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CAIE P2 2023 June Q3
5 marks Standard +0.3
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
7 marks Standard +0.3
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
9 marks Standard +0.3
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
3 marks Standard +0.3
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
5 marks Standard +0.3
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
8 marks Standard +0.3
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
7 marks Standard +0.8
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
9 marks Standard +0.3
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
9 marks Standard +0.3
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
4 marks Standard +0.3
1 Solve the inequality \(| 5 x + 7 | > | 2 x - 3 |\).
CAIE P2 2024 June Q2
4 marks Moderate -0.3
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.
CAIE P2 2024 June Q4
7 marks Standard +0.3
4 A curve is defined by the parametric equations $$x = 4 \cos ^ { 2 } t , \quad y = \sqrt { 3 } \sin 2 t$$ for values of \(t\) such that \(0 < t < \frac { 1 } { 2 } \pi\).
Find the equation of the normal to the curve at the point for which \(t = \frac { 1 } { 6 } \pi\). Give your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\) where \(a , b\) and \(c\) are integers.
CAIE P2 2024 June Q5
8 marks Standard +0.3
5 The polynomial \(\mathrm { p } ( x )\) is defined by \(\mathrm { p } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } + 5 x + 4\).
  1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 3 x + 2 )\), and show that the remainder is 6 .
  2. Find the value of \(\int _ { 0 } ^ { 2 } \frac { \mathrm { p } ( x ) } { 3 x + 2 } \mathrm {~d} x\), giving your answer in the form \(\mathrm { a } + \operatorname { lnb }\) where \(a\) and \(b\) are integers.
    \includegraphics[max width=\textwidth, alt={}, center]{6ee58f43-831d-402c-9f9a-2b247b2f7ffc-10_414_693_276_687} The diagram shows the curve with equation \(\mathrm { y } = \frac { \ln ( 2 \mathrm { x } + 1 ) } { \mathrm { x } + 3 }\). The curve has a maximum point \(M\).
CAIE P2 2024 June Q7
10 marks Standard +0.3
7
  1. Prove that \(2 \sin \theta \operatorname { cosec } 2 \theta \equiv \sec \theta\).
  2. Solve the equation \(\tan ^ { 2 } \theta + 7 \sin \theta \operatorname { cosec } 2 \theta = 8\) for \(- \pi < \theta < \pi\).
  3. Find \(\int 8 \sin ^ { 2 } \frac { 1 } { 2 } x \operatorname { cosec } ^ { 2 } x d x\).
    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
4 marks Moderate -0.3
1 Solve the inequality \(| 5 x + 7 | > | 2 x - 3 |\).
\includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-02_67_1653_333_244}
\includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-02_2715_37_143_2010}
CAIE P2 2024 June Q2
4 marks Standard +0.3
2 Use logarithms to solve the equation \(6 ^ { 2 x - 1 } = 5 \mathrm { e } ^ { 3 x + 2 }\). Give your answer correct to 4 significant figures. [4]
CAIE P2 2024 June Q3
8 marks Moderate -0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-04_776_483_310_769} The diagram shows the curve with equation \(y = 8 \mathrm { e } ^ { - x } - \mathrm { e } ^ { 2 x }\). 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 \(A\).
    \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-04_2715_35_141_2011}
  2. Show that the \(x\)-coordinate of \(B\) is \(\ln 2\) and hence find the area of the shaded region.
CAIE P2 2024 June Q4
7 marks Standard +0.3
4 A curve is defined by the parametric equations $$x = 4 \cos ^ { 2 } t , \quad y = \sqrt { 3 } \sin 2 t ,$$ for values of \(t\) such that \(0 < t < \frac { 1 } { 2 } \pi\) .
Find the equation of the normal to the curve at the point for which \(t = \frac { 1 } { 6 } \pi\) .Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
\includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-06_2718_35_141_2012}
CAIE P2 2024 June Q5
8 marks Standard +0.3
5 The polynomial \(\mathrm { p } ( x )\) is defined by \(\mathrm { p } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } + 5 x + 4\).
  1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 3 x + 2 )\), and show that the remainder is 6 .
    \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-08_2713_33_146_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-09_2723_33_138_20}
  2. Find the value of \(\int _ { 0 } ^ { 2 } \frac { \mathrm { p } ( x ) } { 3 x + 2 } \mathrm {~d} x\) ,giving your answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers.
CAIE P2 2024 June Q6
9 marks Standard +0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-10_417_700_310_685} The diagram shows the curve with equation \(y = \frac { \ln ( 2 x + 1 ) } { x + 3 }\). The curve has a maximum point \(M\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { x + 3 } { \ln ( 2 x + 1 ) } - 0.5\).
  3. Show by calculation that the \(x\)-coordinate of \(M\) lies between 2.5 and 3.0 .
  4. Use an iterative formula based on the equation in part (b) to find the \(x\)-coordinate of \(M\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2024 June Q7
10 marks Standard +0.3
7
  1. Prove that \(2 \sin \theta \operatorname { cosec } 2 \theta \equiv \sec \theta\).
  2. Solve the equation \(\tan ^ { 2 } \theta + 7 \sin \theta \operatorname { cosec } 2 \theta = 8\) for \(- \pi < \theta < \pi\).
    \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-12_2725_37_136_2010}
  3. Find \(\int 8 \sin ^ { 2 } \frac { 1 } { 2 } x \operatorname { cosec } ^ { 2 } x \mathrm {~d} x\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
    \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-14_2715_35_143_2012}
CAIE P2 2020 March Q1
4 marks Standard +0.3
1 Solve the equation \(2 \sin \left( \theta + 30 ^ { \circ } \right) + 5 \cos \theta = 2 \sin \theta\) for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P2 2020 March Q2
6 marks Moderate -0.8
2
  1. Find the quotient when \(4 x ^ { 3 } + 17 x ^ { 2 } + 9 x\) is divided by \(x ^ { 2 } + 5 x + 6\), and show that the remainder is 18 .
  2. Hence solve the equation \(4 x ^ { 3 } + 17 x ^ { 2 } + 9 x - 18 = 0\).
CAIE P2 2020 March Q3
6 marks Standard +0.3
3 It is given that \(\int _ { a } ^ { 3 a } \frac { 2 } { 2 x - 5 } \mathrm {~d} x = \ln \frac { 7 } { 2 }\).
Find the value of the positive constant \(a\).
CAIE P2 2020 March Q4
6 marks Standard +0.3
4 A curve has equation $$3 x ^ { 2 } - y ^ { 2 } - 4 \ln ( 2 y + 3 ) = 26$$ Find the equation of the tangent to the curve at the point \(( 3 , - 1 )\).