A2 MATHEMATICS HOMEWORK C3

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1 Name Teacher A2 MATHEMATICS HOMEWORK C3 Mathematics Department September 2014 Version 1.1

2 Contents Contents... 2 Introduction... 3 HW1 Algebraic Fractions... 4 HW2 Mappings and Functions... 6 HW3 The Modulus Function... 8 HW5 The Exponential Functions HW6 Differentiation: Product and Quotient Rules HW7 Numerical Methods HW8 Trigonometric Equations HW9 Trigonometric Equations HW10 Trigonometric Equations HWX C3 June Page

3 Introduction Aim 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 C3. Your teacher may set homework from this or other tasks. Week Topic Date completed Mark HW0 HW1 HW2 HW3 HW4 Review summer work and revise for test Algebraic fractions and long division Mappings and functions The modulus function and transformations Exponential functions HW5 Differentiation 1 HW6 Differentiation 2 HW7 Numerical methods HW8 Trigonometry 1 HW9 Trigonometry 2 HW10 Trigonometry 3 HWX C3 June Page

4 HW1 Algebraic Fraction Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words Numerator, Denominator, Factorising, Cancelling like terms, improper fractions, polynomial division 1. Simplify the following expressions as much as possible: a) b) c) d) e) f) g) h) i) 2. Combine and simplify the following algebraic fractions a) + b) + c) d) + e) f) + g) h) + i) + 3. Combine and simplify the following algebraic fractions a) b) + c) d) e) f) 4. Express the following as impartial fractions (hint: polynomial division) a) b) c) d) e) 4 Page

5 Exercise B - Exam Questions 1. [C3 June 2007] Given that =, > Show that = (7 marks) 2. [C3 June 2012] Express as a single fraction in its simplest form. (4 marks) Exercise C Extension tasks Queen Mary University Essential Maths Questions 1. a) Add and simplify + b) Solve 4 = c) Simplify + 2. Go to (username: cityisli, password: ask teacher). or 5651 Answers 1a) b) c) d) e) f) g) h) i) 2a) b) c) d) e) f) g) h) i) 3a) 5 b) c) 1 d) e) f) 4a) b) c) 2+ d) e) Exercise B Exam questions check using 2. Exercise C Extension tasks 1a) b) =± c) 5 Page

6 HW2 Mappings and Functions Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words Function, Mapping, Domain, Range, Object, Input, Image, Output, One- One, Many-One, One-Many, Composite Function, Inverse Function Read the chapter on algebra and functions (p14-18, 20-22, 25-29) 1. Evaluate the following mappings a) = 4, R i) 4 ii) 3 iii) 0 b) : i) ii) iii) h c) For a quadratic equation the real solutions are given by the following function: ++= ±, 4. Find: i) +5+6 ii) iii) For each mapping i) Sketch the graph ii) State the range of the mapping iii) Say whether the mapping is one-one, many-one, one-many, or many-many iv) State whether the mapping is a function a) = +3, R b) =, 2 c) h: 3 2, > 1 d) =± 36, Given = +1, =1 3, evaluate: a) b) c) d) 3 6 Page

7 4. a) What has to be true for a function to have an inverse? b) Find the inverse of the following functions: i) = 2+7, R ii) = 1, 0 iii) h:, R iv) :, 3 Exercise B - Exam Questions 1. [C3 Jun 08 Q3] The function f is defined by a) Show that = :,>1.,>1 (4) b) Find. The function g is defined by : +5, R c) Solve =. 2. [C3 Jan 06 Q8] The functions f and g are defined by : a2 + ln 2, R, : a 2x, R. a) Prove that the composite function is : a4 4x, x R. (4) b) Sketch the curve with equation =, and show the coordinates of the point where the curve cuts the -axis. (1) c) Write down the range of. (1) Extension 1. Go to (username: cityisli, password: ask teacher). C3 Sequences. You can try some of the tests. Answers 1ai) 12, ii) 5, iii) -4 bi) 6, ii) 4, iii) undefined, ci) -2,-3, ii) 2, 3/2, ii) undefined. 2aii) 3, iii) many one, (iv) yes, bii) R 0, ii) one-one, iii) function, ci) h R, ii) one-one, iii) function d) 6 6, many-many, no. 3a) 1 3x +1, b) 1 3x +1, c) x +1 +1, d) 25. 4a) function must be one-one mapping b) i) = iii) = +1 iv) Exercise B 1b) =,c) = ± 2 2b) >0,>0, c) ii) h = 7 Page

