 } { 6 } \pi - 1\).
CAIE P2 2023 June Q7
7
  1. Express \(7 \cos \theta + 24 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Solve the equation \(7 \cos \theta + 24 \sin \theta = 18\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
  3. As \(\beta\) varies, the greatest possible value of $$\frac { 150 } { 7 \cos \frac { 1 } { 2 } \beta + 24 \sin \frac { 1 } { 2 } \beta + 50 }$$ is denoted by \(V\).
    Find the value of \(V\) and determine the smallest positive value of \(\beta\) (in degrees) for which the value of \(V\) occurs.
    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 P2 2023 June Q1
1 Solve the equation $$\sec ^ { 2 } \theta + 5 \tan ^ { 2 } \theta = 9 + 17 \sec \theta$$ for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P2 2023 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-03_515_598_260_762} The variables \(x\) and \(y\) satisfy the equation \(y = A \mathrm { e } ^ { ( A - B ) x }\), where \(A\) and \(B\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0.4,3.6 )\) and \(( 2.9,14.1 )\), as shown in the diagram. Find the values of \(A\) and \(B\) correct to 3 significant figures.
CAIE P2 2023 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-04_458_892_269_614} The diagram shows part of the curve \(y = \frac { 6 } { 2 x + 3 }\). The shaded region is bounded by the curve and the lines \(x = 6\) and \(y = 2\). Find the exact area of the shaded region, giving your answer in the form \(a - \ln b\), where \(a\) and \(b\) are integers.
CAIE P2 2023 June Q4
4

  1. \includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-05_753_944_278_630} The diagram shows the graph of \(y = 3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x }\).
    On the diagram, sketch the graph of \(y = | 5 x - 4 |\), and show that the equation \(3 - e ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) has exactly two real roots. It is given that the two roots of \(3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
  2. Show by calculation that \(\alpha\) lies between 0.36 and 0.37 .
  3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 5 } \left( 7 - \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } \right)\) to find \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2023 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-06_526_947_276_591} The diagram shows the curve with equation \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } \left( x ^ { 2 } - 5 x + 4 \right)\). The curve crosses the \(x\)-axis at the points \(A\) and \(B\), and has a maximum at the point \(C\).
  1. Find the exact gradient of the curve at \(B\).
  2. Find the exact coordinates of \(C\).
CAIE P2 2023 June Q6
6
  1. Show that \(4 \sin \left( \theta + \frac { 1 } { 3 } \pi \right) \cos \left( \theta - \frac { 1 } { 3 } \pi \right) \equiv \sqrt { 3 } + 2 \sin 2 \theta\).
  2. Find the exact value of \(4 \sin \frac { 17 } { 24 } \pi \cos \frac { 1 } { 24 } \pi\).
  3. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 8 } \pi } 4 \sin \left( 2 x + \frac { 1 } { 3 } \pi \right) \cos \left( 2 x - \frac { 1 } { 3 } \pi \right) \mathrm { d } x\).
CAIE P2 2023 June Q7
7 A curve has parametric equations $$x = \frac { 2 t + 3 } { t + 2 } , \quad y = t ^ { 2 } + a t + 1$$ where \(a\) is a constant. It is given that, at the point \(P\) on the curve, the gradient is 1 .
  1. Show that the value of \(t\) at \(P\) satisfies the equation $$2 t ^ { 3 } + ( a + 8 ) t ^ { 2 } + ( 4 a + 8 ) t + 4 a - 1 = 0$$
  2. It is given that \(( t + 1 )\) is a factor of $$2 t ^ { 3 } + ( a + 8 ) t ^ { 2 } + ( 4 a + 8 ) t + 4 a - 1$$ Find the value of \(a\).
  3. Hence show that \(P\) is the only point on the curve at which the gradient is 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 P2 2023 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{a1ea242a-c7f4-46b0-b4b8-bd13b3880557-03_515_598_260_762} The variables \(x\) and \(y\) satisfy the equation \(y = A \mathrm { e } ^ { ( A - B ) x }\), where \(A\) and \(B\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0.4,3.6 )\) and \(( 2.9,14.1 )\), as shown in the diagram. Find the values of \(A\) and \(B\) correct to 3 significant figures.
CAIE P2 2023 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{a1ea242a-c7f4-46b0-b4b8-bd13b3880557-04_458_892_269_614} The diagram shows part of the curve \(y = \frac { 6 } { 2 x + 3 }\). The shaded region is bounded by the curve and the lines \(x = 6\) and \(y = 2\). Find the exact area of the shaded region, giving your answer in the form \(a - \ln b\), where \(a\) and \(b\) are integers.
