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CAIE P3 2012 November Q3
5 marks Standard +0.3
3 Solve the equation $$\sin \left( \theta + 45 ^ { \circ } \right) = 2 \cos \left( \theta - 30 ^ { \circ } \right)$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2012 November Q4
7 marks Moderate -0.3
4 When \(( 1 + a x ) ^ { - 2 }\), where \(a\) is a positive constant, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 3 }\) are equal.
  1. Find the exact value of \(a\).
  2. When \(a\) has this value, obtain the expansion up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2012 November Q5
8 marks Standard +0.3
5
  1. By differentiating \(\frac { 1 } { \cos x }\), show that if \(y = \sec x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
  2. Show that \(\frac { 1 } { \sec x - \tan x } \equiv \sec x + \tan x\).
  3. Deduce that \(\frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \equiv 2 \sec ^ { 2 } x - 1 + 2 \sec x \tan x\).
  4. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } ( 8 \sqrt { } 2 - \pi )\).
CAIE P3 2012 November Q6
8 marks Standard +0.3
6 The variables \(x\) and \(y\) are related by the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 1 - y ^ { 2 }$$ When \(x = 2 , y = 0\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2012 November Q7
8 marks Standard +0.3
7 The equation of a curve is \(\ln ( x y ) - y ^ { 3 } = 1\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { x \left( 3 y ^ { 3 } - 1 \right) }\).
  2. Find the coordinates of the point where the tangent to the curve is parallel to the \(y\)-axis, giving each coordinate correct to 3 significant figures.
CAIE P3 2012 November Q8
10 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{7fe27759-d014-4bc6-8391-342d9df8280e-3_397_750_255_699} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x ^ { 2 } } \sqrt { } \left( 1 + 2 x ^ { 2 } \right)\) for \(x \geqslant 0\), and its maximum point \(M\).
  1. Find the exact value of the \(x\)-coordinate of \(M\).
  2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \ln \left( 4 + 8 x _ { n } ^ { 2 } \right) \right) ,$$ with initial value \(x _ { 1 } = 2\), converges to a certain value \(\alpha\). State an equation satisfied by \(\alpha\) and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 0.5\).
  3. Use the iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2012 November Q9
10 marks Standard +0.3
9 The complex number \(1 + ( \sqrt { } 2 ) \mathrm { i }\) is denoted by \(u\). The polynomial \(x ^ { 4 } + x ^ { 2 } + 2 x + 6\) is denoted by \(\mathrm { p } ( x )\).
  1. Showing your working, verify that \(u\) is a root of the equation \(\mathrm { p } ( x ) = 0\), and write down a second complex root of the equation.
  2. Find the other two roots of the equation \(\mathrm { p } ( x ) = 0\).
CAIE P3 2012 November Q10
11 marks Standard +0.3
10 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 5 \\ - 3 \end{array} \right) .$$ The plane \(m\) is parallel to \(\overrightarrow { O C }\) and contains \(A\) and \(B\).
  1. Find the equation of \(m\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the length of the perpendicular from \(C\) to the line through \(A\) and \(B\).
CAIE P3 2012 November Q8
10 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{346e8866-ca23-4ea6-81bf-bf62502a16d1-3_397_750_255_699} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x ^ { 2 } } \sqrt { } \left( 1 + 2 x ^ { 2 } \right)\) for \(x \geqslant 0\), and its maximum point \(M\).
  1. Find the exact value of the \(x\)-coordinate of \(M\).
  2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \ln \left( 4 + 8 x _ { n } ^ { 2 } \right) \right) ,$$ with initial value \(x _ { 1 } = 2\), converges to a certain value \(\alpha\). State an equation satisfied by \(\alpha\) and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 0.5\).
  3. Use the iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2012 November Q10
11 marks Standard +0.3
10 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 5 \\ - 3 \end{array} \right)$$ The plane \(m\) is parallel to \(\overrightarrow { O C }\) and contains \(A\) and \(B\).
  1. Find the equation of \(m\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the length of the perpendicular from \(C\) to the line through \(A\) and \(B\).
CAIE P3 2012 November Q1
3 marks Moderate -0.5
1 Solve the equation $$\ln ( x + 5 ) = 1 + \ln x$$ giving your answer in terms of e.
CAIE P3 2012 November Q2
5 marks Standard +0.3
2
  1. Express \(24 \sin \theta - 7 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence find the smallest positive value of \(\theta\) satisfying the equation $$24 \sin \theta - 7 \cos \theta = 17$$
CAIE P3 2012 November Q3
6 marks Standard +0.3
3 The parametric equations of a curve are $$x = \frac { 4 t } { 2 t + 3 } , \quad y = 2 \ln ( 2 t + 3 )$$
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), simplifying your answer.
