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CAIE P3 2019 June Q9
10 marks Standard +0.3
9 The points \(A\) and \(B\) have position vectors \(\mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and \(3 \mathbf { i } + \mathbf { j } + \mathbf { k }\) respectively. The line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\).
  1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
  2. The plane \(m\) is perpendicular to \(A B\) and passes through the mid-point of \(A B\). The plane \(m\) intersects the line \(l\) at the point \(P\). Find the equation of \(m\) and the position vector of \(P\).
CAIE P3 2019 June Q10
12 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{772393d7-6e81-4b99-913a-63c9f87d1af2-16_524_689_260_726} The diagram shows the curve \(y = \sin 3 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\) and its minimum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  1. By expanding \(\sin ( 3 x + x )\) and \(\sin ( 3 x - x )\) show that $$\sin 3 x \cos x = \frac { 1 } { 2 } ( \sin 4 x + \sin 2 x ) .$$
  2. Using the result of part (i) and showing all necessary working, find the exact area of the region \(R\).
  3. Using the result of part (i), express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\cos 2 x\) and hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 2019 June Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation \(5 ^ { 3 - 2 x } = 4 \left( 7 ^ { x } \right)\), giving your answer correct to 3 decimal places.
CAIE P3 2019 June Q2
5 marks Standard +0.3
2 Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x ^ { 2 } \cos 2 x \mathrm {~d} x = \frac { 1 } { 32 } \left( \pi ^ { 2 } - 8 \right)\).
CAIE P3 2019 June Q3
7 marks Standard +0.3
3 Let \(f ( \theta ) = \frac { 1 - \cos 2 \theta + \sin 2 \theta } { 1 + \cos 2 \theta + \sin 2 \theta }\).
  1. Show that \(\mathrm { f } ( \theta ) = \tan \theta\).
  2. Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \mathrm { f } ( \theta ) \mathrm { d } \theta = \frac { 1 } { 2 } \ln \frac { 3 } { 2 }\).
CAIE P3 2019 June Q4
7 marks Standard +0.3
4 The equation of a curve is \(y = \frac { 1 + \mathrm { e } ^ { - x } } { 1 - \mathrm { e } ^ { - x } }\), for \(x > 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is always negative.
  2. The gradient of the curve is equal to - 1 when \(x = a\). Show that \(a\) satisfies the equation \(\mathrm { e } ^ { 2 a } - 4 \mathrm { e } ^ { a } + 1 = 0\). Hence find the exact value of \(a\).
CAIE P3 2019 June Q5
7 marks Standard +0.3
5 The variables \(x\) and \(y\) satisfy the differential equation $$( x + 1 ) y \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ^ { 2 } + 5$$ It is given that \(y = 2\) when \(x = 0\). Solve the differential equation obtaining an expression for \(y ^ { 2 }\) in terms of \(x\).
CAIE P3 2019 June Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{87392b1c-3683-45b4-8d55-36760b5f0cc1-10_547_531_260_806} The diagram shows the curve \(y = x ^ { 4 } - 2 x ^ { 3 } - 7 x - 6\). The curve intersects the \(x\)-axis at the points \(( a , 0 )\) and \(( b , 0 )\), where \(a < b\). It is given that \(b\) is an integer.
  1. Find the value of \(b\).
  2. Hence show that \(a\) satisfies the equation \(a = - \frac { 1 } { 3 } \left( 2 + a ^ { 2 } + a ^ { 3 } \right)\).
  3. Use an iterative formula based on the equation in part (ii) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2019 June Q7
7 marks Standard +0.3
7 The curve \(y = \sin \left( x + \frac { 1 } { 3 } \pi \right) \cos x\) has two stationary points in the interval \(0 \leqslant x \leqslant \pi\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. By considering the formula for \(\cos ( A + B )\), show that, at the stationary points on the curve, \(\cos \left( 2 x + \frac { 1 } { 3 } \pi \right) = 0\).
  3. Hence find the exact \(x\)-coordinates of the stationary points.
CAIE P3 2019 June Q8
9 marks Standard +0.3
8 Throughout this question the use of a calculator is not permitted.
The complex number \(u\) is defined by $$u = \frac { 4 \mathrm { i } } { 1 - ( \sqrt { } 3 ) \mathrm { i } }$$
  1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
  2. Find the exact modulus and argument of \(u\).
  3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | < 2\) and \(| z - u | < | z |\).
CAIE P3 2019 June Q9
10 marks Standard +0.8
9 Let \(\mathrm { f } ( x ) = \frac { 2 x ( 5 - x ) } { ( 3 + x ) ( 1 - x ) ^ { 2 } }\).
  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 \(x ^ { 3 }\).
CAIE P3 2019 June Q10
11 marks Standard +0.3
10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )\).
