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CAIE P3 2017 June Q6
8 marks Standard +0.3
6 The plane with equation \(2 x + 2 y - z = 5\) is denoted by \(m\). Relative to the origin \(O\), the points \(A\) and \(B\) have coordinates \(( 3,4,0 )\) and \(( - 1,0,2 )\) respectively.
  1. Show that the plane \(m\) bisects \(A B\) at right angles.
    A second plane \(p\) is parallel to \(m\) and nearer to \(O\). The perpendicular distance between the planes is 1 .
  2. Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2017 June Q7
8 marks Standard +0.3
7 Throughout this question the use of a calculator is not permitted.
The complex numbers \(u\) and \(w\) are defined by \(u = - 1 + 7 \mathrm { i }\) and \(w = 3 + 4 \mathrm { i }\).
  1. Showing all your working, find in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(u - 2 w\) and \(\frac { u } { w }\).
    In an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , w\) and \(u - 2 w\) respectively.
  2. Prove that angle \(A O B = \frac { 1 } { 4 } \pi\).
  3. State fully the geometrical relation between the line segments \(O B\) and \(C A\).
CAIE P3 2017 June Q8
8 marks Standard +0.3
8
  1. By first expanding \(2 \sin \left( x - 30 ^ { \circ } \right)\), express \(2 \sin \left( x - 30 ^ { \circ } \right) - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [5]
  2. Hence solve the equation $$2 \sin \left( x - 30 ^ { \circ } \right) - \cos x = 1$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2017 June Q9
9 marks Standard +0.3
9
  1. Express \(\frac { 1 } { x ( 2 x + 3 ) }\) in partial fractions.
  2. The variables \(x\) and \(y\) satisfy the differential equation $$x ( 2 x + 3 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y$$ and it is given that \(y = 1\) when \(x = 1\). Solve the differential equation and calculate the value of \(y\) when \(x = 9\), giving your answer correct to 3 significant figures.
CAIE P3 2017 June Q10
12 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{b00cefad-7c3c-4672-b309-f19aafab8b01-18_324_677_259_734} The diagram shows the curve \(y = \sin x \cos ^ { 2 } 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\) 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. Find the \(x\)-coordinate of \(M\). Give your answer correct to 2 decimal places.
CAIE P3 2017 June Q1
3 marks Standard +0.3
1 Solve the equation \(\ln \left( x ^ { 2 } + 1 \right) = 1 + 2 \ln x\), giving your answer correct to 3 significant figures.
CAIE P3 2017 June Q2
4 marks Standard +0.3
2 Solve the inequality \(| x - 3 | < 3 x - 4\).
CAIE P3 2017 June Q3
5 marks Standard +0.8
3
  1. Express the equation \(\cot \theta - 2 \tan \theta = \sin 2 \theta\) in the form \(a \cos ^ { 4 } \theta + b \cos ^ { 2 } \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants to be determined.
  2. Hence solve the equation \(\cot \theta - 2 \tan \theta = \sin 2 \theta\) for \(90 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2017 June Q4
6 marks Moderate -0.3
4 The parametric equations of a curve are $$x = t ^ { 2 } + 1 , \quad y = 4 t + \ln ( 2 t - 1 )$$
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the equation of the normal to the curve at the point where \(t = 1\). Give your answer in the form \(a x + b y + c = 0\).
CAIE P3 2017 June Q5
8 marks Standard +0.3
5 In a certain chemical process a substance \(A\) reacts with and reduces a substance \(B\). The masses of \(A\) and \(B\) at time \(t\) after the start of the process are \(x\) and \(y\) respectively. It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - 0.2 x y\) and \(x = \frac { 10 } { ( 1 + t ) ^ { 2 } }\). At the beginning of the process \(y = 100\).
  1. Form a differential equation in \(y\) and \(t\), and solve this differential equation.
  2. Find the exact value approached by the mass of \(B\) as \(t\) becomes large. State what happens to the mass of \(A\) as \(t\) becomes large.
CAIE P3 2017 June Q7
9 marks Standard +0.3
7
  1. Prove that if \(y = \frac { 1 } { \cos \theta }\) then \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \sec \theta \tan \theta\).
  2. Prove the identity \(\frac { 1 + \sin \theta } { 1 - \sin \theta } \equiv 2 \sec ^ { 2 } \theta + 2 \sec \theta \tan \theta - 1\).
  3. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 + \sin \theta } { 1 - \sin \theta } \mathrm { d } \theta\). \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } - 7 x + 4 } { ( 3 x + 2 ) \left( x ^ { 2 } + 5 \right) }\).
  4. Express \(\mathrm { f } ( x )\) in partial fractions.
  5. 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 June Q9
11 marks Standard +0.3
9 Relative to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = 9 \mathbf { i } - \mathbf { j } + 8 \mathbf { k } + \mu ( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\).
  1. Find the position vector of the foot of the perpendicular from \(A\) to \(l\). Hence find the position vector of the reflection of \(A\) in \(l\).
  2. Find the equation of the plane through the origin which contains \(l\). Give your answer in the form \(a x + b y + c z = d\).
  3. Find the exact value of the perpendicular distance of \(A\) from this plane.
CAIE P3 2017 June Q10
11 marks Standard +0.8
10 \includegraphics[max width=\textwidth, alt={}, center]{83a6d80b-dc74-4936-ac32-858a517a843c-18_353_675_260_735} The diagram shows the curve \(y = x ^ { 2 } \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The curve has a maximum point at \(M\) where \(x = p\).