8 HW3 The Modulus Function Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words Modulus Function, Absolute value Transformations 1. Calculate the following: a) 3 b) 5 c) 4 7 d) 3+4 e) 4 7 f) True or false; which of the following statements are correct. a) 2 = 2 b) 2 = 2 c) 2+3 = d) 2 = 2 e) 3 2 = Sketch the following graphs: a) = 5 b) = 5 c) = d) = e) = cos f) = +3 g) = 1 4. Solving the following equations (sketching the graphs is very important) a) +5 =3 b) 2 3 =7 c) +1 = 2 5 d) +3 = 6 e) 3+9 = 2+1 Exercise B Exam Questions 1. [C3 Jun 05] Figure 1 y 1 O b 3 x (1, a) Figure 1 shows part of the graph of y = f(x), x R. The graph consists of two line segments that meet at the point (1, a), a < 0. One line meets the x-axis at (3, 0). The other line meets the x-axis at ( 1, 0) and the y-axis at (0, b), b < 0. In separate diagrams, sketch the graph with equation a) y = f(x + 1), (2) b) y = f( x ). Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that f(x) = x 1 2, find c) the value of a and the value of b, (2) 8 Page

9 d) the value of for which = 5. (4) 2. [C3 Jan 06] Figure 1 y M (2, 4) 5 O 5 x Figure 1 shows the graph of =x, 5 5. The point M (2, 4) is the maximum turning point of the graph. Sketch, on separate diagrams, the graphs of a) = + 3, (2) b) y = f(x), (2) c) y = f( x ). 3. [C3 Jun 07] Extension Find the exact values of for which =3. Go to (username: cityisli, password: ask teacher) for further questions. Answers 1a) 3 b) 5 c) 3 d) 1 e) -3 f) 3 2a) T b) T c) T d) F e) T 3 Check using graph plotter or teacher will check. 4a) = 2, 8 b) = 2,5 c) = 4 only sketch the graph d) = e) = 8, 2 For d) & e) you can use the squaring method. See Modulus equations on examsolutions Exercise B Exam questions - see 2. c) = 2, = 1 d) = 3. =3,2 9 Page

10 HW5 The Exponential Functions Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words: Exponentials, Natural logarithms, exponential growth/decay, rates, modelling 1. Sketch the following graphs on different axis a) = and =ln (on the same axis) b) = c) =5 d) = Find the exact solutions of a) ln=3 b) ln+ln2=8 c) ln ln3=5 d) ln+4=ln+2 e) ln3+4=2 3. Find the exact solutions for for the following equations a) 4 =12 b) =5 c) 5 +2=17 d) 0.5= 1 e) 9 +14=0 f) =0 4. [Created by an AS student] A maths class is learning Calculus for the first time. After minutes, their stress levels, micrograms of cortisol per deciliter, is given by: =6 a) What are their stress levels when they enter the class? b) After 10 minutes their stress level is 12 micrograms per deciliter. Show that = to three significant figures. 10 Page

11 Exercise B Exam question 1. [C3 June 2009 Q3] Rabbits were introduced onto an island. The number of rabbits,, years after they were introduced is modelled by the equation Extension =80 R, 0 a) Write down the number of rabbits that were introduced to the island. (1) b) Find the number of years it would take for the number of rabbits to first exceed (2) c) Find (2) d) Find when =50 (Total 8 marks) 1. Create a model for exponential growth/decay for the following examples: (a) A cup of tea cooling down from 80C to room temperature. (b) Population of lizards on an island growing from 10 to 500 over 3 years. Do your models seem accurate for all situations? If not how could you alter it? 2. Try C3 Jan 2010 Q9 3. What is the maximum value of =? try to explain why. Answers 2.a) = b) = c) =3 d) = e) = 3.a) =ln3 b) = c) = d) = e) =ln7 or ln2 f) =ln or ln2 4.a) =6 Exam 1.a) 80 b) =12.6 c) =16 d) Page

