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CAIE P3 2014 November Q10
10 marks Standard +0.8
10 The line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } - 9 \mathbf { j } + 9 \mathbf { k } + \lambda ( - 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\). The point \(A\) has position vector \(3 \mathbf { i } + 8 \mathbf { j } + 5 \mathbf { k }\).
  1. Show that the length of the perpendicular from \(A\) to \(l\) is 15 .
  2. The line \(l\) lies in the plane with equation \(a x + b y - 3 z + 1 = 0\), where \(a\) and \(b\) are constants. Find the values of \(a\) and \(b\).
CAIE P3 2014 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| 3 x - 1 | < | 2 x + 5 |\).
CAIE P3 2014 November Q2
5 marks Standard +0.3
2 A curve is defined for \(0 < \theta < \frac { 1 } { 2 } \pi\) by the parametric equations $$x = \tan \theta , \quad y = 2 \cos ^ { 2 } \theta \sin \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \cos ^ { 5 } \theta - 4 \cos ^ { 3 } \theta\).
CAIE P3 2014 November Q3
7 marks Moderate -0.3
3 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + b x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) and \(( x + 2 )\) are factors of \(\mathrm { p } ( x )\).
  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 \(\left( x ^ { 2 } + 1 \right)\).
CAIE P3 2014 November Q4
7 marks Standard +0.3
4
  1. Show that \(\cos \left( \theta - 60 ^ { \circ } \right) + \cos \left( \theta + 60 ^ { \circ } \right) \equiv \cos \theta\).
  2. Given that \(\frac { \cos \left( 2 x - 60 ^ { \circ } \right) + \cos \left( 2 x + 60 ^ { \circ } \right) } { \cos \left( x - 60 ^ { \circ } \right) + \cos \left( x + 60 ^ { \circ } \right) } = 3\), find the exact value of \(\cos x\).
CAIE P3 2014 November Q5
7 marks Standard +0.3
5 The complex numbers \(w\) and \(z\) are defined by \(w = 5 + 3 \mathrm { i }\) and \(z = 4 + \mathrm { i }\).
  1. Express \(\frac { \mathrm { i } w } { z }\) in the form \(x + \mathrm { i } y\), showing all your working and giving the exact values of \(x\) and \(y\).
  2. Find \(w z\) and hence, by considering arguments, show that $$\tan ^ { - 1 } \left( \frac { 3 } { 5 } \right) + \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right) = \frac { 1 } { 4 } \pi$$
CAIE P3 2014 November Q6
8 marks Moderate -0.3
6 It is given that \(I = \int _ { 0 } ^ { 0.3 } \left( 1 + 3 x ^ { 2 } \right) ^ { - 2 } \mathrm {~d} x\).
  1. Use the trapezium rule with 3 intervals to find an approximation to \(I\), giving the answer correct to 3 decimal places.
  2. For small values of \(x , \left( 1 + 3 x ^ { 2 } \right) ^ { - 2 } \approx 1 + a x ^ { 2 } + b x ^ { 4 }\). Find the values of the constants \(a\) and \(b\). Hence, by evaluating \(\int _ { 0 } ^ { 0.3 } \left( 1 + a x ^ { 2 } + b x ^ { 4 } \right) \mathrm { d } x\), find a second approximation to \(I\), giving the answer correct to 3 decimal places.
CAIE P3 2014 November Q7
8 marks Standard +0.8
7 The equations of two straight lines are $$\mathbf { r } = \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = a \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } + 3 a \mathbf { k } )$$ where \(a\) is a constant.
  1. Show that the lines intersect for all values of \(a\).
  2. Given that the point of intersection is at a distance of 9 units from the origin, find the possible values of \(a\).
CAIE P3 2014 November Q8
9 marks Standard +0.3
8 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 5 } x y ^ { \frac { 1 } { 2 } } \sin \left( \frac { 1 } { 3 } x \right)$$
  1. Find the general solution, giving \(y\) in terms of \(x\).
  2. Given that \(y = 100\) when \(x = 0\), find the value of \(y\) when \(x = 25\).
CAIE P3 2014 November Q9
10 marks Standard +0.3
9
  1. Sketch the curve \(y = \ln ( x + 1 )\) and hence, by sketching a second curve, show that the equation $$x ^ { 3 } + \ln ( x + 1 ) = 40$$ has exactly one real root. State the equation of the second curve.
  2. Verify by calculation that the root lies between 3 and 4 .
  3. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 40 - \ln \left( x _ { n } + 1 \right) \right)$$ with a suitable starting value, to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
  4. Deduce the root of the equation $$\left( \mathrm { e } ^ { y } - 1 \right) ^ { 3 } + y = 40$$ giving the answer correct to 2 decimal places.
