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CAIE P3 2011 June Q1
4 marks Moderate -0.8
1 Expand \(\sqrt [ 3 ] { } ( 1 - 6 x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2011 June Q2
4 marks Moderate -0.8
2 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:
  1. \(y = \ln ( 1 + \sin 2 x )\),
  2. \(y = \frac { \tan x } { x }\).
CAIE P3 2011 June Q3
7 marks Moderate -0.3
3 Points \(A\) and \(B\) have coordinates \(( - 1,2,5 )\) and \(( 2 , - 2,11 )\) respectively. The plane \(p\) passes through \(B\) and is perpendicular to \(A B\).
  1. Find an equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the acute angle between \(p\) and the \(y\)-axis.
CAIE P3 2011 June Q4
7 marks Standard +0.3
4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 12 x ^ { 3 } + 25 x ^ { 2 } - 4 x - 12$$
  1. Show that \(\mathrm { f } ( - 2 ) = 0\) and factorise \(\mathrm { f } ( x )\) completely.
  2. Given that $$12 \times 27 ^ { y } + 25 \times 9 ^ { y } - 4 \times 3 ^ { y } - 12 = 0$$ state the value of \(3 ^ { y }\) and hence find \(y\) correct to 3 significant figures.
CAIE P3 2011 June Q5
7 marks Standard +0.3
5 The curve with equation $$6 \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { y } + \mathrm { e } ^ { 2 y } = c$$ where \(k\) and \(c\) are constants, passes through the point \(P\) with coordinates \(( \ln 3 , \ln 2 )\).
  1. Show that \(58 + 2 k = c\).
  2. Given also that the gradient of the curve at \(P\) is - 6 , find the values of \(k\) and \(c\).
CAIE P3 2011 June Q6
8 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{cc85b13a-7f15-4025-a545-373cda454de8-3_456_495_255_824} The diagram shows a circle with centre \(O\) and radius 10 cm . The chord \(A B\) divides the circle into two regions whose areas are in the ratio \(1 : 4\) and it is required to find the length of \(A B\). The angle \(A O B\) is \(\theta\) radians.
  1. Show that \(\theta = \frac { 2 } { 5 } \pi + \sin \theta\).
  2. Showing all your working, use an iterative formula, based on the equation in part (i), with an initial value of 2.1 , to find \(\theta\) correct to 2 decimal places. Hence find the length of \(A B\) in centimetres correct to 1 decimal place.
CAIE P3 2011 June Q7
8 marks Standard +0.8
7 The integral \(I\) is defined by \(I = \int _ { 0 } ^ { 2 } 4 t ^ { 3 } \ln \left( t ^ { 2 } + 1 \right) \mathrm { d } t\).
  1. Use the substitution \(x = t ^ { 2 } + 1\) to show that \(I = \int _ { 1 } ^ { 5 } ( 2 x - 2 ) \ln x \mathrm {~d} x\).
  2. Hence find the exact value of \(I\).
CAIE P3 2011 June Q8
10 marks Challenging +1.2
8 The complex number \(u\) is defined by \(u = \frac { 6 - 3 \mathrm { i } } { 1 + 2 \mathrm { i } }\).
  1. Showing all your working, find the modulus of \(u\) and show that the argument of \(u\) is \(- \frac { 1 } { 2 } \pi\).
  2. For complex numbers \(z\) satisfying \(\arg ( z - u ) = \frac { 1 } { 4 } \pi\), find the least possible value of \(| z |\).
  3. For complex numbers \(z\) satisfying \(| z - ( 1 + \mathrm { i } ) u | = 1\), find the greatest possible value of \(| z |\).
CAIE P3 2011 June Q9
10 marks Standard +0.8
9
  1. Prove the identity \(\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3\).
  2. Hence
    (a) solve the equation \(\cos 4 \theta + 4 \cos 2 \theta = 1\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\),
    (b) find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta \mathrm {~d} \theta\).
CAIE P3 2011 June Q10
10 marks Standard +0.8
10 The number of birds of a certain species in a forested region is recorded over several years. At time \(t\) years, the number of birds is \(N\), where \(N\) is treated as a continuous variable. The variation in the number of birds is modelled by $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ( 1800 - N ) } { 3600 }$$ It is given that \(N = 300\) when \(t = 0\).
