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CAIE P3 2013 November Q6
8 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{dd7b2aee-4318-48e8-97c0-541e47f2e83a-2_551_567_1416_788} 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 \(\theta\) radians. The shaded region is bounded by the circumference of the circle and the arc with centre \(A\) joining \(B\) and \(C\). The area of the shaded region is equal to half the area of the circle.
  1. Show that \(\cos 2 \theta = \frac { 2 \sin 2 \theta - \pi } { 4 \theta }\).
  2. Use the iterative formula $$\theta _ { n + 1 } = \frac { 1 } { 2 } \cos ^ { - 1 } \left( \frac { 2 \sin 2 \theta _ { n } - \pi } { 4 \theta _ { n } } \right)$$ with initial value \(\theta _ { 1 } = 1\), to determine \(\theta\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places. \(7 \quad\) Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } - 7 x - 1 } { ( x - 2 ) \left( x ^ { 2 } + 3 \right) }\).
  3. Express \(\mathrm { f } ( x )\) in partial fractions.
  4. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2013 November Q9
11 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{dd7b2aee-4318-48e8-97c0-541e47f2e83a-3_704_714_1272_717} The diagram shows three points \(A , B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 1 \\ 2 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 0 \\ 3 \\ 1 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 3 \\ 0 \\ 4 \end{array} \right)\). The point \(D\) lies on \(B C\), between \(B\) and \(C\), and is such that \(C D = 2 D B\).
  1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the position vector of \(D\).
  3. Show that the length of the perpendicular from \(A\) to \(O D\) is \(\frac { 1 } { 3 } \sqrt { } ( 65 )\).
CAIE P3 2013 November Q10
11 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{dd7b2aee-4318-48e8-97c0-541e47f2e83a-4_335_875_262_635} A tank containing water is in the form of a cone with vertex \(C\). The axis is vertical and the semivertical angle is \(60 ^ { \circ }\), as shown in the diagram. At time \(t = 0\), the tank is full and the depth of water is \(H\). At this instant, a tap at \(C\) is opened and water begins to flow out. The volume of water in the tank decreases at a rate proportional to \(\sqrt { } h\), where \(h\) is the depth of water at time \(t\). The tank becomes empty when \(t = 60\).
  1. Show that \(h\) and \(t\) satisfy a differential equation of the form $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - A h ^ { - \frac { 3 } { 2 } } ,$$ where \(A\) is a positive constant.
  2. Solve the differential equation given in part (i) and obtain an expression for \(t\) in terms of \(h\) and \(H\).
  3. Find the time at which the depth reaches \(\frac { 1 } { 2 } H\).
    [0pt] [The volume \(V\) of a cone of vertical height \(h\) and base radius \(r\) is given by \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]
CAIE P3 2013 November Q6
8 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{a5f8f007-5176-4686-93d2-4caabeea182e-2_551_569_1416_788} 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 \(\theta\) radians. The shaded region is bounded by the circumference of the circle and the arc with centre \(A\) joining \(B\) and \(C\). The area of the shaded region is equal to half the area of the circle.
  1. Show that \(\cos 2 \theta = \frac { 2 \sin 2 \theta - \pi } { 4 \theta }\).
  2. Use the iterative formula $$\theta _ { n + 1 } = \frac { 1 } { 2 } \cos ^ { - 1 } \left( \frac { 2 \sin 2 \theta _ { n } - \pi } { 4 \theta _ { n } } \right) ,$$ with initial value \(\theta _ { 1 } = 1\), to determine \(\theta\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
CAIE P3 2013 November Q7
10 marks Standard +0.3
7 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } - 7 x - 1 } { ( x - 2 ) \left( x ^ { 2 } + 3 \right) }\).
  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 \(x ^ { 2 }\).
CAIE P3 2013 November Q9
11 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{a5f8f007-5176-4686-93d2-4caabeea182e-3_704_716_1272_717} The diagram shows three points \(A , B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 1 \\ 2 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 0 \\ 3 \\ 1 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 3 \\ 0 \\ 4 \end{array} \right)\). The point \(D\) lies on \(B C\), between \(B\) and \(C\), and is such that \(C D = 2 D B\).
  1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the position vector of \(D\).
