Circle Theorems. Mathswatch. Clip 150. Diagram NOT accurately drawn

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1 Clip 50 ) In the diagram, A, B and C are points on the circumference of a circle, centre O. PA and PB are tangents to the circle. Angle ACB = 7. a) (i) Work out the size of angle AOB. (ii)give a reason for your answer. b) Work out the size of angle APB. Circle Theorems B O C 7 A Diagram NOT accurately drawn P ) P, Q, R and S are points on the circle. PQ is a diameter of the circle. Angle RPQ =. S b R a) (i) Work out the size of angle PQR. (ii) Give reasons for your answer. P a Q b) (i) Work out the size of angle PSR. (ii)give a reason for your answer. ) The diagram shows a circle, centre O. AC is a diameter. Angle BAC =. D is a point on AC such that angle BDA is a right angle. B Diagram NOT accurately drawn a) Work out the size of angle BCA. Give reasons for your answer. A O D C b) Calculate the size of angle DBC. c) Calculate the size of angle BOA. 4) A, B, C and D are four points on the circumference of a circle. ABE and DCE are straight lines. Angle BAC =. Angle EBC = 58. a) Find the size of angle ADC. A B 58 Diagram NOT accurately drawn b) Find the size of angle ADB. Angle CAD = 69. c) Is BD a diameter of the circle? You must eplain your answer. D C E Page 4

2 Clip 5 Cumulative Frequency The heights of 80 plants were measured and can be seen i n the table, below. Height (cm) Frequency 0 < h < 0 0 < h < < h < < h < < h < < h < 60 a) Complete the cumulative frequency table for the plants. Height (cm) 0 < h < 0 0 < h < 0 0 < h < 0 0 < h < 40 0 < h < 50 Cumulative Frequency CF 80 0 < h < 60 b) Draw a cumulative frequency graph for your table c) Use your graph to find an estimate for (i) the median height of a plant. (ii) the interquartile range of the heights of the plants. 40 d) Use your graph to estimate how many plants had a height that was greater than 45cm Height (cm) Page 44

3 Clip 5 Bo Plots ) The ages of 0 teachers are listed below.,, 4, 5, 7, 7, 8, 9, 9, 9, 4, 5, 4, 4, 44, 49, 55, 57, 58, 58 a) On the grid below, draw a boplot to show the information about the teachers b) What is the interquartile range of the ages of the teachers? ) A warehouse has 60 employees working in it. The age of the youngest employee is 6 years. The age of the oldest employee is 55 years. The median age is 7 years. The lower quartile age is 9 years. The upper quartile age is 4 years. On the grid below, draw a boplot to show information about the ages of the employees Page 45

4 Clip 5 Moving Averages ) The table shows the number of board games sold in a supermarket each month from January to June. Jan Feb Mar Apr May Jun Work out the -month moving averages for this information.,,, ) The table shows the number of computers sold in a shop in the first five months of 007. Jan Feb Mar Apr May June a) Work out the first two -month moving averages for this information., b) Work out the first 4-month moving average for this information. The third 4-month moving average of the number of computers sold in 007 is 96. The number of computers sold in the shop in June was. c) Work out the value of. Page 46

5 Clip 54 Tree Diagrams ) Lucy throws a biased dice twice. Complete the probability tree diagram to show the outcomes. Label clearly the branches of the tree diagram. st Throw nd Throw 6 Si Not Si ) A bag contains 0 coloured balls. 7 of the balls are blue and of the balls are green. Nathan is going to take a ball, replace it, and then take a second ball. a) Complete the tree diagram. st Ball nd Ball Blue Blue Green Green Blue Green b) Work out the probability that Nathan will take two blue balls. c) Work out the probability that Nathan will take one of each coloured balls. d) Work out the probability that Nathan will take two balls of the same colour. Page 47

