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CAIE P3 2016 November Q10
10 marks Standard +0.8
10 The line \(l\) has vector equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\).
  1. Find the position vectors of the two points on the line whose distance from the origin is \(\sqrt { } ( 10 )\).
  2. The plane \(p\) has equation \(a x + y + z = 5\), where \(a\) is a constant. The acute angle between the line \(l\) and the plane \(p\) is equal to \(\sin ^ { - 1 } \left( \frac { 2 } { 3 } \right)\). Find the possible values of \(a\).
CAIE P3 2017 November Q1
3 marks Moderate -0.8
1 Find the quotient and remainder when \(x ^ { 4 }\) is divided by \(x ^ { 2 } + 2 x - 1\).
CAIE P3 2017 November Q2
5 marks Moderate -0.3
2 Two variable quantities \(x\) and \(y\) are believed to satisfy an equation of the form \(y = C \left( a ^ { x } \right)\), where \(C\) and \(a\) are constants. An experiment produced four pairs of values of \(x\) and \(y\). The table below gives the corresponding values of \(x\) and \(\ln y\).
\(x\)0.91.62.43.2
\(\ln y\)1.71.92.32.6
By plotting \(\ln y\) against \(x\) for these four pairs of values and drawing a suitable straight line, estimate the values of \(C\) and \(a\). Give your answers correct to 2 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{be0fb208-2ef1-4fae-84ff-ad2e8bf2dcc5-03_759_944_749_596}
CAIE P3 2017 November Q3
6 marks Standard +0.3
3 The equation \(x ^ { 3 } = 3 x + 7\) has one real root, denoted by \(\alpha\).
  1. Show by calculation that \(\alpha\) lies between 2 and 3 .
    Two iterative formulae, \(A\) and \(B\), derived from this equation are as follows: $$\begin{aligned} & x _ { n + 1 } = \left( 3 x _ { n } + 7 \right) ^ { \frac { 1 } { 3 } } \\ & x _ { n + 1 } = \frac { x _ { n } ^ { 3 } - 7 } { 3 } \end{aligned}$$ Each formula is used with initial value \(x _ { 1 } = 2.5\).
  2. Show that one of these formulae produces a sequence which fails to converge, and use the other formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2017 November Q4
7 marks Standard +0.3
4
  1. Prove the identity \(\tan \left( 45 ^ { \circ } + x \right) + \tan \left( 45 ^ { \circ } - x \right) \equiv 2 \sec 2 x\).
  2. Hence sketch the graph of \(y = \tan \left( 45 ^ { \circ } + x \right) + \tan \left( 45 ^ { \circ } - x \right)\) for \(0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).
CAIE P3 2017 November Q5
8 marks Standard +0.3
5 The equation of a curve is \(2 x ^ { 4 } + x y ^ { 3 } + y ^ { 4 } = 10\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 x ^ { 3 } + y ^ { 3 } } { 3 x y ^ { 2 } + 4 y ^ { 3 } }\).
  2. Hence show that there are two points on the curve at which the tangent is parallel to the \(x\)-axis and find the coordinates of these points.
CAIE P3 2017 November Q6
8 marks Moderate -0.3
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 \cos ^ { 2 } y \tan x ,$$ for \(0 \leqslant x < \frac { 1 } { 2 } \pi\), and \(x = 0\) when \(y = \frac { 1 } { 4 } \pi\). Solve this differential equation and find the value of \(x\) when \(y = \frac { 1 } { 3 } \pi\).
CAIE P3 2017 November Q7
9 marks Moderate -0.3
7
  1. The complex number \(u\) is given by \(u = 8 - 15 \mathrm { i }\). Showing all necessary working, find the two square roots of \(u\). Give answers in the form \(a + \mathrm { i } b\), where the numbers \(a\) and \(b\) are real and exact.
  2. On an Argand diagram, shade the region whose points represent complex numbers satisfying both the inequalities \(| z - 2 - i | \leqslant 2\) and \(0 \leqslant \arg ( z - i ) \leqslant \frac { 1 } { 4 } \pi\).
    [0pt] [4] \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 9 x - 8 } { ( x + 2 ) ( 2 x - 1 ) }\).
