Questions P3 (1203 questions)

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CAIE P3 2017 November Q7
4 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
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
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
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
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
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
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
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.
CAIE P3 2018 November Q5
Moderate -0.3
5 The coordinates \(( x , y )\) of a general point on a curve satisfy the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 2 - x ^ { 2 } \right) y$$ The curve passes through the point \(( 1,1 )\). Find the equation of the curve, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2018 November Q6
Challenging +1.2
6
  1. Show that the equation ( \(\sqrt { } 2\) ) \(\operatorname { cosec } x + \cot x = \sqrt { } 3\) can be expressed in the form \(R \sin ( x - \alpha ) = \sqrt { } 2\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence solve the equation \(( \sqrt { } 2 ) \operatorname { cosec } x + \cot x = \sqrt { } 3\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2018 November Q7
4 marks Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{c861e691-66da-4269-9057-4a343be9835e-12_357_565_260_790} The diagram shows the curve \(y = 5 \sin ^ { 2 } x \cos ^ { 3 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.
  2. Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\). [4]
CAIE P3 2018 November Q8
Standard +0.3
8
  1. Showing all necessary working, express the complex number \(\frac { 2 + 3 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the values of \(r\) and \(\theta\) correct to 3 significant figures.
  2. On an Argand diagram sketch the locus of points representing complex numbers \(z\) satisfying the equation \(| z - 3 + 2 i | = 1\). Find the least value of \(| z |\) for points on this locus, giving your answer in an exact form.
CAIE P3 2018 November Q9
Standard +0.3
9 Let \(\mathrm { f } ( x ) = \frac { 6 x ^ { 2 } + 8 x + 9 } { ( 2 - x ) ( 3 + 2 x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence, showing all necessary working, show that \(\int _ { - 1 } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x = 1 + \frac { 1 } { 2 } \ln \left( \frac { 3 } { 4 } \right)\).
CAIE P3 2018 November Q10
Standard +0.3
10 The planes \(m\) and \(n\) have equations \(3 x + y - 2 z = 10\) and \(x - 2 y + 2 z = 5\) respectively. The line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\).
  1. Show that \(l\) is parallel to \(m\).
  2. Calculate the acute angle between the planes \(m\) and \(n\).
  3. A point \(P\) lies on the line \(l\). The perpendicular distance of \(P\) from the plane \(n\) is equal to 2 . Find the position vectors of the two possible positions of \(P\).
    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 2018 November Q1
Standard +0.3
1 Solve the inequality \(3 | 2 x - 1 | > | x + 4 |\).
CAIE P3 2018 November Q2
4 marks Moderate -0.3
2 Showing all necessary working, solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) + \cos \theta = 2 \sin \theta\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\). [4]
CAIE P3 2018 November Q3
Moderate -0.3
3
  1. Find \(\int \frac { \ln x } { x ^ { 3 } } \mathrm {~d} x\).
  2. Hence show that \(\int _ { 1 } ^ { 2 } \frac { \ln x } { x ^ { 3 } } \mathrm {~d} x = \frac { 1 } { 16 } ( 3 - \ln 4 )\).
CAIE P3 2018 November Q4
Moderate -0.3
4 Showing all necessary working, solve the equation $$\frac { \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { \mathrm { e } ^ { x } + 1 } = 4$$ giving your answer correct to 3 decimal places.
CAIE P3 2018 November Q5
Standard +0.3
5 The equation of a curve is \(y = x \ln ( 8 - x )\). The gradient of the curve is equal to 1 at only one point, when \(x = a\).
  1. Show that \(a\) satisfies the equation \(x = 8 - \frac { 8 } { \ln ( 8 - x ) }\).
  2. Verify by calculation that \(a\) lies between 2.9 and 3.1.
  3. Use an iterative formula based on the equation in part (i) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2018 November Q6
Standard +0.3
6 A certain curve is such that its gradient at a general point with coordinates \(( x , y )\) is proportional to \(\frac { y ^ { 2 } } { x }\). The curve passes through the points with coordinates \(( 1,1 )\) and (e, 2). By setting up and solving a differential equation, find the equation of the curve, expressing \(y\) in terms of \(x\).
CAIE P3 2018 November Q7
Standard +0.3
7 A curve has equation \(y = \frac { 3 \cos x } { 2 + \sin x }\), for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  1. Find the exact coordinates of the stationary point of the curve.
  2. The constant \(a\) is such that \(\int _ { 0 } ^ { a } \frac { 3 \cos x } { 2 + \sin x } \mathrm {~d} x = 1\). Find the value of \(a\), giving your answer correct to 3 significant figures.
    \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 7 x ^ { 2 } - 15 x + 8 } { ( 1 - 2 x ) ( 2 - x ) ^ { 2 } }\).
CAIE P3 2018 November Q9
Standard +0.3
9
    1. Without using a calculator, express the complex number \(\frac { 2 + 6 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    2. Hence, without using a calculator, express \(\frac { 2 + 6 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - 3 \mathrm { i } | \leqslant 1\) and \(\operatorname { Re } z \leqslant 0\), where \(\operatorname { Re } z\) denotes the real part of \(z\). Find the greatest value of \(\arg z\) for points in this region, giving your answer in radians correct to 2 decimal places.
CAIE P3 2018 November Q10
Standard +0.3
10 The line \(l\) has equation \(\mathbf { r } = 5 \mathbf { i } - 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )\). The plane \(p\) has equation $$( \mathbf { r } - \mathbf { i } - 2 \mathbf { j } ) \cdot ( 3 \mathbf { i } + \mathbf { j } + \mathbf { k } ) = 0$$ The line \(l\) intersects the plane \(p\) at the point \(A\).
  1. Find the position vector of \(A\).
  2. Calculate the acute angle between \(l\) and \(p\).
  3. Find the equation of the line which lies in \(p\) and intersects \(l\) at right angles.
    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 2018 November Q5
Moderate -0.3
5 The coordinates \(( x , y )\) of a general point on a curve satisfy the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 2 - x ^ { 2 } \right) y .$$ The curve passes through the point \(( 1,1 )\). Find the equation of the curve, obtaining an expression for \(y\) in terms of \(x\).