  3. Show that the length of the perpendicular from \(A\) to \(O D\) is \(\frac { 1 } { 3 } \sqrt { } ( 65 )\).
CAIE P3 2013 November Q10
11 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{a5f8f007-5176-4686-93d2-4caabeea182e-4_328_867_264_639} A tank containing water is in the form of a cone with vertex \(C\). The axis is vertical and the semivertical angle is \(60 ^ { \circ }\), as shown in the diagram. At time \(t = 0\), the tank is full and the depth of water is \(H\). At this instant, a tap at \(C\) is opened and water begins to flow out. The volume of water in the tank decreases at a rate proportional to \(\sqrt { } h\), where \(h\) is the depth of water at time \(t\). The tank becomes empty when \(t = 60\).
  1. Show that \(h\) and \(t\) satisfy a differential equation of the form $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - A h ^ { - \frac { 3 } { 2 } } ,$$ where \(A\) is a positive constant.
  2. Solve the differential equation given in part (i) and obtain an expression for \(t\) in terms of \(h\) and \(H\).
  3. Find the time at which the depth reaches \(\frac { 1 } { 2 } H\).
    [0pt] [The volume \(V\) of a cone of vertical height \(h\) and base radius \(r\) is given by \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]
CAIE P3 2013 November Q1
4 marks Standard +0.3
1 Given that \(2 \ln ( x + 4 ) - \ln x = \ln ( x + a )\), express \(x\) in terms of \(a\).
CAIE P3 2013 November Q2
4 marks Moderate -0.3
2 Use the substitution \(u = 3 x + 1\) to find \(\int \frac { 3 x } { 3 x + 1 } \mathrm {~d} x\).
CAIE P3 2013 November Q3
5 marks Moderate -0.8
3 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 14$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).
  1. Find the value of \(a\).
  2. Show that, when \(a\) has this value, the equation \(\mathrm { f } ( x ) = 0\) has only one real root.
CAIE P3 2013 November Q4
5 marks Standard +0.3
4 A curve has equation \(3 \mathrm { e } ^ { 2 x } y + \mathrm { e } ^ { x } y ^ { 3 } = 14\). Find the gradient of the curve at the point \(( 0,2 )\).
CAIE P3 2013 November Q5
8 marks Challenging +1.2
5 It is given that \(\int _ { 0 } ^ { p } 4 x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x = 9\), where \(p\) is a positive constant.
  1. Show that \(p = 2 \ln \left( \frac { 8 p + 16 } { 7 } \right)\).
  2. Use an iterative process based on the equation in part (i) to find the value of \(p\) correct to 3 significant figures. Use a starting value of 3.5 and give the result of each iteration to 5 significant figures.
CAIE P3 2013 November Q6
9 marks Standard +0.3
6 Two planes have equations \(3 x - y + 2 z = 9\) and \(x + y - 4 z = - 1\).
  1. Find the acute angle between the planes.
  2. Find a vector equation of the line of intersection of the planes.
CAIE P3 2013 November Q7
10 marks Standard +0.8
7
  1. Given that \(\sec \theta + 2 \operatorname { cosec } \theta = 3 \operatorname { cosec } 2 \theta\), show that \(2 \sin \theta + 4 \cos \theta = 3\).
  2. Express \(2 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  3. Hence solve the equation \(\sec \theta + 2 \operatorname { cosec } \theta = 3 \operatorname { cosec } 2 \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P3 2013 November Q8
10 marks Standard +0.3
8
  1. Express \(\frac { 7 x ^ { 2 } + 8 } { ( 1 + x ) ^ { 2 } ( 2 - 3 x ) }\) in partial fractions.
  2. Hence expand \(\frac { 7 x ^ { 2 } + 8 } { ( 1 + x ) ^ { 2 } ( 2 - 3 x ) }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2013 November Q9
10 marks Standard +0.3
9
  1. Without using a calculator, use the formula for the solution of a quadratic equation to solve $$( 2 - \mathrm { i } ) z ^ { 2 } + 2 z + 2 + \mathrm { i } = 0$$ Give your answers in the form \(a + b \mathrm { i }\).
  2. The complex number \(w\) is defined by \(w = 2 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }\). In an Argand diagram, the points \(A , B\) and \(C\) represent the complex numbers \(w , w ^ { 3 }\) and \(w ^ { * }\) respectively (where \(w ^ { * }\) denotes the complex conjugate of \(w\) ). Draw the Argand diagram showing the points \(A , B\) and \(C\), and calculate the area of triangle \(A B C\).
