C4 HW Pack 2014 V1.0

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A2 MATHEMATICS HOMEWORK C4

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Mathematics DepartmentSeptember 2014Version 1.0Contents

Contents2Introduction3HW1 Partial Fractions4HW2 Parametric equations (including differentiation)6HW3 Binomial Expansions8HW4 Differentiation Implicit functions10HW5 Integration 1 Introduction12HW6 Integration 2 - by Substitution14HW7 Integration 3 - by parts16HW8 Integration 4 Partial fractions18HW9 Integration 5 Trig identities19HW10 Integration 6 Differential equations21HW11 Integration 7 Numerical Methods and Volumes of Revolution24HW12 C4 Vectors27HWX C4 June 201030

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

To learn effectively you should check your work carefully and mark answers ? If you have 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 for the next edition!

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

TopicDate completedComment

HW1Partial Fractions

HW2Parametric Equations (inc differentiation)

HW3Binomial Expansions

HW4Differential Implicit functions

HW5Integration 1 Introduction

HW6Integration 2 by substitution

HW7Integration 3 by parts

HW8Integration 4 Partial Fractions

HW9Integration 5 Trig Identities

HW10Integration 6 Differential Equations

HW11Integration 7 Numerical Methods & Volumes of Revolution

HW12Vectors

HWXC4 June 2010

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 A1.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 partial fractions:

a) b)

c)

Exercise B - Exam Questions

1. [C4 Jun 2011 Q1]

= + +

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

2.[C4 June 2012 Q1]

= + + .

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

3. [C4 Jan 2009 Q3]

, .

Given that can be expressed in the form

+ + ,

find the values of and and show that .(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 A1.a)b)c)d)e)f)2.a)b)c)3.a)b)c)

Exercise B Exam questions 1.2.3.

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 the parameter

equation of parabola equation of a circle where where Read pages 175-185, 215-221

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

Exercise A1.Find in terms of when and are related by the following pairs of parametric equations.a) b)

2.Change to following parametric equations into Cartesian form by eliminating the parametera) b)

3.Find the equation of the tangent to the curve at the point givena) b)

As an extension try to sketch the graphs.4.Calculate the area between the of the parametric equations

Exercise B - Exam Questions 1.[C4 Jan2009 Q7] The curve C shown opposite has parametric equations

x = t 3 8t, y = t 2

where t is a parameter. Given that the point A has parameter 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 2x 5y 9 = 0.(5)The line l also intersects the curve at the point B.

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

2.[C4 Jun 2008 Q8]

The graph opposite shows the curve C with parametric equations

x = 8 cos t, y = 4 sin 2t, 0 t .The point P lies on C and has coordinates (4, 23).

(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 + 63.(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 are used in electronics.

AnswersExercise A1a) b)

2a) b)

3a)

b)

4. Integral Area

Exercise B Exam questions May 08 June 09

HW3 Binomial ExpansionsComplete on a separate sheet of paper. Show clear working. Mark your answers.Key words Expand, Binomial theorem, Factorial, Limit, Ascending, CoefficientRead the chapter on binomial expansion (p14- 18, 20-22, 25-29)Exercise A1. Rewrite each expression in the form example: a)b)c)d)e)f)g)h)2.Find the binomial expansions of parts a to d from the previous question in ascending powers of as far as the term. State the range of values of for which the expansions are valid.

Exercise B - Exam Questions1.[C4 Jun 11 Q2] Find the first three non-zero terms of the binomial expansion of in ascending 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 each term.(6)b) Show that, when , the exact value of , is . (2)c)Substitute into the binomial expansion in part(a) and hence obtain an approximation to . Give your answer to 5 decimal places. (3)

3.[C4 Jan 11 Q5]a) Use the binomial theorem to expand

, ,

in ascending powers of , up to and including the term in . Give each coefficient as a simplified fraction.(5)

, , where and are constants.

