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SUEC 高中 Adv Maths Revision (Earth as Sphere, Locus, Permutation, Combination, Binominial).pptx

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?? ?? 1000 ?? ? 60 ° ? ? = 1000 60 ° ? ? = 6370 ??? 60° ? 360 ° × 2? × 6370 ??? 60° = 1000 ? = 18 ° 15° → 1 ?? 18 ° → 18 15 ?? = 1 ?? 12 ???
?? ?? 1000 ?? ? 60 ° ? ? 60 ° ? 1.853 ?? → 1 海里 1000 ?? → 1000 1.853 海里 = 539.67 海里 ? = ???????? 60 ??? 60° = 539.67 60 ??? 60° = 18 ° 15° → 1 ?? 18 ° → 18 15 ?? = 1 ?? 12 ???
?? ?? 500 ?? ? 60 ° ? ? 60 ° ? ? = ???????? 60 ??? 60° = 500 60 ??? 60° = 16.67 ° 15° → 1 ?? 16.67 ° → 16.67 15 ?? = 1 ?? 7 ??? 1 ?? ?, ???? =? ? ????? ???? = 13 ?? ? 1 ?? 7 ??? ? ?? ? ?? ?, ?? ???????? ???? = 11 ?? 53 ???
?? ?? ?????????? ?? ??? = ????? ??? + ???????? ??? = 35.67 ° + 34.92 ° = 70.59 ° ????? (35 ° 40’ ?, 139 ° ?) ???????? (34 ° 55’ ?, 139 ° ?) = 70.59 × 60 = 4,235 海里 ???????? = 70.59 ° 360 ° × 2? × 6,370 ?? = 7,849 ?? (?) (?)
?? ?? ? (43 ° 40’ ?) ? (9 ° ?) ?????????? ?? ????????? = ? ???????? ? ? ???????? = 43 ° 40’ ? 9 ° = 34 ° 40’ = 34.67 ° = 34.67 × 60 = 2,080 海里
?? ?? ? (15 ° ?, 30 ° ?) ? (?, ?? ° ?) = 2000 海里 ? = 33.33 ° ?????????? ?? ????????? = ? ???????? + ? ???????? 33.33 ° = 15 ° + ? ???????? ? ???????? = 33 ° 20’ ? 15 ° = 18 ° 20’ ∴ ? (18 ° 20’ ?, 30 ° ?)
?? ?? 3000 ?? ? 15 ° ? ? 15 ° ? ? = ???????? 60 ??? 15° = 3000 60 ??? 15° = 51.76 ° ? ? (15 ° ?, ? ) ? (15 ° ?, 30 ° ?) 51.76 ° = 30 ° ? ? ????????? ?????????? ?? ?????????? = ? ????????? ? ? ????????? ? ????????? = 81 ° 46’ ?
?? ?? = 1000 海里 ? = 16 ° 40’ ? ∴ ? (16 ° 40’ ?, 10 ° ?) ???????? ????????? = 600 海里 ? = 10 ° ?
?? ?? = (22 + 30) × 60 = 3,120 海里 ? ?
?? ??? ?2 + ?2 + 4? ? 6? ? 3 = 0 ?? = 4 ? + 2 2 + ? ? 3 2 = 4 ?2 + 4? + 4 + ?2 ? 6? + 9 = 16 ? = (?2, 3) 5? + 2? ? 16 = 0 ??? ? = (?, ?) ? = (?3, 1) ? = (7, 5) ?? = ?? ? + 3 2 + ? ? 1 2 = ? ? 7 2 + ? ? 5 2 ?2 + 6? + 9 + ?2 ? 2? + 1 = ?2 ? 14? + 49 + ?2 ? 10? + 25 ? + 3 2 + ? ? 1 2 = ? ? 7 2 + ? ? 5 2 ??? ? = (?, ?)
