BHASVIC Maths Y2 Learning Pack September – October 2021 0 A LEVEL MATHEMATICS YEAR 2 LEARNING PACK CONTENTS Your teacher will tell you which lesson to watch the videos by. If they don’t then assume all videos need to be watched by the first lesson of the week. Week -1 4th Sept 0 13th Sept 1 20th Sept 2 27th 3 Sept 4th Oct Ref in spec P5 P5 P5 P5 P2 P2 P2 P2 P2 P2 P8 P8 P8 P5 P5 P5 P7 P1 P1 P1 P1 P1 P5 P5 P7 P7 Videos Page Watch by lesson Prior Knowledge – Compound Angles Compound Angle Proofs and Solves Double Angle Proofs and Solves Trig Proofs Prior Knowledge -Partial Fractions Partial Fractions – 2 linear factors Partial Fractions – 3 linear factors Partial Fractions – Repeated Root Partial Fractions – Alternative Method Partial Fractions – Top Heavy Integration Using Partial Fractions Prior Knowledge - Integration using Trig Identities Integration using Trig Identities Prior Knowledge -Small Angle Approximations Small Angle Approximations - proof Small Angle Approximations - applications Differentiation of sin x and cos x from 1st principles Prior Knowledge - Proof Proof by Deduction Proof by Exhaustion Proof by Counter Example Proof by Contradiction Prior Knowledge - Inverse Trig – graphs and equations Inverse Trig – graphs and equations Prior Knowledge - Implicit differentiation Implicit Differentiation 3 4 7 9 10 11 12 13 13 14 15 16 Mini lesson 17 23 24 25 26 1 2 3 1 2 3 27 28 29 30 31 32 1 1 1 1 1 1 33 38 39 1 2 3 1 2 3 1 2 3 1 Mini lesson 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 1 2 3 1 2 3 2 2 2 2 2 2 3 3 3 3 3 3 Tick when watched 4 11th Oct P2 P2 P2 P2 P2 5 18th Oct SM9 SM9 SM9 SM9 25TH Oct Prior Knowledge - Binomial Expansion Binomial expansion - Introduction Finite Binomial Expansion where n is a positive integer Infinite Binomial Expansion where n is fractional or negative Binomial Expansion using Partial Fractions 43 44 45 1 2 3 1 2 3 1 2 3 46 1 2 3 48 1 2 3 Prior Knowledge - Horizontal Moments Horizontal Moments - Introduction Horizontal Moments - Beams Horizontal Moments - Tilting HALF TERM 49 50 52 53 1 1 1 1 2 2 2 2 3 3 3 3 Reference to specification Reference to specification P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 Topic Reference to specification SM1 SM2 SM3 SM4 SM5 SM6 SM7 SM8 SM9 SM1 Proof Algebra and functions Coordinate geometry in the (x,y) plane Sequences and series Trigonometry Exponentials and logarithms Differentiation Integration Numerical methods Vectors 2 Topic Sampling Data Presentation and interpretation Probability Statistical distributions Statistical hypothesis testing Quantities and units in mechanics Kinematics Forces and Newton’s laws Moments Sampling Compound Angle Prior Knowledge Work 1. Find the exact values of: (a) sin 45° (b) cos π (c) tan 6 π 3 2. Solve the following equations in the interval 0 ≤ π₯ ≤ 360°. (a) sin(π₯ + 50°) = −0.9 (b) cos(2π₯ − 30°) = 1 2 (c) 2 sin2 π₯ − sin π₯ − 3 = 0 3. Prove the following: (a) cos π₯ + sin π₯ tan π₯ ≡ sec π₯ (b) cot π₯ sec π₯ sin π₯ ≡ 1 (c) cos2 π₯+sin2 π₯ 1+cot2 π₯ ≡ sec 2 π₯ Answers: 1. (a) 1 √2 (b) √3 2 (c) √3 2. (a) 194.2°, 245.8° (b) 45°, 165°, 225°, 345° (c) 270° 3. Proofs 3 P5 – Trigonometry The Addition formulae (Compound