Questions — CAIE P2 (709 questions)

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CAIE P2 2008 November Q6
7 marks Standard +0.3
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
8 marks Standard +0.3
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
9 marks Standard +0.8
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
4 marks Standard +0.3
1 Solve the inequality \(| 2 x + 3 | < | x - 3 |\).
CAIE P2 2009 November Q2
4 marks Moderate -0.3
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
6 marks Moderate -0.8
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
6 marks Moderate -0.3
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
6 marks Moderate -0.3
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
7 marks Moderate -0.8
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
8 marks Standard +0.3
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
9 marks Standard +0.3
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
4 marks Standard +0.3
1 Solve the inequality \(| x + 3 | > | 2 x |\).
CAIE P2 2009 November Q2
4 marks Moderate -0.5
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
5 marks Moderate -0.3
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).
CAIE P2 2009 November Q4
5 marks Moderate -0.3
4 The parametric equations of a curve are $$x = 1 - \mathrm { e } ^ { - t } , \quad y = \mathrm { e } ^ { t } + \mathrm { e } ^ { - t }$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 2 t } - 1\).
  2. Hence find the exact value of \(t\) at the point on the curve at which the gradient is 2 .
CAIE P2 2009 November Q5
7 marks Moderate -0.8
5 The polynomial \(a x ^ { 3 } + b x ^ { 2 } - 5 x + 2\), 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 2009 November Q6
7 marks Moderate -0.3
6
  1. Express \(3 \cos x + 4 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the exact value of \(R\) and giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$3 \cos x + 4 \sin x = 4.5$$ giving all solutions in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P2 2009 November Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{729aa2f6-2b62-445f-a2aa-a63b45cb6b64-3_604_971_262_587} The diagram shows the curve \(y = x ^ { 2 } \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Show by differentiation that the \(x\)-coordinate of \(M\) satisfies the equation $$\tan x = \frac { 2 } { x }$$
  2. Verify by calculation that this equation has a root (in radians) between 1 and 1.2.
  3. Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { x _ { n } } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2009 November Q8
9 marks Moderate -0.3
8
  1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \sin 2 x + \sec ^ { 2 } x \right) \mathrm { d } x\).
  2. Show that \(\int _ { 1 } ^ { 4 } \left( \frac { 1 } { 2 x } + \frac { 1 } { x + 1 } \right) \mathrm { d } x = \ln 5\).
CAIE P2 2010 November Q1
3 marks Standard +0.3
1 Solve the inequality \(| x + 1 | > | x - 4 |\).
CAIE P2 2010 November Q2
4 marks Moderate -0.8
2 Use logarithms to solve the equation \(5 ^ { x } = 2 ^ { 2 x + 1 }\), giving your answer correct to 3 significant figures.
CAIE P2 2010 November Q3
5 marks Moderate -0.8
3 Show that \(\int _ { 0 } ^ { 1 } \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } + 2 \mathrm { e } - \frac { 3 } { 2 }\).
CAIE P2 2010 November Q4
6 marks Standard +0.3
4 The parametric equations of a curve are $$x = 1 + \ln ( t - 2 ) , \quad y = t + \frac { 9 } { t } , \quad \text { for } t > 2$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( t ^ { 2 } - 9 \right) ( t - 2 ) } { t ^ { 2 } }\).
  2. Find the coordinates of the only point on the curve at which the gradient is equal to 0 .
CAIE P2 2010 November Q5
6 marks Standard +0.3
5 Solve the equation \(8 + \cot \theta = 2 \operatorname { cosec } ^ { 2 } \theta\), giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2010 November Q6
7 marks Moderate -0.3
6 The curve with equation \(y = \frac { 6 } { x ^ { 2 } }\) intersects the line \(y = x + 1\) at the point \(P\).
  1. Verify by calculation that the \(x\)-coordinate of \(P\) lies between 1.4 and 1.6.
  2. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \sqrt { } \left( \frac { 6 } { x + 1 } \right)$$
  3. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 6 } { x _ { n } + 1 } \right)$$ with initial value \(x _ { 1 } = 1.5\), to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.