In the binomial expansion of , 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)

AnswersExercise A1.(a)(b)(c)(d)(e)(f)(g)(h)2.(a)(b)(c)(d)Exercise B1.2.(a)(c)3.(a)(b)(c)

A bit of historyThe binomial formula and the binomial coefficients are often attributed toBlaise Pascal, who described them in the 17th century, but they were known to many mathematicians who preceded him. A more general binomial theorem and the so-called "Pascal's triangle" were known in the 10th century A.D. to Indian mathematicianHalayudha.Arabian mathematicianAl-Karaji,in the 11th century was aware of a more general binomial theorem, and in the 13th century toChinese mathematicianYang Hui, also derived similar results. Sir Isaac Newton is generally credited with the generalised binomial theorem, valid for any rational exponent.

Isaac Newton 1642 - 1726HW4 Differentiation Implicit functions


chain ruleproduct rule

ExplicitImplicit

Read pages 209 215Note

Exercise A1.Practice the chain rule with Explicit functions. Differentiate to find a) b) c)

d) e) f)

2.Differentiate the following Implicit functions with respect to . Give answers in terms of a) b) c)

d) e) f)

3.Differentiate the following Implicit functions with respect to , and find .

a) b) c)

d) e) f)

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

5.The ellipse has equation Find the stationary points.


1.[C4 Jan 12 Q1] The curve C has the equation 2x + 3y2 + 3x2 y = 4x2.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 form ax+by+c=0, where a, b and c are integers.(3)

2. [C4 Jun 2008 Q4] A curve has equation 3x2 y2 + xy = 4. The points P and Q lie on the curve. The gradient of the tangent to the curve is at P and at Q.(a) Use implicit differentiation to show that y 2x = 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 differentiation

Answers

Exercise A

1a) b) c)

d) e) f)

2a) b) c)

d) e) f)

3a) b) c)

d) e) f)

4. 5.

Exercise B Exam questions 1. a) ,

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

HW5 Integration 1 Introduction


Key words Integral, Standard functions, Factorising, Cancelling like terms, improper fractions, polynomial divisionRead pages 234 252. This homework links with Exercise 10A 10F. Use these for further practice.

For more formulae see page 308 and formulae booklet

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)b)c) d)e)f)

4.Find the following indefinite integrals:a)b)c)d)e) f)

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

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

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

a)b)

Exercise B - Exam Questions2.[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 , 0 x .

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

Exercise C Extension tasks1.a) Show that

b) Show that

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) e) f)

4a) b) c)

d) e) f)

5a) b) c)

d) e) f)

6a) b)

Exercise B Exam questions9 unitsHW6 Integration 2 - by SubstitutionComplete 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

A1

B2

C3

D4

E5

F6

G7

H8

I9

J10

11

2.Find the following integrals using the given substitution or otherwise.a) b) c) d) e) f)

3.Evaluate the following integrals using the given substitution or otherwise.a) b) c) d)

Exercise B - Exam Questions 1. [C4 Jan 2013Q4 (adapted)]

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

2. [C4 Jan 2011Q7 (adapted)]

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

4. [C4 Jan 2012Q6 (adapted)]

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

5. [C4 June 2013 (R)Q3]

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

dx

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, J12a) b) c) d) e) f)

3a) b) c) d)

Exercise B Exam questions

2) 3) 4) 5) 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

Exercise A1.Integrate the following by partsa) b)

c) d)

2.Integrate the following by partsa)b)

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

4.By writing show that

Exercise B - Exam Questions1. [C4 Jun 2008 Q2]

(a) Use integration by parts to find .(3)

(b) Hence find .(3)2.[C4 Jan 2012 Q2]

(a) Use integration by parts to find (3)

(b) Using your answer to part (a), find (3)

Answers Exercise A1a) b) c) d)

2a) b) 3a) b) Exercise B Exam questions 1. a) b)

2a) b) 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 practicea) b) c) d)

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

d) Write in the form , then integrate

Exercise B - Exam Question 1. June 2012 Q1

f(x) = = + + .

