ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„ĸī…„ī„˜ī„Žī„§ī…„ī„´ī…† ī„¯ī…Ĩī„šī„Ž ī„¯ī…žī„Ēī„ĩ-2021
ī„ąī…„.ī„—. ī„¯ī…ƒī„ģī„ļī…‚ ī„ēī„ŗī…‰ī„ēī„Ĩī„ĩ
ī„ŗī…‚ī„Ēī„ĩī…‚ī„Ēī„ŗī„› ī„Ēī„›ī„Žī…„ī„›ī…‚ī„-I
ī„ēī„ŗī…‚ī„: ī…ƒī„Ēī…œī„Ž ī„žī…œī„Ĩī…‡ ī„¸ī…žī„­ ī„Ēī…Œ ī„¸ī…žī„­ ī„‘ī…œī„›:100
ī„Žī…‹ī„Ĩ: ī„ģī„ĩī…‡ī„› ī„˛ī…‚ī„ ī…ƒī„¸ī…žī„ ī…Œ ī„žī…žī„Ĩī…‹-ī„žī…žī„Ĩ ī„“ī…žī„› ī„¯ī…Ĩī„šī„Ž ī„ ī……ī„Š ī„›ī…‡, ī„›ī……ī…žī„ļ ī„¯ī…œī„ĸ ī„¯ī…Ĩī„šī„Žī…‚ī„ ī„Ŧī…„ ī„›ī…‹ī…ƒī„šī„š ī„›ī„ĩī…‹îĨĄ ī„¯ī…œī„ĸī„¸ī…‚ī„ ī„¯ī…Ĩī„šī„Ž
ī…ƒī„›ī„ēī…‡ ī„¸ī…„ ī„˛ī…‚ī„ ī…ƒī„¸ī…žī„ ī…Œ ī„›ī…„ī„Ēī…‚ ī„ĸī…‚ ī„ēī„›ī„Ŧī…‚ ī„ģī…‰îĨĄ ī„ēī…‚ī„ĩī…‡ ī„¯ī…Ĩī„šī„Žī…‚ī„ ī„Ŧī…‡ ī„‘ī…œī„› ī„ąī„ĩī…‚ī„ąī„ĩ ī„ģī„ŽîĨĄ
ī„˛ī…‚ī„-ī„—
1. (1) ī„ģī…žī„ļ ī„›ī„ĩī…‹
îēžîēŸ
îēžī„žîēŸ
īŒ
īƒš
īƒĸ

îēžîē 
îēžī„žîē 
īŒ
īƒš
īƒšīƒš
îē‡îē”îēŠî˜ƒ
îē îēž
îē ī„žîēž
īŒ
īƒš
īƒšīƒ™
(ii) ī„ģī…žī„ļ ī„›ī„ĩī…‹:
îĻž
îēž ī…
īƒĄ
îĻž
îēž
īŒ īƒŸ
(iii) ī„ŗī…œī„ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„›ī…Ĩī„ŗī„¸ī…‚ī„ĩ p2 + q2 = 20 ī„‘ī„Ēī…‡ 2p + q = 8 ī„Ŧī…‡ ī„ĩī…†ī„¯ ī…ƒī„¸ī…žī„  ī…ƒī„Ŧī…žī„Ēī…‡ ī„ī„— ī„ģī„Ž ī…ƒī„ĸī…žī„Ģī…‡ p ī„›ī…„ī„ŗī„Ē
ī„ģī…‰ ī„‘ī„Ēī…‡ q ī„ŗī…‚ī„Ēī„ĩī…‚ ī„ģī…‰îĨĄ ī„ēī…œī„Ēī……ī„ļī„Ž ī„›ī…„ī„ŗī„Ē ī„‘ī„Ēī…‡ ī„ŗī…‚ī„Ēī„ĩī…‚ ī„¯ī„Ēī…‚ ī„›ī„ĩī…‹îĨĄ
2. (i) 3 ī„‘ī„Ēī…‡ 24 ī„Ŧī…‡ ī…ƒī„¸ī„ ī„›ī…‚ī„ĩ 6 ī„‘ī…œī„›ī„ī…ƒī„Šī„Ē ī„Ŧī…‡ ī„ēī…‚ī„­ī„Ž ī„¯ī…‚ī„™îĨĄ
(ii) ī„ļī„ŋī…„ ī„Ŧī…‚ ī…ƒī„›ī„ģī„ŋī…‚ ī„¯ī„Ŧ 1 + 1/2 + 1/4 + 1/8 1/512
(iii) ī„ļī„ŋī…„ ī„Ŧī…‡ n ī„¯ī„Ŧī…‚ī„ ī„Ēī…žī„› ī„Ŧī…‚ ī„ĸī…‹ī„ŋ ī„¯ī„Ēī…‚ ī„›ī„ĩī…‹îĨĄ
ī„˛ī…‚ī„-ī„ąī…„
3. (i) ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ ī…ƒī„› y = 5x - 7 ī„‘ī„Ēī…‡ 2y = 10x + 5 ī„ēī„ŗī…‚ī„ī„Ēī„ĩ ī„ģī„ŽîĨĄ
(ii) ī„•ī„ē ī„ēī…ˆī„Ĩ ī„ĩī…‡ī„œī…‚ ī„Ŧī…‚ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„ļī…žī„˛ī…‹ ī„ĸī…‹ (1, 3, 5) ī…ƒī„¸ī…žī„ ī…Œ ī„ļī…œī„žī„Ŧī…„ ī„ģī…‰ ī„‘ī„Ēī…‡ ī…ƒī„Žī„ĩī„Ŧī…‡ī„šī…‚ī„ī„› ī„­ī……ī…ƒī„ĩī„’ī„ 'ī„Ēī…‡ ī„“ī„ēī„Ŧī…‡
ī„‘ī…œī„Ēī„ĩī…‚ī„ļī…‚ī„ ī„Ŧī…‚ ī„ĸī…‹ī„ŋ 9 ī„ģī…‰îĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
(ii) ī„Ŧī……ī…žī„­ ī„Ŧī…„ ī„ŗī…œī„ ī„“ī„ē ī„Ēī„ĩī…§ī…‚ī„ ī…ƒī„Ŧī…žī„Ēī…„ ī„ĸī…‚ī„ī„Ŧī…„ ī„ģī…‰:
ī„›ī…„ī„ŗī„Ē ī„ĩī……ī„¯ī„—/ī„ļī…„ī„Ĩī„ĩ
ī„ļī…„ī„Ĩī„ĩī…‚ī„ ī…ƒī„¸ī…žī„  ī„ŗī…œī„
1
100
2
50
3
0
ī„ĩī…‡ī…ƒī„œī„› ī„ŗī…œī„ ī„°ī…œī„›ī„šī„Ž ī„ļī…žī„˛ī…‹îĨĄ
4. (i) ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Žī…†ī…œ ī„¯ī…ƒī„ĩī„˛ī…‚ī…ƒī„šī„Ē ī„›ī„ĩī…‹îĨĄ ī„¸ī…žī„œ-ī„¸ī…žī„œ ī…ƒī„›ī„ēī„ŗī…‚ī„ ī„Ŧī…‡ ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…„ ī…ƒī„¸ī„’ī…ƒī„œī„’ ī„›ī„ĩī…‹îĨĄ
(ii) ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…‡ ī„ēī…œī„ž, ī„ļī…‚ī„ī„žī…‡, ī„‘ī…œī„Ēī„ĩ ī„‘ī„Ēī…‡ ī„ēī„ŗī…ƒī„ŗī„Ēī…„ ī„‘ī…œī„Ēī„ĩ ī„Ŧī…„ ī…ƒī„¸ī„’ī…ƒī„œī„’ ī„›ī„ĩī…‹îĨĄ
(iii) 70 ī…ƒī„¸ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Ŧī…„ ī„“ī…žī„› ī„›ī„ļī…‚ī„ē, ī…ƒī„ĸī„Žī…§ī…‚ī„ ī…ƒī„¸ī…žī„ ī…Œ 30 ī„Žī…‡ ī„ī…ƒī„Šī„Ē ī„Ŧī…‚ ī…ƒī„¸ī„šī…‚ ī…ƒī„ļī„’ ī„ģī…‰ ī„‘ī„Ēī…‡ 20 ī„Žī…‡
ī„ī…ƒī„Šī„Ē ī„Ŧī…‚ ī…ƒī„¸ī„šī…‚ ī…ƒī„ļī„’ ī„ģī…‰ ī„¯ī„ĩ ī„‘ī…œī„›ī„ŋī…‚ ī„Žī„ģī…„
ī„
ī…ƒī„ļī„’îĨĄ ī„ī…ƒī„Šī„Ē ī„‘ī„Ēī…‡ ī„‘ī…œī„›ī„ŋī…‚ ī…ƒī„¸ī…ƒī„ī„’ī„Ž ī„ļī…‰ī„Š ī„¸ī…‚ī„ļī…‡
ī…ƒī„¸ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Ŧī…„ ī…ƒī„ī„Šī„Ēī…„ ī„ļī…žī„˛ī…‹ ī„‘ī„Ēī…‡ ī…ƒī„ĸī„Žī…§ī…‚ī„ ī„Žī…‡ ī„‘ī…œī„›ī„ŋī…‚ ī…ƒī„¸ī…ƒī„ī„’ī„Ž ī…ƒī„ļī„’ ī„ģī…‰ ī„¯ī„ĩ ī„ī…ƒī„Šī„Ē ī„Žī„ģī…„
ī„
ī…ƒī„ļī„’
ī„ģī…‰îĨĄ
ī„˛ī…‚ī„-ī„ēī…„
