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CAIE P3 2010 November Q9
10 marks Standard +0.3
9 A biologist is investigating the spread of a weed in a particular region. At time \(t\) weeks after the start of the investigation, the area covered by the weed is \(A \mathrm {~m} ^ { 2 }\). The biologist claims that the rate of increase of \(A\) is proportional to \(\sqrt { } ( 2 A - 5 )\).
  1. Write down a differential equation representing the biologist's claim.
  2. At the start of the investigation, the area covered by the weed was \(7 \mathrm {~m} ^ { 2 }\) and, 10 weeks later, the area covered was \(27 \mathrm {~m} ^ { 2 }\). Assuming that the biologist's claim is correct, find the area covered 20 weeks after the start of the investigation.
CAIE P3 2010 November Q10
13 marks Standard +0.3
10 The polynomial \(\mathrm { p } ( z )\) is defined by $$\mathrm { p } ( z ) = z ^ { 3 } + m z ^ { 2 } + 24 z + 32$$ where \(m\) is a constant. It is given that \(( z + 2 )\) is a factor of \(\mathrm { p } ( z )\).
  1. Find the value of \(m\).
  2. Hence, showing all your working, find
    (a) the three roots of the equation \(\mathrm { p } ( z ) = 0\),
    (b) the six roots of the equation \(\mathrm { p } \left( z ^ { 2 } \right) = 0\).
CAIE P3 2011 November Q1
4 marks Moderate -0.8
1 Using the substitution \(u = \mathrm { e } ^ { x }\), or otherwise, solve the equation $$\mathrm { e } ^ { x } = 1 + 6 \mathrm { e } ^ { - x }$$ giving your answer correct to 3 significant figures.
CAIE P3 2011 November Q2
5 marks Moderate -0.3
2 The parametric equations of a curve are $$x = 3 \left( 1 + \sin ^ { 2 } t \right) , \quad y = 2 \cos ^ { 3 } t$$ Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), simplifying your answer as far as possible.
CAIE P3 2011 November Q3
6 marks Standard +0.3
3 The polynomial \(x ^ { 4 } + 3 x ^ { 3 } + a x + 3\) is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(x ^ { 2 } - x + 1\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, find the real roots of the equation \(\mathrm { p } ( x ) = 0\).
CAIE P3 2011 November Q4
7 marks Standard +0.3
4 The variables \(x\) and \(\theta\) are related by the differential equation $$\sin 2 \theta \frac { \mathrm {~d} x } { \mathrm {~d} \theta } = ( x + 1 ) \cos 2 \theta$$ where \(0 < \theta < \frac { 1 } { 2 } \pi\). When \(\theta = \frac { 1 } { 12 } \pi , x = 0\). Solve the differential equation, obtaining an expression for \(x\) in terms of \(\theta\), and simplifying your answer as far as possible.
CAIE P3 2011 November Q5
8 marks Standard +0.3
5
  1. By sketching a suitable pair of graphs, show that the equation $$\sec x = 3 - \frac { 1 } { 2 } x ^ { 2 }$$ where \(x\) is in radians, has a root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 1 and 1.4.
  3. Show that this root also satisfies the equation $$x = \cos ^ { - 1 } \left( \frac { 2 } { 6 - x ^ { 2 } } \right)$$
  4. Use an iterative formula based on the equation in part (iii) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2011 November Q6
8 marks Moderate -0.3
6
  1. Express \(\cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), 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 \(\cos 2 \theta + 3 \sin 2 \theta = 2\), for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P3 2011 November Q7
8 marks Standard +0.8
7 With respect to the origin \(O\), the position vectors of two points \(A\) and \(B\) are given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + 4 \mathbf { j }\). The point \(P\) lies on the line through \(A\) and \(B\), and \(\overrightarrow { A P } = \lambda \overrightarrow { A B }\).
  1. Show that \(\overrightarrow { O P } = ( 1 + 2 \lambda ) \mathbf { i } + ( 2 + 2 \lambda ) \mathbf { j } + ( 2 - 2 \lambda ) \mathbf { k }\).
  2. By equating expressions for \(\cos A O P\) and \(\cos B O P\) in terms of \(\lambda\), find the value of \(\lambda\) for which \(O P\) bisects the angle \(A O B\).
  3. When \(\lambda\) has this value, verify that \(A P : P B = O A : O B\).
CAIE P3 2011 November Q8
9 marks Standard +0.8
8 Let \(f ( x ) = \frac { 12 + 8 x - x ^ { 2 } } { ( 2 - x ) \left( 4 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 2 - x } + \frac { B x + C } { 4 + x ^ { 2 } }\).
  2. Show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \ln \left( \frac { 25 } { 2 } \right)\).
CAIE P3 2011 November Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{f421f03c-57c9-4feb-91b9-a7f9b12f96ce-3_666_956_1231_593} The diagram shows the curve \(y = x ^ { 2 } \ln x\) and its minimum point \(M\).
  1. Find the exact values of the coordinates of \(M\).
  2. Find the exact value of the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).
CAIE P3 2011 November Q10
10 marks Standard +0.3
10
  1. Showing your working, find the two square roots of the complex number \(1 - ( 2 \sqrt { } 6 ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact.
