Questions — CAIE P3 (1110 questions)

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CAIE P3 2006 November Q9
10 marks Standard +0.3
9 The complex number \(u\) is given by $$u = \frac { 3 + \mathrm { i } } { 2 - \mathrm { i } }$$
  1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Find the modulus and argument of \(u\).
  3. Sketch an Argand diagram showing the point representing the complex number \(u\). Show on the same diagram the locus of the point representing the complex number \(z\) such that \(| z - u | = 1\).
  4. Using your diagram, calculate the least value of \(| z |\) for points on this locus.
CAIE P3 2006 November Q10
11 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{9e3b4c96-0989-4ffb-bd74-0e73b79ca45a-3_430_807_1375_667} The diagram shows the curve \(y = x \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The point \(M\) is a maximum point.
  1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(1 = 2 x \tan 2 x\).
  2. The equation in part (i) can be rearranged in the form \(x = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x } \right)\). Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x _ { n } } \right) ,$$ with initial value \(x _ { 1 } = 0.4\), to calculate the \(x\)-coordinate of \(M\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
  3. Use integration by parts to find the exact area of the region enclosed between the curve and the \(x\)-axis from 0 to \(\frac { 1 } { 4 } \pi\).
CAIE P3 2007 November Q1
4 marks Moderate -0.3
1 Find the exact value of the constant \(k\) for which \(\int _ { 1 } ^ { k } \frac { 1 } { 2 x - 1 } \mathrm {~d} x = 1\).
CAIE P3 2007 November Q2
4 marks Standard +0.3
2 The polynomial \(x ^ { 4 } + 3 x ^ { 2 } + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(x ^ { 2 } + x + 2\) is a factor of \(\mathrm { p } ( x )\). Find the value of \(a\) and the other quadratic factor of \(\mathrm { p } ( x )\).
CAIE P3 2007 November Q3
4 marks Standard +0.3
3 Use integration by parts to show that $$\int _ { 2 } ^ { 4 } \ln x \mathrm {~d} x = 6 \ln 2 - 2$$
CAIE P3 2007 November Q4
6 marks Standard +0.3
4 The curve with equation \(y = \mathrm { e } ^ { - x } \sin x\) has one stationary point for which \(0 \leqslant x \leqslant \pi\).
  1. Find the \(x\)-coordinate of this point.
  2. Determine whether this point is a maximum or a minimum point.
CAIE P3 2007 November Q5
7 marks Moderate -0.3
5
  1. Show that the equation $$\tan \left( 45 ^ { \circ } + x \right) - \tan x = 2$$ can be written in the form $$\tan ^ { 2 } x + 2 \tan x - 1 = 0$$
  2. Hence solve the equation $$\tan \left( 45 ^ { \circ } + x \right) - \tan x = 2$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P3 2007 November Q6
8 marks Moderate -0.3
6
  1. By sketching a suitable pair of graphs, show that the equation $$2 - x = \ln x$$ has only one root.
  2. Verify by calculation that this root lies between 1.4 and 1.7.
  3. Show that this root also satisfies the equation $$x = \frac { 1 } { 3 } ( 4 + x - 2 \ln x )$$
  4. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( 4 + x _ { n } - 2 \ln x _ { n } \right)$$ with initial value \(x _ { 1 } = 1.5\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2007 November Q7
10 marks Standard +0.3
7 The number of insects in a population \(t\) days after the start of observations is denoted by \(N\). The variation in the number of insects is modelled by a differential equation of the form $$\frac { \mathrm { d } N } { \mathrm {~d} t } = k N \cos ( 0.02 t )$$ where \(k\) is a constant and \(N\) is taken to be a continuous variable. It is given that \(N = 125\) when \(t = 0\).
  1. Solve the differential equation, obtaining a relation between \(N , k\) and \(t\).
  2. Given also that \(N = 166\) when \(t = 30\), find the value of \(k\).
  3. Obtain an expression for \(N\) in terms of \(t\), and find the least value of \(N\) predicted by this model.
CAIE P3 2007 November Q8
10 marks Standard +0.3
8
  1. The complex number \(z\) is given by \(z = \frac { 4 - 3 \mathrm { i } } { 1 - 2 \mathrm { i } }\).
    1. Express \(z\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    2. Find the modulus and argument of \(z\).
  2. Find the two square roots of the complex number 5-12i, giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2007 November Q9
10 marks Standard +0.3
9
  1. Express \(\frac { 2 - x + 8 x ^ { 2 } } { ( 1 - x ) ( 1 + 2 x ) ( 2 + x ) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 2 - x + 8 x ^ { 2 } } { ( 1 - x ) ( 1 + 2 x ) ( 2 + x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2007 November Q10
12 marks Standard +0.3
10 The straight line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 6 \mathbf { j } - 3 \mathbf { k } + s ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )\). The plane \(p\) has equation \(( \mathbf { r } - 3 \mathbf { i } ) \cdot ( 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k } ) = 0\). The line \(l\) intersects the plane \(p\) at the point \(A\).
