Questions — CAIE (7646 questions)

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CAIE P2 2018 November Q5
9 marks Standard +0.3
5 A curve has parametric equations $$x = t + \ln ( t + 1 ) , \quad y = 3 t \mathrm { e } ^ { 2 t }$$
  1. Find the equation of the tangent to the curve at the origin.
  2. Find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. [4]
CAIE P2 2018 November Q6
8 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{6bf7ba66-8362-4ac0-8e5c-3f88a3ccdf86-10_351_488_264_826} The diagram shows the curve with equation \(y = \sqrt { } \left( 1 + 3 \cos ^ { 2 } \left( \frac { 1 } { 2 } x \right) \right)\) for \(0 \leqslant x \leqslant \pi\). The region \(R\) is bounded by the curve, the axes and the line \(x = \pi\).
  1. Use the trapezium rule with two intervals to find an approximation to the area of \(R\), giving your answer correct to 3 significant figures.
  2. The region \(R\) is rotated completely about the \(x\)-axis. Without using a calculator, find the exact volume of the solid produced.
CAIE P2 2018 November Q7
10 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{6bf7ba66-8362-4ac0-8e5c-3f88a3ccdf86-12_424_488_260_826} The diagram shows the curve with equation \(y = \sin 2 x + 3 \cos 2 x\) for \(0 \leqslant x \leqslant \pi\). At the points \(P\) and \(Q\) on the curve, the gradient of the curve is 3 .
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. By first expressing \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in the form \(R \cos ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), find the \(x\)-coordinates of \(P\) and \(Q\), giving your answers correct to 4 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 2018 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| 3 x - 5 | < 2 | x |\).
CAIE P2 2018 November Q2
5 marks Moderate -0.3
2 Given that \(9 ^ { x } + 3 ^ { x } = 240\), find the value of \(3 ^ { x }\) and hence, using logarithms, find the value of \(x\) correct to 4 significant figures.
CAIE P2 2018 November Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{cc7e798e-0817-405c-bae0-b24b9f451fbf-04_378_486_260_826} The diagram shows the curve with equation $$y = 5 \sin 2 x - 3 \tan 2 x$$ for values of \(x\) such that \(0 \leqslant x < \frac { 1 } { 4 } \pi\). Find the \(x\)-coordinate of the stationary point \(M\), giving your answer correct to 3 significant figures.
CAIE P2 2018 November Q4
5 marks Standard +0.3
4 Find the gradient of the curve $$4 x + 3 y \mathrm { e } ^ { 2 x } + y ^ { 2 } = 10$$ at the point \(( 0,2 )\).
CAIE P2 2018 November Q5
10 marks Standard +0.3
5 The curve with equation $$y = 5 \mathrm { e } ^ { 2 x } - 8 x ^ { 2 } - 20$$ crosses the \(x\)-axis at only one point. This point has coordinates \(( p , 0 )\).
  1. Show that \(p\) satisfies the equation \(x = \frac { 1 } { 2 } \ln \left( 1.6 x ^ { 2 } + 4 \right)\).
  2. Show by calculation that \(0.75 < p < 0.85\).
  3. Use an iterative formula based on the equation in part (i) to find the value of \(p\) correct to 5 significant figures. Give the result of each iteration to 7 significant figures.
  4. Find the gradient of the curve at the point \(( p , 0 )\).
CAIE P2 2018 November Q6
11 marks Standard +0.3
6
  1. Show that \(\int _ { 1 } ^ { 6 } \frac { 12 } { 3 x + 2 } \mathrm {~d} x = \ln 256\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( 8 \sin ^ { 2 } x + \tan ^ { 2 } 2 x \right) \mathrm { d } x\), showing all necessary working.
CAIE P2 2018 November Q7
10 marks Standard +0.3
7
  1. Use the factor theorem to show that ( \(2 x + 3\) ) is a factor of $$8 x ^ { 3 } + 4 x ^ { 2 } - 10 x + 3$$
  2. Show that the equation \(2 \cos 2 \theta = \frac { 6 \cos \theta - 5 } { 2 \cos \theta + 1 }\) can be expressed as $$8 \cos ^ { 3 } \theta + 4 \cos ^ { 2 } \theta - 10 \cos \theta + 3 = 0 .$$
  3. Solve the equation \(2 \cos 2 \theta = \frac { 6 \cos \theta - 5 } { 2 \cos \theta + 1 }\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
    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 2019 November Q1
5 marks Standard +0.3
1
  1. Solve the inequality \(| 2 x - 7 | < | 2 x - 9 |\).
  2. Hence find the largest integer \(n\) satisfying the inequality \(| 2 \ln n - 7 | < | 2 \ln n - 9 |\).
CAIE P2 2019 November Q2
5 marks Moderate -0.8
2 Find the exact value of \(\int _ { 1 } ^ { 2 } \left( 2 \mathrm { e } ^ { 2 x } - 1 \right) ^ { 2 } \mathrm {~d} x\). Show all necessary working.
