Questions — CAIE P3 (1070 questions)

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CAIE P3 2002 November Q1
1 Solve the inequality \(| 9 - 2 x | < 1\).
CAIE P3 2002 November Q2
2 Find the exact value of \(\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x\).
CAIE P3 2002 November Q3
3
  1. Show that the equation $$\log _ { 10 } ( x + 5 ) = 2 - \log _ { 10 } x$$ may be written as a quadratic equation in \(x\).
  2. Hence find the value of \(x\) satisfying the equation $$\log _ { 10 } ( x + 5 ) = 2 - \log _ { 10 } x$$
CAIE P3 2002 November Q4
4 The curve \(y = \mathrm { e } ^ { x } + 4 \mathrm { e } ^ { - 2 x }\) has one stationary point.
  1. Find the \(x\)-coordinate of this point.
  2. Determine whether the stationary point is a maximum or a minimum point.
CAIE P3 2002 November Q5
5
  1. Express \(4 \sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the value of \(\alpha\) correct to 2 decimal places. Hence
  2. solve the equation $$4 \sin \theta - 3 \cos \theta = 2$$ giving all values of \(\theta\) such that \(0 ^ { \circ } < \theta < 360 ^ { \circ }\),
  3. write down the greatest value of \(\frac { 1 } { 4 \sin \theta - 3 \cos \theta + 6 }\).
CAIE P3 2002 November Q6
6 Let \(f ( x ) = \frac { 6 + 7 x } { ( 2 - x ) \left( 1 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that, when \(x\) is sufficiently small for \(x ^ { 4 }\) and higher powers to be neglected, $$f ( x ) = 3 + 5 x - \frac { 1 } { 2 } x ^ { 2 } - \frac { 15 } { 4 } x ^ { 3 }$$
CAIE P3 2002 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{b89c016e-dc56-48f4-b4c4-b432418e1b28-3_435_672_273_684} The diagram shows a curved rod \(A B\) of length 100 cm which forms an arc of a circle. The end points \(A\) and \(B\) of the rod are 99 cm apart. The circle has radius \(r \mathrm {~cm}\) and the arc \(A B\) subtends an angle of \(2 \alpha\) radians at \(O\), the centre of the circle.
  1. Show that \(\alpha\) satisfies the equation \(\frac { 99 } { 100 } x = \sin x\).
  2. Given that this equation has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\), verify by calculation that this root lies between 0.1 and 0.5.
  3. Show that if the sequence of values given by the iterative formula $$x _ { n + 1 } = 50 \sin x _ { n } - 48.5 x _ { n }$$ converges, then it converges to a root of the equation in part (i).
  4. Use this iterative formula, with initial value \(x _ { 1 } = 0.25\), to find \(\alpha\) correct to 3 decimal places, showing the result of each iteration.
CAIE P3 2002 November Q8
8
  1. Find the two square roots of the complex number \(- 3 + 4 \mathrm { i }\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. The complex number \(z\) is given by $$z = \frac { - 1 + 3 \mathrm { i } } { 2 + \mathrm { i } } .$$
    1. Express \(z\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    2. Show on a sketch of an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) representing the complex numbers \(- 1 + 3 \mathrm { i } , 2 + \mathrm { i }\) and \(z\) respectively.
    3. State an equation relating the lengths \(O A , O B\) and \(O C\).
CAIE P3 2002 November Q9
9 In an experiment to study the spread of a soil disease, an area of \(10 \mathrm {~m} ^ { 2 }\) of soil was exposed to infection. In a simple model, it is assumed that the infected area grows at a rate which is proportional to the product of the infected area and the uninfected area. Initially, \(5 \mathrm {~m} ^ { 2 }\) was infected and the rate of growth of the infected area was \(0.1 \mathrm {~m} ^ { 2 }\) per day. At time \(t\) days after the start of the experiment, an area \(a \mathrm {~m} ^ { 2 }\) is infected and an area \(( 10 - a ) \mathrm { m } ^ { 2 }\) is uninfected.
  1. Show that \(\frac { \mathrm { d } a } { \mathrm {~d} t } = 0.004 a ( 10 - a )\).
  2. By first expressing \(\frac { 1 } { a ( 10 - a ) }\) in partial fractions, solve this differential equation, obtaining an expression for \(t\) in terms of \(a\).
  3. Find the time taken for \(90 \%\) of the soil area to become infected, according to this model.
CAIE P3 2002 November Q10
10 With respect to the origin \(O\), the points \(A , B , C , D\) have position vectors given by $$\overrightarrow { O A } = 4 \mathbf { i } + \mathbf { k } , \quad \overrightarrow { O B } = 5 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } , \quad \overrightarrow { O C } = \mathbf { i } + \mathbf { j } , \quad \overrightarrow { O D } = - \mathbf { i } - 4 \mathbf { k }$$
  1. Calculate the acute angle between the lines \(A B\) and \(C D\).
  2. Prove that the lines \(A B\) and \(C D\) intersect.
  3. The point \(P\) has position vector \(\mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\). Show that the perpendicular distance from \(P\) to the line \(A B\) is equal to \(\sqrt { } 3\).