  1. Find the position vector of \(A\).
  2. Find the acute angle between \(l\) and \(p\).
  3. Find a vector equation for the line which lies in \(p\), passes through \(A\) and is perpendicular to \(l\).
CAIE P3 2008 November Q1
3 marks Moderate -0.5
1 Solve the equation $$\ln ( x + 2 ) = 2 + \ln x$$ giving your answer correct to 3 decimal places.
CAIE P3 2008 November Q2
4 marks Moderate -0.3
2 Expand \(( 1 + x ) \sqrt { } ( 1 - 2 x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2008 November Q3
5 marks Standard +0.3
3 The curve \(y = \frac { \mathrm { e } ^ { x } } { \cos x }\), for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\), has one stationary point. Find the \(x\)-coordinate of this point.
CAIE P3 2008 November Q4
5 marks Moderate -0.5
4 The parametric equations of a curve are $$x = a ( 2 \theta - \sin 2 \theta ) , \quad y = a ( 1 - \cos 2 \theta )$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta\).
CAIE P3 2008 November Q5
6 marks Standard +0.3
5 The polynomial \(4 x ^ { 3 } - 4 x ^ { 2 } + 3 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(2 x ^ { 2 } - 3 x + 3\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, solve the inequality \(\mathrm { p } ( x ) < 0\), justifying your answer.
CAIE P3 2008 November Q6
8 marks Standard +0.3
6
  1. Express \(5 \sin x + 12 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$5 \sin 2 \theta + 12 \cos 2 \theta = 11$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2008 November Q7
10 marks Standard +0.3
7 Two planes have equations \(2 x - y - 3 z = 7\) and \(x + 2 y + 2 z = 0\).
  1. Find the acute angle between the planes.
  2. Find a vector equation for their line of intersection.
CAIE P3 2008 November Q8
10 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{c687888e-bef0-4ea9-b5b3-e614028cc07c-3_654_805_274_671} An underground storage tank is being filled with liquid as shown in the diagram. Initially the tank is empty. At time \(t\) hours after filling begins, the volume of liquid is \(V \mathrm {~m} ^ { 3 }\) and the depth of liquid is \(h \mathrm {~m}\). It is given that \(V = \frac { 4 } { 3 } h ^ { 3 }\). The liquid is poured in at a rate of \(20 \mathrm {~m} ^ { 3 }\) per hour, but owing to leakage, liquid is lost at a rate proportional to \(h ^ { 2 }\). When \(h = 1 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = 4.95\).
  1. Show that \(h\) satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 5 } { h ^ { 2 } } - \frac { 1 } { 20 } .$$
  2. Verify that \(\frac { 20 h ^ { 2 } } { 100 - h ^ { 2 } } \equiv - 20 + \frac { 2000 } { ( 10 - h ) ( 10 + h ) }\).
  3. Hence solve the differential equation in part (i), obtaining an expression for \(t\) in terms of \(h\).
CAIE P3 2008 November Q9
12 marks Challenging +1.2
9 The constant \(a\) is such that \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x = 6\).
  1. Show that \(a\) satisfies the equation $$x = 2 + \mathrm { e } ^ { - \frac { 1 } { 2 } x } .$$
  2. By sketching a suitable pair of graphs, show that this equation has only one root.
  3. Verify by calculation that this root lies between 2 and 2.5.
  4. Use an iterative formula based on the equation in part (i) to calculate the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2008 November Q10
12 marks Standard +0.8
10 The complex number \(w\) is given by \(w = - \frac { 1 } { 2 } + \mathrm { i } \frac { \sqrt { } 3 } { 2 }\).
  1. Find the modulus and argument of \(w\).
  2. The complex number \(z\) has modulus \(R\) and argument \(\theta\), where \(- \frac { 1 } { 3 } \pi < \theta < \frac { 1 } { 3 } \pi\). State the modulus and argument of \(w z\) and the modulus and argument of \(\frac { z } { w }\).
  3. Hence explain why, in an Argand diagram, the points representing \(z , w z\) and \(\frac { z } { w }\) are the vertices of an equilateral triangle.
  4. In an Argand diagram, the vertices of an equilateral triangle lie on a circle with centre at the origin. One of the vertices represents the complex number \(4 + 2 \mathrm { i }\). Find the complex numbers represented by the other two vertices. Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
CAIE P3 2009 November Q1
4 marks Moderate -0.3
1 Solve the inequality \(2 - 3 x < | x - 3 |\).
CAIE P3 2009 November Q2
4 marks Moderate -0.8
2 Solve the equation \(3 ^ { x + 2 } = 3 ^ { x } + 3 ^ { 2 }\), giving your answer correct to 3 significant figures.
CAIE P3 2009 November Q3
5 marks Standard +0.3
3 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 3 x _ { n } } { 4 } + \frac { 15 } { x _ { n } ^ { 3 } }$$ with initial value \(x _ { 1 } = 3\), converges to \(\alpha\).
  1. Use this iterative formula to find \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
  2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).