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CAIE P2 2003 June Q3
6 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{a31a4b4e-83a6-47d9-9679-3471b3da1b6e-2_488_664_863_737} The diagram shows the curve \(y = \mathrm { e } ^ { 2 x }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = p\).
  1. Find, in terms of \(p\), the area of \(R\).
  2. Hence calculate the value of \(p\) for which the area of \(R\) is equal to 5 . Give your answer correct to 2 significant figures.
CAIE P2 2003 June Q4
7 marks Standard +0.3
4
  1. Show that the equation $$\tan \left( 45 ^ { \circ } + x \right) = 4 \tan \left( 45 ^ { \circ } - x \right)$$ can be written in the form $$3 \tan ^ { 2 } x - 10 \tan x + 3 = 0$$
  2. Hence solve the equation $$\tan \left( 45 ^ { \circ } + x \right) = 4 \tan \left( 45 ^ { \circ } - x \right)$$ for \(0 ^ { \circ } < x < 90 ^ { \circ }\).
CAIE P2 2003 June Q5
8 marks Moderate -0.3
5
  1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 2 - x ^ { 2 }$$ has exactly one root.
  2. Verify by calculation that the root lies between 1.0 and 1.4 .
  3. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)$$ to determine the root correct to 2 decimal places, showing the result of each iteration.
CAIE P2 2003 June Q6
8 marks Moderate -0.3
6 The equation of a curve is \(y = \frac { 1 } { 1 + \tan x }\).
  1. Show, by differentiation, that the gradient of the curve is always negative.
  2. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { 1 + \tan x } \mathrm {~d} x$$ giving your answer correct to 2 significant figures.
  3. \includegraphics[max width=\textwidth, alt={}, center]{a31a4b4e-83a6-47d9-9679-3471b3da1b6e-3_556_802_1384_708} The diagram shows a sketch of the curve for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).
CAIE P2 2003 June Q7
11 marks Standard +0.3
7 The parametric equations of a curve are $$x = 2 \theta - \sin 2 \theta , \quad y = 2 - \cos 2 \theta$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta\).
  2. Find the equation of the tangent to the curve at the point where \(\theta = \frac { 1 } { 4 } \pi\).
  3. For the part of the curve where \(0 < \theta < 2 \pi\), find the coordinates of the points where the tangent is parallel to the \(x\)-axis.
CAIE P2 2004 June Q1
3 marks Easy -1.2
1 Given that \(2 ^ { x } = 5 ^ { y }\), use logarithms to find the value of \(\frac { x } { y }\) correct to 3 significant figures.
CAIE P2 2004 June Q2
5 marks Standard +0.3
2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 5 } \left( 4 x _ { n } + \frac { 306 } { x _ { n } ^ { 4 } } \right)$$ with initial value \(x _ { 1 } = 3\), converges to \(\alpha\).
  1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places, showing the result of each iteration.
  2. State an equation satisfied by \(\alpha\), and hence show that the exact value of \(\alpha\) is \(\sqrt [ 5 ] { 306 }\).
CAIE P2 2004 June Q3
6 marks Moderate -0.8
3 The cubic polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 13 x - 6\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, solve the equation \(\mathrm { f } ( x ) = 0\).
CAIE P2 2004 June Q4
8 marks Moderate -0.3
4
  1. Express \(3 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$3 \sin \theta + 4 \cos \theta = 4.5$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), correct to 1 decimal place.
  3. Write down the least value of \(3 \sin \theta + 4 \cos \theta + 7\) as \(\theta\) varies.
CAIE P2 2004 June Q5
8 marks Moderate -0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{34177829-f05d-449e-8881-5ab4d852c4ce-3_458_643_285_751} The diagram shows the part of the curve \(y = x \mathrm { e } ^ { - x }\) for \(0 \leqslant x \leqslant 2\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Use the trapezium rule with two intervals to estimate the value of $$\int _ { 0 } ^ { 2 } x \mathrm { e } ^ { - x } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
  3. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).
CAIE P2 2004 June Q6
10 marks Moderate -0.3
6 The parametric equations of a curve are $$x = 2 t + \ln t , \quad y = t + \frac { 4 } { t }$$ where \(t\) takes all positive values.
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { t ^ { 2 } - 4 } { t ( 2 t + 1 ) }\).
  2. Find the equation of the tangent to the curve at the point where \(t = 1\).
  3. The curve has one stationary point. Find the \(y\)-coordinate of this point, and determine whether this point is a maximum or a minimum.
