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CAIE P3 2015 June Q3
6 marks Standard +0.3
3 A curve has equation \(y = \cos x \cos 2 x\). Find the \(x\)-coordinate of the stationary point on the curve in the interval \(0 < x < \frac { 1 } { 2 } \pi\), giving your answer correct to 3 significant figures.
CAIE P3 2015 June Q4
6 marks Moderate -0.3
4
  1. Express \(3 \sin \theta + 2 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), 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 $$3 \sin \theta + 2 \cos \theta = 1$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2015 June Q5
8 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{d1377d66-73c8-4d97-9cae-d784b41fb0a8-2_519_800_1359_669} The diagram shows a circle with centre \(O\) and radius \(r\). The tangents to the circle at the points \(A\) and \(B\) meet at \(T\), and the angle \(A O B\) is \(2 x\) radians. The shaded region is bounded by the tangents \(A T\) and \(B T\), and by the minor \(\operatorname { arc } A B\). The perimeter of the shaded region is equal to the circumference of the circle.
  1. Show that \(x\) satisfies the equation $$\tan x = \pi - x .$$
  2. This equation has one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Verify by calculation that this root lies between 1 and 1.3.
  3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \pi - x _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2015 June Q6
8 marks Standard +0.3
6 Let \(I = \int _ { 0 } ^ { 1 } \frac { \sqrt { } x } { 2 - \sqrt { } x } \mathrm {~d} x\).
  1. Using the substitution \(u = 2 - \sqrt { } x\), show that \(I = \int _ { 1 } ^ { 2 } \frac { 2 ( 2 - u ) ^ { 2 } } { u } \mathrm {~d} u\).
  2. Hence show that \(I = 8 \ln 2 - 5\).
CAIE P3 2015 June Q7
9 marks
7 The complex number \(u\) is given by \(u = - 1 + ( 4 \sqrt { } 3 ) \mathrm { i }\).
  1. Without using a calculator and showing all your working, find the two square roots of \(u\). Give your answers in the form \(a + \mathrm { i } b\), where the real numbers \(a\) and \(b\) are exact.
  2. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying the relation \(| z - u | = 1\). Determine the greatest value of \(\arg z\) for points on this locus. \(8 \quad\) Let \(f ( x ) = \frac { 5 x ^ { 2 } + x + 6 } { ( 3 - 2 x ) \left( x ^ { 2 } + 4 \right) }\).
CAIE P3 2015 June Q9
10 marks Standard +0.8
9 The number of organisms in a population at time \(t\) is denoted by \(x\). Treating \(x\) as a continuous variable, the differential equation satisfied by \(x\) and \(t\) is $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x \mathrm { e } ^ { - t } } { k + \mathrm { e } ^ { - t } }$$ where \(k\) is a positive constant.
  1. Given that \(x = 10\) when \(t = 0\), solve the differential equation, obtaining a relation between \(x , k\) and \(t\).
  2. Given also that \(x = 20\) when \(t = 1\), show that \(k = 1 - \frac { 2 } { \mathrm { e } }\).
  3. Show that the number of organisms never reaches 48, however large \(t\) becomes.
CAIE P3 2015 June Q10
11 marks Standard +0.3
10 The points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { O B } = \mathbf { i } + \mathbf { j } + 5 \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - \mathbf { k } )\).
  1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
  2. Find the equation of the plane containing the line \(l\) and the point \(A\). Give your answer in the form \(a x + b y + c z = d\).
CAIE P3 2015 June Q1
4 marks Moderate -0.3
1 Solve the equation \(\ln ( x + 4 ) = 2 \ln x + \ln 4\), giving your answer correct to 3 significant figures.
CAIE P3 2015 June Q2
4 marks Moderate -0.5
2 Solve the inequality \(| x - 2 | > 2 x - 3\).
CAIE P3 2015 June Q3
6 marks Standard +0.8
3 Solve the equation \(\cot 2 x + \cot x = 3\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2015 June Q4
6 marks Standard +0.3
4 The curve with equation \(y = \frac { \mathrm { e } ^ { 2 x } } { 4 + \mathrm { e } ^ { 3 x } }\) has one stationary point. Find the exact values of the coordinates of this point.
CAIE P3 2015 June Q5
8 marks Standard +0.8
5 The parametric equations of a curve are $$x = a \cos ^ { 4 } t , \quad y = a \sin ^ { 4 } t$$ where \(a\) is a positive constant.
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Show that the equation of the tangent to the curve at the point with parameter \(t\) is $$x \sin ^ { 2 } t + y \cos ^ { 2 } t = a \sin ^ { 2 } t \cos ^ { 2 } t$$
  3. Hence show that if the tangent meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\), then $$O P + O Q = a$$ where \(O\) is the origin.
