Questions P3 (1203 questions)

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CAIE P3 2009 November Q6
6 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \mathbf { i } - \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$$ The mid-point of \(A B\) is \(M\). The point \(N\) lies on \(A C\) between \(A\) and \(C\) and is such that \(A N = 2 N C\).
  1. Find a vector equation of the line \(M N\).
  2. It is given that \(M N\) intersects \(B C\) at the point \(P\). Find the position vector of \(P\).
CAIE P3 2009 November Q7
7 The complex number \(- 2 + \mathrm { i }\) is denoted by \(u\).
  1. Given that \(u\) is a root of the equation \(x ^ { 3 } - 11 x - k = 0\), where \(k\) is real, find the value of \(k\).
  2. Write down the other complex root of this equation.
  3. Find the modulus and argument of \(u\).
  4. Sketch an Argand diagram showing the point representing \(u\). Shade the region whose points represent the complex numbers \(z\) satisfying both the inequalities $$| z | < | z - 2 | \quad \text { and } \quad 0 < \arg ( z - u ) < \frac { 1 } { 4 } \pi$$
CAIE P3 2009 November Q8
8
  1. Express \(\frac { 5 x + 3 } { ( x + 1 ) ^ { 2 } ( 3 x + 2 ) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 5 x + 3 } { ( x + 1 ) ^ { 2 } ( 3 x + 2 ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2009 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{8d134c65-af23-4508-acef-49b6ab49e374-3_504_910_625_614} The diagram shows the curve \(y = \frac { \ln x } { \sqrt { } x }\) and its maximum point \(M\). The curve cuts the \(x\)-axis at the point \(A\).
  1. State the coordinates of \(A\).
  2. Find the exact value of the \(x\)-coordinate of \(M\).
  3. Using integration by parts, show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 4\) is equal to \(8 \ln 2 - 4\).
CAIE P3 2009 November Q10
10 In a model of the expansion of a sphere of radius \(r \mathrm {~cm}\), it is assumed that, at time \(t\) seconds after the start, the rate of increase of the surface area of the sphere is proportional to its volume. When \(t = 0\), \(r = 5\) and \(\frac { \mathrm { d } r } { \mathrm {~d} t } = 2\).
  1. Show that \(r\) satisfies the differential equation $$\frac { \mathrm { d } r } { \mathrm {~d} t } = 0.08 r ^ { 2 }$$ [The surface area \(A\) and volume \(V\) of a sphere of radius \(r\) are given by the formulae \(A = 4 \pi r ^ { 2 }\), \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\).]
  2. Solve this differential equation, obtaining an expression for \(r\) in terms of \(t\).
  3. Deduce from your answer to part (ii) the set of values that \(t\) can take, according to this model.
CAIE P3 2009 November Q1
1 Solve the equation $$\ln ( 5 - x ) = \ln 5 - \ln x$$ giving your answers correct to 3 significant figures.
CAIE P3 2009 November Q2
2 The equation \(x ^ { 3 } - 8 x - 13 = 0\) has one real root.
  1. Find the two consecutive integers between which this root lies.
  2. Use the iterative formula $$x _ { n + 1 } = \left( 8 x _ { n } + 13 \right) ^ { \frac { 1 } { 3 } }$$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2009 November Q3
3 The equation of a curve is \(x ^ { 3 } - x ^ { 2 } y - y ^ { 3 } = 3\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Find the equation of the tangent to the curve at the point \(( 2,1 )\), giving your answer in the form \(a x + b y + c = 0\).
CAIE P3 2009 November Q4
4 The angles \(\alpha\) and \(\beta\) lie in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\), and are such that $$\tan \alpha = 2 \tan \beta \quad \text { and } \quad \tan ( \alpha + \beta ) = 3 .$$ Find the possible values of \(\alpha\) and \(\beta\).
CAIE P3 2009 November Q5
5 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x - 4\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). The result of differentiating \(\mathrm { p } ( x )\) with respect to \(x\) is denoted by \(\mathrm { p } ^ { \prime } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and of \(\mathrm { p } ^ { \prime } ( x )\).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P3 2009 November Q6
6
  1. Use the substitution \(x = 2 \tan \theta\) to show that $$\int _ { 0 } ^ { 2 } \frac { 8 } { \left( 4 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { 2 } \frac { 8 } { \left( 4 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x$$
CAIE P3 2009 November Q7
7 The complex numbers \(- 2 + \mathrm { i }\) and \(3 + \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.
  1. Find, in the form \(x + \mathrm { i } y\), the complex numbers
    (a) \(u + v\),
    (b) \(\frac { u } { v }\), showing all your working.