    1. Express \(\mathrm { f } ( x )\) in the form \(A + \frac { B } { x + 2 } + \frac { C } { 2 x - 1 }\).
    2. Hence show that \(\int _ { 1 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = 6 + \frac { 1 } { 2 } \ln \left( \frac { 16 } { 7 } \right)\).
CAIE P3 2017 November Q9
9 marks Standard +0.8
9 \includegraphics[max width=\textwidth, alt={}, center]{be0fb208-2ef1-4fae-84ff-ad2e8bf2dcc5-16_446_956_260_593} The diagram shows the curve \(y = \left( 1 + x ^ { 2 } \right) \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) for \(x \geqslant 0\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 2\).
  1. Find the exact values of the \(x\)-coordinates of the stationary points of the curve.
  2. Show that the exact value of the area of \(R\) is \(18 - \frac { 42 } { \mathrm { e } }\).
CAIE P3 2017 November Q10
11 marks Standard +0.3
10 The equations of two lines \(l\) and \(m\) are \(\mathbf { r } = 3 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + \lambda ( - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\) respectively.
  1. Show that the lines do not intersect.
  2. Calculate the acute angle between the directions of the lines.
  3. Find the equation of the plane which passes through the point \(( 3 , - 2 , - 1 )\) and which is parallel to both \(l\) and \(m\). Give your answer in the form \(a x + b y + c z = d\).
CAIE P3 2017 November Q1
4 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{746d2c39-7d78-4478-bc36-15ea5e3ba72a-02_460_807_258_667} The diagram shows a sketch of the curve \(y = \frac { 3 } { \sqrt { } \left( 9 - x ^ { 3 } \right) }\) for values of \(x\) from - 1.2 to 1.2 .
  1. Use the trapezium rule, with two intervals, to estimate the value of $$\int _ { - 1.2 } ^ { 1.2 } \frac { 3 } { \sqrt { \left( 9 - x ^ { 3 } \right) } } \mathrm { d } x$$ giving your answer correct to 2 decimal places.
  2. Explain, with reference to the diagram, why the trapezium rule may be expected to give a good approximation to the true value of the integral in this case.
CAIE P3 2017 November Q2
5 marks Standard +0.3
2 Showing all necessary working, solve the equation \(2 \log _ { 2 } x = 3 + \log _ { 2 } ( x + 1 )\), giving your answer correct to 3 significant figures.
CAIE P3 2017 November Q3
5 marks Standard +0.8
3 By expressing the equation \(\tan \left( \theta + 60 ^ { \circ } \right) + \tan \left( \theta - 60 ^ { \circ } \right) = \cot \theta\) in terms of \(\tan \theta\) only, solve the equation for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P3 2017 November Q4
7 marks Standard +0.3
4 The curve with equation \(y = \frac { 2 - \sin x } { \cos x }\) has one stationary point in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
  1. Find the exact coordinates of this point.
  2. Determine whether this point is a maximum or a minimum point.
CAIE P3 2017 November Q5
7 marks Moderate -0.3
5 The variables \(x\) and \(y\) satisfy the differential equation $$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( x + 2 )$$ and it is given that \(y = 2\) when \(x = 1\). Solve the differential equation and obtain an expression for \(y\) in terms of \(x\).
CAIE P3 2017 November Q6
8 marks Standard +0.8
6 The equation of a curve is \(x ^ { 3 } y - 3 x y ^ { 3 } = 2 a ^ { 4 }\), where \(a\) is a non-zero constant.
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 x ^ { 2 } y - 3 y ^ { 3 } } { 9 x y ^ { 2 } - x ^ { 3 } }\).
  2. Hence show that there are only two points on the curve at which the tangent is parallel to the \(x\)-axis and find the coordinates of these points.
CAIE P3 2017 November Q7
8 marks Standard +0.3
7 Throughout this question the use of a calculator is not permitted.
The complex number \(1 - ( \sqrt { } 3 ) \mathrm { i }\) is denoted by \(u\).