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

(b)(i)Hence find .

(ii)Find , leaving your answer in the form a + ln b, where a and b are constants.(6)AnswersExercise A1a) b) c) d)

2a) b) c)

d)

Exercise B Exam questions 1a) b) c) HW9 Integration 5 Trig identities


Key words IntegrationTextbook pages 275-279

Exercise A

1.Find each of the following integrals as introductory practice.a) b) c)

d) e) f)

2.a) Use the identity to find [even powers]

b) Use the identity to find

c) Use the identity to find

d) Use the identity to find [even power]

3.Find the following integrals [ odd powers]

a) b) c)


1. [C4 Jan 2013 Q6]

Figure 3Figure 3 shows a sketch of part of the curve with equation y = 1 2 cos x, where x is measured in radians. The curve crosses the x-axis at the point A and at the point B.

(a) Find, in terms of , the x coordinate of the point A and the x coordinate of the point B.(3)The finite region S enclosed by the curve and the x-axis is shown shaded in Figure 3. The region S is rotated 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 you may find a substitution helpful.

Answers

Exercise A1a) b) c) d) e) f)

2a) b) c) d)

Tips: 2d) then substitute 3a) b) c)

Exercise B Exam questions 1a) b) This is a difficult question

Exercise C Extension tasksAsk your teacherHW10 Integration 6 Differential equations


Key words Differential equation, general solution, particular solution.

Exercise A1.Find the particular solutions of the following differential equations DEsa)

b)

c)

2.Find the general solutions of the following DEsa)Sketch the solution curve which passes through

b)Sketch the solution curve which passes through

c)

d)

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

a)the rate of increase of the surface areab)the rate of increase of the volume

2.If a hemispherical bowl of radius contains water to a depth of , the volume of water is

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

Exercise C - Exam Questions

1.[C4 Jan/Jun 2008 Q7]

(a) Express in partial fractions.(3)(b) Hence obtain the solution of

2 cot x = (4 y2)

for which y = 0 at x = , giving your answer in the form sec2 x = g( y).(8)2. [C4 June 2008 Q3]

Figure 1

Figure 1 shows a right circular cylindrical metal rod which is expanding as it is heated. After tseconds the radius of the rod is x cm and the length of the rod is 5x cm.

The cross-sectional area of the rod is increasing at the constant rate of 0.032 cm2 s1.

(a) Find 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 right circular cone. The height of the container is 24cm and the radius is 16cm, as shown in Figure 2. Water is flowing into the container. When the height of water is hcm, the surface of the water has radius rcm and the volume of water is V cm3.

(a) Show that V = .(2)

[The volume V of a right circular cone with vertical height h and base radius r is given by V = r 2h .]

Water flows into the container at a rate of 8 cm3 s1.

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

Exercise D Extension taskA murder victim was discovered by the police at 6:00 am. The body temperature of the victim was measured and found to be . A doctor arrived on the scene of the crime 30 minutes later and measured the body temperature again. It was found to be . The temperature of the room had remained constant at . The doctor, knowing normal body temperature to e , 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 the object and its surroundings.

Exercise A Answers1a) b) c) 2a)

b) , solution curve is a circle through circle centre 0, radius 5

c) d) e)

Exercise B 1a) b) 2

Exercise C Exam questions 1a) b) 2a) b) 3b)

Exercise D Extension taskSee your teacherHW11 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 A1.a)Estimate using the trapezium rule with five ordinates. Would you expect 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

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 beingdistance from bank (m)024681012

depth (m)01.82.633.420

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

2a) Find the volume of the solid of revolution formed when the region enclosed by each of the following curves and the , between the given values of x, is rotated through radians about the .

