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CAIE P3 2017 March Q4
7 marks Standard +0.3
4
  1. Express \(8 \cos \theta - 15 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), 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 $$8 \cos 2 x - 15 \sin 2 x = 4$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2017 March Q5
7 marks Challenging +1.2
5 The curve with equation \(y = \mathrm { e } ^ { - a x } \tan x\), where \(a\) is a positive constant, has only one point in the interval \(0 < x < \frac { 1 } { 2 } \pi\) at which the tangent is parallel to the \(x\)-axis. Find the value of \(a\) and state the exact value of the \(x\)-coordinate of this point.
CAIE P3 2017 March Q6
8 marks Standard +0.8
6 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\). The plane \(p\) has equation \(3 x + y - 5 z = 20\).
  1. Show that the line \(l\) lies in the plane \(p\).
  2. A second plane is parallel to \(l\), perpendicular to \(p\) and contains the point with position vector \(3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\). Find the equation of this plane, giving your answer in the form \(a x + b y + c z = d\). [5]
CAIE P3 2017 March Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{e26f21c5-3776-4c86-8440-6959c5e37486-12_444_382_258_886} A water tank has vertical sides and a horizontal rectangular base, as shown in the diagram. The area of the base is \(2 \mathrm {~m} ^ { 2 }\). At time \(t = 0\) the tank is empty and water begins to flow into it at a rate of \(1 \mathrm {~m} ^ { 3 }\) per hour. At the same time water begins to flow out from the base at a rate of \(0.2 \sqrt { } h \mathrm {~m} ^ { 3 }\) per hour, where \(h \mathrm {~m}\) is the depth of water in the tank at time \(t\) hours.
  1. Form a differential equation satisfied by \(h\) and \(t\), and show that the time \(T\) hours taken for the depth of water to reach 4 m is given by $$T = \int _ { 0 } ^ { 4 } \frac { 10 } { 5 - \sqrt { } h } \mathrm {~d} h$$
  2. Using the substitution \(u = 5 - \sqrt { } h\), find the value of \(T\).
CAIE P3 2017 March Q9
10 marks Standard +0.8
9 Let \(\mathrm { f } ( x ) = \frac { x ( 6 - x ) } { ( 2 + x ) \left( 4 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2017 March Q10
10 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{e26f21c5-3776-4c86-8440-6959c5e37486-18_337_529_260_808} The diagram shows the curve \(y = ( \ln x ) ^ { 2 }\). The \(x\)-coordinate of the point \(P\) is equal to e, and the normal to the curve at \(P\) meets the \(x\)-axis at \(Q\).
  1. Find the \(x\)-coordinate of \(Q\).
  2. Show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\), where \(c\) is a constant.
  3. Using integration by parts, or otherwise, find the exact value of the area of the shaded region between the curve, the \(x\)-axis and the normal \(P Q\).
CAIE P3 2019 March Q1
5 marks Moderate -0.8
1
  1. Show that the equation \(\log _ { 10 } ( x - 4 ) = 2 - \log _ { 10 } x\) can be written as a quadratic equation in \(x\).
  2. Hence solve the equation \(\log _ { 10 } ( x - 4 ) = 2 - \log _ { 10 } x\), giving your answer correct to 3 significant figures.
CAIE P3 2019 March Q2
5 marks Standard +0.3
2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 6 } + 12 x _ { n } } { 3 x _ { n } ^ { 5 } + 8 }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).
  1. Use the formula to calculate \(\alpha\) correct to 4 decimal places. Give the result of each iteration to 6 decimal places.
  2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).
CAIE P3 2019 March Q3
6 marks Standard +0.3
3
  1. Given that \(\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta\), find the exact value of \(\tan \theta\) in a form involving surds. You need not simplify your answer.
  2. Hence solve the equation \(\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P3 2019 March Q4
5 marks Standard +0.3
4 Show that \(\int _ { 1 } ^ { 4 } x ^ { - \frac { 3 } { 2 } } \ln x \mathrm {~d} x = 2 - \ln 4\).
CAIE P3 2019 March Q5
5 marks Standard +0.8
5 The variables \(x\) and \(y\) satisfy the relation \(\sin y = \tan x\), where \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\). Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \cos x \sqrt { } ( \cos 2 x ) } .$$
CAIE P3 2019 March Q6
7 marks Moderate -0.5
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k y ^ { 3 } \mathrm { e } ^ { - x }$$ where \(k\) is a constant. It is given that \(y = 1\) when \(x = 0\), and that \(y = \sqrt { } \mathrm { e }\) when \(x = 1\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2019 March Q7
10 marks Standard +0.3
7
  1. Showing all working and without using a calculator, solve the equation $$( 1 + \mathrm { i } ) z ^ { 2 } - ( 4 + 3 \mathrm { i } ) z + 5 + \mathrm { i } = 0$$ Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. The complex number \(u\) is given by $$u = - 1 - \mathrm { i }$$ On a sketch of an Argand diagram show the point representing \(u\). Shade the region whose points represent complex numbers satisfying the inequalities \(| z | < | z - 2 \mathrm { i } |\) and \(\frac { 1 } { 4 } \pi < \arg ( z - u ) < \frac { 1 } { 2 } \pi\).
