Questions P2 (856 questions)

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CAIE P2 2008 June Q4
4 The polynomial \(2 x ^ { 3 } + 7 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, 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 5 . Find the values of \(a\) and \(b\).
CAIE P2 2008 June Q5
5
  1. Express \(5 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), 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 $$5 \cos \theta - \sin \theta = 4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2008 June Q6
6 It is given that the curve \(y = ( x - 2 ) \mathrm { e } ^ { x }\) has one stationary point.
  1. Find the exact coordinates of this point.
  2. Determine whether this point is a maximum or a minimum point.
CAIE P2 2008 June Q7
7 The equation of a curve is $$x ^ { 2 } + y ^ { 2 } - 4 x y + 3 = 0$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y - x } { y - 2 x }\).
  2. Find the coordinates of each of the points on the curve where the tangent is parallel to the \(x\)-axis.
CAIE P2 2008 June Q8
8 The constant \(a\), where \(a > 1\), is such that \(\int _ { 1 } ^ { a } \left( x + \frac { 1 } { x } \right) \mathrm { d } x = 6\).
  1. Find an equation satisfied by \(a\), and show that it can be written in the form $$a = \sqrt { } ( 13 - 2 \ln a )$$
  2. Verify, by calculation, that the equation \(a = \sqrt { } ( 13 - 2 \ln a )\) has a root between 3 and 3.5.
  3. Use the iterative formula $$a _ { n + 1 } = \sqrt { } \left( 13 - 2 \ln a _ { n } \right)$$ with \(a _ { 1 } = 3.2\), to calculate the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2009 June Q1
1 Given that \(( 1.25 ) ^ { x } = ( 2.5 ) ^ { y }\), use logarithms to find the value of \(\frac { x } { y }\) correct to 3 significant figures.
CAIE P2 2009 June Q2
2 Solve the inequality \(| 3 x + 2 | < | x |\).
CAIE P2 2009 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{b9556031-871d-4dd3-9523-e3438a41339f-2_451_775_559_683} The diagram shows the curve \(y = \frac { 1 } { 1 + \sqrt { } x }\) for values of \(x\) from 0 to 2 .
  1. Use the trapezium rule with two intervals to estimate the value of $$\int _ { 0 } ^ { 2 } \frac { 1 } { 1 + \sqrt { } x } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
  2. 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 (i).
CAIE P2 2009 June Q4
4 The parametric equations of a curve are $$x = 4 \sin \theta , \quad y = 3 - 2 \cos 2 \theta$$ where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\). Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\), simplifying your answer as far as possible.
CAIE P2 2009 June Q5
5 Solve the equation \(\sec x = 4 - 2 \tan ^ { 2 } x\), giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P2 2009 June Q6
6 The polynomial \(x ^ { 3 } + a x ^ { 2 } + b x + 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x - 1 )\) the remainder is 4 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the other two linear factors of \(\mathrm { p } ( x )\).
CAIE P2 2009 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{b9556031-871d-4dd3-9523-e3438a41339f-3_655_685_262_730} The diagram shows the curve \(y = x \mathrm { e } ^ { 2 x }\) and its minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Show that the curve intersects the line \(y = 20\) at the point whose \(x\)-coordinate is the root of the equation $$x = \frac { 1 } { 2 } \ln \left( \frac { 20 } { x } \right)$$
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 20 } { x _ { n } } \right)$$ with initial value \(x _ { 1 } = 1.3\), to calculate the root correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