6 Clip 55 Recurring Decimals ) a) Convert the recurring decimal 06. to a fraction in its simplest form. 8 b) Prove that the recurring decimal 07. = ) a) Change 4 9 to a decimal. 9 b) Prove that the recurring decimal 057. = ) a) Change to a decimal. 5 b) Prove that the recurring decimal 045. = 4) a) Change 6 to a decimal. 5 b) Prove that the recurring decimal 0. 5 = 7 5) a) Convert the recurring decimal 06. to a fraction in its simplest form. b) Prove that the recurring decimal 07. = 5 8 6) a) Convert the recurring decimal 5. to a fraction in its simplest form. b) Prove that the recurring decimal 06. = Page 48

7 Clip 56 Fractional and Negative Indices a a y a a y = a +y = a y (a ) y = a y a 0 = a - = a y y = ( a ) a = a y ( a ) y ) Simplify a) (p 5 ) 5 c) 5 e) (m -5 ) - b) k k d) (p ) - f) (y ) ) Without using a calculator, find the eact value of the following. a) c) e) (8 5 ) b) d) f) ( ) ) Work out each of these, leaving your answers as eact fractions when needed. a) 4 0 e) 4 i) 49 m) 49 b) 7 0 f) 8 j) 5 n) 5 c) 5 0 g) 5 k) 7 o) 7 d) 9 0 h) 0 5 l) 6 p) 6 4) 5 5 can be written in the form 5 n. Find the value of n. 5) 8 = m Find the value of m. 6) Find the value of when 5 = 5 7) Find the value of y when 8 = y 8) a =, b = y a) Epress in terms of a and b i) + y ii) iii) + y ab = 6 and ab = 6 b) Find the value of and the value of y. Page 49

8 Clips 57, 58 Surds 5 is not a surd because it is equal to eactly 5. is a surd because you can only ever approimate the answer. We don t like surds as denominators. When we rationalise the denominator it means that we transfer the surd epression to the numerator. ) Simplify the following: a) 7 7 b) c) 0 d) 4 e) 7 f) 00 g) 5 ) Simplify the following: a) 8 b) 8 c) 99 d) 45 0 e) 8 8 f) 8 75 ) Epand and simplify where possible: a) ( ) b) ( 6+ ) c) 7( + 7) d) ( 8) 4) Epand and simplify where possble: a) ( + )( ) b) ( + 5)( 5) c) ( + )( + 4) d) ( 5 )( 5+ ) e) ( + 7)( 7) f) ( 6 ) 5) Rationalise the denominator, simplifying where possible: a) b) c) d) e) f) g) ) 7 = n Find the value of n 7) Epress 8 8 in the form m where m is an integer. 8) Rationalise the denominator of giving the answer in 8 8 the form 9) Work out the following, giving your answer in its simplest form: a) b) c) d) e) f) ( 5+ )( 5 ) ( 4 5)( 4+ 5) ( )( + ) 4 ( + ) ( 5+ ) 0 p ( 5 5)( + 5) 0 Page 50

9 Clip 59 Direct and Inverse Proportion ) is directly proportional to y. When =, then y =. a) Epress in terms of y. b) Find the value of when y is equal to: (i) (ii) (iii) 0 ) a is inversely proportional to b. When a =, then b = 4. a) Find a formula for a in terms of b. b) Find the value of a when b is equal to: (i) (ii) 8 (iii) 0 c) Find the value of b when a is equal to: (i) 4 (ii) 4 (iii). ) The variables u and v are in inverse proportion to one another. When u =, then v = 8. Find the value of u when v =. 4) p is directly proportional to the square of q. p = 75 when q = 5 a) Epress p in terms q. b) Work out the value of p when q = 7. c) Work out the positive value of q when p = 7. 5) y is directly proportional to. When =, then y = 6. a) Epress y in terms of. z is inversely proportional to. When = 4, z =. b) Show that z = c y n, where c and n are numbers and c > 0. You must find the values of c and n. Page 5

Standard Form Calculation

Clip Standard Form Calculation ) Work out the following, giving your answer in standard form. a) (6 0 ) (8 0 ) c) 0 6 0 - b) ( 0 ) + ( 0 ) d) (9. 0 ) ( 0 ) ) A spaceship travelled for 0 hours at a speed