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CAIE P3 2018 November Q5
7 marks Moderate -0.3
5 The coordinates \(( x , y )\) of a general point on a curve satisfy the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 2 - x ^ { 2 } \right) y$$ The curve passes through the point \(( 1,1 )\). Find the equation of the curve, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2018 November Q6
8 marks Challenging +1.2
6
  1. Show that the equation ( \(\sqrt { } 2\) ) \(\operatorname { cosec } x + \cot x = \sqrt { } 3\) can be expressed in the form \(R \sin ( x - \alpha ) = \sqrt { } 2\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence solve the equation \(( \sqrt { } 2 ) \operatorname { cosec } x + \cot x = \sqrt { } 3\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2018 November Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{c861e691-66da-4269-9057-4a343be9835e-12_357_565_260_790} The diagram shows the curve \(y = 5 \sin ^ { 2 } x \cos ^ { 3 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.
  2. Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\). [4]
CAIE P3 2018 November Q8
9 marks Standard +0.3
8
  1. Showing all necessary working, express the complex number \(\frac { 2 + 3 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the values of \(r\) and \(\theta\) correct to 3 significant figures.
  2. On an Argand diagram sketch the locus of points representing complex numbers \(z\) satisfying the equation \(| z - 3 + 2 i | = 1\). Find the least value of \(| z |\) for points on this locus, giving your answer in an exact form.
CAIE P3 2018 November Q9
10 marks Standard +0.3
9 Let \(\mathrm { f } ( x ) = \frac { 6 x ^ { 2 } + 8 x + 9 } { ( 2 - x ) ( 3 + 2 x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence, showing all necessary working, show that \(\int _ { - 1 } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x = 1 + \frac { 1 } { 2 } \ln \left( \frac { 3 } { 4 } \right)\).
CAIE P3 2018 November Q10
10 marks Standard +0.3
10 The planes \(m\) and \(n\) have equations \(3 x + y - 2 z = 10\) and \(x - 2 y + 2 z = 5\) respectively. The line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\).
  1. Show that \(l\) is parallel to \(m\).
  2. Calculate the acute angle between the planes \(m\) and \(n\).
  3. A point \(P\) lies on the line \(l\). The perpendicular distance of \(P\) from the plane \(n\) is equal to 2 . Find the position vectors of the two possible positions of \(P\).
    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 2018 November Q1
4 marks Standard +0.3
1 Solve the inequality \(3 | 2 x - 1 | > | x + 4 |\).
CAIE P3 2018 November Q2
4 marks Moderate -0.3
2 Showing all necessary working, solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) + \cos \theta = 2 \sin \theta\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\). [4]
CAIE P3 2018 November Q3
5 marks Moderate -0.3
3
  1. Find \(\int \frac { \ln x } { x ^ { 3 } } \mathrm {~d} x\).
  2. Hence show that \(\int _ { 1 } ^ { 2 } \frac { \ln x } { x ^ { 3 } } \mathrm {~d} x = \frac { 1 } { 16 } ( 3 - \ln 4 )\).
CAIE P3 2018 November Q4
5 marks Moderate -0.3
4 Showing all necessary working, solve the equation $$\frac { \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { \mathrm { e } ^ { x } + 1 } = 4$$ giving your answer correct to 3 decimal places.
CAIE P3 2018 November Q5
8 marks Standard +0.3
5 The equation of a curve is \(y = x \ln ( 8 - x )\). The gradient of the curve is equal to 1 at only one point, when \(x = a\).
  1. Show that \(a\) satisfies the equation \(x = 8 - \frac { 8 } { \ln ( 8 - x ) }\).
  2. Verify by calculation that \(a\) lies between 2.9 and 3.1.
  3. Use an iterative formula based on the equation in part (i) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2018 November Q6
8 marks Standard +0.3
6 A certain curve is such that its gradient at a general point with coordinates \(( x , y )\) is proportional to \(\frac { y ^ { 2 } } { x }\). The curve passes through the points with coordinates \(( 1,1 )\) and (e, 2). By setting up and solving a differential equation, find the equation of the curve, expressing \(y\) in terms of \(x\).
CAIE P3 2018 November Q7
10 marks Standard +0.3
7 A curve has equation \(y = \frac { 3 \cos x } { 2 + \sin x }\), for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  1. Find the exact coordinates of the stationary point of the curve.
  2. The constant \(a\) is such that \(\int _ { 0 } ^ { a } \frac { 3 \cos x } { 2 + \sin x } \mathrm {~d} x = 1\). Find the value of \(a\), giving your answer correct to 3 significant figures. \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 7 x ^ { 2 } - 15 x + 8 } { ( 1 - 2 x ) ( 2 - x ) ^ { 2 } }\).
CAIE P3 2018 November Q9
10 marks Standard +0.3
9
    1. Without using a calculator, express the complex number \(\frac { 2 + 6 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    2. Hence, without using a calculator, express \(\frac { 2 + 6 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - 3 \mathrm { i } | \leqslant 1\) and \(\operatorname { Re } z \leqslant 0\), where \(\operatorname { Re } z\) denotes the real part of \(z\). Find the greatest value of \(\arg z\) for points in this region, giving your answer in radians correct to 2 decimal places.
CAIE P3 2018 November Q10
11 marks Standard +0.3
10 The line \(l\) has equation \(\mathbf { r } = 5 \mathbf { i } - 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )\). The plane \(p\) has equation $$( \mathbf { r } - \mathbf { i } - 2 \mathbf { j } ) \cdot ( 3 \mathbf { i } + \mathbf { j } + \mathbf { k } ) = 0$$ The line \(l\) intersects the plane \(p\) at the point \(A\).
  1. Find the position vector of \(A\).
  2. Calculate the acute angle between \(l\) and \(p\).
  3. Find the equation of the line which lies in \(p\) and intersects \(l\) at right angles.
    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 2018 November Q5
7 marks Moderate -0.3
5 The coordinates \(( x , y )\) of a general point on a curve satisfy the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 2 - x ^ { 2 } \right) y .$$ The curve passes through the point \(( 1,1 )\). Find the equation of the curve, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2018 November Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{2a3df76c-2323-470c-8586-009753a4c1e3-12_357_565_260_790} The diagram shows the curve \(y = 5 \sin ^ { 2 } x \cos ^ { 3 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.
  2. Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\). [4]
CAIE P3 2018 November Q9
10 marks Standard +0.3
9 Let \(f ( x ) = \frac { 6 x ^ { 2 } + 8 x + 9 } { ( 2 - x ) ( 3 + 2 x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence, showing all necessary working, show that \(\int _ { - 1 } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x = 1 + \frac { 1 } { 2 } \ln \left( \frac { 3 } { 4 } \right)\).
CAIE P3 2019 November Q1
3 marks Standard +0.3
1 Given that \(\ln \left( 1 + \mathrm { e } ^ { 2 y } \right) = x\), express \(y\) in terms of \(x\).
CAIE P3 2019 November Q2
4 marks Standard +0.8
2 Solve the inequality \(| 2 x - 3 | > 4 | x + 1 |\).
CAIE P3 2019 November Q3
5 marks
3 The parametric equations of a curve are $$x = 2 t + \sin 2 t , \quad y = \ln ( 1 - \cos 2 t )$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosec } 2 t\).
CAIE P3 2019 November Q4
8 marks Standard +0.3
4 The number of insects in a population \(t\) weeks after the start of observations is denoted by \(N\). The population is decreasing at a rate proportional to \(N \mathrm { e } ^ { - 0.02 t }\). The variables \(N\) and \(t\) are treated as continuous, and it is given that when \(t = 0 , N = 1000\) and \(\frac { \mathrm { d } N } { \mathrm {~d} t } = - 10\).
  1. Show that \(N\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = - 0.01 \mathrm { e } ^ { - 0.02 t } N .$$
  2. Solve the differential equation and find the value of \(t\) when \(N = 800\).
  3. State what happens to the value of \(N\) as \(t\) becomes large.
CAIE P3 2019 November Q5
8 marks Standard +0.8
5 The curve with equation \(y = \mathrm { e } ^ { - 2 x } \ln ( x - 1 )\) has a stationary point when \(x = p\).
  1. Show that \(p\) satisfies the equation \(x = 1 + \exp \left( \frac { 1 } { 2 ( x - 1 ) } \right)\), where \(\exp ( x )\) denotes \(\mathrm { e } ^ { x }\).
  2. Verify by calculation that \(p\) lies between 2.2 and 2.6.
  3. Use an iterative formula based on the equation in part (i) to determine \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2019 November Q6
8 marks Standard +0.3
6
  1. By differentiating \(\frac { \cos x } { \sin x }\), show that if \(y = \cot x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
  2. Show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } x \operatorname { cosec } ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 4 } ( \pi + \ln 4 )\). \(7 \quad\) Two lines \(l\) and \(m\) have equations \(\mathbf { r } = a \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\) respectively, where \(a\) is a constant. It is given that the lines intersect.
CAIE P3 2019 November Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + x + 6 } { x ^ { 2 } ( x + 2 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence, showing full working, show that the exact value of \(\int _ { 1 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x\) is \(\frac { 9 } { 4 }\).