  1. Show that \(p\) satisfies the equation \(p = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { p } \right)\).
  2. Use the iterative formula \(p _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { p _ { n } } \right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
  3. Find, showing all necessary working, the exact area of the region bounded by the curve and the \(x\)-axis.
CAIE P3 2017 June Q1
3 marks Standard +0.3
1 Prove the identity \(\frac { \cot x - \tan x } { \cot x + \tan x } \equiv \cos 2 x\).
CAIE P3 2017 June Q2
4 marks Moderate -0.8
2 Expand \(( 3 + 2 x ) ^ { - 3 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2017 June Q3
4 marks Moderate -0.8
3 Using the substitution \(u = \mathrm { e } ^ { x }\), solve the equation \(4 \mathrm { e } ^ { - x } = 3 \mathrm { e } ^ { x } + 4\). Give your answer correct to 3 significant figures.
CAIE P3 2017 June Q4
4 marks Moderate -0.3
4 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \theta \sin \frac { 1 } { 2 } \theta \mathrm {~d} \theta\).
CAIE P3 2017 June Q5
6 marks Standard +0.3
5 A curve has equation \(y = \frac { 2 } { 3 } \ln \left( 1 + 3 \cos ^ { 2 } x \right)\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan x\).
  2. Hence find the \(x\)-coordinate of the point on the curve where the gradient is - 1 . Give your answer correct to 3 significant figures.
CAIE P3 2017 June Q6
7 marks Standard +0.8
6 The equation \(\cot x = 1 - x\) has one root in the interval \(0 < x < \pi\), denoted by \(\alpha\).
  1. Show by calculation that \(\alpha\) is greater than 2.5.
  2. Show that, if a sequence of values in the interval \(0 < x < \pi\) given by the iterative formula \(x _ { n + 1 } = \pi + \tan ^ { - 1 } \left( \frac { 1 } { 1 - x _ { n } } \right)\) 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.
CAIE P3 2017 June Q7
8 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{7f6f82c3-37d3-48da-9958-e4ef366a6467-10_389_488_258_831} The diagram shows a sketch of the curve \(y = \frac { \mathrm { e } ^ { \frac { 1 } { 2 } x } } { x }\) for \(x > 0\), and its minimum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Use the trapezium rule with two intervals to estimate the value of $$\int _ { 1 } ^ { 3 } \frac { \mathrm { e } ^ { \frac { 1 } { 2 } x } } { x } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
  3. The estimate found in part (ii) is denoted by \(E\). Explain, without further calculation, whether another estimate found using the trapezium rule with four intervals would be greater than \(E\) or less than \(E\).
CAIE P3 2017 June Q8
9 marks Standard +0.3
8 In a certain chemical reaction, a compound \(A\) is formed from a compound \(B\). The masses of \(A\) and \(B\) at time \(t\) after the start of the reaction are \(x\) and \(y\) respectively and the sum of the masses is equal to 50 throughout the reaction. At any time the rate of increase of the mass of \(A\) is proportional to the mass of \(B\) at that time.
  1. Explain why \(\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 50 - x )\), where \(k\) is a constant.
    It is given that \(x = 0\) when \(t = 0\), and \(x = 25\) when \(t = 10\).
  2. Solve the differential equation in part (i) and express \(x\) in terms of \(t\).
CAIE P3 2017 June Q9
10 marks Standard +0.3
9 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 2 } - 4 } { x ^ { 2 } ( 3 x + 2 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \ln \left( \frac { 25 } { 8 } \right) - 1\).
CAIE P3 2017 June Q10
10 marks Standard +0.3
10 The points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + \mathbf { j } + \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + m \mathbf { k } + \mu ( \mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k } )\), where \(m\) is a constant.
  1. Given that the line \(l\) intersects the line passing through \(A\) and \(B\), find the value of \(m\).
  2. Find the equation of the plane which is parallel to \(\mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k }\) and contains the points \(A\) and \(B\). Give your answer in the form \(a x + b y + c z = d\).
CAIE P3 2017 June Q11
10 marks Standard +0.8
11 Throughout this question the use of a calculator is not permitted.
  1. The complex numbers \(z\) and \(w\) satisfy the equations $$z + ( 1 + \mathrm { i } ) w = \mathrm { i } \quad \text { and } \quad ( 1 - \mathrm { i } ) z + \mathrm { i } w = 1$$ Solve the equations for \(z\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. The complex numbers \(u\) and \(v\) are given by \(u = 1 + ( 2 \sqrt { 3 } ) \mathrm { i }\) and \(v = 3 + 2 \mathrm { i }\). In an Argand diagram, \(u\) and \(v\) are represented by the points \(A\) and \(B\). A third point \(C\) lies in the first quadrant and is such that \(B C = 2 A B\) and angle \(A B C = 90 ^ { \circ }\). Find the complex number \(z\) represented by \(C\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
CAIE P3 2018 June Q5
7 marks Moderate -0.5
5 \includegraphics[max width=\textwidth, alt={}, center]{e835a60b-fbeb-49fb-ba6b-ac12c702d487-08_558_785_258_680} The diagram shows a kite \(O A B C\) in which \(A C\) is the line of symmetry. The coordinates of \(A\) and \(C\) are \(( 0,4 )\) and \(( 8,0 )\) respectively and \(O\) is the origin.
  1. Find the equations of \(A C\) and \(O B\).
  2. Find, by calculation, the coordinates of \(B\).