  1. Find an expression for \(N\) in terms of \(t\).
  2. According to the model, how many birds will there be after a long time?
CAIE P3 2011 June Q1
3 marks Standard +0.3
1 Solve the inequality \(| x | < | 5 + 2 x |\).
CAIE P3 2011 June Q2
5 marks Moderate -0.8
2
  1. Show that the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$ can be written as a quadratic equation in \(x\).
  2. Hence solve the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$
CAIE P3 2011 June Q3
5 marks Moderate -0.3
3 Solve the equation $$\cos \theta + 4 \cos 2 \theta = 3$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2011 June Q4
6 marks Standard +0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{76371b0f-0145-4cc4-a147-27bcd749816a-2_339_1395_1089_374} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). The tangent at \(C\) meets \(A B\) produced at \(T\). The angle \(B O C\) is \(x\) radians. The area of the shaded region is equal to the area of the semicircle.
  1. Show that \(x\) satisfies the equation $$\tan x = x + \pi$$
  2. Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } + \pi \right)\) to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2011 June Q5
7 marks Standard +0.3
5 The parametric equations of a curve are $$x = \ln ( \tan t ) , \quad y = \sin ^ { 2 } t$$ where \(0 < t < \frac { 1 } { 2 } \pi\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the equation of the tangent to the curve at the point where \(x = 0\).
CAIE P3 2011 June Q6
9 marks Moderate -0.3
6 A certain curve is such that its gradient at a point \(( x , y )\) is proportional to \(x y\). At the point \(( 1,2 )\) the gradient is 4 .
  1. By setting up and solving a differential equation, show that the equation of the curve is \(y = 2 \mathrm { e } ^ { x ^ { 2 } - 1 }\).
  2. State the gradient of the curve at the point \(( - 1,2 )\) and sketch the curve.
CAIE P3 2011 June Q7
9 marks Standard +0.3
7
  1. The complex number \(u\) is defined by \(u = \frac { 5 } { a + 2 \mathrm { i } }\), where the constant \(a\) is real.
    1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    2. Find the value of \(a\) for which \(\arg \left( u ^ { * } \right) = \frac { 3 } { 4 } \pi\), where \(u ^ { * }\) denotes the complex conjugate of \(u\).
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z | < 2\) and \(| z | < | z - 2 - 2 \mathrm { i } |\).
CAIE P3 2011 June Q8
10 marks Standard +0.3
8
  1. Express \(\frac { 5 x - x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 5 x - x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
CAIE P3 2011 June Q9
10 marks Standard +0.3
9 Two planes have equations \(x + 2 y - 2 z = 7\) and \(2 x + y + 3 z = 5\).
  1. Calculate the acute angle between the planes.
  2. Find a vector equation for the line of intersection of the planes.
CAIE P3 2011 June Q10
11 marks Standard +0.8
10 \includegraphics[max width=\textwidth, alt={}, center]{76371b0f-0145-4cc4-a147-27bcd749816a-3_451_933_1777_605} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).
  1. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 3\) is equal to \(2 - \frac { 17 } { \mathrm { e } ^ { 3 } }\).
  2. Find the \(x\)-coordinate of the maximum point \(M\) on the curve.
  3. Find the \(x\)-coordinate of the point \(P\) at which the tangent to the curve passes through the origin.
CAIE P3 2011 June Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation \(5 ^ { 2 x - 1 } = 2 \left( 3 ^ { x } \right)\), giving your answer correct to 3 significant figures.
CAIE P3 2011 June Q2
4 marks Moderate -0.3
2 The curve \(y = \frac { \ln x } { x ^ { 3 } }\) has one stationary point. Find the \(x\)-coordinate of this point.
CAIE P3 2011 June Q3
5 marks Moderate -0.3
3 Show that \(\int _ { 0 } ^ { 1 } ( 1 - x ) \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x = 4 \mathrm { e } ^ { - \frac { 1 } { 2 } } - 2\).
CAIE P3 2011 June Q4
7 marks Standard +0.3
4
  1. Show that the equation $$\tan \left( 60 ^ { \circ } + \theta \right) + \tan \left( 60 ^ { \circ } - \theta \right) = k$$ can be written in the form $$( 2 \sqrt { } 3 ) \left( 1 + \tan ^ { 2 } \theta \right) = k \left( 1 - 3 \tan ^ { 2 } \theta \right)$$
  2. Hence solve the equation $$\tan \left( 60 ^ { \circ } + \theta \right) + \tan \left( 60 ^ { \circ } - \theta \right) = 3 \sqrt { } 3$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2011 June Q5
7 marks Moderate -0.3
5 The polynomial \(a x ^ { 3 } + b x ^ { 2 } + 5 x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 12 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the quadratic factor of \(\mathrm { p } ( x )\).