(i) from to (ii) from to (i) from to (i) from to

b) Obtain the formula for the volume of a hemisphere by rotating the part of the circle in the first quadrant about the . Deduce the formula for the volume of a sphere.

c) The region between and 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 , 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 The region under the 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 = , 0 x .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 = .

x0

y01.171571.022800

(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 estimate for the area of R, giving your answer to 4 decimal places.(3)(c) Using the substitution u = 1 + cos x, or otherwise, show that

= 4 ln (1 + cos x) 4 cos x + k,where k is a constant.(5)(d) Hence calculate the error of the estimate in part (b), giving your answer to 2 significant figures.(3)2.[C4 Jan 2012 Q4]

Figure 1

Figure 1 shows the curve with equation

y = , x 0.

The finite region S, shown shaded in Figure 1, is bounded by the curve, the x-axis and the linex=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 answer in 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 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 largeb) 3.75c) 2.97m 0.8%d)

2a(i) (ii) (iii) (iv)

b)c) 0.818, 0.6%Exercise B Exam questions 1. a) 0.73508 b) 1.1504 d)0.0772.

Exercise C Extension tasksSee Teacher

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

Exercise A1.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 2.Find the vector equations of the lines joining the points given.

a) b)

c) d)

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

a)

b)

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

5.The line and have equations 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 acute angle between AB and the line

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 = (9i + 10k) + (2i + j k)

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

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 5i + 7j + 3k.

(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)

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 . 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)

Exercise C Extension questions

Vectors have many applications in advanced mathematics and engineering. Find out about finite 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) b) or

c) d)

3a) Intersection b)

4a) b)

Exercise B 1.(a) 3i +3j + 7k (d) 11i j + 11k2.(b) p = 1 (c) (d)

HWX C4 June 20101.Figure 1

Figure 1 shows part of the curve with equation y = (0.75 + cos2 x). The finite region R, shown shaded in Figure 1, is bounded by the curve, the y-axis, the x-axis and the line with equationx=.

(a) Copy and complete the table with values of y corresponding to x = and x = .

x0

y1.32291.29731

(2)(b) Use the trapezium rule

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

= e(e 1). (6)

3. A curve C has equation2x + y2 = 2xy.

Find the exact value of at the point on C with coordinates (3, 2).(7)4.A curve C has parametric equations

x = sin2 t, y = 2 tan t , 0 t < .

(a) Find in terms of t.(4)

The tangent to C at the point where t = cuts the x-axis at the point P.

(b) Find the x-coordinate of P.(6)

5. A + + .

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

(b) Hence, or otherwise, expand in ascending powers of x, as far as the term in x2. Give each coefficient as a simplified fraction.(7)

6.f() = 4 cos2 3sin2

(a) Show that f() = + cos 2.(3)

(b) Hence, using calculus, find the exact value of . (7)

7.The line has equation r = , where is a scalar parameter.

The line has equation r = , where is a scalar parameter.

Given that and meet at the point C, find

(a) the coordinates of C.(3)

The point A is the point on where = 0 and the point B is the point on where = 1.

(b) Find the size of the angle ACB. Give your answer in degrees to 2 decimal places.(4)(c) Hence, or otherwise, find the area of the triangle ABC.(5)

8.

Figure 2

Figure 2 shows a cylindrical water tank. The diameter of a circular cross-section of the tank is6m. Water is flowing into the tank at a constant rate of 0.48 m3 min1. At time t minutes, the depth of the water in the tank is h metres. There is a tap at a point T at the bottom of the tank. When the tap is open, water leaves the tank at a rate of 0.6h m3 min1.

(a) Show that, t minutes after the tap has been opened,

75 = (4 5h).

(5)

When t = 0, h = 0.2

(b) Find the value of t when h = 0.5(6)TOTAL FOR PAPER: 75 MARKSEND

page 18

C4 HW Pack 2014 V1.0

Documents

technique of partial

improper fractions

answers exercise

exercise b exam questions

introduction12hw6 integration

parts16hw8 integration

substitution14hw7 integration

exercise a1