CAIE P3 2013 November Q10
10 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{3621a7e5-a3fb-42c1-828d-7068fddbf2f9-3_677_691_781_724} A particular solution of the differential equation $$3 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = 4 \left( y ^ { 3 } + 1 \right) \cos ^ { 2 } x$$ is such that \(y = 2\) when \(x = 0\). The diagram shows a sketch of the graph of this solution for \(0 \leqslant x \leqslant 2 \pi\); the graph has stationary points at \(A\) and \(B\). Find the \(y\)-coordinates of \(A\) and \(B\), giving each coordinate correct to 1 decimal place.
CAIE P3 2014 November Q1
3 marks Moderate -0.8
1 Use logarithms to solve the equation \(\mathrm { e } ^ { x } = 3 ^ { x - 2 }\), giving your answer correct to 3 decimal places.
CAIE P3 2014 November Q2
5 marks Standard +0.3
2
  1. Use the trapezium rule with 3 intervals to estimate the value of $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 2 } { 3 } \pi } \operatorname { cosec } x d x$$ giving your answer correct to 2 decimal places.
  2. Using a sketch of the graph of \(y = \operatorname { cosec } x\), explain whether the trapezium rule gives an overestimate or an underestimate of the true value of the integral in part (i).
CAIE P3 2014 November Q3
5 marks Moderate -0.5
3 The polynomial \(a x ^ { 3 } + b x ^ { 2 } + x + 3\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 3 x + 1 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 21 . Find the values of \(a\) and \(b\).
CAIE P3 2014 November Q4
7 marks Standard +0.3
4 The parametric equations of a curve are $$x = \frac { 1 } { \cos ^ { 3 } t } , \quad y = \tan ^ { 3 } t$$ where \(0 \leqslant t < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin t\).
  2. Hence show that the equation of the tangent to the curve at the point with parameter \(t\) is \(y = x \sin t - \tan t\).
CAIE P3 2014 November Q6
9 marks Standard +0.8
6 It is given that \(\int _ { 1 } ^ { a } \ln ( 2 x ) \mathrm { d } x = 1\), where \(a > 1\).
  1. Show that \(a = \frac { 1 } { 2 } \exp \left( 1 + \frac { \ln 2 } { a } \right)\), where \(\exp ( x )\) denotes \(\mathrm { e } ^ { x }\).
  2. Use the iterative formula $$a _ { n + 1 } = \frac { 1 } { 2 } \exp \left( 1 + \frac { \ln 2 } { a _ { n } } \right)$$ to determine the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2014 November Q7
9 marks Standard +0.3
7 In a certain country the government charges tax on each litre of petrol sold to motorists. The revenue per year is \(R\) million dollars when the rate of tax is \(x\) dollars per litre. The variation of \(R\) with \(x\) is modelled by the differential equation $$\frac { \mathrm { d } R } { \mathrm {~d} x } = R \left( \frac { 1 } { x } - 0.57 \right)$$ where \(R\) and \(x\) are taken to be continuous variables. When \(x = 0.5 , R = 16.8\).
  1. Solve the differential equation and obtain an expression for \(R\) in terms of \(x\).
  2. This model predicts that \(R\) cannot exceed a certain amount. Find this maximum value of \(R\).
CAIE P3 2014 November Q8
9 marks Standard +0.3
8
  1. By first expanding \(\sin ( 2 \theta + \theta )\), show that $$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$
  2. Show that, after making the substitution \(x = \frac { 2 \sin \theta } { \sqrt { 3 } }\), the equation \(x ^ { 3 } - x + \frac { 1 } { 6 } \sqrt { } 3 = 0\) can be written in the form \(\sin 3 \theta = \frac { 3 } { 4 }\).
  3. Hence solve the equation $$x ^ { 3 } - x + \frac { 1 } { 6 } \sqrt { } 3 = 0$$ giving your answers correct to 3 significant figures.
CAIE P3 2014 November Q9
10 marks Standard +0.3
9 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } - 8 x + 9 } { ( 1 - x ) ( 2 - x ) ^ { 2 } }\).
  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 \(x ^ { 2 }\).