Questions — CAIE (7646 questions)

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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.
CAIE P2 2024 November Q2
6 marks Moderate -0.3
2 Let \(\mathrm { f } ( x ) = 4 \sin ^ { 2 } 3 x\).
  1. Find the value of \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 4 } \pi \right)\).
  2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\). \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-05_2723_35_101_20}
CAIE P2 2024 November Q3
6 marks Standard +0.3
3 A curve has equation \(6 \mathrm { e } ^ { - x } y ^ { 2 } + \mathrm { e } ^ { 2 x } - 12 y + 7 = 0\).
Find the gradient of the curve at the point \(( \ln 3,2 )\).
CAIE P2 2024 November Q5
10 marks Standard +0.3
5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - a x + 8$$ where \(a\) and \(b\) are constants.It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) ,and that the remainder is 24 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) .
  1. Find the values of \(a\) and \(b\) . \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-09_2723_35_101_20}
  2. Factorise \(\mathrm { p } ( x )\) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.
  3. Solve the equation \(\mathrm { p } \left( \frac { 1 } { 2 } \operatorname { cosec } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_499_696_264_680} The diagram shows the curves with equations \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and \(y = \frac { 27 } { 2 x + 5 }\) for \(x \geqslant 0\).
    The curves meet at the point \(( 2,3 )\).
    Region \(A\) is bounded by the curve \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and the straight lines \(x = 0 , x = 2\) and \(y = 0\).
    Region \(B\) is bounded by the two curves and the straight line \(x = 0\).
CAIE P2 2024 November Q7
9 marks Standard +0.3
7
  1. Express \(4 \sin \theta \sin \left( \theta + 60 ^ { \circ } \right)\) in the form $$a + R \sin ( 2 \theta - \alpha ) ,$$ where \(a\) and \(R\) are positive integers and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-13_2723_33_99_21}
  2. Hence find the smallest positive value of \(\theta\) satisfying the equation $$\frac { 1 } { 5 } + 4 \sin \theta \sin \left( \theta + 60 ^ { \circ } \right) = 0 .$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-14_2714_38_109_2010}
CAIE P2 2020 Specimen Q1
4 marks Moderate -0.8
1
  1. The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - a x - 12\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\). Find the value of \(a\).
  2. When \(a\) has this value, find the remainder when \(\mathrm { p } ( x )\) is divided by \(( x + 3 )\).
CAIE P2 2020 Specimen Q2
4 marks Standard +0.3
2 Solve the equation \(3 \sin 2 \theta \tan \theta = 2\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2020 Specimen Q3
6 marks Moderate -0.8
3 It is given that \(a\) is a positive constant.
    1. Sketch on a single diagram the graphs of \(y = | 2 x - 3 a |\) and \(y = | 2 x + 4 a |\).
    2. State the coordinates of each of the points where each graph meets an axis.
  2. Solve the inequality \(| 2 x - 3 a | < | 2 x + 4 a |\).
CAIE P2 2020 Specimen Q4
8 marks Moderate -0.3
4
  1. Solve the equation \(5 ^ { 2 x } + 5 ^ { x } = 12\), giving your answer correct to 3 significant figures.
  2. It is given that \(\ln ( y + 5 ) - \ln y = 2 \ln x\). Express \(y\) in terms of \(x\), in a form not involving logarithms.
CAIE P2 2020 Specimen Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{0af2714b-d3eb-4112-a869-eda5cf266cd8-08_410_977_274_543} The diagram shows the curve \(y = \frac { \sin 2 x } { x + 2 }\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The \(x\)-coordinate of the maximum point \(M\) is denoted by \(\alpha\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(\alpha\) satisfies the equation \(\tan 2 x = 2 x + 4\).
  2. Show by calculation that \(\alpha\) lies between 0.6 and 0.7 .
  3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( 2 x _ { n } + 4 \right)\) to find the value of \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2020 Specimen Q6
8 marks Standard +0.3
6 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } , \quad y = 4 t \mathrm { e } ^ { t } .$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 ( t + 1 ) } { \mathrm { e } ^ { t } }\).
  2. Find the equation of the normal to the curve at the point where \(t = 0\).
CAIE P2 2020 Specimen Q7
11 marks Standard +0.3
7
  1. Show that \(\tan ^ { 2 } x + \cos ^ { 2 } x \equiv \sec ^ { 2 } x + \frac { 1 } { 2 } \cos 2 x - \frac { 1 } { 2 }\) and hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 2 } x + \cos ^ { 2 } x \right) \mathrm { d } x$$
  2. \includegraphics[max width=\textwidth, alt={}, center]{0af2714b-d3eb-4112-a869-eda5cf266cd8-13_535_771_274_648} The region enclosed by the curve \(y = \tan x + \cos x\) and the lines \(x = 0 , x = \frac { 1 } { 4 } \pi\) and \(y = 0\) is shown in the diagram. Find the exact volume of the solid produced when this region is rotated completely about the \(x\)-axis.
CAIE P2 2002 June Q1
4 marks Standard +0.3
1 Solve the inequality \(| x + 2 | < | 5 - 2 x |\).
CAIE P2 2002 June Q2
5 marks Moderate -0.8
2 The cubic polynomial \(3 x ^ { 3 } + a x ^ { 2 } - 2 x - 8\) is denoted by \(\mathrm { f } ( x )\).
  1. Given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
  2. When \(a\) has this value, factorise \(\mathrm { f } ( x )\) completely.
CAIE P2 2002 June Q3
5 marks Moderate -0.8
3 Two variable quantities \(x\) and \(y\) are related by the equation $$y = A x ^ { n }$$ where \(A\) and \(n\) are constants. \includegraphics[max width=\textwidth, alt={}, center]{9b103197-7ba0-427a-b983-34edb51b6cca-2_422_697_977_740} When a graph is plotted showing values of \(\ln y\) on the vertical axis and values of \(\ln x\) on the horizontal axis, the points lie on a straight line. This line crosses the vertical axis at the point ( \(0,2.3\) ) and also passes through the point (4.0,1.7), as shown in the diagram. Find the values of \(A\) and \(n\).