12 HW6 Differentiation: Product and Quotient Rules Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words: derivative, function, gradient function, natural log, exponential, chain rule, product rule, quotient, substitution Read pages ; and make sure you understand. Try to help this consolidate what you ve already learnt about differentiation 1. Differentiate the following using the product rule: a) = b) =cossin c) =ln d) =1+3 e) =3 3 1 f) = Differentiate the following using the quotient rule: a) =tan b) =sec c) = d) = e) = f) = Exercise B Exam questions 1. [C3 Jun 05] a) Differentiate with respect to i) 3sin +sec2 ii) + 2 b) Given that =, 1, show that 2. [C3 Jan 06] Answers a) Differentiate with respect to = (6) i), (4) ii) b) Given that =4sin2+6 find in terms of. (5) 1a) 3 +, b) cos sin =1 2cos c) +ln d) e) a) = b) Exam Questions 1a) =3 2a) = =sin c) d) e) f) 2a) 6sincos+2sec2tan2 b) 3+ln2^2 1+ leading to = ai) =3 +2 aii) = b) = =± (4) 12 Page

13 HW7 Numerical Methods Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words: roots, interval, algebraic, accuracy, continuous function, domain, iteration, converge, diverge 1. Sketch the following curves and say for which ones the change on sign method will work. If it doesn t work can you give a reason? a) = 2, 1,3 Figure 1 b) =, 2,3 c) =sin, 0,2 2. Figure 1 shows a sketch of = 5 a) How many roots does the equation 5=0 have? b) Find an interval of unit length (e.g 8, 7) containing each root. c) Show that one root is 2.54 to 2 d.p. and find the other root. 3. What do you need to be true for the iteration method to work? 4. Find all the roots of 5+2=0 to 1 d.p. [hint; it might be useful to make a sketch using and note all the roots are in the range [-3,3]] Exercise B Exam questions 1. [C3 Jan 07] = 4 8 a) Show that there is a root of =0 in the interval 2, 1 b) Sketch the graph of = 2. [C3 Jun 08 Q7] = a) Show that =0 has a root,, between =1.4 and =1.45. (2) b) Show that the equation =0 can be written as = +, 0. c) Starting with =1.43, use the iteration = to calculate the values of, and, giving your answers to 4 decimal places. d) By choosing a suitable interval, show that =1.435 is correct to 3 d.p. 13 Page

14 3. [C3Jan 06 Q5] =2 4 a) Show that the equation =0 can be written as = + The equation 2 4=0 has a root between 1.35 and 1.4. b) Use the iteration formula = with =1.35, to find, to 2 decimal places, the value of, and. The only real root of =0 is α. c) By choosing a suitable interval, prove that α=1.392, to 3 decimal places. 4. [C3 Jun 05 Q4] Consider =3 2,>0. The iterative formula =, =1, is used to find an approximate value for α. a) Calculate the values of,, and, giving your answers to 4 decimal places. (2) b) By considering the change of sign of in a suitable interval, prove that α= correct to 4 decimal places. (2) Extension Find the solutions to the simultaneous equations =2 and =. ANSWERS 2.b) 2, c) 0,1, 2,3, d) 2.535<0, 2.545>0 and continuous 3. successive iteration must converge, and must converge to the root you are looking for , 0.4, 2. Exercise B 1. a) 2= =16>0 1=1+4 8 = 3<0. Change of sign (and continuity) root in interval 2, 1 2. a) 1.4= 0.568<0,1.45=0.245>0 1.4,1.45 < 0 > 0 Change of sign (and continuity) b) =1.4371, =1.4347, =1.4347, c) Choosing the interval , d) = 0.01,1.4355=0.003 =1.435 Due to change of sign (and continuity) 3. b) =1.41, =1.39, =1.39. c) Choosing (1.3915, ), , , Change of sign (and continuity) , c) =0.0613, =0.1568, =0.1425, = d) Using = = , = Page

15 HW8 Trigonometric Equations 1 Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words principal value, secondary value, repeat period, radians, interval 1. Read pages and make sure you understand. Try to link this with what you are learning about mappings and functions. Learn the following = 1 1 = 1 = sin + cos 1 tan +1 sec 1+cot 2. Starting with sin + cos 1, prove the following identities a) 1+cot cosec 2 b) tan +1 sec 3. Sketch the graphs of a) = b) = c) = 4. Solve each of the following equations giving your answers in the specified interval. a) sec=2 0 3 b) cosec =5 < c) cot=0.5 0 <2 5. Solve each of the following equations giving your answers in the range specified. a) sec =3+tan 0 <360 b) cot2+ = 3 <2 Exercise B - Exam questions 1. [C3 Jun 06 Q6] a) Using sin +cos 1 show that cosec 2 cot 1. (2) b) Hence, or otherwise, prove that cosec 4 cot cosec 2 +cot (2) c) Solve, for 90 <<180, cosec cot =2 cot (6) 2. [C3 Jan 07 Q8] Prove that sec cosec 2 tan cot Exercise C - Extension 1. Solve cot=0 0,2 15 Page

