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CAIE P3 2011 June Q6
7 marks Standard +0.3
6
  1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 1 + x ^ { 2 }$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 0.5 and 0.8.
  3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + x _ { n } ^ { 2 } } \right)$$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2011 June Q7
8 marks Standard +0.3
7
  1. Find the roots of the equation $$z ^ { 2 } + ( 2 \sqrt { } 3 ) z + 4 = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. State the modulus and argument of each root.
  3. Showing all your working, verify that each root also satisfies the equation $$z ^ { 6 } = - 64$$
CAIE P3 2011 June Q8
10 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{5b219e1c-e5a0-4f75-910d-fca9761e5088-3_435_895_799_625} The diagram shows the curve \(y = 5 \sin ^ { 3 } x \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Using the substitution \(u = \cos x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.
CAIE P3 2011 June Q9
11 marks Standard +0.3
9 In a chemical reaction, a compound \(X\) is formed from two compounds \(Y\) and \(Z\). The masses in grams of \(X , Y\) and \(Z\) present at time \(t\) seconds after the start of the reaction are \(x , 10 - x\) and \(20 - x\) respectively. At any time the rate of formation of \(X\) is proportional to the product of the masses of \(Y\) and \(Z\) present at the time. When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.01 ( 10 - x ) ( 20 - x )$$
  2. Solve this differential equation and obtain an expression for \(x\) in terms of \(t\).
  3. State what happens to the value of \(x\) when \(t\) becomes large.
CAIE P3 2011 June Q10
12 marks Standard +0.3
10 With respect to the origin \(O\), the lines \(l\) and \(m\) have vector equations \(\mathbf { r } = 2 \mathbf { i } + \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { j } + 6 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )\) respectively.
  1. Prove that \(l\) and \(m\) do not intersect.
  2. Calculate the acute angle between the directions of \(l\) and \(m\).
  3. Find the equation of the plane which is parallel to \(l\) and contains \(m\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2012 June Q1
3 marks Moderate -0.3
1 Solve the equation \(\left| 4 - 2 ^ { x } \right| = 10\), giving your answer correct to 3 significant figures.
CAIE P3 2012 June Q2
5 marks Standard +0.3
2
  1. Expand \(\frac { 1 } { \sqrt { } ( 1 - 4 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { 1 + 2 x } { \sqrt { } ( 4 - 16 x ) }\).
CAIE P3 2012 June Q3
7 marks Standard +0.3
3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } - 3 a x + 4 a$$ where \(a\) is a constant.
  1. Given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
  2. When \(a\) has this value,
    (a) factorise \(\mathrm { p } ( x )\) completely,
    (b) find all the roots of the equation \(\mathrm { p } \left( x ^ { 2 } \right) = 0\).
CAIE P3 2012 June Q4
7 marks Moderate -0.3
4 The complex number \(u\) is defined by \(u = \frac { ( 1 + 2 \mathrm { i } ) ^ { 2 } } { 2 + \mathrm { i } }\).
  1. Without using a calculator and showing your working, express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Sketch an Argand diagram showing the locus of the complex number \(z\) such that \(| z - u | = | u |\).
CAIE P3 2012 June Q5
7 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{4c71f68a-efb9-4408-bf03-874e0d4426d5-2_458_807_1786_667} The diagram shows the curve $$y = 8 \sin \frac { 1 } { 2 } x - \tan \frac { 1 } { 2 } x$$ for \(0 \leqslant x < \pi\). The \(x\)-coordinate of the maximum point is \(\alpha\) and the shaded region is enclosed by the curve and the lines \(x = \alpha\) and \(y = 0\).
  1. Show that \(\alpha = \frac { 2 } { 3 } \pi\).
  2. Find the exact value of the area of the shaded region.
CAIE P3 2012 June Q6
7 marks Standard +0.3
6 The equation of a curve is \(3 x ^ { 2 } - 4 x y + y ^ { 2 } = 45\).
  1. Find the gradient of the curve at the point \(( 2 , - 3 )\).
  2. Show that there are no points on the curve at which the gradient is 1 .
CAIE P3 2012 June Q7
8 marks Moderate -0.3
7 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x \mathrm { e } ^ { 3 x } } { y ^ { 2 } } .$$ It is given that \(y = 2\) when \(x = 0\). Solve the differential equation and hence find the value of \(y\) when \(x = 0.5\), giving your answer correct to 2 decimal places.
CAIE P3 2012 June Q8
9 marks Standard +0.3
8 The point \(P\) has coordinates \(( - 1,4,11 )\) and the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ 3 \\ - 4 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right)\).
  1. Find the perpendicular distance from \(P\) to \(l\).
  2. Find the equation of the plane which contains \(P\) and \(l\), giving your answer in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers.