angle) https://drive.explaineverything.com/thecode/AMLHFXQ What is sin (10 + 20)? Why? 4 22min In the formula book sin (A+B) = cos (A+B) = sin (A-B) = cos (A-B) = tan (A+B) = tan (A-B) = The Proof of tan (A+B) (this needs learning off by heart) Prove that tan (A+B) = 5 Example 1 Express the following as a single sine, cosine or tangent: π‘ππ 60 − π‘ππ 45 1 + π‘ππ 60 π‘ππ45 Example 2 Calculate the exact value of tan π₯ πππ 1 sin(π₯ − 30) = cos π₯ 2 Example 3 6 P5 – Trigonometry The Double Angle formulae https://drive.explaineverything.com/thecode/CGETHCB In the formula book The Double angle formulae are: sin (2A) = cos (2A) = cos (2A) = cos (2A) = tan (2A) = 7 22 min Example 1 Solve 5 π ππ2π + 4 π πππ = 0 πππ 0 ≤ π ≤ 360 Example 2 Solve 3 cos 2π₯ − cos π₯ + 2 = 0 πππ 0 ≤ π₯ ≤ 360 Something clever (manipulating the formulae): Example 3 Solve sin 4 π = πππ 2π πππ 0 ≤ π ≤ π Example 4 π Solve 3πππ π − π ππ (2 ) −1 = 0 πππ 0 ≤ π ≤ π 8 P5 – Trigonometry Trig proofs https://drive.explaineverything.com/thecode/BLKLKSX Example 1 cos 2π΄ Prove πππ 2 π΄−πππ π΄π πππ΄ ≡ 1 + tan π΄ Example 2 Prove πππ 3π π πππ + π ππ3π πππ π ≡ 2 cot 2π Example 3 Prove cos 3π΄ ≡ 4πππ 3 π΄ − 3 cos π΄ 9 10min Partial Fractions Prior Knowledge Work: To prepare for this week’s topic, you need a sound understanding of algebraic fractions and algebraic manipulation. Try these questions and seek out help if you need it from your teacher or other students. 1. Write each of this expression as a single fraction. 4 3 2 + + π₯ + 2 π₯ + 3 (π₯ + 2)2 Answer in this video: https://youtu.be/NvdT5wKP160 2. Integrate the following expressions: 3 (a) ∫ 3π₯−1 ππ₯ 6 π₯ (b) ∫ 2π₯−5 ππ₯ (c) ∫ π₯ 2 +4 ππ₯ Answers: 2. (a) ln(3π₯ − 1) + π (b) 3 ln(2π₯ − 5) + π 10 1 (d) 2 ln(π₯ 2 + 4) + π P1 – Algebra and Functions 1. Partial Fractions – 2 Linear Factors https://youtu.be/NUuJlgmb8PA 3 mins π₯+3 (π₯ + 1)(π₯ + 2) Now you try this: Check your answer at: https://youtu.be/nWSnWNgjUy8 11 P1- Algebra and Functions 2. Partial Fractions – 3 Linear Factors https://youtu.be/dCmrPqIfWW0 5 mins 5π₯ 2 + 3π₯ + 4 (π₯ + 1)(2π₯ + 1)(π₯ − 2) Now you try this: Check your answer at: https://youtu.be/TB1d8TXx50U 12 P1 – Algebra and Functions 3. Partial Fractions – Repeated Factors 6 mins https://youtu.be/qztRKnUnuC4 4π₯ 2 − 3π₯ + 2 (π₯ + 1)(π₯ − 2)2 4. Alternative Method https://youtu.be/HDzfnWelx3o 10 mins 2π₯ 2 − 3 (π₯ − 2)(π₯ + 3)2 13 P1 Algebra and Functions 5. Partial Fractions – Top Heavy https://youtu.be/SO8GfX9taJA 4 mins 2π₯ 2 + 5 (π₯ + 4)(π₯ − 2) 14 P8- Integration using Partial Fractions 6. Partial Fractions – Use of Partial Fractions in Integration https://estream.bhasvic.ac.uk/View.aspx?id=15928~5m~eYAyXjcqPj 4 mins ∫ π₯2 3π₯ ππ₯ − 4π₯ − 5 15 Prior Knowledge: Integration Using Trig Identities To prepare for this week’s topic, you need to remember the trig identities that we have met thus far in the course. You also need to be able to integrate trig functions using the reverse chain rule (GDA). 1. Match up the following trig identities. 