16 2. [C3 Jan 08 Q7] Given that =arccos, 1 1 and 0 a) Express arcsin in terms of b) Hence evaluate arccos + arcsin. give your answer in terms of Answers 3) Sec x cot x 4. a) (- ),,,, b) 0.201, 2.94 c) 1.11, 4.25, 7.39, a) 63.4, 135, 243.4, 315 b),,,,, Exercise B 1 =135 Extension 1 =, 2 a) = =cos,=sin arcsin=, b) arccos+arcsin=+ = 16 Page

17 HW9 Trigonometric Equations 2 Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words compound angle formula, double angle formula sin+=+ sin = cos+= cos =+ tan+= + 1 tan = 1+ 2=2 2=cos sin =2cos 1 =1 2sin 2= 2 1 tan 1. Read pages and make sure you understand [key points on p79,80] Learn the identities. Sketch a graph of cos and show that cos =cos Sketch a graph of sin and show that sin = sin 2. Use the compound angle formula to expand simplifying your answers without a calculator: a) cos60 b) sin2+30 c) tan45 3. Solve the following; a) sin=cos+120 in the range b) 2cos =cos+ 4. Start with the compound angle formulae and prove the following: a) cos2 1 2sin [eg start with cos+= ] b) cos2 2cos 1 c) tan2 d) sin2cos cos2sin sin 17 Page

18 Exercise B - Exam questions 1. [C3 Jun 10 Q1]. a) Show that =tan (2) b) Hence find, for 180 <180, all the solutions of Give your answers to 1 decimal place. =1 2. [C3 Jan 2011 Q3]. Find all the solutions of 2cos2=1 2sin in the interval 0 <360 (6) 3. [C3 Jan 2012 Q8] a) Starting from the formulae for sin + and cos +, prove that tan+= (4) b) Deduce that tan+ = c) Hence, or otherwise, solve, for 0, 1+ 3tan= 3 tantan Give your answers as multiples of. (6) Exercise C - Extension 1. Prove sin3 3sin 4 Answers: 2. a) cos+ 3 sin b) cos2+ 3 sin2 c) 3. a) 165 b) 2.79 radians (This is a hard question expand both sides then use =/ Exercise B Exam questions 1. =26.6, 154.4, 2. =54,126,198, = = 18 Page

19 HW10 Trigonometric Equations 3 Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words compound angle formula, double angle formula Read pages and have a look at the examples. +=+ = += =+ 1. Express each of the following in the form shown, where >0 and 0<< a) sin+3cos=sin+ b) 3sin 4cos=sin c) 2cos+7sin=cos d) cos2 2sin2=cos2+ 2. Sketch the graph of =5sin+30 and mark on this the maximum and minimum points on the graph. Use this graph to help solve the equation 5sin+ =4,, 3. For the following functions find the max value of the function and the value of theta for which it occurs a) =13sin+67.4 b) = 3cos 35.5 c) = 3cos+41.8 Exercise B Exam questions 1. [C3 Jun 07 Q6] a) Express 3sin+2cos in the form sin+α where >0 and 0<α<. (4) b) Hence find the greatest value 3 sin x + 2 cos x. (2) c) Solve, for 0 < <2, the equation 3sin+2cos =1, giving your answers to 3 decimal places. (5) 19 Page

