2011 Λύσεις Σχ. β&I

Ôìóˆ ë âíûˆûë âè ïûë ìfióô ó 0 0

This preview shows 34 out of 37 pages.

∂Ô̤ӈ˜ Ë Â͛ۈÛË ¤¯ÂÈ Ï‡ÛË ÌfiÓÔ ·Ó · ≠ 0, ‚ ≠ 0 Î·È · ≠ ‚. x = ·‚ ‚ – · . x · x = 1 ‚x – ·x ·‚ = 1 (‚ – ·)x = ·‚. x = (· – ‚)(· + ‚) · – ‚ = · + ‚. x – · = x – ‚ · x = (· + ‚) 2 2(· + ‚) = · + ‚ 2 . ∫∂º∞§∞π√ 3: ∂•π™ø™∂π™ 34
Image of page 34

Subscribe to view the full document.

3. i) ™Ù· 200 ml ‰È¿Ï˘Ì· ÂÚȤ¯ÔÓÙ·È 30 ml ηı·Úfi ÔÈÓfiÓÂ˘Ì·. ∞Ó ÚÔ- Ûı¤ÛÔ˘Ì x ml ηı·Úfi ÔÈÓfiÓÂ˘Ì· ÙfiÙ ÙÔ ‰È¿Ï˘Ì· Ô˘ ı· ÚÔ·„ÂÈ ı· Â›Ó·È (200 + x) ml Î·È ı· ÂÚȤ¯ÂÈ (30 + x) ml ηı·Úfi ÔÈÓfiÓÂ˘Ì· ÔfiÙ ÚÔ·ÙÂÈ Ë Â͛ۈÛË ∂Ô̤ӈ˜ Ô Ê·ÚÌ·ÎÔÔÈfi˜ Ú¤ÂÈ Ó· ÚÔÛı¤ÛÂÈ 50 ml ηı·Úfi ÔÈÓfi- ÓÂ˘Ì·. 4. ŒÛÙˆ fiÙÈ x ÒÚ˜ ÌÂÙ¿ ÙËÓ ÚÔÛ¤Ú·ÛË Ù· ‰‡Ô ·˘ÙÔΛÓËÙ· ı· ·¤¯Ô˘Ó ÌÂ- ٷ͇ ÙÔ˘˜ 1 km. ΔÔ ‰È¿ÛÙËÌ· Ô˘ ‰È·Ó‡ÂÈ ÙÔ ∞ ÛÙȘ x ÒÚ˜ Â›Ó·È 100x ÂÓÒ ÙÔ ·ÓÙ›ÛÙÔÈ¯Ô ‰È¿ÛÙËÌ· ÁÈ· ÙÔ μ Â›Ó·È 120x. ŒÙÛÈ ¤¯Ô˘Ì ÙËÓ Â͛ۈÛË 120x – 100x = 1 20x = 1 x = 1 20 ÒÚ˜, ÔfiÙ x = 1 20 Ø 60 = 3 ÏÂÙ¿. √fiÙ ٷ ·˘ÙÔΛÓËÙ· ı· ·¤¯Ô˘Ó 1km ÙÚ›· ÏÂÙ¿ ÌÂÙ¿ ÙËÓ ÚÔÛ¤Ú·ÛË. 5. ∏ Â͛ۈÛË ·˘Ù‹ Â›Ó·È ÔÚÈṲ̂ÓË ÁÈ· x ≠ · Î·È x ≠ –·. ªÂ ·˘ÙÔ‡˜ ÙÔ˘˜ Â- ÚÈÔÚÈÛÌÔ‡˜ ¤¯Ô˘ÌÂ: (x + ·) 2 = x 2 x + · = x x + · = –x 0x = · 2x = –·. ñ ∞Ó · = 0, ÙfiÙÂ Ë Â͛ۈÛË ¤¯ÂÈ ˆ˜ χÛË Î¿ı ·ÚÈıÌfi x ≠ 0. ñ ∞Ó · ≠ 0, ÙfiÙÂ Ë Â͛ۈÛË ¤¯ÂÈ ÌÔÓ·‰È΋ χÛË ÙÔÓ ·ÚÈıÌfi . 6. ∏ Â͛ۈÛË ·˘Ù‹ Â›Ó·È ÔÚÈṲ̂ÓË ÁÈ· x ≠ 2. ªÂ ·˘Ùfi ÙÔÓ ÂÚÈÔÚÈÛÌfi ¤¯Ô˘ÌÂ: x 3 – 8 = x 3 – 2x 2 + 4x – 8 2x 2 – 4x = 0 2x(x – 2) = 0 x = 0 x = 2. ∞fi ÙȘ ÙÈ̤˜ ·˘Ù¤˜ ‰ÂÎÙ‹ Â›Ó·È ÌfiÓÔ Ë x = 0 ∂Ô̤ӈ˜ Ë Â͛ۈÛË ¤¯ÂÈ ÌÔÓ·‰È΋ χÛË, ÙÔÓ ·ÚÈıÌfi x = 0. x 3 – 8 x – 2 = x 2 + 4 x = –· 2 x + · x – · = x 2 x 2 – · 2 x + · x – · = x 2 (x + ·)(x – ·) x = 3400 68 x = 50. 3000 + 100x = 6400 + 32x 68x = 3400 30 + x 200 + x = 32 100 100(30 + x) = 32(200 + x) 3.1. ∂ÍÈÛÒÛÂȘ 1Ô˘ ‚·ıÌÔ‡ 35
Image of page 35
7. |2|x| – 1| = 3 2|x| – 1 = 3 2|x| – 1 = –3 2|x| = 4 2|x| = –2. ∏ ‰Â‡ÙÂÚË Â›Ó·È ·‰‡Ó·ÙË ÔfiÙ ¤¯Ô˘Ì 2|x| = 4 |x| = 2 x = –2 x = 2. ∂Ô̤ӈ˜ ÔÈ Ï‡ÛÂȘ Ù˘ Â͛ۈÛ˘ Â›Ó·È ÔÈ ·ÚÈıÌÔ› –2 Î·È 2.
Image of page 36

Subscribe to view the full document.

Image of page 37
You've reached the end of this preview.
  • Winter '09
  • Nikos

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern