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Name

Teacher

A2 MATHEMATICS HOMEWORK C4

Sierpinskis Gasket

Mathematics DepartmentSeptember 2014Version 1.0

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Contents

Contents ............................................................................................................. 2

Introduction ......................................................................................................... 3

HW1 Partial Fractions ............................................................................................ 4

HW2 Parametric equations (including differentiation) ............................................ 6

HW3 Binomial Expansions ...................................................................................... 8

HW4 Differentiation Implicit functions............................................................... 10

HW5 Integration 1 Introduction ............................................................................ 12

HW6 Integration 2 - by Substitution ..................................................................... 14

HW7 Integration 3 - by parts ............................................................................... 16

HW8 Integration 4 Partial fractions .................................................................... 17

HW9 Integration 5 Trig identities ........................................................................ 18

HW10 Integration 6 Differential equations .......................................................... 20

HW11 Integration 7 Numerical Methods and Volumes of Revolution ....................... 23

HW12 C4 Vectors ............................................................................................... 26 HWX C4 June 2010 ............................................................................................ 29

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IntroductionAim to complete all the questions. If you find the work difficult then get help [lunchtimeworkshops in room 216, online, friends, teacher etc].

To learn effectively you should check your work carefully and mark answers ? If youhave questions or comments, please write these on your homework. Your teacher will then

review and mark your mathematics.

If you spot an error in this pack please let your teacher know so we can make changes forthe next edition!

Homework Tasks These cover the main topics in C4. Your teacher may set homeworkfrom this or other tasks. www.examsolutions.net has video solutions to exam questionsand clear explanations of many topics.

TopicDatecompleted

Comment

HW1 Partial Fractions

HW2 Parametric Equations(inc differentiation)

HW3 Binomial Expansions

HW4 Differential Implicitfunctions

HW5Integration 1 Introduction

HW6 Integration 2 bysubstitution

HW7 Integration 3 byparts

HW8 Integration 4 PartialFractions

HW9 Integration 5 TrigIdentities

HW10 Integration 6 Differential Equations

HW11Integration 7 Numerical Methods &Volumes of Revolution

HW12 Vectors

HWX C4 June 2010

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HW1 Partial Fractions

Complete on a separate sheet of paper. Show clear working. Mark your answers.

Key words: Numerator, Denominator, Equating, Coefficient, Proper fraction,Improper fraction,

Read text book pages 162-174

Exercise A

1. Express the following as partial fractions:

a) b)

c) d)

e) f)

2. Express the following as partial fractions (be careful, repeated factors):

a) b)

c)

3. By using long division (or otherwise) express these improper fractions as partialfractions:

a) b)

c)

Exercise B - Exam Questions

1. [C4 Jun 2011 Q1]

)12()1(9

2

2

+ x x x

=)1( x

A +

2)1( x B

+)12( + x

C

Find the values of the constants , and .(4)

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2. [C4 June 2012 Q1]

= 2)13(1 x x

= x A

+)13( x

B +

2)13( xC

.

Find the values of the constants , and .(4)

3. [C4 Jan 2009 Q3]

= )1()23(163227

2

2

x x x x

+++

, < .

Given that can be expressed in the form

=

)23( + x

A +

2

)23( + x

B +

)1( x

C

,

find the values of and and show that = 0.(4)

Exercise C - Extension

1. Express = as the sum of partial factions.2. Find out more about how we use the technique of Partial Fractions in Mathematics.

Answers

Exercise A

1. a) b) c) + d) e) + + f) +

2. a) b) c) + 3. a) 1 + 12 1+1 b) 1 c) 41+3 Exercise B Exam questions

1. = 4, = 3, = 1 2. = 1, = 3, = 3 3. = 4, = 3

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HW2 Parametric equations (including differentiation) Complete on a separate sheet of paper. Show clear working. Mark your answers.

Key words Parameter, parametric equation, Cartesian equation, eliminate theparameter

= , = 2equation of parabola = , = equation of a circle = where = 1 = where = , =

Read pages 175-185, 215-221

Starter: write down the main trig identities. See page 80 if you cant remember

Exercise A

1. Find in terms of when and are related by the following pairs ofparametric equations.

a) = 3 + , = 2 + b) = , = 2. Change to following parametric equations into Cartesian form by eliminating the

parameter

a) = +2, = 3 b) = 3 1, = 3 +2 3. Find the equation of the tangent to the curve at the point given

a) = +1, = 3 , = 2 b) = 4 , = 3 , = As an extension try to sketch the graphs.

4. Calculate the area between the , = 0 = 2 of the parametric equations = + , = 1

Exercise B - Exam Questions1. [C4 Jan2009 Q7] The curve C shown opposite has parametric equations

x = t 3 8 t , y = t 2

where t is a parameter. Given that the point A hasparameter t = 1,

(a ) find the coordinates of A.(1)

The line l is the tangent to C at A.

(b) Show that an equation for l is 2 x 5 y 9 = 0.(5)

The line l also intersects the curve at the point B.

(c) Find the coordinates of B.(6)

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2. [C4 Jun 2008 Q8]

The graph opposite shows the curve C with parametricequations

x = 8 cos t , y = 4 sin 2 t , 0 t

2

.

The point P lies on C and has coordinates (4, 2 3).

(a ) Find the value of t at the point P .(2)

The line l is a normal to C at P .

(b) Show that an equation for l is y = x3 + 6 3.(6)

Exercise C Extension tasks

1. Sketch some of the graphs in Exercise A Q 1 and 2

2. Find out about Lissajou figures and how they areused in electronics.

AnswersExercise A

1a) = b) = = 2a) = b) 3a)

b)

4. Integral Area

Exercise B Exam questionsMay 08 June 09

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HW3 Binomial Expansions Complete on a separate sheet of paper. Show clear working. Mark your answers.

Key words Expand, Binomial theorem, Factorial, Limit, Ascending, Coefficient

!

!

Read the chapter on binomial expansion (p14- 18, 20-22, 25-29)

Exercise A

1. Rewrite each expression in the form example : 4 4 4 4

a) b)

c) d)

4

e) f) g) h)

2. Find the binomial expansions of parts a to d from the previous question in ascendingpowers of as far as the term. State the range of values of for which theexpansions are valid.


1. [C4 Jun 11 Q2]

f < Find the first three non-zero terms of the binomial expansion of inascending powers of . Give each coefficient as a simplified fraction.

(6)

2. [C4 Jan 10 Q1]

a) Find the binomial expansion of

, < ,in ascending powers of up to and including the term in , simplifying eachterm.

(6)

b) Show that, when , the exact value of , is .(2)

c) Substitute into the binomial expansion in part (a) and hence obtain anapproximation to . Give your answer to 5 decimal places.

(3)

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3. [C4 Jan 11 Q5]

a) Use the binomial theorem to expand

, < ,in ascending powers of , up to and including the term in . Give eachcoefficient as a simplified fraction.

(5)

f , < , where and are constants.In the binomial expansion of f , in ascending powers of , the coefficient of is and the coefficient of is .

Find

b) the value of and the value of ,(5)

c) the coefficient of , giving your answer as a simplified fraction.(3)

Answers

Exercise A

1. (a) (b) (c) (d) (e)

(f) (g) (h) 2. (a) (b) (c)

(d) Exercise B

1. 2. (a) 4 (c) 4. 4 3. (a) (b) (c)

A bit of history

The binomial formula and the binomial coefficients are often attributedto Blaise Pascal , who described them in the 17th century, but they wereknown to many mathematicians who preceded him. A more generalbinomial theorem and the so-called "Pascal's triangle" were known in the10th century A.D. to Indian mathematician Halayudha . Arabianmathematician Al-Karaji , in the 11th century was aware of a moregeneral binomial theorem, and in the 13th century to Chinesemathematician Yang Hui , also derived similar results. Sir IsaacNewton is generally credited with the generalised binomial theorem,valid for any rational exponent.

Isaac Newton 1642 - 1726

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HW4 Differentiation Implicit functions


chain rule product rule

Explicit Implicit Read pages 209 215 Note i i Exercise A

1. Practice the chain rule with Explicit functions. Differentiate to find

a)

b)

i c)

l

d) e) e f) l 2. Differentiate the following Implicit functions with respect to . Give answers in terms

of a) b) c)

d) e) c f) 3. Differentiate the following Implicit functions with respect to , and find .

a) b) c)

d) 4 e) f)

4. Find the equation of the tangent to the circle at the point 4

5. The ellipse has equation Findthe stationary points.

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1. [C4 Jan 12 Q1] The curve C has the equation 2 x + 3 y2 + 3 x2 y = 4 x2.

The point P on the curve has coordinates (1, 1).

(a ) Find the gradient of the curve at P .(5)

(b) Hence find the equation of the normal to C at P , giving your answer in the formax + by + c = 0, where a , b and c are integers.

(3)

2. [C4 Jun 2008 Q4] A curve has equation 3 x2 y2 + xy = 4. The points P and Q lieon the curve. The gradient of the tangent to the curve is 3

8 at P and at Q .

(a ) Use implicit differentiation to show that y 2 x = 0 at P and at Q .(6)

(b) Find the coordinates of P and Q.(3)

Exercise C Extension tasks1. Find out about ellipses and conic sections

2. Show how gives using Implicit differentiationAnswers

Exercise A

1a) b) c) d) e) f) 2a) b) c) d) e) i f) 3a)

b)

c)

d) e) f) 4. 4 5. Exercise B Exam questions

1. a) , 4 2. b) P=(2,4), Q=(-2,-4)

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HW5 Integration 1 Introduction


Key words Integral, Standard functions, Factorising, Cancelling like terms,improper fractions, polynomial division

Read pages 234 252. This homework links with Exercise 10A 10F. Use these for furtherpractice.

For more formulae see page 308 and formulae booklet

ec

l l Exercise A

1. Sketch the graph and find the area given by the integral

2. Find the particular solution of the differential equation which passes

through the point

3. Find the following indefinite integrals:

a) i d b) d c) c d d) c d e) 4 d f) ec d

4. Find the following indefinite integrals: l

a) d b) d c) d) d e) d f) d

5. Find the following indefinite integrals:[think of chain rule in reverse]

a) d b) ec d c) d d) i d e) f)

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6. Evaluate the following definite integrals and give your answers as surds

a) b) ec Exercise B - Exam Questions

2. [June 2009 Q2 adapted]

Figure 1

Figure 1 shows the finite region R bounded by the x-axis, the y-axis and the curve with equation

y = 3 cos

3 x

, 0 x 2

3 .

Use integration to find the exact area of R. (3)

Exercise C Extension tasks1. a) Show that l

b) Show that ec c) Prove that

2. See also questions in the text book Exercise A-F

AnswersExercise A

1. Sketch try completing the square. Area = 2. 3a) b) c) d) l e) 4 f) 4a) l b) l c) l d) l e) l c f) l 5a) i b) c) d) c e) c f) 6a) b) Exercise B Exam questions9 units

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HW6 Integration 2 - by Substitution Complete on a separate sheet of paper. Show clear working. Mark your answers.