?? ??? 16?2 + 7?2 ? 64? ? 48 = 0 ?? + ?? = 8 ? ? 2 2 + ? ? 3 2 + ? ? 2 2 + ? + 3 2 = 8 ??? ? = (?, ?) ? = (2, 3) ? = (2, ?3) ? ? 2 2 + ? ? 3 2 = 64 ? 16 ? ? 2 2 + ? + 3 2 + ? ? 2 2 + ? + 3 2 ? ? 2 2 + ? ? 3 2 = 8 ? ? ? 2 2 + ? + 3 2 ?2 ? 6? + 9 = 64 ? 16 ? ? 2 2 + ? + 3 2 + ?2 + 6? + 9 12? + 64 = 16 ? ? 2 2 + ? + 3 2 3 4 ? + 4 2 = ? ? 2 2 + ? + 3 2 9 16 ?2 + 6? + 16 = ?2 ? 4? + 4 + ?2 + 6? + 9
?? ??? ?? = 6 ?2 + ?2 = 6 ?2 + ?2 = 36 ??? ? & ? = ??????????? ?? ??? ? & ? ????. ? = (?, 0) , ? = (0, ?) ? = (?, ?) = ???????? ?? ?? ∴ ? = (2?, 0), ? = (0, 2?)
?? ??? ???????? ???? ?? ∶ ? ? ?1 = ? (? ? ?1) ? = ? (? ? 4) ? = (4,0) = (?1 , ?1) ? = ???????? ?? ?? = (? , ?) ??? ? = (?? , ??) ? = (?? , ??) ?? 2 + ?? 2 = 4 ? ?① ?? 2 + ?? 2 = 4 ? ?② ① ? ② (?? + ??)(?? ? ??) + (?? + ??)(?? ? ??) = 0 2? (?? ? ??) + 2? (?? ? ??) = 0 (?? 2 ? ?? 2 ) + (?? 2 ? ?? 2 ) = 0 2? (?? ? ??) = ? 2? (?? ? ??) ? ? = ? (?? ? ??) (?? ? ??) ? = ? (? ? 4) = ? ? ?2 + ?2 ? 4? = 0
?? ??? (?) (? + ?1)(? + ?2) = ?2 + ?2? + ?1? + ?1?2 = ?2 + (?1 + ?2) ? + ?1?2 (?) (? + ?1)(? + ?2)(? + ?3) = ?2 + (?1 + ?2) ? + ?1?2 (? + ?3) = ?3 + (?1 + ?2) ?2 + ?1?2 ? + ?3 ?2 + (?1 + ?2) ?3 ? + ?1?2?3 = ?3 + (?1 + ?2 + ?3) ?2 + (?1?2 + ?1?3 + ?2?3) ? + ?1?2?3
?? ??? (?) (? + ?1)(? + ?2)(? + ?3)(? + ?4) = ?3 + (?1 + ?2 + ?3) ?2 + (?1?2 + ?1?3 + ?2?3) ? + ?1?2?3 (? + ?4) = ?4 +(?1 + ?2 + ?3) ?3 + (?1?2 + ?1?3 + ?2?3) ?2 + ?1?2?3? + ?4?3 + (?1 + ?2 + ?3) ?4 ?2 + (?1?2 + ?1?3 + ?2?3) ?4? + ?1?2?3?4 = ?4 +(?1 + ?2 + ?3 + ?4) ?3 + (?1?2 + ?1?3 + ?1?4 + ?2?3 + ?2?4 + ?3?4) ?2 + (?1?2?3 + ?1?2?4 + ?1?3?4 + ?2?3?4)? + ?1?2?3?4 (?) (? ? 1)(? + 3)(? ? 5)(? + 7) = ?4 +(?1 + 3 ? 5 + 7) ?3 + (?3 + 5 ? 7 ? 15 + 21 ? 35) ?2 + (15 ? 21 ? 35 ? 105)? + (?1)(3)(?5)(7) = ?4 +4?3 ? 34?2 ? 76? + 105
