Pre-IB Course - Blast From The Past PDF

PRE-IB COURSE – BLAST FROM THE PAST MARINOS PAPADOPOULOS MSc International Baccalaureate LECTURE NOTES MATHEMATICS...

PRE-IB COURSE – BLAST FROM THE PAST MARINOS PAPADOPOULOS MSc International Baccalaureate LECTURE NOTES MATHEMATICS PRE-IB COURSE Marinos Papadopoulos, Math Teacher - MSc SECTION: BLAST FROM THE PAST UNITS 1. ROUNDING OFF................................................................................................. 3 2. SIGNIFICANT FIGURES................................................................................... 3 3. DECIMAL PLACES............................................................................................ 4 4. UPPER & LOWER BOUNDS............................................................................. 5 5. THE REAL NUMBER SYSTEM, SETS & INTERVALS............................... 6 6. PROPERTIES OF REAL NUMBERS............................................................... 8 7. FORMULA REARRANGEMENT................................................................... 14 8. SIMPLE APPLICATIONS OF PERCENTAGE............................................ 14 9. COMMAND TERMS......................................................................................... 26 10. VOCABULARY.................................................................................................. 28 www.mathacademy.gr -1- PRE-IB COURSE – BLAST FROM THE PAST MARINOS PAPADOPOULOS MSc www.mathacademy.gr -2- PRE-IB COURSE – BLAST FROM THE PAST MARINOS PAPADOPOULOS MSc 1. ROUNDING OFF If the digit after the one being rounded off is less than 5, (i.e. 0,1,2,3 or 4) we round down. If the digit after the one being rounded off is 5 or more, (i.e. 5,6,7,8 or 9) we round up. To round off to nth digit, we look at the (n+1)th digit.  If the (n+1)th digit is 0,1,2,3 or 4 we do not change the nth digit but  If the (n+1)th digit is 5,6,7,8 or 9 we increase the nth digit by 1.  We delete all digits after the nth digit, replacing by 0s if necessary. 2. SIGNIFICANT FIGURES The first s.f is the first non-zero digit in the number, counting from the left. e.g. 1: 35123 e.g. 2: 0.2365 e.g. 3: 0.00178 If we want a number to be rounded to a suitable degree of accuracy, for example to 2 s.f, we look the third s.f. If this is greater than or equal to 5 then round the second figure up. If rounding to 3 s.f, we look at the fourth s.f. and so on. e.g. 1: Write 35123 correct to 4 s.f. Ans: 5th s.f is 3. 3 a ⇔............... α x + y =0 ⇔.......... =....................., if............... b E. 0.0064579622 2 d.p =......................... 2 s.f =......................... 4 d.p =......................... 4 s.f =......................... 5 d.p =......................... 5 s.f =......................... 6 d.p =......................... 6 s.f =......................... 7 d.p =......................... 7 s.f =......................... 8 d.p =......................... 8 s.f =......................... 3. Match Column A with Column B. Answers must be written in all CAPITAL letters, and can be selected only once. www.mathacademy.gr -20- PRE-IB COURSE – BLAST FROM THE PAST MARINOS PAPADOPOULOS MSc Α. Column (Α) Column (Β) 1 Α. 9 ( −1) 2009 1. ….. Β. 1 −2 2. −3 ….. 1 C. − −4 8 9 3. 9 ⋅ 3 ….. D. −1 4. ( 3 : 3) : 3 ….. 5 6 Ε. 9 Β. Column (Α) Column (Β) 1. ( −3) 2 …. Α. 9 Β. 6 3 2. −2 …. C. −9 2 3. −3 …. D. −8 4. − ( −2 ) …. 