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CAIE P2 2022 November Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{68f4b2dc-a05d-4061-aaf0-de15cfe186a9-10_657_792_269_664} The diagram shows the curve with parametric equations $$x = 3 \cos 2 \theta , \quad y = 4 \sin \theta ,$$ for \(\pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\). Points \(P\) and \(Q\) lie on the curve. The gradient of the curve at \(P\) is 2 . The straight line \(3 x + y = 0\) meets the curve at \(Q\).
  1. Find the value of \(\theta\) at \(P\), giving your answer correct to 3 significant figures.
  2. Find the gradient of the curve at \(Q\), giving your answer correct to 3 significant figures.
    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 2022 November Q1
4 marks Moderate -0.8
1 Solve the inequality \(| 2 x - 5 | > x\).
CAIE P2 2022 November Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{1cd04df5-3fe3-4573-b880-d49262afd16a-10_456_598_269_762} The diagram shows the curve with equation \(y = \frac { 2 \ln x } { 3 x + 1 }\). The curve crosses the \(x\)-axis at the point \(A\) and has a maximum point \(B\). The shaded region is bounded by the curve and the lines \(x = 3\) and \(y = 0\).
  1. Find the gradient of the curve at \(A\).
  2. Show by calculation that the \(x\)-coordinate of \(B\) lies between 3.0 and 3.1.
  3. Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 decimal places.
CAIE P2 2023 November Q1
3 marks Moderate -0.8
1 It is given that \(\theta\) is an acute angle in degrees such that \(\sin \theta = \frac { 2 } { 3 }\).
Find the exact value of \(\sin \left( \theta + 60 ^ { \circ } \right)\).
CAIE P2 2023 November Q2
5 marks Standard +0.3
2 A curve has equation \(y = 3 \tan \frac { 1 } { 2 } x \cos 2 x\).
Find the gradient of the curve at the point for which \(x = \frac { 1 } { 3 } \pi\).
CAIE P2 2023 November Q3
6 marks Moderate -0.8
3
  1. Find \(\int _ { 4 } ^ { 10 } \frac { 4 } { 2 x - 5 } \mathrm {~d} x\), giving your answer in the form \(\ln a\), where \(a\) is an integer.
  2. Find the exact value of \(\int _ { 4 } ^ { 10 } \mathrm { e } ^ { 2 x - 5 } \mathrm {~d} x\).
CAIE P2 2023 November Q4
7 marks Standard +0.8
4
  1. Sketch, on the same diagram, the graphs of \(y = | 3 x - 5 |\) and \(y = 2 x + 7\).
  2. Solve the equation \(| 3 x - 5 | = 2 x + 7\).
  3. Hence solve the equation \(\left| 3 ^ { y + 1 } - 5 \right| = 2 \times 3 ^ { y } + 7\), giving your answer correct to 3 significant figures.
CAIE P2 2023 November Q5
9 marks Moderate -0.3
5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x - 20$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is - 11 when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).
  1. Find the values of \(a\) and \(b\).
  2. Hence factorise \(\mathrm { p } ( x )\), and determine the exact roots of the equation \(\mathrm { p } ( 3 x ) = 0\).
CAIE P2 2023 November Q6
9 marks
6
  1. Show that \(\operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) \equiv 4 + 6 \cos \theta - 4 \cos ^ { 2 } \theta\).
  2. Solve the equation $$\operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) + 3 = 0$$ for \(- \pi < \theta < 0\).
  3. Find \(\int \operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) \mathrm { d } \theta\).
CAIE P2 2023 November Q7
11 marks Challenging +1.2
7 The curve with equation \(\mathrm { e } ^ { 2 x } - 18 x + y ^ { 3 } + y = 11\) has a stationary point at \(( p , q )\).
  1. Find the exact value of \(p\).
  2. Show that \(q = \sqrt [ 3 ] { 2 + 18 \ln 3 - q }\).
  3. Show by calculation that the value of \(q\) lies between 2.5 and 3.0.
  4. Use an iterative formula, based on the equation in (b), to find the value of \(q\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
    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 November Q1
2 marks Easy -1.2
1 When the polynomial $$a x ^ { 3 } + 4 a x ^ { 2 } - 7 x - 5$$ is divided by \(( x + 2 )\), the remainder is 33 .
Find the value of the constant \(a\).
CAIE P2 2023 November Q2
5 marks Standard +0.3
2 Solve the equation \(\sec \theta \cos \left( \theta - 60 ^ { \circ } \right) = 4\) for \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2023 November Q3
5 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-04_652_392_274_872} The diagram shows the curve with equation \(y = 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x }\). The points on the curve with \(x\)-coordinates 0 and 2 are denoted by \(A\) and \(B\) respectively. The shaded region is enclosed by the curve, the line through \(A\) parallel to the \(x\)-axis and the line through \(B\) parallel to the \(y\)-axis.
  1. Find the exact gradient of the curve at \(B\).
  2. Find the exact area of the shaded region.
CAIE P2 2023 November Q4
9 marks Moderate -0.3
4
  1. Sketch, on the same diagram, the graphs of \(y = | 3 - x |\) and \(y = 9 - 2 x\).
  2. Solve the inequality \(| 3 - x | > 9 - 2 x\).
  3. Use logarithms to solve the inequality \(2 ^ { 3 x - 10 } < 500\). Give your answer in the form \(x < a\), where the value of \(a\) is given correct to 3 significant figures.
  4. List the integers that satisfy both of the inequalities \(| 3 - x | > 9 - 2 x\) and \(2 ^ { 3 x - 10 } < 500\).
CAIE P2 2023 November Q5
8 marks Moderate -0.3
5
  1. Find the quotient when \(6 x ^ { 3 } - 5 x ^ { 2 } - 24 x - 4\) is divided by ( \(2 x + 1\) ), and show that the remainder is 6 .