  1. The point \(P\) has position vector \(4 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\). Find the length of the perpendicular from \(P\) to \(l\).
  2. It is given that \(l\) lies in the plane with equation \(a x + b y + 2 z = 13\), where \(a\) and \(b\) are constants. Find the values of \(a\) and \(b\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 2016 March Q1
3 marks Standard +0.3
1 Solve the equation \(\ln \left( x ^ { 2 } + 4 \right) = 2 \ln x + \ln 4\), giving your answer in an exact form.
CAIE P3 2016 March Q2
6 marks Standard +0.3
2 Express the equation \(\tan \left( \theta + 45 ^ { \circ } \right) - 2 \tan \left( \theta - 45 ^ { \circ } \right) = 4\) as a quadratic equation in \(\tan \theta\). Hence solve this equation for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2016 March Q3
6 marks Standard +0.3
3 The equation \(x ^ { 5 } - 3 x ^ { 3 } + x ^ { 2 } - 4 = 0\) has one positive root.
  1. Verify by calculation that this root lies between 1 and 2 .
  2. Show that the equation can be rearranged in the form $$\left. x = \sqrt [ 3 ] { ( } 3 x + \frac { 4 } { x ^ { 2 } } - 1 \right)$$
  3. Use an iterative formula based on this rearrangement to determine the positive root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2016 March Q4
7 marks Moderate -0.5
4 The polynomial \(4 x ^ { 3 } + a x + 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value,
    (a) factorise \(\mathrm { p } ( x )\),
    (b) solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.
CAIE P3 2016 March Q5
7 marks Standard +0.8
5 Let \(I = \int _ { 0 } ^ { 1 } \frac { 9 } { \left( 3 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x\).
  1. Using the substitution \(x = ( \sqrt { } 3 ) \tan \theta\), show that \(I = \sqrt { } 3 \int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta\).
  2. Hence find the exact value of \(I\).
CAIE P3 2016 March Q6
8 marks Challenging +1.2
6 A curve has equation $$\sin y \ln x = x - 2 \sin y$$ for \(- \frac { 1 } { 2 } \pi \leqslant y \leqslant \frac { 1 } { 2 } \pi\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Hence find the exact \(x\)-coordinate of the point on the curve at which the tangent is parallel to the \(x\)-axis.
CAIE P3 2016 March Q7
8 marks Standard +0.3
7 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { x + y }$$ and it is given that \(y = 0\) when \(x = 0\).
  1. Solve the differential equation and obtain an expression for \(y\) in terms of \(x\).
  2. Explain briefly why \(x\) can only take values less than 1 .
CAIE P3 2016 March Q8
9 marks Standard +0.3
8 The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right)\). The plane \(p\) has equation \(\mathbf { r } \cdot \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) = 6\).
  1. Show that \(l\) is parallel to \(p\).
  2. A line \(m\) lies in the plane \(p\) and is perpendicular to \(l\). The line \(m\) passes through the point with coordinates (5, 3, 1). Find a vector equation for \(m\).
CAIE P3 2016 March Q9
10 marks Standard +0.3
9 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 3 } + 6 x - 8 } { x \left( x ^ { 2 } + 2 \right) }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(A + \frac { B } { x } + \frac { C x + D } { x ^ { 2 } + 2 }\).
  2. Show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 3 - \ln 4\).
CAIE P3 2016 March Q10
11 marks Standard +0.3
10
  1. Find the complex number \(z\) satisfying the equation \(z ^ { * } + 1 = 2 \mathrm { i } z\), where \(z ^ { * }\) denotes the complex conjugate of \(z\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z + 1 - 3 \mathrm { i } | \leqslant 1\) and \(\operatorname { Im } z \geqslant 3\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\).
    2. Determine the difference between the greatest and least values of \(\arg z\) for points lying in this region.
CAIE P3 2017 March Q1
3 marks Moderate -0.5
1 Solve the equation \(\ln \left( 1 + 2 ^ { x } \right) = 2\), giving your answer correct to 3 decimal places.
CAIE P3 2017 March Q2
4 marks Standard +0.8
2 Solve the inequality \(| x - 4 | < 2 | 3 x + 1 |\).
CAIE P3 2017 March Q3
7 marks Moderate -0.3
3
  1. By sketching suitable graphs, show that the equation \(\mathrm { e } ^ { - \frac { 1 } { 2 } x } = 4 - x ^ { 2 }\) has one positive root and one negative root.
  2. Verify by calculation that the negative root lies between - 1 and - 1.5 .
  3. Use the iterative formula \(x _ { n + 1 } = - \sqrt { } \left( 4 - e ^ { - \frac { 1 } { 2 } x _ { n } } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.