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CAIE P3 2013 June Q2
4 marks Standard +0.3
2 It is given that \(\ln ( y + 1 ) - \ln y = 1 + 3 \ln x\). Express \(y\) in terms of \(x\), in a form not involving logarithms.
CAIE P3 2013 June Q3
5 marks Standard +0.3
3 Solve the equation \(\tan 2 x = 5 \cot x\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2013 June Q4
7 marks Standard +0.3
4
  1. Express \(( \sqrt { } 3 ) \cos x + \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
  2. Hence show that $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( ( \sqrt { } 3 ) \cos x + \sin x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } \sqrt { } 3$$
CAIE P3 2013 June Q5
8 marks Moderate -0.3
5 The polynomial \(8 x ^ { 3 } + a x ^ { 2 } + b x + 3\), 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 ( \(2 x - 1\) ) the remainder is 1 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the remainder when \(\mathrm { p } ( x )\) is divided by \(2 x ^ { 2 } - 1\).
CAIE P3 2013 June Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{436d891d-92ee-4076-8369-db756d413979-2_435_597_1516_776} The diagram shows the curves \(y = \mathrm { e } ^ { 2 x - 3 }\) and \(y = 2 \ln x\). When \(x = a\) the tangents to the curves are parallel.
  1. Show that \(a\) satisfies the equation \(a = \frac { 1 } { 2 } ( 3 - \ln a )\).
  2. Verify by calculation that this equation has a root between 1 and 2 .
  3. Use the iterative formula \(a _ { n + 1 } = \frac { 1 } { 2 } \left( 3 - \ln a _ { n } \right)\) to calculate \(a\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
CAIE P3 2013 June Q7
8 marks Standard +0.3
7 The complex number \(z\) is defined by \(z = a + \mathrm { i } b\), where \(a\) and \(b\) are real. The complex conjugate of \(z\) is denoted by \(z ^ { * }\).
  1. Show that \(| z | ^ { 2 } = z z ^ { * }\) and that \(( z - k \mathrm { i } ) ^ { * } = z ^ { * } + k \mathrm { i }\), where \(k\) is real. In an Argand diagram a set of points representing complex numbers \(z\) is defined by the equation \(| z - 10 \mathrm { i } | = 2 | z - 4 \mathrm { i } |\).
  2. Show, by squaring both sides, that $$z z ^ { * } - 2 \mathrm { i } z ^ { * } + 2 \mathrm { i } z - 12 = 0$$ Hence show that \(| z - 2 i | = 4\).
  3. Describe the set of points geometrically.
CAIE P3 2013 June Q8
10 marks Standard +0.8
8 The variables \(x\) and \(t\) satisfy the differential equation $$t \frac { \mathrm {~d} x } { \mathrm {~d} t } = \frac { k - x ^ { 3 } } { 2 x ^ { 2 } }$$ for \(t > 0\), where \(k\) is a constant. When \(t = 1 , x = 1\) and when \(t = 4 , x = 2\).
  1. Solve the differential equation, finding the value of \(k\) and obtaining an expression for \(x\) in terms of \(t\).
  2. State what happens to the value of \(x\) as \(t\) becomes large.
CAIE P3 2013 June Q9
10 marks Challenging +1.2
9 \includegraphics[max width=\textwidth, alt={}, center]{436d891d-92ee-4076-8369-db756d413979-3_307_601_1553_772} The diagram shows the curve \(y = \sin ^ { 2 } 2 x \cos x\) 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 \(u = \sin x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.
CAIE P3 2013 June Q10
11 marks Standard +0.8
10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + \lambda ( a \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\), where \(a\) is a constant. The plane \(p\) has equation \(x + 2 y + 2 z = 6\). Find the value or values of \(a\) in each of the following cases.
  1. The line \(l\) is parallel to the plane \(p\).
  2. The line \(l\) intersects the line passing through the points with position vectors \(3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and \(\mathbf { i } + \mathbf { j } - \mathbf { k }\).
  3. The acute angle between the line \(l\) and the plane \(p\) is \(\tan ^ { - 1 } 2\).
CAIE P3 2014 June Q1
5 marks Moderate -0.3
1
  1. Simplify \(\sin 2 \alpha \sec \alpha\).
  2. Given that \(3 \cos 2 \beta + 7 \cos \beta = 0\), find the exact value of \(\cos \beta\).
CAIE P3 2014 June Q2
5 marks Standard +0.3
2 Use the substitution \(u = 1 + 3 \tan x\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sqrt { } ( 1 + 3 \tan x ) } { \cos ^ { 2 } x } d x$$
CAIE P3 2014 June Q3
6 marks Standard +0.3
3 The parametric equations of a curve are $$x = \ln ( 2 t + 3 ) , \quad y = \frac { 3 t + 2 } { 2 t + 3 }$$ Find the gradient of the curve at the point where it crosses the \(y\)-axis.
CAIE P3 2014 June Q4
6 marks Moderate -0.3
4 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 y \mathrm { e } ^ { 3 x } } { 2 + \mathrm { e } ^ { 3 x } }$$ Given that \(y = 36\) when \(x = 0\), find an expression for \(y\) in terms of \(x\).