20 2. [C3 Jun 2008 Q2] = 5cos+12sin Given that =cos where >0 and 0<< a) find the value of R and the value of α to 3 decimal places. (4) b) Hence solve the equation 5cos+12sin=6 for 0 2 (5) c) i) Write down the maximum value of 5cos+12sin. (1) ii) Find the smallest positive value of for which this maximum value occurs. (2) 3. [C3 Jan 09 Q8] a) Express 3+4 in the form, where and are constants, > 0 and 0 < < 90. (4) b) Hence find the maximum value of and the smallest positive value of θ for which this maximum occurs. The temperature,, of a warehouse is modelled using the equation = where t is the time in hours from midday and 0 < 24. c) Calculate the minimum temperature of the warehouse as given by this model. (2) d) Find the value of t when this minimum temperature occurs. Extension For = find the max and minimum value of this function Also see page 77 in text book for more questions. Answers: 1a 10sin b 5sin c 53cos 74.1 d 5cos( =0.404, a f(x) max = 13, =22.6 b f(x) max = 3, =35.5 c y max = 3, =228.2 Jun 07, 6a =, α = (Allow 33.7 ), b 169, c x = or x = (awrt)both (radians only) Jun 08, 2a R =13, = b 2.3, awrt or c R max =13 at max =1.176 Jun 09, 8a R=5, =53. b max value = 5 and this occurs at =53. c min temp is 5. D t = 15.5 Extension 5 max = 2, min = 1 / 3 20 Page

21 HWX C3 June a) Show that sin 2θ 1+ cos 2θ = tan θ. (2) b) Hence find, for 180 θ < 180, all the solutions of 2sin 2θ 1+ cos 2θ Give your answers to 1 decimal place. 2. A curve C has equation y = 3 ( 5 3x) 2 = 1., x 3 5. The point P on C has x-coordinate 2. Find an equation of the normal to C at P in the form ax + by + c = 0, where a, b and c are integers. (7) 3. f(x) = 4 cosec x 4x +1, where x is in radians. a) Show that there is a root α of f(x) = 0 in the interval [1.2, 1.3]. b) Show that the equation f(x) = 0 can be written in the form 1 1 x = + sin x 4 c) Use the iterative formula 1 1 x n+ 1 = +, x0 = 1.25, sin x n 4 to calculate the values of x 1, x 2 and x 3, giving your answers to 4 decimal places. d) By considering the change of sign of f(x) in a suitable interval, verify that α = correct to 3 decimal places. (2) 4. The function f is defined by f : x 2x 5, x R. a) Sketch the graph with equation y = f(x), showing the coordinates of the points where the graph cuts or meets the axes. (2) b) Solve f(x) =15 + x. The function g is defined by c) Find fg(2). d) Find the range of g. g : x x 2 4x + 1, x R, 0 x 5. (2) (2) (2) 21 Page

22 5. Figure 1 Figure 1 shows a sketch of the curve C with the equation y = (2x 2 5x + 2)e x. a) Find the coordinates of the point where C crosses the y-axis. b) Show that C crosses the x-axis at x = 2 and find the x-coordinate of the other point where C crosses the x-axis. d y c) Find. dx d) Hence find the exact coordinates of the turning points of C. (5) (1) 7. a) Express 2 sin θ 1.5 cos θ in the form R sin (θ α), where R > 0 and 0 < α < 2 π. Give the value of α to 4 decimal places. b) (i) Find the maximum value of 2 sin θ 1.5 cos θ. (ii) Find the value of θ, for 0 θ < π, at which this maximum occurs. Tom models the height of sea water, H metres, on a particular day by the equation 4π t H = sin 25 where t hours is the number of hours after midday. 4π t 1.5 cos, 0 t <12, 25 c) Calculate the maximum value of H predicted by this model and the value of t, to 2 decimal places, when this maximum occurs. d) Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres. (6) 22 Page

23 6. Figure 2 Figure 2 shows a sketch of the curve with the equation y = f(x), x R. The curve has a turning point at A(3, 4) and also passes through the point (0, 5). a) Write down the coordinates of the point to which A is transformed on the curve with equation i) y = f(x), ii) y = 2f( 1 2 x). b) Sketch the curve with equation y = f( x ). (4) On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the y-axis. The curve with equation y = f(x) is a translation of the curve with equation y = x 2. c) Find f(x). d) Explain why the function f does not have an inverse. (2) (1) 8. a) Simplify fully Given that b) find x in terms of e. 2 2x + 9x 5. 2 x + 2x 15 ln (2x 2 + 9x 5) = 1 + ln (x 2 + 2x 15), x 5, (4) TOTAL FOR PAPER: 75 MARKS END 23 Page

A2 MATHEMATICS HOMEWORK C3

Name Teacher A2 MATHEMATICS HOMEWORK C3 Mathematics Department September 2016 Version 1 Contents Contents... 2 Introduction... 3 Week 1 Trigonometric Equations 1... 4 Week 2 Trigonometric Equations 2...