  2. State the argument of \(\frac { u } { v }\). In an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(u + v\) respectively.
  3. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).
  4. State fully the geometrical relationship between the line segments \(O A\) and \(B C\).
CAIE P3 2009 November Q8
8
  1. Express \(\frac { 1 + x } { ( 1 - x ) \left( 2 + x ^ { 2 } \right) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 1 + x } { ( 1 - x ) \left( 2 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2009 November Q9
9 The temperature of a quantity of liquid at time \(t\) is \(\theta\). The liquid is cooling in an atmosphere whose temperature is constant and equal to \(A\). The rate of decrease of \(\theta\) is proportional to the temperature difference \(( \theta - A )\). Thus \(\theta\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - A )$$ where \(k\) is a positive constant.
  1. Find, in any form, the solution of this differential equation, given that \(\theta = 4 A\) when \(t = 0\).
  2. Given also that \(\theta = 3 A\) when \(t = 1\), show that \(k = \ln \frac { 3 } { 2 }\).
  3. Find \(\theta\) in terms of \(A\) when \(t = 2\), expressing your answer in its simplest form.
CAIE P3 2009 November Q10
10 The plane \(p\) has equation \(2 x - 3 y + 6 z = 16\). The plane \(q\) is parallel to \(p\) and contains the point with position vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\).
  1. Find the equation of \(q\), giving your answer in the form \(a x + b y + c z = d\).
  2. Calculate the perpendicular distance between \(p\) and \(q\).
  3. The line \(l\) is parallel to the plane \(p\) and also parallel to the plane with equation \(x - 2 y + 2 z = 5\). Given that \(l\) passes through the origin, find a vector equation for \(l\).
CAIE P3 2010 November Q1
1 Solve the inequality \(2 | x - 3 | > | 3 x + 1 |\).
CAIE P3 2010 November Q2
2 Solve the equation $$\ln \left( 1 + x ^ { 2 } \right) = 1 + 2 \ln x$$ giving your answer correct to 3 significant figures.
CAIE P3 2010 November Q3
3 Solve the equation $$\cos \left( \theta + 60 ^ { \circ } \right) = 2 \sin \theta$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2010 November Q4
4
  1. By sketching suitable graphs, show that the equation $$4 x ^ { 2 } - 1 = \cot x$$ has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 0.6 and 1 .
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \sqrt { } \left( 1 + \cot x _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2010 November Q5
5 Let \(I = \int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \sqrt { } \left( 4 - x ^ { 2 } \right) } \mathrm { d } x\).
  1. Using the substitution \(x = 2 \sin \theta\), show that $$I = \int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 4 \sin ^ { 2 } \theta \mathrm {~d} \theta$$
  2. Hence find the exact value of \(I\).
CAIE P3 2010 November Q6
6 The complex number \(z\) is given by $$z = ( \sqrt { } 3 ) + \mathrm { i } .$$
  1. Find the modulus and argument of \(z\).
  2. The complex conjugate of \(z\) is denoted by \(z ^ { * }\). Showing your working, express in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real,
    (a) \(2 z + z ^ { * }\),
    (b) \(\frac { \mathrm { i } z ^ { * } } { z }\).
  3. On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z\) and \(\mathrm { i } z ^ { * }\) respectively. Prove that angle \(A O B = \frac { 1 } { 6 } \pi\).
CAIE P3 2010 November Q7
7 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + 4 \mathbf { j }\). The point \(P\) lies on the line \(A B\) and \(O P\) is perpendicular to \(A B\).
  1. Find a vector equation for the line \(A B\).
  2. Find the position vector of \(P\).
  3. Find the equation of the plane which contains \(A B\) and which is perpendicular to the plane \(O A B\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2010 November Q8
8 Let \(\mathrm { f } ( x ) = \frac { 3 x } { ( 1 + x ) \left( 1 + 2 x ^ { 2 } \right) }\).
  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 2010 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{bbc19395-6f88-4a7c-b5d4-59ced9ccdcf2-4_597_895_258_625} The diagram shows the curve \(y = x ^ { 3 } \ln x\) and its minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\).
CAIE P3 2010 November Q10
10 A certain substance is formed in a chemical reaction. The mass of substance formed \(t\) seconds after the start of the reaction is \(x\) grams. At any time the rate of formation of the substance is proportional to \(( 20 - x )\). When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.05 ( 20 - x ) .$$
  2. Find, in any form, the solution of this differential equation.
  3. Find \(x\) when \(t = 10\), giving your answer correct to 1 decimal place.
  4. State what happens to the value of \(x\) as \(t\) becomes very large. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }