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CAIE P2 2020 March Q5
9 marks Standard +0.8
5
  1. Sketch, on the same diagram, the graphs of \(y = | x + 2 k |\) and \(y = | 2 x - 3 k |\), where \(k\) is a positive constant. Give, in terms of \(k\), the coordinates of the points where each graph meets the axes.
  2. Find, in terms of \(k\), the coordinates of each of the two points where the graphs intersect.
  3. Find, in terms of \(k\), the largest value of \(t\) satisfying the inequality $$\left| 2 ^ { t } + 2 k \right| \geqslant \left| 2 ^ { t + 1 } - 3 k \right| .$$
CAIE P2 2020 March Q6
9 marks Standard +0.3
6 A curve has equation \(y = x ^ { 3 } \mathrm { e } ^ { 0.2 x }\) where \(x \geqslant 0\). At the point \(P\) on the curve, the gradient of the curve is 15 .
  1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \sqrt { \frac { 75 \mathrm { e } ^ { - 0.2 x } } { 15 + x } }\).
  2. Use the equation in part (a) to show by calculation that the \(x\)-coordinate of \(P\) lies between 1.7 and 1.8.
  3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(P\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2020 March Q7
10 marks Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{78a9b100-c3bd-4054-b539-ec8304440063-10_551_641_260_751} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3 ,$$ where \(x\) is measured in radians. The curve crosses the \(x\)-axis at the point \(A\) and the shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\).
  1. Find the exact \(x\)-coordinate of \(A\).
  2. Find the exact gradient of the curve at \(A\).
  3. Find the exact area of the shaded region.
    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 2021 March Q1
5 marks Moderate -0.8
1
  1. Sketch, on the same diagram, the graphs of \(y = | 3 x - 5 |\) and \(y = x + 2\).
  2. Solve the equation \(| 3 x - 5 | = x + 2\).
CAIE P2 2021 March Q2
5 marks Standard +0.3
2 Solve the equation \(\sec ^ { 2 } \theta \cot \theta = 8\) for \(0 < \theta < \pi\).
CAIE P2 2021 March Q3
5 marks Moderate -0.3
3 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } \cos 4 t , \quad y = 3 \sin 2 t$$ Find the gradient of the curve at the point for which \(t = 0\).
CAIE P2 2021 March Q4
8 marks Standard +0.3
4
\includegraphics[max width=\textwidth, alt={}, center]{9cf008d5-c15f-4491-9e4d-4bd070f896d5-06_446_832_260_653} The diagram shows part of the curve with equation \(y = \frac { 5 x } { 4 x ^ { 3 } + 1 }\). The shaded region is bounded by the curve and the lines \(x = 1 , x = 3\) and \(y = 0\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the \(x\)-coordinate of the maximum point.
  2. Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 significant figures.
  3. State, with a reason, whether your answer to part (b) is an over-estimate or under-estimate of the exact area of the shaded region.
CAIE P2 2021 March Q5
8 marks Moderate -0.3
5
  1. Given that \(2 \ln ( x + 1 ) + \ln x = \ln ( x + 9 )\), show that \(x = \sqrt { \frac { 9 } { x + 2 } }\).
  2. It is given that the equation \(x = \sqrt { \frac { 9 } { x + 2 } }\) has a single root. Show by calculation that this root lies between 1.5 and 2.0.
  3. Use an iterative formula, based on the equation in part (b), to find the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2021 March Q6
10 marks Moderate -0.3
6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + a x + b$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 5 when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\).
  1. Find the values of \(a\) and \(b\).
  2. Hence find the exact root of the equation \(\mathrm { p } \left( \mathrm { e } ^ { 2 y } \right) = 0\).
CAIE P2 2021 March Q7
9 marks Standard +0.8
7
  1. Express \(5 \sqrt { 3 } \cos x + 5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
  2. As \(x\) varies, find the least possible value of $$4 + 5 \sqrt { 3 } \cos x + 5 \sin x$$ and determine the corresponding value of \(x\) where \(- \pi < x < \pi\).
  3. Find \(\int \frac { 1 } { ( 5 \sqrt { 3 } \cos 3 \theta + 5 \sin 3 \theta ) ^ { 2 } } d \theta\).
    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 March Q1
3 marks Moderate -0.8
1 Solve the equation \(| 5 x - 2 | = | 4 x + 9 |\).
CAIE P2 2022 March Q2
5 marks Moderate -0.8
2 A curve has equation \(y = 7 + 4 \ln ( 2 x + 5 )\).
Find the equation of the tangent to the curve at the point ( \(- 2,7\) ), giving your answer in the form \(y = m x + c\).
CAIE P2 2022 March Q3
5 marks Moderate -0.8
3 The variables \(x\) and \(y\) satisfy the equation \(y = 3 ^ { 2 a } a ^ { x }\), where \(a\) is a constant. The graph of \(\ln y\) against \(x\) is a straight line with gradient 0.239 .