2 cos2 π΄ − 1 sin π₯ cos π₯ sec 2 π₯ sin2 π₯ + cos 2 π₯ tan π΄ ± tan π΅ 1 β tan π΄ tan π΅ tan2 π₯ + 1 sin π΄ cos π΅ ± cos π΄ sin π΅ 1 + cot 2 π₯ cos2 π΄ − sin2 π΄ sin(π΄ ± π΅) tan π₯ cos(π΄ ± π΅) cos π΄ cos π΅ β sin π΄ sin π΅ tan(π΄ ± π΅) 2 sin π΄ cos π΄ sin 2π΄ 1 cos 2π΄ 2 tan π΄ 1 − tan2 π΄ tan 2π΄ cosec 2 π₯ 1 − 2 sin2 π΄ Check your answers using the online textbook or your learning pack. 2. Integrate the following functions: π₯ (b) ∫ sec 2 2 ππ₯ (a) ∫ sin 5π₯ ππ₯ π π₯ 3. Evaluate ∫0 cos 2 ππ₯ Answers: 1 2. (a) 5 cos 5π₯ + π π₯ (b) 2 tan 2 + π 3. 2 16 P8 – Integration – Integration Using Trig Identities Note that the following standard integrals, that can be found in the differentiation and/or integration sections of the formula booklet, are regularly used in the integrals in this section. 1. https://youtu.be/mkgOwMD140Y ∫ 3 mins sin 2π₯ ππ₯ cos π₯ 17 2. https://youtu.be/3RIxvN-7IFI 5 mins ∫ tan2 π₯ + 1 ππ₯ ∫ 1 + tan2 5π ππ ∫ 3 + 3 tan2 ππ₯ ππ₯ 18 3. https://youtu.be/yc8Kepa1s20 5 mins ∫ sin π₯ cos π₯ ππ₯ ∫ 5 cos 3π₯ 3π₯ sin ππ₯ 2 2 19 4. https://youtu.be/KrWPb6JTVRc 7 mins ∫ sin2 π₯ ππ₯ ∫ 3 sin2 5π ππ 5. https://youtu.be/7cT0lHvE-fM 7 mins ∫ cos 2 π₯ ππ₯ ∫ 5 cos 2 8π ππ 20 6. https://drive.explaineverything.com/thecode/RKVAWPF 14 mins ∫ π ππ2 3π₯ ππ₯ ∫ πππ 4 π₯ ππ₯ To integrate an even power of sin or cos I must use……… 21 ∫ πππ 5 π₯ ππ₯ To integrate an odd power of sin or cos I must use……… 22 Prior Knowledge Work: Small Angle Approximations and the Differentiation of sin x and cos x: To prepare for this week’s topic, we are going to explore the trig graphs close to the y axis, ie. when x or θ is very small. 1. Use DESMOS (making sure it is in radians) to plot the graphs of y = sin x and y = x. Zoom in towards the origin. What do you notice about the two graphs? What does this tell us about sin x and x when x is very small? 2. Use DESMOS (making sure it is in radians) to plot the graphs of y = tan x and y = x. Zoom in towards the origin. What do you notice about the two graphs? What does this tell us about tan x and x when x is very small? 3. Use DESMOS (making sure it is in radians) to plot the graphs of y = cos x and y = 1 − Zoom in towards the point (0, 1). What do you notice about the two graphs? What does this tell us about cos x and 1 − π₯2 2 when x is very small? You also need to remember how to differentiate from first principals. 4. Prove, from first principals, that the derivative of 4x2 is 8x. 23 π₯2 . 2 P5 - Trigonometry 1. Small Angle Approximations - Proof https://youtu.be/-IRHJmBPWw4 11 mins The small angle approximations are: sin π ≈ cos π ≈ tan π ≈ This is a really nice proof of where these approximations come from. Try to follow the proof and take notes. However, you will not be asked for this proof in an exam. 24 P5 - Trigonometry 1. Small Angle Approximations - Applications https://youtu.be/ozZINFNndiY 5 mins Try this question: Check your solutions at: https://youtu.be/lp32QiaNbAI 25 P7 - Differentiation 1. Proof of the Derivative of sin x and cos x https://youtu.be/6D1Zrzgq4t4 lim sin β β→0 β 10 mins cos β − 1 = β→0 β = lim π¦ = sin π₯ π¦ = πππ π₯ 26 Prior Knowledge – Proof To prepare for this week’s topic you need a good understanding of some of the proofs you have already covered in the course. If you are stuck refer to Chapter 7.3 (page 174) in the Year 2 Pure textbook Try these questions: 1. Using completing the square, prove that π2 − 8π + 17 is positive for