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CAIE P3 2019 November Q9
10 marks Standard +0.3
9
  1. By first expanding \(\cos ( 2 x + x )\), show that \(\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x\).
  2. Hence solve the equation \(\cos 3 x + 3 \cos x + 1 = 0\), for \(0 \leqslant x \leqslant \pi\).
  3. Find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 3 } x \mathrm {~d} x\).
CAIE P3 2019 November Q10
10 marks Standard +0.3
10
  1. The complex number \(u\) is given by \(u = - 3 - ( 2 \sqrt { } 10 )\) i. Showing all necessary working and without using a calculator, find the square roots of \(u\). Give your answers in the form \(a + \mathrm { i } b\), where the numbers \(a\) and \(b\) are real and exact.
  2. On a sketch of an Argand diagram shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - \mathrm { i } | \leqslant 3 , \arg z \geqslant \frac { 1 } { 4 } \pi\) and \(\operatorname { Im } z \geqslant 2\), where \(\operatorname { Im } z\) denotes the imaginary part of the complex number \(z\).
    [0pt] [5] If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 2019 November Q1
3 marks Moderate -0.3
1 Solve the equation \(5 \ln \left( 4 - 3 ^ { x } \right) = 6\). Show all necessary working and give the answer correct to 3 decimal places.
CAIE P3 2019 November Q2
5 marks Standard +0.3
2 The curve with equation \(y = \frac { \mathrm { e } ^ { - 2 x } } { 1 - x ^ { 2 } }\) has a stationary point in the interval \(- 1 < x < 1\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the \(x\)-coordinate of this stationary point, giving the answer correct to 3 decimal places.
CAIE P3 2019 November Q3
5 marks Standard +0.3
3 The polynomial \(x ^ { 4 } + 3 x ^ { 3 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + x - 1\) the remainder is \(2 x + 3\). Find the values of \(a\) and \(b\).
CAIE P3 2019 November Q4
7 marks Moderate -0.3
4
  1. Express \(( \sqrt { } 6 ) \sin x + \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 3 decimal places.
  2. Hence solve the equation \(( \sqrt { } 6 ) \sin 2 \theta + \cos 2 \theta = 2\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2019 November Q5
7 marks Standard +0.8
5 The equation of a curve is \(2 x ^ { 2 } y - x y ^ { 2 } = a ^ { 3 }\), where \(a\) is a positive constant. Show that there is only one point on the curve at which the tangent is parallel to the \(x\)-axis and find the \(y\)-coordinate of this point.
CAIE P3 2019 November Q6
8 marks Standard +0.3
6 The variables \(x\) and \(\theta\) satisfy the differential equation $$\sin \frac { 1 } { 2 } \theta \frac { d x } { d \theta } = ( x + 2 ) \cos \frac { 1 } { 2 } \theta$$ for \(0 < \theta < \pi\). It is given that \(x = 1\) when \(\theta = \frac { 1 } { 3 } \pi\). Solve the differential equation and obtain an expression for \(x\) in terms of \(\cos \theta\).
CAIE P3 2019 November Q7
9 marks Standard +0.3
7
  1. Find the complex number \(z\) satisfying the equation $$z + \frac { \mathrm { i } z } { z ^ { * } } - 2 = 0$$ where \(z ^ { * }\) denotes the complex conjugate of \(z\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On a single Argand diagram sketch the loci given by the equations \(| z - 2 \mathrm { i } | = 2\) and \(\operatorname { Im } z = 3\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\).
    2. In the first quadrant the two loci intersect at the point \(P\). Find the exact argument of the complex number represented by \(P\).
CAIE P3 2019 November Q8
10 marks Standard +0.8
8 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + x + 8 } { ( 2 x - 1 ) \left( x ^ { 2 } + 2 \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence, showing full working, find \(\int _ { 1 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x\), giving the answer in the form \(\ln c\), where \(c\) is an integer.
CAIE P3 2019 November Q9
10 marks Challenging +1.2
9 It is given that \(\int _ { 0 } ^ { a } x \cos \frac { 1 } { 3 } x \mathrm {~d} x = 3\), where the constant \(a\) is such that \(0 < a < \frac { 3 } { 2 } \pi\).
  1. Show that \(a\) satisfies the equation $$a = \frac { 4 - 3 \cos \frac { 1 } { 3 } a } { \sin \frac { 1 } { 3 } a }$$
  2. Verify by calculation that \(a\) lies between 2.5 and 3 .
  3. Use an iterative formula based on the equation in part (i) to calculate \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2019 November Q10
11 marks Standard +0.3
10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )\). The plane \(p\) has equation \(2 x + y - 3 z = 5\).
  1. Find the position vector of the point of intersection of \(l\) and \(p\).
  2. Calculate the acute angle between \(l\) and \(p\).
  3. A second plane \(q\) is perpendicular to the plane \(p\) and contains the line \(l\). Find the equation of \(q\), giving your answer in the form \(a x + b y + c z = d\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 2019 November Q1
4 marks Standard +0.3
1 Solve the inequality \(2 | x + 2 | > | 3 x - 1 |\).
CAIE P3 2019 November Q2
5 marks Moderate -0.8
2 The polynomial \(6 x ^ { 3 } + a x ^ { 2 } + b x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is - 24 . Find the values of \(a\) and \(b\).