CAIE P3 2012 June Q9
10 marks Standard +0.8
9 By first expressing \(\frac { 4 x ^ { 2 } + 5 x + 3 } { 2 x ^ { 2 } + 5 x + 2 }\) in partial fractions, show that $$\int _ { 0 } ^ { 4 } \frac { 4 x ^ { 2 } + 5 x + 3 } { 2 x ^ { 2 } + 5 x + 2 } \mathrm {~d} x = 8 - \ln 9$$
CAIE P3 2012 June Q10
12 marks Standard +0.3
10
  1. It is given that \(2 \tan 2 x + 5 \tan ^ { 2 } x = 0\). Denoting \(\tan x\) by \(t\), form an equation in \(t\) and hence show that either \(t = 0\) or \(t = \sqrt [ 3 ] { } ( t + 0.8 )\).
  2. It is given that there is exactly one real value of \(t\) satisfying the equation \(t = \sqrt [ 3 ] { } ( t + 0.8 )\). Verify by calculation that this value lies between 1.2 and 1.3 .
  3. Use the iterative formula \(t _ { n + 1 } = \sqrt [ 3 ] { } \left( t _ { n } + 0.8 \right)\) to find the value of \(t\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
  4. Using the values of \(t\) found in previous parts of the question, solve the equation $$2 \tan 2 x + 5 \tan ^ { 2 } x = 0$$ for \(- \pi \leqslant x \leqslant \pi\).
CAIE P3 2012 June Q1
4 marks Moderate -0.3
1 Solve the equation $$\ln ( 3 x + 4 ) = 2 \ln ( x + 1 )$$ giving your answer correct to 3 significant figures.
CAIE P3 2012 June Q2
5 marks Standard +0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-2_300_767_518_689} In the diagram, \(A B C\) is a triangle in which angle \(A B C\) is a right angle and \(B C = a\). A circular arc, with centre \(C\) and radius \(a\), joins \(B\) and the point \(M\) on \(A C\). The angle \(A C B\) is \(\theta\) radians. The area of the sector \(C M B\) is equal to one third of the area of the triangle \(A B C\).
  1. Show that \(\theta\) satisfies the equation $$\tan \theta = 3 \theta .$$
  2. This equation has one root in the interval \(0 < \theta < \frac { 1 } { 2 } \pi\). Use the iterative formula $$\theta _ { n + 1 } = \tan ^ { - 1 } \left( 3 \theta _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2012 June Q3
5 marks Moderate -0.3
3 Expand \(\sqrt { } \left( \frac { 1 - x } { 1 + x } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2012 June Q4
6 marks Standard +0.8
4 Solve the equation $$\operatorname { cosec } 2 \theta = \sec \theta + \cot \theta$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P3 2012 June Q5
6 marks Moderate -0.8
5 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 2 x + y }$$ and \(y = 0\) when \(x = 0\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2012 June Q6
8 marks Standard +0.8
6 The equation of a curve is \(y = 3 \sin x + 4 \cos ^ { 3 } x\).
  1. Find the \(x\)-coordinates of the stationary points of the curve in the interval \(0 < x < \pi\).
  2. Determine the nature of the stationary point in this interval for which \(x\) is least.
CAIE P3 2012 June Q8
10 marks Standard +0.8
8 Let \(I = \int _ { 2 } ^ { 5 } \frac { 5 } { x + \sqrt { } ( 6 - x ) } \mathrm { d } x\).
  1. Using the substitution \(u = \sqrt { } ( 6 - x )\), show that $$I = \int _ { 1 } ^ { 2 } \frac { 10 u } { ( 3 - u ) ( 2 + u ) } \mathrm { d } u$$
  2. Hence show that \(I = 2 \ln \left( \frac { 9 } { 2 } \right)\).
CAIE P3 2012 June Q9
11 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-3_421_767_1567_689} The diagram shows the curve \(y = x ^ { \frac { 1 } { 2 } } \ln x\). The shaded region between the curve, the \(x\)-axis and the line \(x = \mathrm { e }\) is denoted by \(R\).
  1. Find the equation of the tangent to the curve at the point where \(x = 1\), giving your answer in the form \(y = m x + c\).
  2. Find by integration the volume of the solid obtained when the region \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and e.
CAIE P3 2012 June Q10
12 marks Standard +0.8
10 Two planes, \(m\) and \(n\), have equations \(x + 2 y - 2 z = 1\) and \(2 x - 2 y + z = 7\) respectively. The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } - \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\).
  1. Show that \(l\) is parallel to \(m\).
  2. Find the position vector of the point of intersection of \(l\) and \(n\).
  3. A point \(P\) lying on \(l\) is such that its perpendicular distances from \(m\) and \(n\) are equal. Find the position vectors of the two possible positions for \(P\) and calculate the distance between them.
    [0pt] [The perpendicular distance of a point with position vector \(x _ { 1 } \mathbf { i } + y _ { 1 } \mathbf { j } + z _ { 1 } \mathbf { k }\) from the plane \(a x + b y + c z = d\) is \(\frac { \left| a x _ { 1 } + b y _ { 1 } + c z _ { 1 } - d \right| } { \sqrt { } \left( a ^ { 2 } + b ^ { 2 } + c ^ { 2 } \right) }\).] \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P3 2012 June Q1
4 marks Moderate -0.3
1 Expand \(\frac { 1 } { \sqrt { } ( 4 + 3 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.