Key Words integration by substitution, recognition, natural log, exponential. Read pages 253-262

Exercise A1. Draw lines between the matching boxes

A i 1 B c 2 C l 3 i D

i c 4

i

E 5 c F c 6 ec G i 7 H 8

I 9 i c J 10

11 ec 2. Find the following integrals using the given substitution or otherwise.

a) i b) c) d) c i i e) f)

3. Evaluate the following integrals using the given substitution or otherwise.

a) b) c) ec d)

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Exercise B - Exam Questions1. [C4 Jan 2013 Q4 (adapted)]

Use the substitution u = 1 + to find the exact value of

2. [C4 Jan 2011 Q7 (adapted)]

Use the substitution x = (u 4) 2 + 1 to find the exact value of

4. [C4 Jan 2012 Q6 (adapted)]

Use the substitution u = 1 + cos x to find the exact value of

ic 5. [C4 June 2013 (R) Q3]

Using the substitution u = 2 + (2 x + 1), or other suitable substitutions, find the exactvalue of

4

0

12 (2 1) x+ + d x

giving your answer in the form A + 2ln B, where A is an integer and B is a positive constant.

Answers:Exercise A

1) A9, B4, C2, D8, E11, F3, G10, H6, I7, J1

2a) i2 +3 + b) + c) 7 + 3 + d) i e) f) 3 + 1 + 3a) b) c) d)

Exercise B Exam questions

2) +2 3) 2 +8 4) 4 4 2 5) 2+ 2 0.6

+

5

2

d)1(4

1 x

x

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HW7 Integration 3 - by parts

Complete on a separate sheet of paper. Show clear working. Mark your answers.Key words Integration Textbook pages 262-271

= = 1 = + Exercise A

1. Integrate the following by parts

a) b) 4 c) d) c 5

2. Integrate the following by parts

a) 3 b) 3. Evaluate the following definite integrals using integration by parts

a) 2 / b) 4. By writing 1 show that = ln + Exercise B - Exam Questions1. [C4 Jun 2008 Q2]

(a ) Use integration by parts to find

x x x de .

(3)

(b) Hence find

x x x de2 .

(3)2. [C4 Jan 2012 Q2]

(a ) Use integration by parts to find

.d3sin x x x

(3)

(b) Using your answer to part ( a ), find

.d3cos2 x x x

(3)

Answers Exercise A

1a) +s n + b) + c) + d) + 2a) cos3 + 3 + 3 + b) + + 3a) +s n/ = / b) ln = 3 Exercise B Exam questions1. a)

+ b)

() +

2a) cos3 + cos3 + b) s n3 +cos3 s n3 +

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HW8 Integration 4 Partial fractions

Complete on a separate sheet of paper. Show clear working. Mark your answers.Key words Integration, repeated factors, quadratic factors, see also HW1 Partial FractionsTextbook pages 271-274

Exercise A1. Integrate the following for practice

a) b) c) d) ( ) 2. Convert the following to partial fractions then integrate.

a) ( +1)( +3) b) +5( +3) c) ( )() d) Write in the form + + , then integrateExercise B - Exam Question

1. June 2012 Q1

f( x) =2)13(

1 x x

= x A

+)13( x

B +

2)13( xC

.

(a ) Find the values of the constants A, B and C .(4)

(b) (i) Hence find

x x d)(f .

(ii) Find 2

1

d)(f x x , leaving your answer in the form a + ln b, where a and b are constants.

(6)AnswersExercise A

1a)

5ln + + b)

ln3 + + c)

5ln + d)

+

2a) ln+ b) ln +3+ + c) ln + + d) = 1, = 3, = 1 = +3ln ln + 3 + Exercise B Exam questions

1a) = 1, = 3, = 3 b) ln + c) + ln

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HW9 Integration 5 Trig identities


Key words IntegrationTextbook pages 275-279

s n +cos 1 n +1 s c 1+ co cos s n 1 1 Exercise A

1. Find each of the following integrals as introductory practice.

a)

b)

s n3 c)

d) s c 5 e) s n f) s c n 2. a) Use the identity 1 s n to find s n [even powers]

b) Use the identity to find c) Use the identity

1 + n s cto find

n 5

d) Use the identity cos 1 to find cos [even power]3. Find the following integrals [ odd powers]

a) s n b) cos c) cos Exercise B - Exam Questions1. [C4 Jan 2013 Q6]

Figure 3Figure 3 shows a sketch of part of thecurve with equation y = 1 2 cos x, where

x is measured in radians. The curvecrosses the x-axis at the point A and at thepoint B.

(a ) Find, in terms of , the x coordinateof the point A and the x coordinate of thepoint B.

(3)

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The finite region S enclosed by the curve and the x-axis is shown shaded in Figure 3. The region S isrotated through 2 radians about the x-axis.

(b) Find, by integration, the exact value of the volume of the solid generated.

Note Volume of revolution =

(6)


Show that cos = cos + , you may find a substitution helpful.Answers

Exercise A

1a)

s n + b)

3 + c)

cos+

d) n + e) s n + f) n + 2a) ( s n cos ) b) cos + c) n5 + d) + ++ Tips: 2d) cos = = ( ) then substitute cos = 3a) cos +cos + b) s n + c) s n +s n + Exercise B Exam questions

1a) = , = b) = +3 3 This is a difficult questionExercise C Extension tasksAsk your teacher

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HW10 Integration 6 Differential equations


Key words Differential equation, general solution, particular solution.

Exercise A1. Find the particular solutions of the following differential equations DEs

a) = , = 0 n = 0 b) = , 0, = n c) = cos, = 0 n =

2. Find the general solutions of the following DEs

a) = ( ), Sketch the solution curve which passes through (0,3) b) = Sketch the solution curve which passes through ( ,3) c) = d) cos = , 0e)

= Exercise B (involving rates of change) 1. The length of the edge of a cube is increasing at a constant rate of 0.5 . At theinstant when the length of the edge is 6 , find

a) the rate of increase of the surface area

b) the rate of increase of the volume

2. If a hemispherical bowl of radius 6 contains water to adepth of , the volume of water is(1 )

Water is poured into the bowl at a rate of 3 . Find therate at which the water level is rising when the depth is .

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Exercise C - Exam Questions

1. [C4 Jan/Jun 2008 Q7]

(a ) Express24

2 y

in partial fractions.

(3)

(b) Hence obtain the solution of

2 cot x x y

dd

= (4 y2)

for which y = 0 at x =3

, giving your answer in the form sec 2 x = g( y).

(8)

2. [C4 June 2008 Q3]

Figure 1

Figure 1 shows a right circular cylindrical metal rod which isexpanding as it is heated. After t seconds the radius of the rod is x cmand the length of the rod is 5 x cm.

The cross-sectional area of the rod is increasing at the constant rate of0.032 cm 2 s1.

(a ) Findt

xdd

when the radius of the rod is 2 cm, giving your answer to 3 significant figures.

(4)

(b) Find the rate of increase of the volume of the rod when x = 2.(4)

3. [C4 Jan 2009 Q5]

Figure 2

A container is made in the shape of a hollow inverted rightcircular cone. The height of the container is 24 cm and theradius is 16 cm, as shown in Figure 2. Water is flowinginto the container. When the height of water is h cm, thesurface of the water has radius r cm and the volume ofwater is V cm 3.

(a ) Show that V =27

4 3h .

(2)

[The volume V of a right circular cone with vertical height

h and base radius r is given by V = 31

r 2h .]

Water flows into the container at a rate of 8 cm 3 s1.

(b) Find, in terms of , the rate of change of h when h = 12. (5)

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Exercise D Extension taskA murder victim was discovered by the police at 6:00 am. The body temperature of the victim wasmeasured and found to be 5. A doctor arrived on the scene of the crime 30 minutes later andmeasured the body temperature again. It was found to be . The temperature of the room hadremained constant at 15. The doctor, knowingnormal body temperature to e

3, was able to

estimate the time of death of the victim.What would be your estimate for the time of death?What assumptions have you made?

The cooling of an object which is hotter than its surroundings is described by Newtons law of cooling.

The rate of cooling at any instant is directly proportional to the difference in temperature between theobject and its surroundings.