all values of n 2. Prove that π πππ₯π ππ2π₯ + πππ π₯πππ 2π₯ ≡ πππ π₯ Answers 1. π2 − 8π + 17 = (π − 4)2 + 1 (π − 4)2 ≥ 0 (as a square number is always greater than or equal to zero) π π (π − 4)2 ≥ 1 (which means it is always a positive number) Therefore π2 − 8π + 17 is positive for all values of n 2. πΏπ»π = π πππ₯π ππ2π₯ + πππ π₯πππ 2π₯ = π πππ₯ (2 π πππ₯πππ π₯) + πππ π₯ (2πππ π₯ 2 − 1) = 2π ππ2 π₯πππ π₯ + 2πππ 3 π₯ − πππ π₯ = 2 cos π₯ (1 − πππ 2 π₯) + 2πππ 3 π₯ − πππ π₯ = 2πππ π₯ − 2πππ 3 π₯ + 2πππ 3 π₯ − πππ π₯ = πππ π₯ = π π»π ππΈπ· 27 P1 – Proof by deduction Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion. This is the most commonly used method of proof throughout this specification. For example, you could be asked to prove by deduction, using completion of the square, that 3n2 − 18n + 30 is positive for all values of n. Other examples where you would use this type of proof is if you were asked to prove a simple derivative from first principles or proving results for arithmetic and geometric series. https://www.youtube.com/watch?v=SBvurpUG81w&feature=youtu.be 8 mins 1. Prove that for any four consecutive integers, the difference between the product of the last two and the product of the first two of these numbers is equal to their sum. 2. k3 – k is divisible by 6 for all k > 1 28 P1 – Proof by exhaustion Proof by exhaustion This involves trying all the options. For example, suppose x and y are odd positive integers less than 7, prove that their sum is divisible by 2. To prove this we would go through all the options of adding together 2 positive integers, both of which are less than 7, then showing for each of our sums, it is divisible by 2. https://www.youtube.com/watch?v=0PgXAzJ2QNo&feature=youtu.be Prove that 53 is a prime n umber 29 6 mins P1 – Proof by counter example For example, suppose that we are asked to show that the statement “n2 – n + 1 is a prime number for all values of n” is untrue. To prove this by counterexample we only have to find a single example of n where n2 – n + 1 is not a prime number. https://www.youtube.com/watch?v=mxcGpNji4ik&feature=youtu.be 6 mins Prove the following are not true. It is sufficient to find one example where they are not true. 1. 4n + 4 is always a multiple of 8 2. If x > y then x/y > 1 3. π − π ≤ π2 − π 2 for all values of p and q 4. 2n2 – 16 + 31 is always positive 30 P1 – Proof by contradiction Proof by contradiction is performed by initially assuming that the given statement we are asked to prove is not true. Then using logical deduction we reach a conclusion that contradicts our assumption. Thus we conclude that the assumption must be incorrect - in which case the initial statement must be true. Examples of when we use this method of proof is to prove irrationality of √p where p is prime. We also use it to show that there are an infinite number of prime numbers. This method is often the one to choose when the other methods do not lend themselves to easily starting the proof we are asked to do. https://www.youtube.com/watch?v=VNZoB0qao1U&feature=youtu.be Prove that √2 is irrational 31 7 mins Prior Knowledge Work: Inverse Trig Functions To prepare for this week’s topic, you need a sound understanding of inverse functions, and solving trig equations. Try these questions and seek out help if you need it from your teacher or other students. 