3 Ε. 8 C. Column (Α) Column (Β) Α. − ( α − b ) 3 ( −α + b ) 2 1. …. Β. − ( α − b ) 2 ( −α − b ) 3 2. …. C. ( α − b ) 2 ( −α − b ) 2 3. …. D. ( α + b ) 2 ( −α + b ) 3 4. …. Ε. − ( α + b ) 3 www.mathacademy.gr -21- PRE-IB COURSE – BLAST FROM THE PAST MARINOS PAPADOPOULOS MSc D. Column (Α) Column (Β) 1. −b 2 − ( α + b )( α − b ) …. Α. α 2 Β. 0 ( 2x + y ) − ( 2y 2 2 + x ) …. 2 2. 2 2 + xy C. 3. ( 4x 2 + 4xy + y 2 ) − ( 2x + y ) 3 …. xy 2x + y 4x 2 + 4xy + y 2 2 2 D. −α 2 1 1 1 1  x + y  −  x − y  − xy 4.     …. Ε. 3 ( x 2 − y 4 ) 2 − xy 4. Simplify the following expressions: −2 −1 1 −4 −1 4  x 2   2y 2  5 3 2 i) 2α b c 4α b c 2 −3 5 α b c ii)  3  ⋅  3  ⋅ 4xy −4 , x, y ≠ 0 16  y   5x  5. Rationalise the denominator of each of the following expressions: 1 1 3 5 i) + ii ) −. 8 2 3 5 6. Show that ( 3− 2 )( ) 1. Hence, rationalize the denominator of 3+ 2 = 1 the fraction. 3− 2 ( )( ) 6. If 2 + a ⋅ 5 − 3 a = 22 + k a , where α and k are integers, find the values of α and k. 20 − 2 8 + 3 12 7. Simplify the following expression: Α =. 45 − 2 18 + 3 27 8. Prove that: 3 7 5  8 11   12 71  i) + = ii)  +  − =31. 7− 3 7+ 3 2  5 −1 4 + 5   5 −1 3 + 4 5  www.mathacademy.gr -22- PRE-IB COURSE – BLAST FROM THE PAST MARINOS PAPADOPOULOS MSc 9. Express as a single fraction 1 1 x y 1 1 (a) + (b) − (c) 2 + 2 x y y x ab a b 1 3 2 x 2 x 1 (d) − + (e) 2 + (f) 1 + − x x −1 x − 2 x −1 x +1 x + 1 ( x + 1)( x + 2 ) 10. Factorize the following expressions: (a) ax + ay + az (b) a 2b − ab 2 (c) 6 x 2 y + 18 xy 3 (d) cx − cy + dy − dx (e) x 2 − x ( y + z ) + yz 11. Factorize using the difference of two squares: (b) ( x − y ) − z 2 (c) 9 − 4 ( a − b ) 2 2 (a) z 2 − 36 (d) x 4 − 16 (e) x 2 − y 2 − kx + ky (f) 27v 2 − 3 12. Factorize the following quadratics (a) x 2 + 11x + 18 (b) t 2 − 7t + 12 (c) y 2 + 2 y − 35 (d) 3n 2 − 7 m − 6 (e) 6 − q − q 2 (f) k 4 + 10k 2 + 24 (g) 2a 2 − ab − 6b 2 (h) 25 + 20 x + 4 x 2 (i) a 6 + 4a 3 + 4 13. Simplify each algebraic fraction: x 3 − 4x x 3 − 2x 2 + x (x 2 − x) + 2x − 2 i) 2 ii) iii) 2x − 4x x2 − x x2 −1 2 x(x − 2) + 1 x2 + x +1 x2 −1  1  x3 + x 2 iv) v) ⋅ 3 vi)  x −  ⋅ (x − 2)(x − 1) x +1 x −1  x  ( x + 1)3 x 2 − 3x + 2 x 2 + 2x α 2 + β2 + 2αb αx + αy + bx + by vii) ⋅ 2 viii) ix) x2 − x x +x−2 3(α + b) x 2 − y2 36x 2 − 12x + 1 x 2 − 9y 2 164α 2 b3 − 8α 3 b3 x) xi) 2 xii) 48x − 8 2x − 12xy + 18y 2 4α 2 b 2 (x + h) x2 − (α + b) 2 2 − x2 x2 −1 xiiii) xiv) xv) (1 − x ) α2 − ( x + b ) 2 2 h 14. Write in the standard form the following numbers: i )30.000.000 ii )5600000 iii )243000000 iv)0, 00002 v)0, 00000034 vi )0, 000000000735 15. Solve for…. www.mathacademy.gr -23-  PRE-IB COURSE – BLAST FROM THE PAST MARINOS PAPADOPOULOS MSc i ) xy z=(Solve for x) ii )a xy (Solve for y) α x α iii ) z= (Solve for b) iv) (Solve for k) b y k x− y =v)α + bx c (Solve= for b) vi )α (Solve for x) b k x vii ) x = (Solve for α) viii )α = 1 − , α ≠ 0 (Solve for y) α −b y 16. A rectangle is 1350 cm long and 2275 cm wide. a) Find the perimeter of the rectangle, giving your answer in the form α × 10k , where 1 ≤ α < 10 and k ∈ Z. b) Find the area of the rectangle, giving your answer correct to the nearest thousand square centimeters. 17. Given x = 1.5 × 104 and y = 3 × 10-8, calculate the value of w= x ⋅ y , x v = and = q x −2 + y. y a) Give your answers in the form α × 10k , where 1 ≤ α < 10 and k ∈ Z. b) Which two of the following statements about the nature of x, y and w above are incorrect? i) x∈ N ii) y∈Z iii) y ∈Q 1 iv) w < y v) x+ y∈R vi)