  2. Find the gradient of the curve at the point for which \(x = 1\).
CAIE P3 2012 November Q4
6 marks Moderate -0.3
4 The variables \(x\) and \(y\) are related by the differential equation $$\left( x ^ { 2 } + 4 \right) \frac { d y } { d x } = 6 x y$$ It is given that \(y = 32\) when \(x = 0\). Find an expression for \(y\) in terms of \(x\).
CAIE P3 2012 November Q5
8 marks Moderate -0.8
5 The expression \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 3 x \mathrm { e } ^ { - 2 x }\).
  1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( - \frac { 1 } { 2 } \right)\).
  2. Find the exact value of \(\int _ { - \frac { 1 } { 2 } } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x\).
CAIE P3 2012 November Q6
8 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{adbef77f-e2ac-40ce-a56b-cf6776534ec1-3_561_732_255_705} The diagram shows the curve \(y = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 4 x - 16\), which crosses the \(x\)-axis at the points \(( \alpha , 0 )\) and \(( \beta , 0 )\) where \(\alpha < \beta\). It is given that \(\alpha\) is an integer.
  1. Find the value of \(\alpha\).
  2. Show that \(\beta\) satisfies the equation \(x = \sqrt [ 3 ] { } ( 8 - 2 x )\).
  3. Use an iteration process based on the equation in part (ii) to find the value of \(\beta\) correct to 2 decimal places. Show the result of each iteration to 4 decimal places.
CAIE P3 2012 November Q7
8 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{adbef77f-e2ac-40ce-a56b-cf6776534ec1-3_543_1091_1402_529} The diagram shows part of the curve \(y = \sin ^ { 3 } 2 x \cos ^ { 3 } 2 x\). The shaded region shown is bounded by the curve and the \(x\)-axis and its exact area is denoted by \(A\).
  1. Use the substitution \(u = \sin 2 x\) in a suitable integral to find the value of \(A\).
  2. Given that \(\int _ { 0 } ^ { k \pi } \left| \sin ^ { 3 } 2 x \cos ^ { 3 } 2 x \right| \mathrm { d } x = 40 A\), find the value of the constant \(k\).
CAIE P3 2012 November Q8
10 marks Standard +0.3
8 Two lines have equations $$\mathbf { r } = \left( \begin{array} { r } 5 \\ 1 \\ - 4 \end{array} \right) + s \left( \begin{array} { r } 1 \\ - 1 \\ 3 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } p \\ 4 \\ - 2 \end{array} \right) + t \left( \begin{array} { r } 2 \\ 5 \\ - 4 \end{array} \right) ,$$ where \(p\) is a constant. It is given that the lines intersect.
  1. Find the value of \(p\) and determine the coordinates of the point of intersection.
  2. Find the equation of the plane containing the two lines, giving your answer in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers.
CAIE P3 2012 November Q9
10 marks Standard +0.3
9
  1. Express \(\frac { 9 - 7 x + 8 x ^ { 2 } } { ( 3 - x ) \left( 1 + x ^ { 2 } \right) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 9 - 7 x + 8 x ^ { 2 } } { ( 3 - x ) \left( 1 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
CAIE P3 2012 November Q10
11 marks Standard +0.3
10
  1. Without using a calculator, solve the equation \(\mathrm { i } w ^ { 2 } = ( 2 - 2 \mathrm { i } ) ^ { 2 }\).
    1. Sketch an Argand diagram showing the region \(R\) consisting of points representing the complex numbers \(z\) where $$| z - 4 - 4 i | \leqslant 2$$
    2. For the complex numbers represented by points in the region \(R\), it is given that $$p \leqslant | z | \leqslant q \quad \text { and } \quad \alpha \leqslant \arg z \leqslant \beta$$ Find the values of \(p , q , \alpha\) and \(\beta\), giving your answers correct to 3 significant figures. \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 2013 November Q1
3 marks Easy -1.2
1 The equation of a curve is \(y = \frac { 1 + x } { 1 + 2 x }\) for \(x > - \frac { 1 } { 2 }\). Show that the gradient of the curve is always negative.
CAIE P3 2013 November Q2
4 marks Standard +0.3
2 Solve the equation \(2 \left| 3 ^ { x } - 1 \right| = 3 ^ { x }\), giving your answers correct to 3 significant figures.
CAIE P3 2013 November Q3
5 marks Standard +0.3
3 Find the exact value of \(\int _ { 1 } ^ { 4 } \frac { \ln x } { \sqrt { } x } \mathrm {~d} x\).
CAIE P3 2013 November Q4
6 marks Standard +0.3
4 The parametric equations of a curve are $$x = \mathrm { e } ^ { - t } \cos t , \quad y = \mathrm { e } ^ { - t } \sin t$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \tan \left( t - \frac { 1 } { 4 } \pi \right)\).
CAIE P3 2013 November Q5
7 marks
5
  1. Prove that \(\cot \theta + \tan \theta \equiv 2 \operatorname { cosec } 2 \theta\).
  2. Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \operatorname { cosec } 2 \theta \mathrm {~d} \theta = \frac { 1 } { 2 } \ln 3\).