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CAIE P2 2020 June Q8
10 marks Standard +0.3
8
  1. Show that \(3 \sin 2 \theta \cot \theta \equiv 6 \cos ^ { 2 } \theta\).
  2. Solve the equation \(3 \sin 2 \theta \cot \theta = 5\) for \(0 < \theta < \pi\).
  3. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } 3 \sin x \cot \frac { 1 } { 2 } x \mathrm {~d} x\).
    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 2020 June Q4
5 marks Moderate -0.5
4
\includegraphics[max width=\textwidth, alt={}, center]{ad833f8c-80de-42ae-a186-93091a6fdf1e-06_659_828_262_660} The variables \(x\) and \(y\) satisfy the equation \(y = A x ^ { - 2 p }\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( - 0.68,3.02 )\) and \(( 1.07 , - 1.53 )\), as shown in the diagram. Find the values of \(A\) and \(p\).
CAIE P2 2021 June Q1
4 marks Standard +0.3
1 Solve the inequality \(| 3 x - 7 | < | 4 x + 5 |\).
CAIE P2 2021 June Q2
6 marks Moderate -0.3
2 By first expanding \(\sin \left( \theta + 30 ^ { \circ } \right)\), solve the equation \(\sin \left( \theta + 30 ^ { \circ } \right) \operatorname { cosec } \theta = 2\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P2 2021 June Q3
6 marks Standard +0.3
3
  1. Show that \(( \sec x + \cos x ) ^ { 2 }\) can be expressed as \(\sec ^ { 2 } x + a + b \cos 2 x\), where \(a\) and \(b\) are constants to be determined.
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \sec x + \cos x ) ^ { 2 } \mathrm {~d} x\).
CAIE P2 2021 June Q4
8 marks Standard +0.3
4 A curve has parametric equations $$x = \ln ( 2 t + 6 ) - \ln t , \quad y = t \ln t$$
  1. Find the value of \(t\) at the point \(P\) on the curve for which \(x = \ln 4\).
  2. Find the exact gradient of the curve at \(P\).
CAIE P2 2021 June Q5
8 marks Standard +0.3
5
\includegraphics[max width=\textwidth, alt={}, center]{2d6fc4c5-70ec-4cd8-9b48-59d5ce0e39b7-08_575_618_262_762} The diagram shows the curve with equation \(y = \frac { 3 x + 2 } { \ln x }\). The curve has a minimum point \(M\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { 3 x + 2 } { 3 \ln x }\). [3]
  2. Use the equation in part (a) to show by calculation that the \(x\)-coordinate of \(M\) lies between 3 and 4.
  3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(M\) correct to 5 significant figures. Give the result of each iteration to 7 significant figures.
CAIE P2 2021 June Q6
9 marks Moderate -0.3
6
  1. Use the trapezium rule with three intervals to find an approximation to \(\int _ { 1 } ^ { 4 } \frac { 6 } { 1 + \sqrt { x } } \mathrm {~d} x\). Give your answer correct to 5 significant figures.
  2. Find the exact value of \(\int _ { 1 } ^ { 4 } 2 \mathrm { e } ^ { \frac { 1 } { 2 } x - 2 } \mathrm {~d} x\).

  3. \includegraphics[max width=\textwidth, alt={}, center]{2d6fc4c5-70ec-4cd8-9b48-59d5ce0e39b7-11_556_805_262_705} The diagram shows the curves \(y = \frac { 6 } { 1 + \sqrt { x } }\) and \(y = 2 \mathrm { e } ^ { \frac { 1 } { 2 } x - 2 }\) which meet at a point with \(x\)-coordinate 4. The shaded region is bounded by the two curves and the line \(x = 1\). Use your answers to parts (a) and (b) to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures.
  4. State, with a reason, whether your answer to part (c) is an over-estimate or under-estimate of the exact area of the shaded region.
CAIE P2 2021 June Q7
9 marks Standard +0.3
7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } - 19 x - a$$ where \(a\) is a constant. It is given that \(( x - 3 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.
