Questions P2 (867 questions)

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CAIE P2 2020 June Q1
3 marks Moderate -0.5
1 Solve the equation $$\ln ( x + 1 ) - \ln x = 2 \ln 2$$
CAIE P2 2020 June Q2
5 marks Moderate -0.8
2 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + 9 x + b$$ where \(a\) and \(b\) are constants. It is given that \(( x - 2 )\) and \(( 2 x + 1 )\) are factors of \(\mathrm { p } ( x )\).
Find the values of \(a\) and \(b\).
CAIE P2 2020 June Q3
5 marks Moderate -0.3
3 A curve has parametric equations $$x = \mathrm { e } ^ { t } - 2 \mathrm { e } ^ { - t } , \quad y = 3 \mathrm { e } ^ { 2 t } + 1$$ Find the equation of the tangent to the curve at the point for which \(t = 0\).
CAIE P2 2020 June Q4
7 marks Moderate -0.3
4
  1. Sketch, on the same diagram, the graphs of \(y = | 3 x + 2 a |\) and \(y = | 3 x - 4 a |\), where \(a\) is a positive constant. Give the coordinates of the points where each graph meets the axes.
  2. Find the coordinates of the point of intersection of the two graphs.
  3. Deduce the solution of the inequality \(| 3 x + 2 a | < | 3 x - 4 a |\).
CAIE P2 2020 June Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{8bdd1285-9e39-465a-8c09-bbe410504f9d-06_442_698_260_721} The diagram shows part of the curve with equation \(y = x ^ { 3 } \cos 2 x\). The curve has a maximum at the point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \sqrt [ 3 ] { 1.5 x ^ { 2 } \cot 2 x }\).
  2. Use the equation in part (a) to show by calculation that the \(x\)-coordinate of \(M\) lies between 0.59 and 0.60.
  3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2020 June Q6
10 marks Standard +0.8
6
  1. Prove that $$\sin 2 \theta ( \operatorname { cosec } \theta - \sec \theta ) \equiv \sqrt { 8 } \cos \left( \theta + \frac { 1 } { 4 } \pi \right)$$
  2. Solve the equation $$\sin 2 \theta ( \operatorname { cosec } \theta - \sec \theta ) = 1$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\). Give the answer correct to 3 significant figures.
  3. Find \(\int \sin x \left( \operatorname { cosec } \frac { 1 } { 2 } x - \sec \frac { 1 } { 2 } x \right) \mathrm { d } x\).
CAIE P2 2020 June Q7
12 marks Standard +0.3
7
  1. Find the quotient when \(9 x ^ { 3 } - 6 x ^ { 2 } - 20 x + 1\) is divided by ( \(3 x + 2\) ), and show that the remainder is 9 .
  2. Hence find \(\int _ { 1 } ^ { 6 } \frac { 9 x ^ { 3 } - 6 x ^ { 2 } - 20 x + 1 } { 3 x + 2 } \mathrm {~d} x\), giving the answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers.
  3. Find the exact root of the equation \(9 e ^ { 9 y } - 6 e ^ { 6 y } - 20 e ^ { 3 y } - 8 = 0\).
    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 2020 June Q1
3 marks Moderate -0.8
1 Given that \(2 ^ { y } = 9 ^ { 3 x }\), use logarithms to show that \(y = k x\) and find the value of \(k\) correct to 3 significant figures.
CAIE P2 2020 June Q2
5 marks Moderate -0.3
2 Find the exact coordinates of the stationary point on the curve with equation \(y = 5 x \mathrm { e } ^ { \frac { 1 } { 2 } x }\).
CAIE P2 2020 June Q3
5 marks Moderate -0.3
3 The equation of a curve is \(\cos 3 x + 5 \sin y = 3\).
Find the gradient of the curve at the point \(\left( \frac { 1 } { 9 } \pi , \frac { 1 } { 6 } \pi \right)\).
CAIE P2 2020 June Q4
5 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{01f2de2b-3482-4694-889e-7fcd016b57e3-06_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 2020 June Q5
5 marks Moderate -0.8
5
  1. Sketch, on the same diagram, the graphs of \(y = | 2 x - 3 |\) and \(y = 3 x + 5\).
  2. Solve the inequality \(3 x + 5 < | 2 x - 3 |\).
CAIE P2 2020 June Q6
8 marks Standard +0.3
6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } - 4 x - 3$$ 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. Using this value of \(a\), factorise \(\mathrm { p } ( x )\) completely.
  3. Hence solve the equation \(\mathrm { p } ( \operatorname { cosec } \theta ) = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P2 2020 June Q7
9 marks Standard +0.3
7 It is given that \(\int _ { 0 } ^ { a } \left( \frac { 4 } { 2 x + 1 } + 8 x \right) \mathrm { d } x = 10\), where \(a\) is a positive constant.
  1. Show that \(a = \sqrt { 2.5 - 0.5 \ln ( 2 a + 1 ) }\).
  2. Using the equation in part (a), show by calculation that \(1 < a < 2\).
  3. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
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 \(| 3 a - 1 | + | 7 b - 1 |\).