CAIE P3 2014 November Q10
10 marks Standard +0.3
10 By first using the substitution \(u = \mathrm { e } ^ { x }\), show that $$\int _ { 0 } ^ { \ln 4 } \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { 2 x } + 3 \mathrm { e } ^ { x } + 2 } \mathrm {~d} x = \ln \left( \frac { 8 } { 5 } \right)$$
CAIE P3 2015 November Q1
4 marks Standard +0.8
1 Solve the inequality \(| 2 x - 5 | > 3 | 2 x + 1 |\).
CAIE P3 2015 November Q2
5 marks Moderate -0.3
2 Using the substitution \(u = 3 ^ { x }\), solve the equation \(3 ^ { x } + 3 ^ { 2 x } = 3 ^ { 3 x }\) giving your answer correct to 3 significant figures.
CAIE P3 2015 November Q3
6 marks Standard +0.8
3 The angles \(\theta\) and \(\phi\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), and are such that $$\tan ( \theta - \phi ) = 3 \quad \text { and } \quad \tan \theta + \tan \phi = 1$$ Find the possible values of \(\theta\) and \(\phi\).
CAIE P3 2015 November Q4
7 marks Standard +0.3
4 The equation \(x ^ { 3 } - x ^ { 2 } - 6 = 0\) has one real root, denoted by \(\alpha\).
  1. Find by calculation the pair of consecutive integers between which \(\alpha\) lies.
  2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( x _ { n } + \frac { 6 } { x _ { n } } \right)$$ converges, then it converges to \(\alpha\).
  3. Use this iterative formula to determine \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2015 November Q5
7 marks Standard +0.8
5 The equation of a curve is \(y = \mathrm { e } ^ { - 2 x } \tan x\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
  1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that it can be written in the form \(\mathrm { e } ^ { - 2 x } ( a + b \tan x ) ^ { 2 }\), where \(a\) and \(b\) are constants.
  2. Explain why the gradient of the curve is never negative.
  3. Find the value of \(x\) for which the gradient is least.
CAIE P3 2015 November Q6
8 marks Moderate -0.3
6 The polynomial \(8 x ^ { 3 } + a x ^ { 2 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) 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, factorise \(\mathrm { p } ( x )\) completely.
CAIE P3 2015 November Q7
9 marks Standard +0.3
7 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 4 \end{array} \right)$$ The plane \(m\) is perpendicular to \(A B\) and contains the point \(C\).
  1. Find a vector equation for the line passing through \(A\) and \(B\).
  2. Obtain the equation of the plane \(m\), giving your answer in the form \(a x + b y + c z = d\).
  3. The line through \(A\) and \(B\) intersects the plane \(m\) at the point \(N\). Find the position vector of \(N\) and show that \(C N = \sqrt { } ( 13 )\).
CAIE P3 2015 November Q8
9 marks Standard +0.3
8 The variables \(x\) and \(\theta\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} \theta } = ( x + 2 ) \sin ^ { 2 } 2 \theta$$ and it is given that \(x = 0\) when \(\theta = 0\). Solve the differential equation and calculate the value of \(x\) when \(\theta = \frac { 1 } { 4 } \pi\), giving your answer correct to 3 significant figures.
CAIE P3 2015 November Q9
10 marks Standard +0.8
9 The complex number 3 - i is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).
  1. On an Argand diagram with origin \(O\), show the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. What type of quadrilateral is \(O A B C\) ?
  2. Showing your working and without using a calculator, express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  3. By considering the argument of \(\frac { u ^ { * } } { u }\), prove that $$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)$$
CAIE P3 2015 November Q10
10 marks Standard +0.8
10 \includegraphics[max width=\textwidth, alt={}, center]{5d76d7cb-c2cb-4fa4-8133-d5d43702b293-3_366_764_1914_687} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).
  1. Find the exact value of the \(x\)-coordinate of \(M\).
  2. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.
CAIE P3 2015 November Q10
10 marks Standard +0.8
10 \includegraphics[max width=\textwidth, alt={}, center]{efa44efb-9350-4d54-b5e1-4f722781a5f3-3_366_764_1914_687} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).
  1. Find the exact value of the \(x\)-coordinate of \(M\).
  2. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.
CAIE P3 2015 November Q1
2 marks Easy -1.2
1 Sketch the graph of \(y = \mathrm { e } ^ { a x } - 1\) where \(a\) is a positive constant.
CAIE P3 2015 November Q2
3 marks Moderate -0.8
2 Given that \(\sqrt [ 3 ] { } ( 1 + 9 x ) \approx 1 + 3 x + a x ^ { 2 } + b x ^ { 3 }\) for small values of \(x\), find the values of the coefficients \(a\) and \(b\).
CAIE P3 2015 November Q3
6 marks Standard +0.3
3 A curve has equation $$y = \frac { 2 - \tan x } { 1 + \tan x }$$ Find the equation of the tangent to the curve at the point for which \(x = \frac { 1 } { 4 } \pi\), giving the answer in the form \(y = m x + c\) where \(c\) is correct to 3 significant figures.