CAIE P3 2019 November Q3
4 marks Standard +0.3
3 Showing all necessary working, solve the equation \(\frac { 3 ^ { 2 x } + 3 ^ { - x } } { 3 ^ { 2 x } - 3 ^ { - x } } = 4\). Give your answer correct to 3 decimal places.
CAIE P3 2019 November Q4
7 marks Standard +0.8
4
  1. By first expanding \(\tan ( 2 x + x )\), show that the equation \(\tan 3 x = 3 \cot x\) can be written in the form \(\tan ^ { 4 } x - 12 \tan ^ { 2 } x + 3 = 0\).
  2. Hence solve the equation \(\tan 3 x = 3 \cot x\) for \(0 ^ { \circ } < x < 90 ^ { \circ }\).
CAIE P3 2019 November Q5
7 marks Moderate -0.3
5
  1. By sketching a suitable pair of graphs, show that the equation \(\ln ( x + 2 ) = 4 \mathrm { e } ^ { - x }\) has exactly one real root.
  2. Show by calculation that this root lies between \(x = 1\) and \(x = 1.5\).
  3. Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 4 } { \ln \left( x _ { n } + 2 \right) } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2019 November Q7
9 marks Standard +0.3
7 The plane \(m\) has equation \(x + 4 y - 8 z = 2\). The plane \(n\) is parallel to \(m\) and passes through the point \(P\) with coordinates \(( 5,2 , - 2 )\).
  1. Find the equation of \(n\), giving your answer in the form \(a x + b y + c z = d\).
  2. Calculate the perpendicular distance between \(m\) and \(n\).
  3. The line \(l\) lies in the plane \(n\), passes through the point \(P\) and is perpendicular to \(O P\), where \(O\) is the origin. Find a vector equation for \(l\).
CAIE P3 2019 November Q8
10 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{5b5ed7d1-028e-4f9a-ae9e-26071d0df678-14_604_497_262_822} The diagram shows the graph of \(y = \sec x\) for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
  1. Use the trapezium rule with 2 intervals to estimate the value of \(\int _ { 0 } ^ { 1.2 } \sec x \mathrm {~d} x\), giving your answer correct to 2 decimal places.
  2. Explain, with reference to the diagram, whether the trapezium rule gives an overestimate or an underestimate of the true value of the integral in part (i).
  3. \(P\) is the point on the part of the curve \(y = \sec x\) for \(0 \leqslant x < \frac { 1 } { 2 } \pi\) at which the gradient is 2 . By first differentiating \(\frac { 1 } { \cos x }\), find the \(x\)-coordinate of \(P\), giving your answer correct to 3 decimal places.
CAIE P3 2019 November Q9
10 marks Standard +0.3
9 The variables \(x\) and \(t\) satisfy the differential equation \(5 \frac { \mathrm {~d} x } { \mathrm {~d} t } = ( 20 - x ) ( 40 - x )\). It is given that \(x = 10\) when \(t = 0\).
  1. Using partial fractions, solve the differential equation, obtaining an expression for \(x\) in terms of \(t\). [9]
  2. State what happens to the value of \(x\) when \(t\) becomes large.
CAIE P3 2019 November Q10
12 marks Challenging +1.2
10 \includegraphics[max width=\textwidth, alt={}, center]{5b5ed7d1-028e-4f9a-ae9e-26071d0df678-18_449_787_262_678} The diagram shows the graph of \(y = \mathrm { e } ^ { \cos x } \sin ^ { 3 } x\) for \(0 \leqslant x \leqslant \pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  1. Find the \(x\)-coordinate of \(M\). Show all necessary working and give your answer correct to 2 decimal places.
  2. By first using the substitution \(u = \cos x\), find the exact value of the area of \(R\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 Specimen Q1
4 marks Standard +0.8
1 Solve the inequality \(| 2 x - 5 | > 3 | 2 x + 1 |\).
CAIE P3 Specimen Q2
5 marks Moderate -0.3
2 Using the substitution \(u = 3 ^ { x }\), solve the equation \(3 ^ { x } + 3 ^ { 2 x } = 3 ^ { 3 x }\) giving your answer correct to 3 significant figures.
CAIE P3 Specimen Q3
6 marks Standard +0.8
3 The angles \(\theta\) and \(\phi\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), and are such that $$\tan ( \theta - \phi ) = 3 \quad \text { and } \quad \tan \theta + \tan \phi = 1$$ Find the possible values of \(\theta\) and \(\phi\).
CAIE P3 Specimen Q4
7 marks Standard +0.3
4 The equation \(x ^ { 3 } - x ^ { 2 } - 6 = 0\) has one real root, denoted by \(\alpha\).
  1. Find by calculation the pair of consecutive integers between which \(\alpha\) lies.
  2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( x _ { n } + \frac { 6 } { x _ { n } } \right)$$ converges, then it converges to \(\alpha\).
  3. Use this iterative formula to determine \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.