Exercise A Answers1a) = ( 1) b) = +3 c) = + 2a)

= + b) + = , solution curve is a circle through ( ,3) circle centre 0, radius 5c) = + d) = e) + = Exercise B1a) 36 b) 5 2 Exercise C Exam questions1a) ( )+( ) b) ec 2a) 0.03 b) = 0.00 55 3b) Exercise D Extension taskSee your teacher

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HW11 Integration 7 Numerical Methods and Volumes of Revolution


Key words Volume, Revolution, rotation, radius, parametric,ordinate (x coordinate)

Volume of Revolution Area ~ + + or ~ + + ( + + ) Read textbook page 279-287

Exercise A

1. a) Estimate using the trapezium rule with five ordinates. Would youexpect you estimate to be too large or too small?b) Use the trapezium rule with six ordinates to estimate the value of c) Use the trapezium rule with six ordinates to estimate the value of

Use integration by parts to show that the exact value is 3 Find the percentage error in your estimate

d) The depth of a river, of width 12m, is measured at intervals of 2m across its

width, the resultant data being

distance from bank (m) 0 2 4 6 8 10 12

depth (m) 0 1.8 2.6 3 3.4 2 0

Estimate the area of cross section of the river. Determine the flow of the river inlitres per minute given that the water has an average velocity of /

2 a) Find the volume of the solid of revolution formed when the region enclosed byeach of the following curves and the , between the given values of x, isrotated through radians about the .(i) = from = 0 to = 3 (ii) = + from = 1 to = 3 (i) = from = 1 to = (i) = s n from = 0 to = b) Obtain the formula for the volume of a hemisphere by rotating the part of thecircle + = in the first quadrant about the . Deduce the formula forthe volume of a sphere.

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c) The region between = 0 and = 3 under the curve = is rotated through radians about the to form a solid of revolution.(i) Use the trapezium rule with four ordinates to estimate the volume of the solid.

(ii) By using the substitution = 3, show that the exact value of the volume is (iii) Find the percentage error in the estimated volume.

d) A curve has the parametric equations = + n The region underthe curve between = = is rotated through radians about the to form a solid of revolution. Show that the volume of the solid is +


1. [C4 Jan 2012 Q6]

Figure 3

Figure 3 shows a sketch of the curve with

equation y =)cos1(

2sin2 x x

+ , 0 x

2 .

The finite region R, shown shaded in

Figure 3, is bounded by the curve and the

x-axis.

The table below shows corresponding values of x and y for y =)cos1(

2sin2 x x

+ .

x 08

4

8

3

2

y 0 1.17157 1.02280 0

(a ) Complete the table above giving the missing value of y to 5 decimal places.(1)

(b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimatefor the area of R, giving your answer to 4 decimal places.

(3)

(c) Using the substitution u = 1 + cos x, or otherwise, show that

x x x

d)cos1(

2sin2

+ = 4 ln (1 + cos x) 4 cos x + k ,

where k is a constant.(5)

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(d ) Hence calculate the error of the estimate in part ( b), giving your answer to 2 significantfigures.

(3)

2. [C4 Jan 2012 Q4]

Figure 1

Figure 1 shows the curve with equation

y =

+ 4322 x x

, x 0.

The finite region S , shown shaded in Figure 1, is bounded by the curve, the x-axis and theline x = 2.

The region S is rotated 360 about the x-axis.

Use integration to find the exact value of the volume of the solid generated, giving your answerin the form k ln a , where k and a are constants.


1. Find the volume of the solid formed when the area bounded by the curve =

1 and the lines = = is rotated about the 2. A cylindrical hole of radius b is boared symmetrically through a sphere of radius . Find the volume remaining. AnswersExercise A1a) 4.37, too large b) 3.75 c) 2.97m 0.8% d) 1 1 1 2a(i) 1 (ii) (iii) (iv) b) = c) 0.818, 0.6% Exercise B Exam questions1. a) 0.73508 b) 1.1504 d)0.077

2.