?? ??? ????? ?? ?3 =? ????? ?? ?3 = ?2 + 3 ? 4 ? 6 + 8 ? 12 + 5(1 ? 2 + 3 ? 4) = ?4 +(?1 + ?2 + ?3 + ?4) ?3 + (?1?2 + ?1?3 + ?1?4 + ?2?3 + ?2?4 + ?3?4) ?2 + (?1?2?3 + ?1?2?4 + ?1?3?4 + ?2?3?4)? + ?1?2?3?4 (? + ?1)(? + ?2)(? + ?3)(? + ?4)(? + 5) = ?4 + (?1 + ?2 + ?3 + ?4) ?3 + (?1?2 + ?1?3 + ?1?4 + ?2?3 + ?2?4 + ?3?4) ?2+ . . . (? + 5) =. . . +(?1?2 + ?1?3 + ?1?4 + ?2?3 + ?2?4 + ?3?4) ?3 + (?1 + ?2 + ?3 + ?4) 5?3 +. . . = ?23
?? ??? (?) (1 ? 2?)5 = 1 + 5?1 (? 2?) + 5?2 ? 2? 2 +5?3 ? 2? 3 + 5?4 ? 2? 4 + 5?5 ? 2? 5 = 1 ? 10? + 40 ?2 ? 80 ?3 + 80 ?4 ? 32 ?5 (?) ?2 ? 1 ? 4 = 4?0 (?2 )4 + 4?1 ?2 3 ? 1 ? +4?2 ?2 2 ? 1 ? 2 + 4?3 ?2 ? 1 ? 3 + 4?4 ? 1 ? 4 = ?8 ? 4 ?5 + 6 ?2 ? 4 ? + 1 ?4
?? ??? (?) (1 + 2?) 1 2 = 1 + 1 2 2? + 1 2 ? 1 2 2 ! 2? 2 + 1 2 ? 1 2 ? 3 2 3 ! 2? 3 + ? = 1 + ? ? 1 2 ?2 + 1 2 ?3 + ?
?? ??? (?) 1 + ? 1 ? 2? = (1 + ?) 1 ? 2? ? 1 = (1 + ?) 1 + ?1 ?2? + ?1 ?2 2 ! ?2? 2 + ?1 ?2 (?3) 3 ! ?2? 3+ . . . = (1 + ?) 1 + 2? + 4?2 + 8?3 . . . = 1 + 2? + 4?2 + 8?3 + ? + 2?2 + 4?3+. . . = 1 + 3? + 6?2 + 12?3 . . .
?? ??? (?) (1 + ?) ?1 + 1 + 2? ? 1 2 = 1 + ?1 (?) + (?1)(?2) 2 ! ? 2 + (?1)(?2)(?3) 3 ! ? 3+. . . + 1 + ? 1 2 (2?) + ? 1 2 ? 3 2 2 ! 2? 2 + ? 1 2 ? 3 2 ? 5 2 3 ! 2? 3 +. . . = (1 ? ? + ?2 ? ?3 +. . . ) + 1 ? ? + 3 2 ?2 ? 15 6 ?3 +. . . = 2 ? 2? + 5 2 ?2 ? 7 2 ?3 +. . .

More Related Content

SUEC 高中 Adv Maths Revision (Earth as Sphere, Locus, Permutation, Combination, Binominial).pptx

  • 1. ?? ?? 1000 ?? ? 60 ° ? ? = 1000 60 ° ? ? = 6370 ??? 60° ? 360 ° × 2? × 6370 ??? 60° = 1000 ? = 18 ° 15° → 1 ?? 18 ° → 18 15 ?? = 1 ?? 12 ???
  • 2. ?? ?? 1000 ?? ? 60 ° ? ? 60 ° ? 1.853 ?? → 1 海里 1000 ?? → 1000 1.853 海里 = 539.67 海里 ? = ???????? 60 ??? 60° = 539.67 60 ??? 60° = 18 ° 15° → 1 ?? 18 ° → 18 15 ?? = 1 ?? 12 ???
  • 3. ?? ?? 500 ?? ? 60 ° ? ? 60 ° ? ? = ???????? 60 ??? 60° = 500 60 ??? 60° = 16.67 ° 15° → 1 ?? 16.67 ° → 16.67 15 ?? = 1 ?? 7 ??? 1 ?? ?, ???? =? ? ????? ???? = 13 ?? ? 1 ?? 7 ??? ? ?? ? ?? ?, ?? ???????? ???? = 11 ?? 53 ???