CAIE P3 2019 March Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 12 + 12 x - 4 x ^ { 2 } } { ( 2 + x ) ( 3 - 2 x ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2019 March Q9
10 marks Standard +0.3
9 Two planes have equations \(2 x + 3 y - z = 1\) and \(x - 2 y + z = 3\).
  1. Find the acute angle between the planes.
  2. Find a vector equation for the line of intersection of the planes.
CAIE P3 2019 March Q10
12 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{dcfbe7af-c212-42b1-8a90-8e0418cf0ffd-16_330_689_264_726} The diagram shows the curve \(y = \sin ^ { 3 } x \sqrt { } ( \cos x )\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Using the substitution \(u = \cos x\), find by integration the exact area of the shaded region bounded by the curve and the \(x\)-axis.
  2. Showing all your working, find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.
    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 2002 November Q1
3 marks Easy -1.2
1 Solve the inequality \(| 9 - 2 x | < 1\).
CAIE P3 2002 November Q2
4 marks Moderate -0.3
2 Find the exact value of \(\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x\).
CAIE P3 2002 November Q3
5 marks Moderate -0.5
3
  1. Show that the equation $$\log _ { 10 } ( x + 5 ) = 2 - \log _ { 10 } x$$ may be written as a quadratic equation in \(x\).
  2. Hence find the value of \(x\) satisfying the equation $$\log _ { 10 } ( x + 5 ) = 2 - \log _ { 10 } x$$
CAIE P3 2002 November Q4
6 marks Moderate -0.3
4 The curve \(y = \mathrm { e } ^ { x } + 4 \mathrm { e } ^ { - 2 x }\) has one stationary point.
  1. Find the \(x\)-coordinate of this point.
  2. Determine whether the stationary point is a maximum or a minimum point.
CAIE P3 2002 November Q5
8 marks Standard +0.3
5
  1. Express \(4 \sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the value of \(\alpha\) correct to 2 decimal places. Hence
  2. solve the equation $$4 \sin \theta - 3 \cos \theta = 2$$ giving all values of \(\theta\) such that \(0 ^ { \circ } < \theta < 360 ^ { \circ }\),
  3. write down the greatest value of \(\frac { 1 } { 4 \sin \theta - 3 \cos \theta + 6 }\).
CAIE P3 2002 November Q6
9 marks Standard +0.3
6 Let \(f ( x ) = \frac { 6 + 7 x } { ( 2 - x ) \left( 1 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that, when \(x\) is sufficiently small for \(x ^ { 4 }\) and higher powers to be neglected, $$f ( x ) = 3 + 5 x - \frac { 1 } { 2 } x ^ { 2 } - \frac { 15 } { 4 } x ^ { 3 }$$
CAIE P3 2002 November Q7
9 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{b89c016e-dc56-48f4-b4c4-b432418e1b28-3_435_672_273_684} The diagram shows a curved rod \(A B\) of length 100 cm which forms an arc of a circle. The end points \(A\) and \(B\) of the rod are 99 cm apart. The circle has radius \(r \mathrm {~cm}\) and the arc \(A B\) subtends an angle of \(2 \alpha\) radians at \(O\), the centre of the circle.
  1. Show that \(\alpha\) satisfies the equation \(\frac { 99 } { 100 } x = \sin x\).
  2. Given that this equation has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\), verify by calculation that this root lies between 0.1 and 0.5.
  3. Show that if the sequence of values given by the iterative formula $$x _ { n + 1 } = 50 \sin x _ { n } - 48.5 x _ { n }$$ converges, then it converges to a root of the equation in part (i).
  4. Use this iterative formula, with initial value \(x _ { 1 } = 0.25\), to find \(\alpha\) correct to 3 decimal places, showing the result of each iteration.
CAIE P3 2002 November Q8
9 marks Moderate -0.3
8
  1. Find the two square roots of the complex number \(- 3 + 4 \mathrm { i }\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. The complex number \(z\) is given by $$z = \frac { - 1 + 3 \mathrm { i } } { 2 + \mathrm { i } } .$$
    1. Express \(z\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    2. Show on a sketch of an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) representing the complex numbers \(- 1 + 3 \mathrm { i } , 2 + \mathrm { i }\) and \(z\) respectively.
    3. State an equation relating the lengths \(O A , O B\) and \(O C\).
CAIE P3 2002 November Q9
10 marks Standard +0.3
9 In an experiment to study the spread of a soil disease, an area of \(10 \mathrm {~m} ^ { 2 }\) of soil was exposed to infection. In a simple model, it is assumed that the infected area grows at a rate which is proportional to the product of the infected area and the uninfected area. Initially, \(5 \mathrm {~m} ^ { 2 }\) was infected and the rate of growth of the infected area was \(0.1 \mathrm {~m} ^ { 2 }\) per day. At time \(t\) days after the start of the experiment, an area \(a \mathrm {~m} ^ { 2 }\) is infected and an area \(( 10 - a ) \mathrm { m } ^ { 2 }\) is uninfected.
  1. Show that \(\frac { \mathrm { d } a } { \mathrm {~d} t } = 0.004 a ( 10 - a )\).
  2. By first expressing \(\frac { 1 } { a ( 10 - a ) }\) in partial fractions, solve this differential equation, obtaining an expression for \(t\) in terms of \(a\).
  3. Find the time taken for \(90 \%\) of the soil area to become infected, according to this model.