  2. Hence find $$\int _ { 2 } ^ { 7 } \frac { 6 x ^ { 3 } - 5 x ^ { 2 } - 24 x - 4 } { 2 x + 1 } d x$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are integers.
CAIE P2 2023 November Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-10_803_394_269_863} The diagram shows the curve with parametric equations $$x = 3 \ln ( 2 t - 3 ) , \quad y = 4 t \ln t$$ The curve crosses the \(y\)-axis at the point \(A\). At the point \(B\), the gradient of the curve is 12 .
  1. Find the exact gradient of the curve at \(A\).
  2. Show that the value of the parameter \(t\) at \(B\) satisfies the equation $$t = \frac { 9 } { 1 + \ln t } + \frac { 3 } { 2 }$$
  3. Use an iterative formula, based on the equation in (b), to find the value of \(t\) at \(B\), giving your answer correct to 3 significant figures. Use an initial value of 5 and give the result of each iteration to 5 significant figures.
CAIE P2 2023 November Q7
11 marks Standard +0.3
7
  1. Prove that \(\sin 2 x ( \cot x + 3 \tan x ) \equiv 4 - 2 \cos 2 x\).
  2. Hence find the exact value of \(\cot \frac { 1 } { 12 } \pi + 3 \tan \frac { 1 } { 12 } \pi\).
  3. \includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-13_796_789_278_708} The diagram shows the curve with equation \(y = 4 - 2 \cos 2 x\), for \(0 < x < 2 \pi\). At the point \(A\), the gradient of the curve is 4 . The point \(B\) is a minimum point. The \(x\)-coordinates of \(A\) and \(B\) are \(a\) and \(b\) respectively. Show that \(\int _ { a } ^ { b } ( 4 - 2 \cos 2 x ) \mathrm { d } x = 3 \pi + 1\).
    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 2024 November Q1
5 marks Moderate -0.3
1 The variables \(x\) and \(y\) satisfy the equation \(a ^ { 2 y } = \mathrm { e } ^ { 3 x + k }\), where \(a\) and \(k\) are constants.
The graph of \(y\) against \(x\) is a straight line.
  1. Use logarithms to show that the gradient of the straight line is \(\frac { 3 } { 2 \ln a }\).
  2. Given that the straight line passes through the points \(( 0.4,0.95 )\) and \(( 3.3,3.80 )\), find the values of \(a\) and \(k\). \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-03_2723_33_99_21}
CAIE P2 2024 November Q2
4 marks Standard +0.3
2 Solve the inequality \(| x - 7 | > 4 x + 3\).
CAIE P2 2024 November Q3
7 marks Moderate -0.3
3 The function f is defined by \(\mathrm { f } ( x ) = \tan ^ { 2 } \left( \frac { 1 } { 2 } x \right)\) for \(0 \leqslant x < \pi\).
  1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( \frac { 2 } { 3 } \pi \right)\). \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-05_2726_33_97_22}
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } ( \mathrm { f } ( x ) + \sin x ) \mathrm { d } x\).
CAIE P2 2024 November Q4
8 marks Standard +0.3
4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - a x ^ { 2 } - 15 x + 18$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. Hence factorise \(\mathrm { p } ( x )\) completely.
  3. Solve the equation \(\mathrm { p } \left( \operatorname { cosec } ^ { 2 } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P2 2024 November Q5
8 marks Standard +0.3
5 It is given that \(\int _ { a } ^ { a ^ { 3 } } \frac { 10 } { 2 x + 1 } \mathrm {~d} x = 7\), where \(a\) is a constant greater than 1 .
  1. Show that \(a = \sqrt [ 3 ] { 0.5 \mathrm { e } ^ { 1.4 } ( 2 a + 1 ) - 0.5 }\). \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-08_2718_35_107_2011} \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-09_2725_35_99_20}
  2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 2 and give the result of each iteration to 5 significant figures.
CAIE P2 2024 November Q6
7 marks Standard +0.3
6 A curve has parametric equations $$x = \frac { \mathrm { e } ^ { 2 t } - 2 } { \mathrm { e } ^ { 2 t } + 1 } , \quad y = \mathrm { e } ^ { 3 t } + 1$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-10_2718_42_107_2007} \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-11_2725_35_99_20}
  2. Find the exact gradient of the curve at the point where the curve crosses the \(y\)-axis.
CAIE P2 2024 November Q7
11 marks Standard +0.3
7
  1. Prove that \(\cos \left( \theta + 30 ^ { \circ } \right) \cos \left( \theta + 60 ^ { \circ } \right) \equiv \frac { 1 } { 4 } \sqrt { 3 } - \frac { 1 } { 2 } \sin 2 \theta\).
  2. Solve the equation \(5 \cos \left( 2 \alpha + 30 ^ { \circ } \right) \cos \left( 2 \alpha + 60 ^ { \circ } \right) = 1\) for \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  3. Show that the exact value of \(\cos 20 ^ { \circ } \cos 50 ^ { \circ } + \cos 40 ^ { \circ } \cos 70 ^ { \circ }\) is \(\frac { 1 } { 2 } \sqrt { 3 }\).
    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]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-14_2714_38_109_2010}
CAIE P2 2024 November Q1
3 marks Moderate -0.8
1 Use logarithms to show that the equation \(5 ^ { 8 y } = 6 ^ { 7 x }\) can be expressed in the form \(y = k x\). Give the value of the constant \(k\) correct to 3 significant figures.