Exercise C Extension tasksSee Teacher

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HW12 C4 Vectors


Key Words direction vector, position vectors, unit vector, vector equation of line,intersection, skew, parallel, scalar [dot] product.Read pages 310 - 353

= + = 1+ = = Exercise A

1. a) Explain the concept of a unit vector using as an example. You should draw

the vector and unit vector accurately.

b) Explain how you find the unit vector of 1 2. Find the vector equations of the lines joining the points given.

a) = = 1 b) = = 111 c) = 1 = d) = =

3. Find the position vectors of the points of intersection of each of these pairs of lines.

a) = + 1 = + 1 b) = 1 + 1 = 1 + 11

4. Determine the acute angles between the pairs of lines in question 3 above.

5. The line and have equations = 1 + 11 = +11 a) Show that lies on

b) Show that and are skew.

c) A is the point on where = . B is the point on where = . Find the acuteangle between AB and the line

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Exercise B - Exam Questions [Draw a diagram to help with understanding not to scale]

1. [June 2008] With respect to a fixed origin O , the lines l1 and l2 are given by the equations

l1 : r = (9 i + 10 k) + (2i + j k )

l2 : r = (3 i + j + 17 k) + (3i j + 5 k)

where and are scalar parameters.

(a ) Show that l1 and l2 meet and find the position vector of their point of intersection.(6)

(b) Show that l1 and l2 are perpendicular to each other.(2)

The point A has position vector 5 i + 7 j + 3 k.

(c) Show that A lies on l1.

(1)

The point B is the image of A after reflection in the line l2.

(d ) Find the position vector of B. [Hint think about the answers to a,b,c and draw a diagram](3)

2. [Jan 2009] With respect to a fixed origin O the lines l1 and l2 are given by the equations

b)

: = 111+1

: = 511 +

where and are parameters and p and q are constants. Given that l1 and l2 are perpendicular,

(a ) show that q = 3.(2)

Given further that l1 and l2 intersect, find

(b) the value of p ,(6)

(c) the coordinates of the point of intersection.(2)

The point A lies on l1 and has position vector 313. The point C lies on l2.Given that a circle, with centre C , cuts the line l1 at the points A and B,

(d ) find the position vector of B.(3)

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Exercise C Extension questions

Vectors have many applications in advanced mathematics and engineering. Find out aboutfinite dimensional vector spaces and the topic of linear algebra.

You have learned about the scalar (dot) product. There is also a vector (cross) product.

Answers

Exercise A2a) = 3 + 3 b) = 5 + 63 or = 5 + 1 c) = 13 + 0 d) = 05 + 116 3a) Intersection 15 b) 1 4a) 0 b) . Exercise B1. (a) 3 i +3 j + 7 k (d ) 11 i j + 11 k

2. (b) p = 1 ( c) 13 (d ) 111

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HWX C4 June 20101.

Figure 1

Figure 1 shows part of the curve with equation y =(0.75 + cos 2 x). The finite region R, shown shaded inFigure 1, is bounded by the curve, the y-axis, the x-axis

and the line with equation x =3

.

(a ) Copy and complete the table with values of y

corresponding to x =6

and x =4

.

x 012

6

4

3

y 1.3229 1.2973 1

(2)

(b) Use the trapezium rule

(i) with the values of y at x = 0, x =

6

and x =

3

to find an estimate of the area of R.

Give your answer to 3 decimal places.

(ii) with the values of y at x = 0, x =12

, x =6

, x =4

and x =3

to find a

further estimate of the area of R. Give your answer to 3 decimal places.

(6)2. Using the substitution u = cos x +1, or otherwise, show that

+2

0

1 dsine

x x xcos = e(e 1).

(6)

3. A curve C has equation

2 x + y2 = 2 xy.

Find the exact value of x ydd

at the point on C with coordinates (3, 2).

(7)

4. A curve C has parametric equations

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x = sin 2 t , y = 2 tan t , 0 t

c4 hw pack 2014

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