CAIE P3 2014 June Q5
8 marks Standard +0.3
5 The complex number \(z\) is defined by \(z = \frac { 9 \sqrt { } 3 + 9 i } { \sqrt { } 3 - i }\). Find, showing all your working,
  1. an expression for \(z\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\),
  2. the two square roots of \(z\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
CAIE P3 2014 June Q6
8 marks Moderate -0.3
6 It is given that \(2 \ln ( 4 x - 5 ) + \ln ( x + 1 ) = 3 \ln 3\).
  1. Show that \(16 x ^ { 3 } - 24 x ^ { 2 } - 15 x - 2 = 0\).
  2. By first using the factor theorem, factorise \(16 x ^ { 3 } - 24 x ^ { 2 } - 15 x - 2\) completely.
  3. Hence solve the equation \(2 \ln ( 4 x - 5 ) + \ln ( x + 1 ) = 3 \ln 3\).
CAIE P3 2014 June Q7
8 marks Standard +0.3
7 The straight line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k } )\). The plane \(p\) passes through the point \(( 4 , - 1,2 )\) and is perpendicular to \(l\).
  1. Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the perpendicular distance from the origin to \(p\).
  3. A second plane \(q\) is parallel to \(p\) and the perpendicular distance between \(p\) and \(q\) is 14 units. Find the possible equations of \(q\).
CAIE P3 2014 June Q8
9 marks Standard +0.3
8
  1. By sketching each of the graphs \(y = \operatorname { cosec } x\) and \(y = x ( \pi - x )\) for \(0 < x < \pi\), show that the equation $$\operatorname { cosec } x = x ( \pi - x )$$ has exactly two real roots in the interval \(0 < x < \pi\).
  2. Show that the equation \(\operatorname { cosec } x = x ( \pi - x )\) can be written in the form \(x = \frac { 1 + x ^ { 2 } \sin x } { \pi \sin x }\).
  3. The two real roots of the equation \(\operatorname { cosec } x = x ( \pi - x )\) in the interval \(0 < x < \pi\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
    (a) Use the iterative formula $$x _ { n + 1 } = \frac { 1 + x _ { n } ^ { 2 } \sin x _ { n } } { \pi \sin x _ { n } }$$ to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
    (b) Deduce the value of \(\beta\) correct to 2 decimal places.
CAIE P3 2014 June Q9
10 marks Standard +0.3
9
  1. Express \(\frac { 4 + 12 x + x ^ { 2 } } { ( 3 - x ) ( 1 + 2 x ) ^ { 2 } }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 4 + 12 x + x ^ { 2 } } { ( 3 - x ) ( 1 + 2 x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2014 June Q10
10 marks Standard +0.8
10 \includegraphics[max width=\textwidth, alt={}, center]{b6bede75-3da4-4dda-9303-a5a692fc2572-3_556_1093_1596_523} The diagram shows the curve \(y = 10 e ^ { - \frac { 1 } { 2 } x } \sin 4 x\) for \(x \geqslant 0\). The stationary points are labelled \(T _ { 1 } , T _ { 2 }\), \(T _ { 3 } , \ldots\) as shown.
  1. Find the \(x\)-coordinates of \(T _ { 1 }\) and \(T _ { 2 }\), giving each \(x\)-coordinate correct to 3 decimal places.
  2. It is given that the \(x\)-coordinate of \(T _ { n }\) is greater than 25 . Find the least possible value of \(n\).
CAIE P3 2014 June Q1
4 marks Standard +0.8
1 Find the set of values of \(x\) satisfying the inequality $$| x + 2 a | > 3 | x - a |$$ where \(a\) is a positive constant.
CAIE P3 2014 June Q2
4 marks Moderate -0.8
2 Solve the equation $$2 \ln \left( 5 - \mathrm { e } ^ { - 2 x } \right) = 1$$ giving your answer correct to 3 significant figures.
CAIE P3 2014 June Q3
5 marks Moderate -0.3
3 Solve the equation $$\cos \left( x + 30 ^ { \circ } \right) = 2 \cos x$$ giving all solutions in the interval \(- 180 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2014 June Q4
7 marks Moderate -0.3
4 The parametric equations of a curve are $$x = t - \tan t , \quad y = \ln ( \cos t )$$ for \(- \frac { 1 } { 2 } \pi < t < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot t\).
  2. Hence find the \(x\)-coordinate of the point on the curve at which the gradient is equal to 2 . Give your answer correct to 3 significant figures.
  3. The polynomial \(\mathrm { f } ( x )\) is of the form \(( x - 2 ) ^ { 2 } \mathrm {~g} ( x )\), where \(\mathrm { g } ( x )\) is another polynomial. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ^ { \prime } ( x )\).
  4. The polynomial \(x ^ { 5 } + a x ^ { 4 } + 3 x ^ { 3 } + b x ^ { 2 } + a\), where \(a\) and \(b\) are constants, has a factor \(( x - 2 ) ^ { 2 }\). Using the factor theorem and the result of part (i), or otherwise, find the values of \(a\) and \(b\). [5]
CAIE P3 2014 June Q6
8 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{326d0ea0-8060-4439-8043-3301b281a30f-3_551_519_260_813} In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(O A B\) is equal to \(x\) radians. The shaded region is bounded by \(A B , A C\) and the circular arc with centre \(A\) joining \(B\) and \(C\). The perimeter of the shaded region is equal to half the circumference of the circle.
  1. Show that \(x = \cos ^ { - 1 } \left( \frac { \pi } { 4 + 4 x } \right)\).
  2. Verify by calculation that \(x\) lies between 1 and 1.5.
  3. Use the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { \pi } { 4 + 4 x _ { n } } \right)$$ to determine the value of \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2014 June Q7
8 marks Standard +0.3
7
  1. It is given that \(- 1 + ( \sqrt { } 5 ) \mathrm { i }\) is a root of the equation \(z ^ { 3 } + 2 z + a = 0\), where \(a\) is real. Showing your working, find the value of \(a\), and write down the other complex root of this equation.
  2. The complex number \(w\) has modulus 1 and argument \(2 \theta\) radians. Show that \(\frac { w - 1 } { w + 1 } = \mathrm { i } \tan \theta\).