  • 4. ?? ?? ?????????? ?? ??? = ????? ??? + ???????? ??? = 35.67 ° + 34.92 ° = 70.59 ° ????? (35 ° 40’ ?, 139 ° ?) ???????? (34 ° 55’ ?, 139 ° ?) = 70.59 × 60 = 4,235 海里 ???????? = 70.59 ° 360 ° × 2? × 6,370 ?? = 7,849 ?? (?) (?)
  • 5. ?? ?? ? (43 ° 40’ ?) ? (9 ° ?) ?????????? ?? ????????? = ? ???????? ? ? ???????? = 43 ° 40’ ? 9 ° = 34 ° 40’ = 34.67 ° = 34.67 × 60 = 2,080 海里
  • 6. ?? ?? ? (15 ° ?, 30 ° ?) ? (?, ?? ° ?) = 2000 海里 ? = 33.33 ° ?????????? ?? ????????? = ? ???????? + ? ???????? 33.33 ° = 15 ° + ? ???????? ? ???????? = 33 ° 20’ ? 15 ° = 18 ° 20’ ∴ ? (18 ° 20’ ?, 30 ° ?)
  • 7. ?? ?? 3000 ?? ? 15 ° ? ? 15 ° ? ? = ???????? 60 ??? 15° = 3000 60 ??? 15° = 51.76 ° ? ? (15 ° ?, ? ) ? (15 ° ?, 30 ° ?) 51.76 ° = 30 ° ? ? ????????? ?????????? ?? ?????????? = ? ????????? ? ? ????????? ? ????????? = 81 ° 46’ ?
  • 8. ?? ?? = 1000 海里 ? = 16 ° 40’ ? ∴ ? (16 ° 40’ ?, 10 ° ?) ???????? ????????? = 600 海里 ? = 10 ° ?
  • 9. ?? ?? = (22 + 30) × 60 = 3,120 海里 ? ?
  • 10. ?? ??? ?2 + ?2 + 4? ? 6? ? 3 = 0 ?? = 4 ? + 2 2 + ? ? 3 2 = 4 ?2 + 4? + 4 + ?2 ? 6? + 9 = 16 ? = (?2, 3) 5? + 2? ? 16 = 0 ??? ? = (?, ?) ? = (?3, 1) ? = (7, 5) ?? = ?? ? + 3 2 + ? ? 1 2 = ? ? 7 2 + ? ? 5 2 ?2 + 6? + 9 + ?2 ? 2? + 1 = ?2 ? 14? + 49 + ?2 ? 10? + 25 ? + 3 2 + ? ? 1 2 = ? ? 7 2 + ? ? 5 2 ??? ? = (?, ?)
  • 11. ?? ??? 16?2 + 7?2 ? 64? ? 48 = 0 ?? + ?? = 8 ? ? 2 2 + ? ? 3 2 + ? ? 2 2 + ? + 3 2 = 8 ??? ? = (?, ?) ? = (2, 3) ? = (2, ?3) ? ? 2 2 + ? ? 3 2 = 64 ? 16 ? ? 2 2 + ? + 3 2 + ? ? 2 2 + ? + 3 2 ? ? 2 2 + ? ? 3 2 = 8 ? ? ? 2 2 + ? + 3 2 ?2 ? 6? + 9 = 64 ? 16 ? ? 2 2 + ? + 3 2 + ?2 + 6? + 9 12? + 64 = 16 ? ? 2 2 + ? + 3 2 3 4 ? + 4 2 = ? ? 2 2 + ? + 3 2 9 16 ?2 + 6? + 16 = ?2 ? 4? + 4 + ?2 + 6? + 9
  • 12. ?? ??? ?? = 6 ?2 + ?2 = 6 ?2 + ?2 = 36 ??? ? & ? = ??????????? ?? ??? ? & ? ????. ? = (?, 0) , ? = (0, ?) ? = (?, ?) = ???????? ?? ?? ∴ ? = (2?, 0), ? = (0, 2?)