CAIE P2 2004 June Q7
10 marks Standard +0.3
7
  1. By expanding \(\cos ( 2 x + x )\), show that $$\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x$$
  2. Hence, or otherwise, show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 3 } x \mathrm {~d} x = \frac { 2 } { 3 }$$
CAIE P2 2005 June Q1
4 marks Standard +0.3
1 Solve the inequality \(| x | > | 3 x - 2 |\).
CAIE P2 2005 June Q2
5 marks Moderate -0.8
2
  1. Use logarithms to solve the equation \(3 ^ { X } = 8\), giving your answer correct to 2 decimal places.
  2. It is given that $$\ln z = \ln ( y + 2 ) - 2 \ln y$$ where \(y > 0\). Express \(z\) in terms of \(y\) in a form not involving logarithms.
CAIE P2 2005 June Q3
5 marks Moderate -0.3
3 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 3 x _ { n } } { 4 } + \frac { 2 } { x _ { n } ^ { 3 } }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).
  1. Use this iteration to calculate \(\alpha\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
  2. State an equation which is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).
CAIE P2 2005 June Q4
7 marks Moderate -0.8
4 The polynomial \(x ^ { 3 } - x ^ { 2 } + a x + b\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 1\) ) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 12 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\).
CAIE P2 2005 June Q5
9 marks Moderate -0.3
5
  1. By differentiating \(\frac { 1 } { \cos \theta }\), show that if \(y = \sec \theta\) then \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \sec \theta \tan \theta\).
  2. The parametric equations of a curve are $$x = 1 + \tan \theta , \quad y = \sec \theta$$ for \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\). Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin \theta\).
  3. Find the coordinates of the point on the curve at which the gradient of the curve is \(\frac { 1 } { 2 }\).
CAIE P2 2005 June Q6
10 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{08210e25-0f0e-405b-b72d-1bf989689b0a-3_641_865_264_641} The diagram shows the part of the curve \(y = \frac { \ln x } { x }\) for \(0 < x \leqslant 4\). The curve cuts the \(x\)-axis at \(A\) and its maximum point is \(M\).
  1. Write down the coordinates of \(A\).
  2. Show that the \(x\)-coordinate of \(M\) is e, and write down the \(y\)-coordinate of \(M\) in terms of e.
  3. Use the trapezium rule with three intervals to estimate the value of $$\int _ { 1 } ^ { 4 } \frac { \ln x } { x } \mathrm {~d} x$$ correct to 2 decimal places.
  4. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (iii).
CAIE P2 2005 June Q7
10 marks Standard +0.3
7
  1. By expanding \(\sin ( 2 x + x )\) and using double-angle formulae, show that $$\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x$$
  2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x d x = \frac { 5 } { 24 }$$
CAIE P2 2006 June Q1
3 marks Easy -1.2
1 Solve the inequality \(| 2 x - 7 | > 3\).
CAIE P2 2006 June Q2
5 marks Moderate -0.3
2
  1. Prove the identity $$\cos \left( x + 30 ^ { \circ } \right) + \sin \left( x + 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \cos x$$
  2. Hence solve the equation $$\cos \left( x + 30 ^ { \circ } \right) + \sin \left( x + 60 ^ { \circ } \right) = 1$$ for \(0 ^ { \circ } < x < 90 ^ { \circ }\).
CAIE P2 2006 June Q3
7 marks Moderate -0.3
3 The equation of a curve is \(y = x + 2 \cos x\). Find the \(x\)-coordinates of the stationary points of the curve for \(0 \leqslant x \leqslant 2 \pi\), and determine the nature of each of these stationary points.
CAIE P2 2006 June Q4
7 marks Moderate -0.8
4 The cubic polynomial \(a x ^ { 3 } + b x ^ { 2 } - 3 x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 1 )\) and \(( x + 2 )\) are factors of \(\mathrm { p } ( x )\).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the other linear factor of \(\mathrm { p } ( x )\).
CAIE P2 2006 June Q5
8 marks Standard +0.3
5 The equation of a curve is \(3 x ^ { 2 } + 2 x y + y ^ { 2 } = 6\). It is given that there are two points on the curve where the tangent is parallel to the \(x\)-axis.
  1. Show by differentiation that, at these points, \(y = - 3 x\).
  2. Hence find the coordinates of the two points.
CAIE P2 2006 June Q6
9 marks Standard +0.3
6
  1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) that is a root of the equation \(x = 9 \mathrm { e } ^ { - 2 x }\).
  2. Verify, by calculation, that this root lies between 1 and 2 .
  3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left( \ln 9 - \ln x _ { n } \right)$$ converges, then it converges to the root of the equation given in part (i).
  4. Use the iterative formula, with \(x _ { 1 } = 1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.