Questions P2 (856 questions)

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CAIE P2 2007 November Q2
2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } } { 3 } + \frac { 4 } { x _ { n } ^ { 2 } }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).
  1. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
  2. State an equation that is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).
CAIE P2 2007 November Q3
3
  1. Solve the inequality \(| y - 5 | < 1\).
  2. Hence solve the inequality \(\left| 3 ^ { x } - 5 \right| < 1\), giving 3 significant figures in your answer.
CAIE P2 2007 November Q4
4 The equation of a curve is \(y = 2 x - \tan x\), where \(x\) is in radians. Find the coordinates of the stationary points of the curve for which \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
CAIE P2 2007 November Q5
5 The polynomial \(3 x ^ { 3 } + 8 x ^ { 2 } + a x - 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, solve the equation \(\mathrm { p } ( x ) = 0\).
CAIE P2 2007 November Q6
6
  1. Express \(8 \sin \theta - 15 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$8 \sin \theta - 15 \cos \theta = 14$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2007 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{8f815127-61b2-4a7f-8687-747950ea6597-3_693_1061_262_541} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\) and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Show that the tangent to the curve at the point where \(x = 1\) passes through the origin.
  3. Use the trapezium rule, with two intervals, to estimate the value of $$\int _ { 1 } ^ { 3 } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
CAIE P2 2008 November Q1
1 Solve the inequality \(| x - 3 | > | 2 x |\).
CAIE P2 2008 November Q2
2 The polynomial \(2 x ^ { 3 } - x ^ { 2 } + a x - 6\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2008 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{733c3711-0429-415d-a8f3-8de86097635a-2_550_843_769_651} The variables \(x\) and \(y\) satisfy the equation \(y = A \left( b ^ { - x } \right)\), where \(A\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(0,1.3\) ) and ( \(1.6,0.9\) ), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 2 decimal places.
CAIE P2 2008 November Q4
4
  1. Show that the equation $$\sin \left( x + 30 ^ { \circ } \right) = 2 \cos \left( x + 60 ^ { \circ } \right)$$ can be written in the form $$( 3 \sqrt { } 3 ) \sin x = \cos x$$
  2. Hence solve the equation $$\sin \left( x + 30 ^ { \circ } \right) = 2 \cos \left( x + 60 ^ { \circ } \right)$$ for \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P2 2008 November Q5
5 Show that \(\int _ { 1 } ^ { 2 } \left( \frac { 1 } { x } - \frac { 4 } { 2 x + 1 } \right) \mathrm { d } x = \ln \frac { 18 } { 25 }\).
CAIE P2 2008 November Q6
6 Find the exact coordinates of the point on the curve \(y = x \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) at which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\).
CAIE P2 2008 November Q7
7
  1. By sketching a suitable pair of graphs, show that the equation $$\cos x = 2 - 2 x$$ where \(x\) is in radians, has only one root for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 0.5 and 1 .
  3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = 1 - \frac { 1 } { 2 } \cos x _ { n }$$ converges, then it converges to the root of the equation in part (i).
  4. Use this iterative formula, with initial value \(x _ { 1 } = 0.6\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2008 November Q8
8
  1. (a) Prove the identity $$\sec ^ { 2 } x + \sec x \tan x \equiv \frac { 1 + \sin x } { \cos ^ { 2 } x }$$ (b) Hence prove that $$\sec ^ { 2 } x + \sec x \tan x \equiv \frac { 1 } { 1 - \sin x }$$
  2. By differentiating \(\frac { 1 } { \cos x }\), show that if \(y = \sec x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
  3. Using the results of parts (i) and (ii), find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { 1 - \sin x } \mathrm {~d} x$$
CAIE P2 2009 November Q1
1 Solve the inequality \(| 2 x + 3 | < | x - 3 |\).
CAIE P2 2009 November Q2
2 Solve the equation \(\ln \left( 3 - x ^ { 2 } \right) = 2 \ln x\), giving your answer correct to 3 significant figures.
CAIE P2 2009 November Q3
3 The polynomial \(4 x ^ { 3 } - 8 x ^ { 2 } + a x - 3\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2009 November Q4
4
  1. Show that the equation \(\sin \left( 60 ^ { \circ } - x \right) = 2 \sin x\) can be written in the form \(\tan x = k\), where \(k\) is a constant.
  2. Hence solve the equation \(\sin \left( 60 ^ { \circ } - x \right) = 2 \sin x\), for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P2 2009 November Q5
5
  1. Express \(\cos ^ { 2 } 2 x\) in terms of \(\cos 4 x\).
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 8 } \pi } \cos ^ { 2 } 2 x \mathrm {~d} x\).
CAIE P2 2009 November Q6
6 The curve with equation \(y = x \ln x\) has one stationary point.
  1. Find the exact coordinates of this point, giving your answers in terms of e .
  2. Determine whether this point is a maximum or a minimum point.
CAIE P2 2009 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{67a12825-d7ce-4853-ada4-b8d3009331b5-3_531_759_262_694} The diagram shows the curve \(y = \mathrm { e } ^ { - x }\). The shaded region \(R\) is bounded by the curve and the lines \(y = 1\) and \(x = p\), where \(p\) is a constant.
  1. Find the area of \(R\) in terms of \(p\).
  2. Show that if the area of \(R\) is equal to 1 then $$p = 2 - \mathrm { e } ^ { - p }$$
  3. Use the iterative formula $$p _ { n + 1 } = 2 - \mathrm { e } ^ { - p _ { n } }$$ with initial value \(p _ { 1 } = 2\), to calculate the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2009 November Q8
8 The equation of a curve is \(y ^ { 2 } + 2 x y - x ^ { 2 } = 2\).
  1. Find the coordinates of the two points on the curve where \(x = 1\).
  2. Show by differentiation that at one of these points the tangent to the curve is parallel to the \(x\)-axis. Find the equation of the tangent to the curve at the other point, giving your answer in the form \(a x + b y + c = 0\).
CAIE P2 2009 November Q1
1 Solve the inequality \(| x + 3 | > | 2 x |\).
CAIE P2 2009 November Q2
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 2009 November Q3
3
  1. Use the trapezium rule with two intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sec x \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
  2. Using a sketch of the graph of \(y = \sec x\) for \(0 \leqslant x \leqslant \frac { 1 } { 3 } \pi\), explain whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (i).