1. For each function f(x), find the inverse function f-1(x). (a) π(π₯) = (π₯ + 3)2 − 2, π₯ ≥ −3 (b) π(π₯) = π₯ 2 − 7π₯ + 1 2. The diagram below shows the graph of π¦ = π(π₯). Add to the diagram the graph of π¦ = π −1 (π₯). π 3. Given that sin π₯ = sin 3 , find all values of x where −2π < π₯ < 2π. Answers: 1. (a) π −1 (π₯) = −3 + √π₯ + 2 7 2 (b) π −1 (π₯) = + √π₯ + 2. Graph reflected in the line y = x 45 4 π 2π 4π 6π ,− 3 ,− 3 3 3. π₯ = 3 , 32 P5- Trigonometry 1. The Inverse Trig Functions https://youtu.be/ZM52fcPp1yw 8 mins arcsin x https://youtu.be/5o2lqjX-p_Q 6 mins arccos x https://youtu.be/1A-IbjIIy_I 6 mins 33 arctan x P5 - Trigonometry 2. Using the Inverse Trig Functions https://youtu.be/rUpnyYkVPwQ 14 mins 34 P5 - Trigonometry 3. Using the Inverse Trig Functions https://youtu.be/k7XlnCoBK28 17 mins 35 36 4. Using the Inverse Trig Functions https://youtu.be/P3owr0cwZbw 13 mins 37 Prior Knowledge - Implicit Differentiation To prepare for this week’s topic you need a good understanding of differentiation and equations of tangents and normals. If you are stuck refer to 12.6 (page 268) in the Pure Year 2 textbook. Try these questions 1. Differentiate a) 3π₯ 2 − 5π₯ b) 2 π₯ − √π₯ c) 4π₯ 2 (1 − π₯ 2 ) 2. Find the equation of the tangent to the curve with equation π¦ = 8 − π₯ 2 at the point (3, −1) Answers 38 P7 – Implicit Differentiation Differentiate simple functions and relations defined implicitly for first derivative only. The finding of equations of tangents and normals to curves given implicitly is required. https://drive.explaineverything.com/thecode/BKCHBXQ 8 mins Example 1 πΉπππ ππ¦ ππ₯ ππ 2π¦ = 4π₯ 2 + 2 Example 2 πΉπππ ππ¦ ππ π₯ 2 + π¦ 2 = 1 ππ₯ So to differentiate implicitly, differentiate everything you can see , left and ππ¦ right with respect to x. If you differentiate a y multiply it by ππ₯ 39 Example 3 a) Find the coordinates of the stationary points for π₯ 2 − 4π₯π¦ = π¦ 2 − 20 b) Show that the gradient is not parallel to the y axis for π₯ 2 − 4π₯π¦ = π¦ 2 − 20 40 Example 4 Find the gradient of the tangent to the curve (π₯ + π¦)2 = 4π₯ when π₯ = 1, π¦ > 0 ππ¦ =0 To find the gradient which is parallel to the x axis use To find the gradient which is parallel to the y axis (or infinite) use 41 ππ₯ ππ₯ ππ¦ =0 Example 5 Prove the differential of π¦ = 2π₯ is 2π₯ ln 2 42 Prior Knowledge - Binomial Expansion To prepare for this week’s topic you need a good understanding of expanding brackets and simplifying indices. If you are stuck refer to 1.2 (page 4) and 1.4 (page 9) in the Pure Year 1 textbook. Try these questions 1. Expand and simplify where possible a) (2x – 4y)(3x + y) b) (x – 4y)(2x + y +5) c) 3x(x – 2y)(2x + 1) 2. Simplify a) (−2π₯)4 b) (−3π₯)−2 2 3 c) (5 π₯ 3 ) 3. Simplify 1 a) (3 π₯) −2 b) (125π₯12 )−2 3 9 c) (4 π₯ ) 4 2 Answers 1. a) 6π₯ 2 − 10π₯π¦ − 4π¦ 2 b) 2π₯ 2 + 5π₯ − 7π₯π¦ − 4π¦ 2 − 20π¦ c) 6π₯ 3 + 15π₯ 2 − 3π₯ 2 π¦ − 18π₯π¦ 2 − 30π₯π¦ 1 2. a) 16π₯ 4 b) π₯ −2 c) 3. a) 9π₯ −2 b) 5 π₯ −6 1 c) 9 43 9 25 π₯6 27 8 π₯6 P2 – Binomial expansion - Introduction Understand and use the binomial expansion of (a + bx)n for