  1. Find the modulus and argument of \(u\).
  2. Show that \(u ^ { 3 } + 8 = 0\).
  3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - u | \leqslant 2\) and \(\operatorname { Re } z \geqslant 2\), where \(\operatorname { Re } z\) denotes the real part of \(z\).
    [0pt] [4] \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 8 x ^ { 2 } + 9 x + 8 } { ( 1 - x ) ( 2 x + 3 ) ^ { 2 } }\).
CAIE P3 2017 November Q9
10 marks Challenging +1.2
9 It is given that \(\int _ { 1 } ^ { a } x ^ { \frac { 1 } { 2 } } \ln x \mathrm {~d} x = 2\), where \(a > 1\).
  1. Show that \(a ^ { \frac { 3 } { 2 } } = \frac { 7 + 2 a ^ { \frac { 3 } { 2 } } } { 3 \ln a }\).
  2. Show by calculation that \(a\) lies between 2 and 4 .
  3. Use the iterative formula $$a _ { n + 1 } = \left( \frac { 7 + 2 a _ { n } ^ { \frac { 3 } { 2 } } } { 3 \ln a _ { n } } \right) ^ { \frac { 2 } { 3 } }$$ to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2017 November Q10
11 marks Standard +0.8
10 Two planes \(p\) and \(q\) have equations \(x + y + 3 z = 8\) and \(2 x - 2 y + z = 3\) respectively.
  1. Calculate the acute angle between the planes \(p\) and \(q\).
  2. The point \(A\) on the line of intersection of \(p\) and \(q\) has \(y\)-coordinate equal to 2 . Find the equation of the plane which contains the point \(A\) and is perpendicular to both the planes \(p\) and \(q\). Give your answer in the form \(a x + b y + c z = d\).
CAIE P3 2017 November Q2
5 marks Standard +0.3
2 Two variable quantities \(x\) and \(y\) are believed to satisfy an equation of the form \(y = C \left( a ^ { x } \right)\), where \(C\) and \(a\) are constants. An experiment produced four pairs of values of \(x\) and \(y\). The table below gives the corresponding values of \(x\) and \(\ln y\).
\(x\)0.91.62.43.2
\(\ln y\)1.71.92.32.6
By plotting \(\ln y\) against \(x\) for these four pairs of values and drawing a suitable straight line, estimate the values of \(C\) and \(a\). Give your answers correct to 2 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{21878d10-7f16-4dbb-86ef-65a7ba5eeafb-03_759_944_749_596}
CAIE P3 2017 November Q9
9 marks Standard +0.8
9 \includegraphics[max width=\textwidth, alt={}, center]{21878d10-7f16-4dbb-86ef-65a7ba5eeafb-16_446_956_260_593} The diagram shows the curve \(y = \left( 1 + x ^ { 2 } \right) \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) for \(x \geqslant 0\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 2\).
  1. Find the exact values of the \(x\)-coordinates of the stationary points of the curve.
  2. Show that the exact value of the area of \(R\) is \(18 - \frac { 42 } { \mathrm { e } }\).
CAIE P3 2018 November Q1
4 marks Standard +0.8
1 Find the set of values of \(x\) satisfying the inequality \(2 | 2 x - a | < | x + 3 a |\), where \(a\) is a positive constant. [4]
CAIE P3 2018 November Q2
4 marks Moderate -0.3
2 Showing all necessary working, solve the equation \(\frac { 2 \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } } = 4\), giving your answer correct to 2 decimal places.
CAIE P3 2018 November Q3
7 marks Standard +0.8
3
  1. By sketching a suitable pair of graphs, show that the equation \(x ^ { 3 } = 3 - x\) has exactly one real root.
  2. Show that if a sequence of real values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 3 } { 3 x _ { n } ^ { 2 } + 1 }$$ converges, then it converges to the root of the equation in part (i).
  3. Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2018 November Q4
7 marks Standard +0.3
4 The parametric equations of a curve are $$x = 2 \sin \theta + \sin 2 \theta , \quad y = 2 \cos \theta + \cos 2 \theta$$ where \(0 < \theta < \pi\).
  1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\).
  2. Hence find the exact coordinates of the point on the curve at which the tangent is parallel to the \(y\)-axis.