CAIE P3 2003 November Q1
1 Solve the inequality \(\left| 2 ^ { x } - 8 \right| < 5\).
CAIE P3 2003 November Q2
2 Expand \(\left( 2 + x ^ { 2 } \right) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\), simplifying the coefficients.
CAIE P3 2003 November Q3
3 Solve the equation $$\cos \theta + 3 \cos 2 \theta = 2$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2003 November Q4
4 The equation of a curve is $$\sqrt { } x + \sqrt { } y = \sqrt { } a$$ where \(a\) is a positive constant.
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. The straight line with equation \(y = x\) intersects the curve at the point \(P\). Find the equation of the tangent to the curve at \(P\).
CAIE P3 2003 November Q5
5
  1. By sketching suitable graphs, show that the equation $$\sec x = 3 - x ^ { 2 }$$ has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x _ { n } ^ { 2 } } \right)$$ converges, then it converges to a root of the equation given in part (i).
  3. Use this iterative formula, with initial value \(x _ { 1 } = 1\), to determine the root in the interval \(0 < x < \frac { 1 } { 2 } \pi\) correct to 2 decimal places, showing the result of each iteration.
CAIE P3 2003 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{79efa364-da5a-4888-85a9-dc4de1e0908e-3_543_825_287_660} The diagram shows the curve \(y = ( 3 - x ) \mathrm { e } ^ { - 2 x }\) and its minimum point \(M\). The curve intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  1. Calculate the \(x\)-coordinate of \(M\).
  2. Find the area of the region bounded by \(O A , O B\) and the curve, giving your answer in terms of e.
CAIE P3 2003 November Q7
7 The complex number \(u\) is given by \(u = \frac { 7 + 4 \mathrm { i } } { 3 - 2 \mathrm { i } }\).
  1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Sketch an Argand diagram showing the point representing the complex number \(u\). Show on the same diagram the locus of the complex number \(z\) such that \(| z - u | = 2\).
  3. Find the greatest value of \(\arg z\) for points on this locus.
CAIE P3 2003 November Q8
8 Let \(\mathrm { f } ( x ) = \frac { x ^ { 3 } - x - 2 } { ( x - 1 ) \left( x ^ { 2 } + 1 \right) }\).
  1. Express \(\mathrm { f } ( x )\) in the form $$A + \frac { B } { x - 1 } + \frac { C x + D } { x ^ { 2 } + 1 }$$ where \(A , B , C\) and \(D\) are constants.
  2. Hence show that \(\int _ { 2 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x = 1\).
CAIE P3 2003 November Q9
9 Compressed air is escaping from a container. The pressure of the air in the container at time \(t\) is \(P\), and the constant atmospheric pressure of the air outside the container is \(A\). The rate of decrease of \(P\) is proportional to the square root of the pressure difference ( \(P - A\) ). Thus the differential equation connecting \(P\) and \(t\) is $$\frac { \mathrm { d } P } { \mathrm {~d} t } = - k \sqrt { } ( P - A )$$ where \(k\) is a positive constant.
  1. Find, in any form, the general solution of this differential equation.
  2. Given that \(P = 5 A\) when \(t = 0\), and that \(P = 2 A\) when \(t = 2\), show that \(k = \sqrt { } A\).
  3. Find the value of \(t\) when \(P = A\).
  4. Obtain an expression for \(P\) in terms of \(A\) and \(t\).
CAIE P3 2003 November Q10
10 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = \mathbf { i } - 2 \mathbf { k } + s ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 6 \mathbf { i } - 5 \mathbf { j } + 4 \mathbf { k } + t ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$ respectively.
  1. Show that \(l\) and \(m\) intersect, and find the position vector of their point of intersection.
  2. Find the equation of the plane containing \(l\) and \(m\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2004 November Q1
1 Expand \(\frac { 1 } { ( 2 + x ) ^ { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2004 November Q2
2 Solve the equation $$\ln ( 1 + x ) = 1 + \ln x$$ giving your answer correct to 2 significant figures.
CAIE P3 2004 November Q3
3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 4\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x - 2\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\). When \(a\) has this value,
  2. factorise \(\mathrm { p } ( x )\),
  3. solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.
CAIE P3 2004 November Q4
4
  1. Show that the equation $$\tan \left( 45 ^ { \circ } + x \right) = 2 \tan \left( 45 ^ { \circ } - x \right)$$ can be written in the form $$\tan ^ { 2 } x - 6 \tan x + 1 = 0$$
  2. Hence solve the equation \(\tan \left( 45 ^ { \circ } + x \right) = 2 \tan \left( 45 ^ { \circ } - x \right)\), for \(0 ^ { \circ } < x < 90 ^ { \circ }\).
CAIE P3 2004 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{8c533469-393c-4e4c-a6ec-eab1303741e7-2_385_476_1653_836} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \frac { 1 } { 2 } \pi\). The point \(N\) on \(O A\) is such that \(B N\) is perpendicular to \(O A\). The area of the triangle \(O N B\) is half the area of the sector \(O A B\).
  1. Show that \(\alpha\) satisfies the equation \(\sin 2 x = x\).
  2. By sketching a suitable pair of graphs, show that this equation has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  3. Use the iterative formula $$x _ { n + 1 } = \sin \left( 2 x _ { n } \right)$$ with initial value \(x _ { 1 } = 1\), to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.