  3. Hence find the exact values of \(y\) that satisfy the equation \(\mathrm { p } \left( \mathrm { e } ^ { y } + \mathrm { e } ^ { - y } \right) = 0\).
    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 June Q1
5 marks Moderate -0.3
1
  1. Solve the equation \(\ln ( 2 + x ) - \ln x = 2 \ln 3\).
  2. Hence solve the equation \(\ln ( 2 + \cot y ) - \ln ( \cot y ) = 2 \ln 3\) for \(0 < y < \frac { 1 } { 2 } \pi\). Give your answer correct to 4 significant figures.
CAIE P2 2021 June Q2
5 marks Moderate -0.3
2 The solutions of the equation \(5 | x | = 5 - 2 x\) are \(x = a\) and \(x = b\), where \(a < b\).
Find the value of \(| 3 a - 1 | + | 7 b - 1 |\).
CAIE P2 2021 June Q3
6 marks Standard +0.3
3 Solve the equation \(\sin \left( 2 \theta + 30 ^ { \circ } \right) = 5 \cos \left( 2 \theta + 60 ^ { \circ } \right)\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2021 June Q4
8 marks Moderate -0.8
4
  1. Find the exact value of \(\int _ { 0 } ^ { 2 } 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x\).
  2. Find \(\int \left( \tan ^ { 2 } x + 4 \sin ^ { 2 } 2 x \right) \mathrm { d } x\).
CAIE P2 2021 June Q5
7 marks Standard +0.3
5
  1. Find the quotient when \(x ^ { 4 } - 32 x + 55\) is divided by \(( x - 2 ) ^ { 2 }\) and show that the remainder is 7 .
  2. Factorise \(x ^ { 4 } - 32 x + 48\).
  3. Hence solve the equation \(\mathrm { e } ^ { - 12 y } - 32 \mathrm { e } ^ { - 3 y } + 48 = 0\), giving your answer in an exact form.
CAIE P2 2021 June Q6
8 marks Standard +0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{388d7076-636c-417d-84cb-e6e2a3e9a6a0-08_451_1086_260_525} The diagram shows the curve with equation $$y = ( \ln x ) ^ { 2 } - 2 \ln x$$ The curve crosses the \(x\)-axis at the points \(A\) and \(B\), and has a minimum point \(M\).
  1. Find the exact value of the gradient of the curve at each of the points \(A\) and \(B\).
  2. Find the exact \(x\)-coordinate of \(M\).
CAIE P2 2021 June Q7
11 marks Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{388d7076-636c-417d-84cb-e6e2a3e9a6a0-10_465_785_260_680} The diagram shows the curve with parametric equations $$x = 4 t + \mathrm { e } ^ { 2 t } , \quad y = 6 t \sin 2 t$$ for \(0 \leqslant t \leqslant 1\). The point \(P\) on the curve has parameter \(p\) and \(y\)-coordinate 3 .
  1. Show that \(p = \frac { 1 } { 2 \sin 2 p }\).
  2. Show by calculation that the value of \(p\) lies between 0.5 and 0.6 .
  3. Use an iterative formula, based on the equation in part (a), to find the value of \(p\) correct to 3 significant figures. Use an initial value of 0.55 and give the result of each iteration to 5 significant figures.
  4. Find the gradient of the curve at \(P\).
    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 June Q6
8 marks Moderate -0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{61df367d-741f-4906-8ab9-2f32e8711aa6-08_451_1086_260_525} The diagram shows the curve with equation $$y = ( \ln x ) ^ { 2 } - 2 \ln x$$ The curve crosses the \(x\)-axis at the points \(A\) and \(B\), and has a minimum point \(M\).
  1. Find the exact value of the gradient of the curve at each of the points \(A\) and \(B\).
  2. Find the exact \(x\)-coordinate of \(M\).
CAIE P2 2021 June Q7
11 marks Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{61df367d-741f-4906-8ab9-2f32e8711aa6-10_465_785_260_680} The diagram shows the curve with parametric equations $$x = 4 t + \mathrm { e } ^ { 2 t } , \quad y = 6 t \sin 2 t$$ for \(0 \leqslant t \leqslant 1\). The point \(P\) on the curve has parameter \(p\) and \(y\)-coordinate 3 .