  1. Find the value of \(a\) correct to 3 significant figures.
  2. Hence find the value of \(x\) when \(y = 36\). Give your answer correct to 3 significant figures.
CAIE P2 2022 March Q4
7 marks Standard +0.3
4
  1. Show that \(\sin 2 \theta \cot \theta - \cos 2 \theta \equiv 1\).
  2. Hence find the exact value of \(\sin \frac { 1 } { 6 } \pi \cot \frac { 1 } { 12 } \pi\).
  3. Find the smallest positive value of \(\theta\) (in radians) satisfying the equation $$\sin 2 \theta \cot \theta - 3 \cos 2 \theta = 1 .$$
CAIE P2 2022 March Q5
8 marks Standard +0.3
5
  1. Given that \(y = \tan ^ { 2 } x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \tan x + 2 \tan ^ { 3 } x\).
  2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \left( \tan x + \tan ^ { 2 } x + \tan ^ { 3 } x \right) \mathrm { d } x\).
CAIE P2 2022 March Q6
10 marks Standard +0.3
6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + 16 x ^ { 2 } + 9 x - 15$$
  1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 2 x + 3 )\), and show that the remainder is - 6 .
  2. Find \(\int \frac { \mathrm { p } ( x ) } { 2 x + 3 } \mathrm {~d} x\).
  3. Factorise \(\mathrm { p } ( x ) + 6\) completely and hence solve the equation $$p ( \operatorname { cosec } 2 \theta ) + 6 = 0$$ for \(0 ^ { \circ } < \theta < 135 ^ { \circ }\).
CAIE P2 2022 March Q7
12 marks Standard +0.8
7 A curve has equation \(\mathrm { e } ^ { 2 x } y - \mathrm { e } ^ { y } = 100\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \mathrm { e } ^ { 2 x } y } { \mathrm { e } ^ { y } - \mathrm { e } ^ { 2 x } }\).
  2. Show that the curve has no stationary points.
    It is required to find the \(x\)-coordinate of \(P\), the point on the curve at which the tangent is parallel to the \(y\)-axis.
  3. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \ln 10 - \frac { 1 } { 2 } \ln ( 2 x - 1 )$$
  4. Use an iterative formula, based on the equation in part (c), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Use an initial value of 2 and give the result of each iteration to 5 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 March Q1
4 marks Standard +0.3
1 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 2 \tan ^ { 2 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).
CAIE P2 2023 March Q2
5 marks Standard +0.8
2 Solve the equation \(\tan \left( \theta - 60 ^ { \circ } \right) = 3 \cot \theta\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P2 2023 March Q3
8 marks Moderate -0.3
3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - a x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\), and that the remainder is 35 when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\).
  1. Find the values of \(a\) and \(b\).
  2. Hence factorise \(\mathrm { p } ( x )\) and show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.
CAIE P2 2023 March Q4
7 marks Standard +0.3
4
  1. Sketch, on the same diagram, the graphs of \(y = | 2 x - 11 |\) and \(y = 3 x - 3\).
  2. Solve the inequality \(| 2 x - 11 | < 3 x - 3\).
  3. Find the smallest integer \(N\) satisfying the inequality \(| 2 \ln N - 11 | < 3 \ln N - 3\).
CAIE P2 2023 March Q5
8 marks Standard +0.3
5 It is given that \(\int _ { 1 } ^ { a } \left( \frac { 4 } { 1 + 2 x } + \frac { 3 } { x } \right) \mathrm { d } x = \ln 10\), where \(a\) is a constant greater than 1 .
  1. Show that \(a = \sqrt [ 3 ] { 90 ( 1 + 2 a ) ^ { - 2 } }\).
  2. Use an iterative formula, based on the equation in (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 1.7 and give the result of each iteration to 5 significant figures.
    \includegraphics[max width=\textwidth, alt={}, center]{ce0d5faa-9428-4afd-829d-7634c5bd150d-10_798_495_269_810} The diagram shows the curve with equation \(y = \frac { 4 \mathrm { e } ^ { 2 x } + 9 } { \mathrm { e } ^ { x } + 2 }\). The curve has a minimum point \(M\) and crosses the \(y\)-axis at the point \(P\).
  3. Find the exact value of the gradient of the curve at \(P\).
  4. Find the exact coordinates of \(M\).
CAIE P2 2023 March Q7
10 marks Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{ce0d5faa-9428-4afd-829d-7634c5bd150d-12_446_613_274_760} The diagram shows the curve with parametric equations $$x = k \tan t , \quad y = 3 \sin 2 t - 4 \sin t ,$$ for \(0 < t < \frac { 1 } { 2 } \pi\). It is given that \(k\) is a positive constant. The curve crosses the \(x\)-axis at the point \(P\).
  1. Find the value of \(\cos t\) at \(P\), giving your answer as an exact fraction.
  2. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(k\) and \(\cos t\).
  3. Given that the normal to the curve at \(P\) has gradient \(\frac { 9 } { 10 }\), find the value of \(k\), giving your answer as an exact fraction.
    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 March Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation \(3 ^ { 4 x + 3 } = 5 ^ { 2 x + 7 }\). Give your answer correct to 3 significant figures. [4]
CAIE P2 2024 March Q2
4 marks Standard +0.3
2
  1. Sketch the graph of \(y = | 3 x - 7 |\), stating the coordinates of the points where the graph meets the axes.
  2. Hence find the set of values of the constant \(k\) for which the equation \(| 3 \mathrm { x } - 7 | = \mathrm { k } ( \mathrm { x } - 4 )\) has exactly two real roots.