CAIE P2 2023 June Q4
4

  1. \includegraphics[max width=\textwidth, alt={}, center]{a1ea242a-c7f4-46b0-b4b8-bd13b3880557-05_753_944_278_630} The diagram shows the graph of \(y = 3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x }\).
    On the diagram, sketch the graph of \(y = | 5 x - 4 |\), and show that the equation \(3 - e ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) has exactly two real roots. It is given that the two roots of \(3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
  2. Show by calculation that \(\alpha\) lies between 0.36 and 0.37 .
  3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 5 } \left( 7 - \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } \right)\) to find \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2023 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{a1ea242a-c7f4-46b0-b4b8-bd13b3880557-06_526_947_276_591} The diagram shows the curve with equation \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } \left( x ^ { 2 } - 5 x + 4 \right)\). The curve crosses the \(x\)-axis at the points \(A\) and \(B\), and has a maximum at the point \(C\).
  1. Find the exact gradient of the curve at \(B\).
  2. Find the exact coordinates of \(C\).
CAIE P2 2024 June Q1
1 A curve has equation \(\mathrm { y } = 2 \tan \mathrm { x } - 5 \sin \mathrm { x }\) for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
Find the \(x\)-coordinate of the stationary point of the curve. Give your answer correct to 3 significant figures.
CAIE P2 2024 June Q2
2 A curve has equation \(x ^ { 2 } \ln y + y ^ { 2 } + 4 x = 9\).
Find the gradient of the curve at the point \(( 2,1 )\).
CAIE P2 2024 June Q3
3
  1. Sketch on the same diagram the graphs of \(y = | 3 x - 8 |\) and \(y = 5 - x\).
  2. Solve the inequality \(| 3 x - 8 | < 5 - x\).
  3. Hence determine the largest integer \(N\) satisfying the inequality \(\left| 3 e ^ { 0.1 N } - 8 \right| < 5 - e ^ { 0.1 N }\).
CAIE P2 2024 June Q4
4
  1. Show that \(3 \tan 2 \theta + \tan \left( \theta + 45 ^ { \circ } \right) \equiv \frac { \tan ^ { 2 } \theta + 8 \tan \theta + 1 } { 1 - \tan ^ { 2 } \theta }\).
  2. Hence solve the equation \(3 \tan 2 \theta + \tan \left( \theta + 45 ^ { \circ } \right) = 4\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2024 June Q5
5 A curve has equation \(\mathrm { y } = \frac { 1 + \mathrm { e } ^ { 2 \mathrm { x } } } { 1 + 3 \mathrm { x } }\). The curve has exactly one stationary point \(P\).
  1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) and hence show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 6 } + \frac { 1 } { 2 } \mathrm { e } ^ { - 2 x }\).
  2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 0.35 and 0.45 .
  3. Use an iterative formula based on the equation in part (a) to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
    \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-10_451_647_258_699} The diagram shows the curve with equation \(\mathrm { y } = \sqrt { \sin 2 \mathrm { x } + \sin ^ { 2 } 2 \mathrm { x } }\) for \(0 \leqslant x \leqslant \frac { 1 } { 6 } \pi\). The shaded region is bounded by the curve and the straight lines \(x = \frac { 1 } { 6 } \pi\) and \(y = 0\).
CAIE P2 2024 June Q7
7 The polynomial \(\mathrm { p } ( x )\) is defined by $$p ( x ) = 9 x ^ { 3 } + 6 x ^ { 2 } + 12 x + k$$ where \(k\) is a constant.
  1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 3 x + 2 )\) and show that the remainder is \(( k - 8 )\).
  2. It is given that \(\int _ { 1 } ^ { 6 } \frac { \mathrm { p } ( \mathrm { x } ) } { 3 \mathrm { x } + 2 } \mathrm { dx } = \mathrm { a } + \ln 64\), where \(a\) is an integer. Find the values of \(a\) and \(k\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P2 2024 June Q1
1 Solve the inequality \(| 5 x + 7 | > | 2 x - 3 |\).
CAIE P2 2024 June Q2
4 marks
2 Use logarithms to solve the equation \(6 ^ { 2 x - 1 } = 5 e ^ { 3 x + 2 }\). Give your answer correct to 4 significant figures. [4]
\includegraphics[max width=\textwidth, alt={}, center]{6ee58f43-831d-402c-9f9a-2b247b2f7ffc-04_778_486_276_769} The diagram shows the curve with equation \(\mathrm { y } = 8 \mathrm { e } ^ { - \mathrm { x } } - \mathrm { e } ^ { 2 \mathrm { x } }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\). The shaded region is bounded by the curve and the two axes.
  1. Find the gradient of the curve at \(A\).
  2. Show that the \(x\)-coordinate of \(B\) is \(\ln 2\) and hence find the area of the shaded region.