  • 13. ?? ??? ???????? ???? ?? ∶ ? ? ?1 = ? (? ? ?1) ? = ? (? ? 4) ? = (4,0) = (?1 , ?1) ? = ???????? ?? ?? = (? , ?) ??? ? = (?? , ??) ? = (?? , ??) ?? 2 + ?? 2 = 4 ? ?① ?? 2 + ?? 2 = 4 ? ?② ① ? ② (?? + ??)(?? ? ??) + (?? + ??)(?? ? ??) = 0 2? (?? ? ??) + 2? (?? ? ??) = 0 (?? 2 ? ?? 2 ) + (?? 2 ? ?? 2 ) = 0 2? (?? ? ??) = ? 2? (?? ? ??) ? ? = ? (?? ? ??) (?? ? ??) ? = ? (? ? 4) = ? ? ?2 + ?2 ? 4? = 0
  • 14. ?? ??? (?) (? + ?1)(? + ?2) = ?2 + ?2? + ?1? + ?1?2 = ?2 + (?1 + ?2) ? + ?1?2 (?) (? + ?1)(? + ?2)(? + ?3) = ?2 + (?1 + ?2) ? + ?1?2 (? + ?3) = ?3 + (?1 + ?2) ?2 + ?1?2 ? + ?3 ?2 + (?1 + ?2) ?3 ? + ?1?2?3 = ?3 + (?1 + ?2 + ?3) ?2 + (?1?2 + ?1?3 + ?2?3) ? + ?1?2?3
  • 15. ?? ??? (?) (? + ?1)(? + ?2)(? + ?3)(? + ?4) = ?3 + (?1 + ?2 + ?3) ?2 + (?1?2 + ?1?3 + ?2?3) ? + ?1?2?3 (? + ?4) = ?4 +(?1 + ?2 + ?3) ?3 + (?1?2 + ?1?3 + ?2?3) ?2 + ?1?2?3? + ?4?3 + (?1 + ?2 + ?3) ?4 ?2 + (?1?2 + ?1?3 + ?2?3) ?4? + ?1?2?3?4 = ?4 +(?1 + ?2 + ?3 + ?4) ?3 + (?1?2 + ?1?3 + ?1?4 + ?2?3 + ?2?4 + ?3?4) ?2 + (?1?2?3 + ?1?2?4 + ?1?3?4 + ?2?3?4)? + ?1?2?3?4 (?) (? ? 1)(? + 3)(? ? 5)(? + 7) = ?4 +(?1 + 3 ? 5 + 7) ?3 + (?3 + 5 ? 7 ? 15 + 21 ? 35) ?2 + (15 ? 21 ? 35 ? 105)? + (?1)(3)(?5)(7) = ?4 +4?3 ? 34?2 ? 76? + 105
  • 16. ?? ??? ????? ?? ?3 =? ????? ?? ?3 = ?2 + 3 ? 4 ? 6 + 8 ? 12 + 5(1 ? 2 + 3 ? 4) = ?4 +(?1 + ?2 + ?3 + ?4) ?3 + (?1?2 + ?1?3 + ?1?4 + ?2?3 + ?2?4 + ?3?4) ?2 + (?1?2?3 + ?1?2?4 + ?1?3?4 + ?2?3?4)? + ?1?2?3?4 (? + ?1)(? + ?2)(? + ?3)(? + ?4)(? + 5) = ?4 + (?1 + ?2 + ?3 + ?4) ?3 + (?1?2 + ?1?3 + ?1?4 + ?2?3 + ?2?4 + ?3?4) ?2+ . . . (? + 5) =. . . +(?1?2 + ?1?3 + ?1?4 + ?2?3 + ?2?4 + ?3?4) ?3 + (?1 + ?2 + ?3 + ?4) 5?3 +. . . = ?23
  • 17. ?? ??? (?) (1 ? 2?)5 = 1 + 5?1 (? 2?) + 5?2 ? 2? 2 +5?3 ? 2? 3 + 5?4 ? 2? 4 + 5?5 ? 2? 5 = 1 ? 10? + 40 ?2 ? 80 ?3 + 80 ?4 ? 32 ?5 (?) ?2 ? 1 ? 4 = 4?0 (?2 )4 + 4?1 ?2 3 ? 1 ? +4?2 ?2 2 ? 1 ? 2 + 4?3 ?2 ? 1 ? 3 + 4?4 ? 1 ? 4 = ?8 ? 4 ?5 + 6 ?2 ? 4 ? + 1 ?4
  • 18. ?? ??? (?) (1 + 2?) 1 2 = 1 + 1 2 2? + 1 2 ? 1 2 2 ! 2? 2 + 1 2 ? 1 2 ? 3 2 3 ! 2? 3 + ? = 1 + ? ? 1 2 ?2 + 1 2 ?3 + ?
