Questions — CAIE (7646 questions)

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CAIE P2 2022 November Q3
4 marks Standard +0.3
3 It is given that \(\sec \theta = \sqrt { 17 }\) where \(0 < \theta < \frac { 1 } { 2 } \pi\).
Find the exact value of \(\tan \left( \theta + \frac { 1 } { 4 } \pi \right)\).
CAIE P2 2022 November Q4
5 marks Standard +0.3
4
  1. By sketching a suitable pair of graphs on the same diagram, show that the equation $$\mathrm { e } ^ { - \frac { 1 } { 2 } x } = x ^ { 5 }$$ has exactly one real root.
  2. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 5 ] { \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } }\) to determine the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2022 November Q5
5 marks Standard +0.3
5 A curve has equation \(4 \mathrm { e } ^ { 2 x } y + y ^ { 2 } = 21\).
Find the gradient of the curve at the point \(( 0 , - 7 )\).
CAIE P2 2022 November Q6
9 marks Standard +0.3
6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 12 x ^ { 3 } - 9 x ^ { 2 } + 8 x - 4$$
  1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 4 x - 3 )\) and show that the remainder is 2 .
  2. Hence find \(\int _ { 2 } ^ { 12 } \left( \frac { \mathrm { p } ( x ) } { 4 x - 3 } - 3 x ^ { 2 } \right) \mathrm { d } x\), giving your answer in the form \(a + \ln b\).
CAIE P2 2022 November Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{389df578-e7a7-4d19-9416-5e580d107717-10_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 2022 November Q8
10 marks Standard +0.3
8 The expression \(\mathrm { f } ( \theta )\) is defined by \(\mathrm { f } ( \theta ) = 12 \sin \theta \cos \theta + 16 \cos ^ { 2 } \theta\).
  1. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( 2 \theta - \alpha ) + k\), where \(R > 0,0 < \alpha < \frac { 1 } { 2 } \pi\) and \(k\) is a constant. State the values of \(R\) and \(k\), and give the value of \(\alpha\) correct to 4 significant figures.
  2. Find the smallest positive value of \(\theta\) satisfying the equation \(\mathrm { f } ( \theta ) = 17\).
  3. Find \(\int f ( \theta ) d \theta\).
    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 November Q1
4 marks Moderate -0.3
1 Solve the equation \(\sec \theta = 5 \operatorname { cosec } \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P2 2022 November Q2
5 marks Moderate -0.3
2 The solutions of the equation \(| 4 x - 1 | = | x + 3 |\) are \(x = p\) and \(x = q\), where \(p < q\).
Find the exact values of \(p\) and \(q\), and hence determine the exact value of \(| p - 2 | - | q - 1 |\).
CAIE P2 2022 November Q3
5 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{68f4b2dc-a05d-4061-aaf0-de15cfe186a9-04_714_515_262_804} The variables \(x\) and \(y\) satisfy the equation \(y = A x ^ { k }\), where \(A\) and \(k\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points ( \(0.56,2.87\) ) and ( \(0.81,3.47\) ), as shown in the diagram. Find the value of \(k\), and the value of \(A\) correct to 2 significant figures.
CAIE P2 2022 November Q4
7 marks Standard +0.3
4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + 23 x ^ { 2 } - a x - 8$$ where \(a\) is a constant. It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\) completely.
  2. Hence solve the equation \(\mathrm { p } \left( \mathrm { e } ^ { 4 y } \right) = 0\), giving your answer correct to 3 significant figures.
CAIE P2 2022 November Q5
9 marks Standard +0.3
5 The curve with equation \(y = x \ln ( 4 x + 1 ) - 3 x\) has one stationary point \(P\).
  1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \frac { 2 x + 0.75 } { \ln ( 4 x + 1 ) } - 0.25$$
  2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 1.8 and 1.9.
  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.
CAIE P2 2022 November Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{68f4b2dc-a05d-4061-aaf0-de15cfe186a9-08_616_531_269_799} The diagram shows the curves \(y = \frac { 6 } { 3 x + 2 }\) and \(y = 3 \mathrm { e } ^ { - x } - 3\) for values of \(x\) between 0 and 4. The shaded region is bounded by the two curves and the lines \(x = 0\) and \(x = 4\). Find the exact area of the shaded region, giving your answer in the form \(\ln a + b + c \mathrm { e } ^ { d }\).
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 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 Standard +0.3
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 \(2 ^ { 3 x - 10 } < 500\).