CAIE P2 2009 June Q8
8
  1. Find the equation of the tangent to the curve \(y = \ln ( 3 x - 2 )\) at the point where \(x = 1\).
    1. Find the value of the constant \(A\) such that $$\frac { 6 x } { 3 x - 2 } \equiv 2 + \frac { A } { 3 x - 2 }$$
    2. Hence show that \(\int _ { 2 } ^ { 6 } \frac { 6 x } { 3 x - 2 } \mathrm {~d} x = 8 + \frac { 8 } { 3 } \ln 2\).
CAIE P2 2010 June Q1
1 Solve the inequality \(| 2 x - 3 | > 5\).
CAIE P2 2010 June Q2
2 Show that \(\int _ { 0 } ^ { 6 } \frac { 1 } { x + 2 } \mathrm {~d} x = 2 \ln 2\).
CAIE P2 2010 June Q3
3
  1. Show that the equation \(\tan \left( x + 45 ^ { \circ } \right) = 6 \tan x\) can be written in the form $$6 \tan ^ { 2 } x - 5 \tan x + 1 = 0$$
  2. Hence solve the equation \(\tan \left( x + 45 ^ { \circ } \right) = 6 \tan x\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P2 2010 June Q4
4 The polynomial \(x ^ { 3 } + 3 x ^ { 2 } + 4 x + 2\) is denoted by \(\mathrm { f } ( x )\).
  1. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(x ^ { 2 } + x - 1\).
  2. Use the factor theorem to show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
CAIE P2 2010 June Q5
5
  1. Given that \(y = 2 ^ { x }\), show that the equation $$2 ^ { x } + 3 \left( 2 ^ { - x } \right) = 4$$ can be written in the form $$y ^ { 2 } - 4 y + 3 = 0$$
  2. Hence solve the equation $$2 ^ { x } + 3 \left( 2 ^ { - x } \right) = 4$$ giving the values of \(x\) correct to 3 significant figures where appropriate.
CAIE P2 2010 June Q6
6 The equation of a curve is $$x ^ { 2 } y + y ^ { 2 } = 6 x$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 - 2 x y } { x ^ { 2 } + 2 y }\).
  2. Find the equation of the tangent to the curve at the point with coordinates ( 1,2 ), giving your answer in the form \(a x + b y + c = 0\).
CAIE P2 2010 June Q7
7
  1. By sketching a suitable pair of graphs, show that the equation $$\mathrm { e } ^ { 2 x } = 2 - x$$ has only one root.
  2. Verify by calculation that this root lies between \(x = 0\) and \(x = 0.5\).
  3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 2 - x _ { n } \right)$$ converges, then it converges to the root of the equation in part (i).
  4. Use this iterative formula, with initial value \(x _ { 1 } = 0.25\), to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
  5. By differentiating \(\frac { \cos x } { \sin x }\), show that if \(y = \cot x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
  6. By expressing \(\cot ^ { 2 } x\) in terms of \(\operatorname { cosec } ^ { 2 } x\) and using the result of part (i), show that $$\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \cot ^ { 2 } x \mathrm {~d} x = 1 - \frac { 1 } { 4 } \pi$$
  7. Express \(\cos 2 x\) in terms of \(\sin ^ { 2 } x\) and hence show that \(\frac { 1 } { 1 - \cos 2 x }\) can be expressed as \(\frac { 1 } { 2 } \operatorname { cosec } ^ { 2 } x\). Hence, using the result of part (i), find $$\int \frac { 1 } { 1 - \cos 2 x } \mathrm {~d} x$$
CAIE P2 2010 June Q1
1 Given that \(13 ^ { x } = ( 2.8 ) ^ { y }\), use logarithms to show that \(y = k x\) and find the value of \(k\) correct to 3 significant figures.
CAIE P2 2010 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{f55e1431-9443-40d7-bec0-6f5c1d9fa2d7-2_531_949_431_598} The diagram shows part of the curve \(y = x \mathrm { e } ^ { - x }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 2 , x = 3\) and \(y = 0\).
  1. Use the trapezium rule with two intervals to estimate the area of \(R\), giving your answer correct to 2 decimal places.
  2. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the area of \(R\).
CAIE P2 2010 June Q3
3 Solve the inequality \(| 2 x - 1 | < | x + 4 |\).
CAIE P2 2010 June Q4
4
  1. Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos 2 x \mathrm {~d} x = \frac { 1 } { 2 }\).
  2. By using an appropriate trigonometrical identity, find the exact value of $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } 3 \tan ^ { 2 } x \mathrm {~d} x$$
CAIE P2 2010 June Q5
5 The equation of a curve is \(y = x ^ { 3 } \mathrm { e } ^ { - x }\).
  1. Show that the curve has a stationary point where \(x = 3\).
  2. Find the equation of the tangent to the curve at the point where \(x = 1\).