  • 19. ?? ??? (?) 1 + ? 1 ? 2? = (1 + ?) 1 ? 2? ? 1 = (1 + ?) 1 + ?1 ?2? + ?1 ?2 2 ! ?2? 2 + ?1 ?2 (?3) 3 ! ?2? 3+ . . . = (1 + ?) 1 + 2? + 4?2 + 8?3 . . . = 1 + 2? + 4?2 + 8?3 + ? + 2?2 + 4?3+. . . = 1 + 3? + 6?2 + 12?3 . . .
  • 20. ?? ??? (?) (1 + ?) ?1 + 1 + 2? ? 1 2 = 1 + ?1 (?) + (?1)(?2) 2 ! ? 2 + (?1)(?2)(?3) 3 ! ? 3+. . . + 1 + ? 1 2 (2?) + ? 1 2 ? 3 2 2 ! 2? 2 + ? 1 2 ? 3 2 ? 5 2 3 ! 2? 3 +. . . = (1 ? ? + ?2 ? ?3 +. . . ) + 1 ? ? + 3 2 ?2 ? 15 6 ?3 +. . . = 2 ? 2? + 5 2 ?2 ? 7 2 ?3 +. . .

Editor's Notes

  • #2: 2 places on latitide circle at 60 N are 1000 km apart. Try to find their time difference.
  • #3: 2 places on latitide circle at 60 N are 1000 km apart. Try to find their time difference.
  • #4: P & Q are at 60N latitude and 500 nm apart. The time at the place of P = 1 pm. If Q is due W of P, try to find the place of Q.
  • #6: How many nm is the distance from Panama (9N, 79 20 W) to Toronto (43 40 N, 79 20 W)?
  • #7: An aircraft flew 2000 nm southward from its base P (15 N, 30 E) to reach B. Find the latitude and longitude of B. Another plane flew away from the same base P, flew 3000 nm eastwards and arrived at C. Find the latitude and longitude of B.
  • #8: An aircraft flew 2000 nm southward from its base P (15 N, 30 E) to reach B. Find the latitude and longitude of B. Another plane flew away from the same base P, flew 3000 nm eastwards and arrived at C. Find the latitude and longitude of B.
  • #9: A is 1000 nm N of the Equator and 600 nm E of Greenwich. Find the latitude and longitude of A.
  • #10: An airplane flies along the equator from A (42 E) to B (20 E) and then N to C (30 N). Find the distance traveled by this aircraft.
  • #11: 1. The distance between a moving point P and a fixed point (-2,3) is equal to 4. Find the trajectory equation of point P. 2. The distance between a moving point and 2 known points (-3,1) and (7.5) is equal. Find the trajectory equation of this moving point.
  • #12: The sum of the distance between a moving point P and 2 fixed points (2,3) and (2,-3) is always 8 and find the trajectory equation of point P.
  • #13: The 2 ends of a line segment with a length of 12 units often move on 2 axes. Find the trajectory equation of its midpoint.
  • #14: The line passing through the point A (4,0) intersects the circle x^2 + y^2 = 4 at the point B, C. Find the trajectory equation of the midpoint M in the chord BC.