  1. Show that \(p = \frac { 1 } { 2 \sin 2 p }\).
  2. Show by calculation that the value of \(p\) lies between 0.5 and 0.6 .
  3. Use an iterative formula, based on the equation in part (a), to find the value of \(p\) correct to 3 significant figures. Use an initial value of 0.55 and give the result of each iteration to 5 significant figures.
  4. Find the gradient of the curve at \(P\).
    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 June Q1
4 marks Moderate -0.3
1
\includegraphics[max width=\textwidth, alt={}, center]{ed12a4fb-e3bf-4d00-ad09-9ba5be941dd5-02_654_396_258_872} The variables \(x\) and \(y\) satisfy the equation \(y = 4 ^ { 2 x - a }\), where \(a\) is an integer. As shown in the diagram, the graph of \(\ln y\) against \(x\) is a straight line passing through the point \(( 0 , - 20.8 )\), where the second coordinate is given correct to 3 significant figures.
  1. Show that the gradient of the straight line is \(\ln 16\).
  2. Determine the value of \(a\).
CAIE P2 2022 June Q2
6 marks Standard +0.3
2
  1. Express the equation \(7 \tan \theta + 4 \cot \theta - 13 \sec \theta = 0\) in terms of \(\sin \theta\) only.
  2. Hence solve the equation \(7 \tan \theta + 4 \cot \theta - 13 \sec \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P2 2022 June Q3
7 marks Standard +0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{ed12a4fb-e3bf-4d00-ad09-9ba5be941dd5-04_531_739_258_703} The diagram shows the curve with equation \(y = 3 \sin x - 3 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\). The curve meets the \(x\)-axis at the origin and at the points with \(x\)-coordinates \(a\) and \(\pi\).
  1. Find the exact value of \(a\).
  2. Find the area of the shaded region.
CAIE P2 2022 June Q4
7 marks Standard +0.3
4 A curve has equation \(x ^ { 2 } y + 2 y ^ { 3 } = 48\).
Find the equation of the normal to the curve at the point ( 4,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
CAIE P2 2022 June Q5
9 marks Standard +0.3
5
  1. By sketching the graphs of $$y = | 5 - 2 x | \quad \text { and } \quad y = 3 \ln x$$ on the same diagram, show that the equation \(| 5 - 2 x | = 3 \ln x\) has exactly two roots.
  2. Show that the value of the larger root satisfies the equation \(x = 2.5 + 1.5 \ln x\).
  3. Show by calculation that the value of the larger root lies between 4.5 and 5.0.
  4. Use an iterative formula, based on the equation in part (b), to find the value of the larger root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2022 June Q6
8 marks Standard +0.8
6 A curve has equation \(y = \frac { 9 \mathrm { e } ^ { 2 x } + 16 } { \mathrm { e } ^ { x } - 1 }\).
  1. Show that the \(x\)-coordinate of any stationary point on the curve satisfies the equation $$\mathrm { e } ^ { x } \left( 3 \mathrm { e } ^ { x } - 8 \right) \left( 3 \mathrm { e } ^ { x } + 2 \right) = 0$$
  2. Hence show that the curve has only one stationary point and find its exact coordinates.
CAIE P2 2022 June Q7
9 marks Standard +0.3
7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + 5 x ^ { 2 } + a x + 2 a$$ where \(a\) is an integer.
  1. Find, in terms of \(x\) and \(a\), the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x + 2\) ), and show that the remainder is 4 .
  2. It is given that \(\int _ { - 1 } ^ { 1 } \frac { \mathrm { p } ( x ) } { x + 2 } \mathrm {~d} x = \frac { 22 } { 3 } + \ln b\), where \(b\) is an integer. Find the values of \(a\) and \(b\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.