  2. On a sketch of an Argand diagram, shade the region whose points represent the complex numbers \(z\) which satisfy the inequality \(| z - 3 i | \leqslant 2\). Find the greatest value of \(\arg z\) for points in this region.
CAIE P3 2011 November Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{9e129863-5994-4e17-81f8-e139515998d1-3_666_956_1231_593} The diagram shows the curve \(y = x ^ { 2 } \ln x\) and its minimum point \(M\).
  1. Find the exact values of the coordinates of \(M\).
  2. Find the exact value of the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).
CAIE P3 2011 November Q1
4 marks Moderate -0.8
1 Expand \(\frac { 16 } { ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2011 November Q2
4 marks Moderate -0.8
2 The equation of a curve is \(y = \frac { \mathrm { e } ^ { 2 x } } { 1 + \mathrm { e } ^ { 2 x } }\). Show that the gradient of the curve at the point for which \(x = \ln 3\) is \(\frac { 9 } { 50 }\).
CAIE P3 2011 November Q3
7 marks Standard +0.3
3
  1. Express \(8 \cos \theta + 15 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(8 \cos \theta + 15 \sin \theta = 12\), giving all solutions in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P3 2011 November Q4
7 marks Standard +0.3
4 During an experiment, the number of organisms present at time \(t\) days is denoted by \(N\), where \(N\) is treated as a continuous variable. It is given that $$\frac { \mathrm { d } N } { \mathrm {~d} t } = 1.2 \mathrm { e } ^ { - 0.02 t } N ^ { 0.5 }$$ When \(t = 0\), the number of organisms present is 100 .
  1. Find an expression for \(N\) in terms of \(t\).
  2. State what happens to the number of organisms present after a long time.
CAIE P3 2011 November Q5
8 marks Standard +0.8
5 It is given that \(\int _ { 1 } ^ { a } x \ln x \mathrm {~d} x = 22\), where \(a\) is a constant greater than 1 .
  1. Show that \(a = \sqrt { } \left( \frac { 87 } { 2 \ln a - 1 } \right)\).
  2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 2 decimal places. Use an initial value of 6 and give the result of each iteration to 4 decimal places.
CAIE P3 2011 November Q6
8 marks Standard +0.3
6 The complex number \(w\) is defined by \(w = - 1 + \mathrm { i }\).
  1. Find the modulus and argument of \(w ^ { 2 }\) and \(w ^ { 3 }\), showing your working.
  2. The points in an Argand diagram representing \(w\) and \(w ^ { 2 }\) are the ends of a diameter of a circle. Find the equation of the circle, giving your answer in the form \(| z - ( a + b \mathrm { i } ) | = k\).
CAIE P3 2011 November Q7
9 marks Moderate -0.3
7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - x ^ { 2 } + 4 x - a$$ where \(a\) is a constant. It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\).
  2. When \(a\) has the value found in part (i), express \(\frac { 8 x - 13 } { \mathrm { p } ( x ) }\) in partial fractions.
CAIE P3 2011 November Q8
9 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{6025cf1d-525e-4f12-9517-f20ef5fff2fa-3_698_1006_758_571} The diagram shows the curve with parametric equations $$x = \sin t + \cos t , \quad y = \sin ^ { 3 } t + \cos ^ { 3 } t$$ for \(\frac { 1 } { 4 } \pi < t < \frac { 5 } { 4 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 3 \sin t \cos t\).
  2. Find the gradient of the curve at the origin.
  3. Find the values of \(t\) for which the gradient of the curve is 1 , giving your answers correct to 2 significant figures.
CAIE P3 2011 November Q9
9 marks Standard +0.3
9 The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { l } a \\ 1 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 4 \\ 3 \\ - 2 \end{array} \right)\), where \(a\) is a constant. The plane \(p\) has equation \(2 x - 2 y + z = 10\).
  1. Given that \(l\) does not lie in \(p\), show that \(l\) is parallel to \(p\).
  2. Find the value of \(a\) for which \(l\) lies in \(p\).
  3. It is now given that the distance between \(l\) and \(p\) is 6 . Find the possible values of \(a\).
CAIE P3 2011 November Q10
10 marks Challenging +1.2
10
  1. Use the substitution \(u = \tan x\) to show that, for \(n \neq - 1\), $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { n + 2 } x + \tan ^ { n } x \right) \mathrm { d } x = \frac { 1 } { n + 1 }$$
  2. Hence find the exact value of
    (a) \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sec ^ { 4 } x - \sec ^ { 2 } x \right) \mathrm { d } x\),
    (b) \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 9 } x + 5 \tan ^ { 7 } x + 5 \tan ^ { 5 } x + \tan ^ { 3 } x \right) \mathrm { d } x\).
CAIE P3 2012 November Q1
4 marks Standard +0.3
1 Find the set of values of \(x\) satisfying the inequality \(3 | x - 1 | < | 2 x + 1 |\).
CAIE P3 2012 November Q2
4 marks Standard +0.3
2 Solve the equation $$5 ^ { x - 1 } = 5 ^ { x } - 5$$ giving your answer correct to 3 significant figures.