CAIE P3 2015 June Q6
9 marks Standard +0.3
6 It is given that \(\int _ { 0 } ^ { a } x \cos x \mathrm {~d} x = 0.5\), where \(0 < a < \frac { 1 } { 2 } \pi\).
  1. Show that \(a\) satisfies the equation \(\sin a = \frac { 1.5 - \cos a } { a }\).
  2. Verify by calculation that \(a\) is greater than 1 .
  3. Use the iterative formula $$a _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 1.5 - \cos a _ { n } } { a _ { n } } \right)$$ to determine the value of \(a\) correct to 4 decimal places, giving the result of each iteration to 6 decimal places.
CAIE P3 2015 June Q7
9 marks Standard +0.3
7 The number of micro-organisms in a population at time \(t\) is denoted by \(M\). At any time the variation in \(M\) is assumed to satisfy the differential equation $$\frac { \mathrm { d } M } { \mathrm {~d} t } = k ( \sqrt { } M ) \cos ( 0.02 t )$$ where \(k\) is a constant and \(M\) is taken to be a continuous variable. It is given that when \(t = 0 , M = 100\).
  1. Solve the differential equation, obtaining a relation between \(M , k\) and \(t\).
  2. Given also that \(M = 196\) when \(t = 50\), find the value of \(k\).
  3. Obtain an expression for \(M\) in terms of \(t\) and find the least possible number of micro-organisms.
CAIE P3 2015 June Q8
9 marks Standard +0.3
8 The complex number 1 - i is denoted by \(u\).
  1. Showing your working and without using a calculator, express $$\frac { \mathrm { i } } { u }$$ in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On an Argand diagram, sketch the loci representing complex numbers \(z\) satisfying the equations \(| z - u | = | z |\) and \(| z - \mathrm { i } | = 2\).
  3. Find the argument of each of the complex numbers represented by the points of intersection of the two loci in part (ii).
CAIE P3 2015 June Q9
10 marks Standard +0.3
9 Two planes have equations \(x + 3 y - 2 z = 4\) and \(2 x + y + 3 z = 5\). The planes intersect in the straight line \(l\).
  1. Calculate the acute angle between the two planes.
  2. Find a vector equation for the line \(l\).
CAIE P3 2015 June Q10
10 marks Standard +0.3
10 Let \(\mathrm { f } ( x ) = \frac { 11 x + 7 } { ( 2 x - 1 ) ( x + 2 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 4 } + \ln \left( \frac { 9 } { 4 } \right)\).
CAIE P3 2016 June Q1
5 marks Moderate -0.3
1
  1. Solve the equation \(2 | x - 1 | = 3 | x |\).
  2. Hence solve the equation \(2 \left| 5 ^ { x } - 1 \right| = 3 \left| 5 ^ { x } \right|\), giving your answer correct to 3 significant figures.
CAIE P3 2016 June Q2
5 marks Moderate -0.8
2 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
CAIE P3 2016 June Q3
5 marks Standard +0.3
3 By expressing the equation \(\operatorname { cosec } \theta = 3 \sin \theta + \cot \theta\) in terms of \(\cos \theta\) only, solve the equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2016 June Q4
6 marks Moderate -0.3
4 The variables \(x\) and \(y\) satisfy the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = y \left( 1 - 2 x ^ { 2 } \right)$$ and it is given that \(y = 2\) when \(x = 1\). Solve the differential equation and obtain an expression for \(y\) in terms of \(x\) in a form not involving logarithms.
CAIE P3 2016 June Q5
6 marks Standard +0.3
5 The curve with equation \(y = \sin x \cos 2 x\) has one stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinate of this point, giving your answer correct to 3 significant figures.
CAIE P3 2016 June Q6
7 marks Standard +0.3
6
  1. By sketching a suitable pair of graphs, show that the equation $$5 \mathrm { e } ^ { - x } = \sqrt { } x$$ has one root.
  2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 25 } { x _ { n } } \right)$$ converges, then it converges to the root of the equation in part (i).
  3. Use this iterative formula, with initial value \(x _ { 1 } = 1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2016 June Q7
9 marks Standard +0.3
7 The equation of a curve is \(x ^ { 3 } - 3 x ^ { 2 } y + y ^ { 3 } = 3\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } - 2 x y } { x ^ { 2 } - y ^ { 2 } }\).
  2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.
CAIE P3 2016 June Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 12 } { ( x + 1 ) ( x - 3 ) ^ { 2 } }\).
  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 \(x ^ { 2 }\).