positive integer n; the notations n! and nCr Use of Pascal’s triangle. Relation between binomial coefficients. π Also be aware of alternative notations such as( ) π https://youtu.be/qZzSWz27ROY 14 mins Finding the coefficients On casio fx-991EX On casio fx-CG50 n For both nCr and x! OPTN > PROB Cr button is shift ÷ x! is shift x-1 44 P2 – Finite Binomial Expansion where n is a positive integer This video shows a vertical layout of the Binomial. https://www.youtube.com/watch?v=EMbUhER6ro&list=UUhk8LXCP7lzhEK2lvyaQ7JQ&index=34&t=0s Vertical layout of binomial expansion 45 11 mins P2 - Infinite Binomial Expansion where n is fractional or negative Extend to any rational n, including its use for approximation; be aware that the ππ₯ expansion is valid for | | < 1 (proof not required) π May be used with partial fractions. Include the range of validity. https://youtu.be/JWSBHTjy2os 17 mins In formula book Next two terms 1. Expand (1 – 3x) −2 Why are the sequences infinite? Validity What does this mean about the validity of the sequence? 46 2 2. Expand (1 – x) 3 up to the term in x3. State the values for which the series is valid. 3 3. Expand (4 + x) 2 up to the term in x3. State the values for which the series is valid. 4. Given √ 1+π₯ 1 −3π₯ 2 = A + Bx + Cx … Find A, B and C 47 P2 - Binomial Expansion using Partial Fractions May be used with the expansion of rational functions by decomposition into partial fractions. May be asked to comment on the range of validity. https://youtu.be/wbZQW3l3G0w Express 5π₯−1 (2−π₯)(1+π₯) 12 mins in ascending powers of x up to the term in π₯ 2 and state the range of values for which it is valid. 48 Prior Knowledge – Horizontal Moments To prepare for this week’s topic you need a good understanding of forces and connected particles. If you are stuck refer to 10.3 (page 162) and 10.5 (page 169) in the Statistics and Mechanics Year 1 textbook. Try these questions 1. Answers 1. 49 SM9 – Horizontal Moments - Introduction Introduction to moments https://youtu.be/izbxpjNg7Q8?t=7 11 mins What does the moment of a force measure? ………………………………………… How do you calculate the moment of a force? ……………………………………… The anti-clockwise MOMENT about O = ……… X ……… [Nm] For a body in equilibrium the CLOCKWISE (CW) moments are ……………… to the ANTI-CLOCKWISE (ACW) moments. Example 1 Taking MOMENTS about O: M(O) : Resultant Moment = …………… x ……………. = …………………… [Nm] If the ruler is now in EQUILIBRIUM then: THE SUM OF THE MOMENTS (IN EITHER DIRECTION) = ZERO 50 Example 2 If we assume the beam is light (no mass), rigid and in EQUILIBRIUM take moments about O to find the force F. Taking moment about O: M(O): 20 ( ………. ) – F ( ………... ) …………. – …………. F F 51 = 0 = 0 = [N] SM9 – Horizontal Moments - Beams Rigid bodies in equilibrium https://youtu.be/DQqF8PpMsHs?t=8 20 mins The most important step in solving MOMENT problems is FIRST to draw a FORCE DIAGRAM. PLEASE NOTE THAT IF THE BEAM IS NOT UNIFORMLY DISTRIBUTED WE CANNOT ASSUME THAT ITS WEIGHT ACTS THROUGH THE CENTRE!! • Draw the forces acting on the beam above - take moments about C - equate forces vertically - take moments about D. 52 SM9 – Horizontal Moments - Tilting Tilting beams! https://youtu.be/h5kSeVwWhCE 7 mins 53 54