CAIE P2 2019 November Q3
5 marks Standard +0.3
3 A curve has equation \(y = \frac { 3 + 2 \ln x } { 1 + \ln x }\). Find the exact gradient of the curve at the point for which \(y = 4\).
CAIE P2 2019 November Q4
7 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. Using this value of \(a\), factorise \(\mathrm { p } ( x )\) completely.
  3. Hence solve the equation \(\mathrm { p } \left( \mathrm { e } ^ { \sqrt { } y } \right) = 0\), giving the answer correct to 2 significant figures.
CAIE P2 2019 November Q5
9 marks Standard +0.3
5 It is given that \(\int _ { 0 } ^ { a } \left( 3 x ^ { 2 } + 4 \cos 2 x - \sin x \right) \mathrm { d } x = 2\), where \(a\) is a constant.
  1. Show that \(a = \sqrt [ 3 ] { } ( 3 - 2 \sin 2 a - \cos a )\).
  2. Using the equation in part (i), show by calculation that \(0.5 < a < 0.75\).
  3. Use an iterative formula, based on the equation in part (i), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2019 November Q6
9 marks Moderate -0.3
6
  1. Showing all necessary working, solve the equation $$\sec \alpha \operatorname { cosec } \alpha = 7$$ for \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Showing all necessary working, solve the equation $$\sin \left( \beta + 20 ^ { \circ } \right) + \sin \left( \beta - 20 ^ { \circ } \right) = 6 \cos \beta$$ for \(0 ^ { \circ } < \beta < 90 ^ { \circ }\).
CAIE P2 2019 November Q7
10 marks Standard +0.3
7 The equation of a curve is \(x ^ { 2 } - 4 x y - 2 y ^ { 2 } = 1\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that the gradient of the curve at the point \(( - 1,2 )\) is \(- \frac { 5 } { 2 }\). [5]
  2. Show that the curve has no stationary points.
  3. Find the \(x\)-coordinate of each of the points on the curve at which the tangent is parallel to the \(y\)-axis.
    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 2019 November Q1
3 marks Moderate -0.5
1 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - 6 x + 11$$ Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(\left( x ^ { 2 } + 2 \right)\).
CAIE P2 2019 November Q2
5 marks Standard +0.3
2
  1. Solve the equation \(| 4 x + 5 | = | x - 7 |\).
  2. Hence, using logarithms, solve the equation \(\left| 2 ^ { y + 2 } + 5 \right| = \left| 2 ^ { y } - 7 \right|\), giving the answer correct to 3 significant figures.
CAIE P2 2019 November Q3
5 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{9c26457d-4b65-4cd4-a9b9-128aba92dbf4-04_586_734_260_701} The variables \(x\) and \(y\) satisfy the equation \(y = k x ^ { a }\), where \(k\) and \(a\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points ( \(0.22,3.96\) ) and ( \(1.32,2.43\) ), as shown in the diagram. Find the values of \(k\) and \(a\) correct to 3 significant figures.
CAIE P2 2019 November Q4
5 marks Standard +0.3
4 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) defined by $$x _ { 1 } = 1 , \quad x _ { n + 1 } = \frac { x _ { n } } { \ln \left( 2 x _ { n } \right) }$$ converges to the value \(\alpha\).
  1. Use the iterative formula to find the value of \(\alpha\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
  2. State an equation satisfied by \(\alpha\) and hence determine the exact value of \(\alpha\).
CAIE P2 2019 November Q5
5 marks Standard +0.3
5 Find the exact coordinates of the stationary point of the curve with equation \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } ( 2 x + 5 )\).
CAIE P2 2019 November Q6
9 marks Standard +0.3
6
  1. Show that \(\int _ { 2 } ^ { 18 } \frac { 3 } { 2 x } \mathrm {~d} x = \ln 27\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 4 \sin ^ { 2 } \left( \frac { 3 } { 2 } x \right) \mathrm { d } x\). Show all necessary working.
CAIE P2 2019 November Q7
8 marks Standard +0.3
7 The parametric equations of a curve are $$x = 3 \sin 2 \theta , \quad y = 1 + 2 \tan 2 \theta$$ for \(0 \leqslant \theta < \frac { 1 } { 4 } \pi\).
  1. Find the exact gradient of the curve at the point for which \(\theta = \frac { 1 } { 6 } \pi\).
  2. Find the value of \(\theta\) at the point where the gradient of the curve is 2 , giving the value correct to 3 significant figures.
CAIE P2 2019 November Q8
10 marks Standard +0.3
8
  1. Express \(0.5 \cos \theta - 1.2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(0.5 \cos \theta - 1.2 \sin \theta = 0.8\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
  3. Determine the greatest and least possible values of \(( 3 - \cos \theta + 2.4 \sin \theta ) ^ { 2 }\) as \(\theta\) varies.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.