5. (1) ī„°ī…œī„›ī„šī„Ž ī„Ŧī…„ ī„­ī…‚ī„ĩī„Žī…‚ ī„‘ī„Ēī…‡ ī„¸ī…žī„œ-ī„¸ī…žī„œ ī…ƒī„›ī„ēī„ŗī…‚ī„ ī„Ŧī…‡ ī„°ī…œī„›ī„šī„Žī…‚ī„ ī„Ŧī…„ ī…ƒī„¸ī„’ī…ƒī„œī„’ ī„›ī„ĩī…‹îĨĄ
(ii) ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ ī…ƒī„›
îē’îēîē“îēîēš
îēžî­œīƒš

îēž
īƒ›
ī„ŋīƒœîēžī„žīƒ›
îēžī„ŋīƒš
īŒ ī†īƒš
6. (i) ī„“ī…žī„› ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„°ī…œī„›ī„šī„Ž ī„‘ī„Ēī…‡ ī„‘ī„ēī…œī„Ēī……ī„šī„Ĩ ī„°ī…œī„›ī„šī„Ž ī…ƒī„¸ī…žī„  ī„‘ī…œī„Ēī„ĩ ī„Ŧī…žī„ēī…‹îĨĄ
(ii) ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ ī…ƒī„› x = 2 ī„Ēī…‡ ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„ģī…‰
îēž
īƒ›
ī„žīƒš
îēž
īƒ›
ī„ŋīƒš

ī„ēī…‰ī„›ī„šī„Ž-ī„§ī…„
7. (i) (7x - 8)4 (5x - 1)3 ī„¯ī„Ŧī…‚ī„ ī„Žī…†ī…œ ī„¸ī…žī„œ ī„›ī„ĩī…‹
(ii) wrtx ī„Ŧī…‚ ī„§ī…‰ī„ĩī…„ī„¸ī…‡ī…ƒī„Ĩī„¸ ī„ļī…žī„˛ī…‹îĨĄ
îēžī„žīƒ›
īƒœī„žîē’îē•îēî˜ƒîēžî˜ƒ

ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
(iii) wr.tx ī„Žī…†ī…œ ī„¸ī…žī„œī„ĩī…‚ ī„›ī„ĩī…‹īƒž
īŽĨ
īƒšī„žîē‹îēž
īƒ›
8. (i) ī„ŗī…œī„ ī„°ī…œī„›ī„šī„Ž p = 50 - 3q ī…ƒī„Ŧī…žī„Ēī…‡ ī„ĸī…‚ī„Š 'ī„Ēī…‡ ī„ŗī…œī„ p = 5 ī„Ŧī…„ ī„ļī„ ī„›ī„Ēī…‚ ī„¯ī„Ēī…‚ ī„›ī„ĩī…‹îĨĄ
(ii) ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ ī…ƒī„› ī„ŗī…œī„ ī„°ī…œī„›ī„šī„Ž ī„Ŧī…„ ī„ļī„ ī„›ī„Ēī…‚ = ī„ģī…‰ ī…ƒī„ĸī…žī„Ģī…‡ ī„ŗī…œī„ p = 50 - 3x ī„‘ī„Ēī…‡ p = 5 ī„ģī…‰îĨĄ
îš­îšž
îš­îšžī„ŋîššîšž
(iii) ī„›ī……ī…žī„ļ ī„ļī…‚ī„ī„Ē ī„°ī…œī„›ī„šī„Ž C = 60 - 12q + 2q2 ī…ƒī„Ŧī…žī„Ēī…‡ ī„ĸī…‚ī„Š 'ī„Ēī…‡ AC, MC ī„ļī…žī„˛ī…‹ ī„‘ī„Ēī…‡ AC ī„Ŧī…„ ī„¨ī„ļī…‚ī„Š = (MC -
AC) ī…ƒī„Ŧī„œī…‚ī„™îĨĄ
īƒš
îē—
ī„ĸī…„ī„˜ī„Žī„§ī…„ī„´ī…† ī„•ī…žī„Ēī„ĩ ī„¯ī…žī„Ēī„ĩ-2021
ī„ąī…„.ī„—. ī„¯ī…ƒī„ģī„ļī…‚ ī„ēī„ŗī…‰ī„ēī„Ĩī„ĩ
ī„ŗī…‚ī„Ēī„ĩī…‚ī„Ēī„ŗī„› ī„Ēī„›ī„Žī…„ī„›ī…‚ī„-I
ī„ēī„ŗī…‚ī„: ī…ƒī„Ēī…œī„Ž ī„žī…œī„Ĩī…‡ ī„¸ī…žī„­ ī„Ēī…Œ ī„¸ī…žī„­ ī„‘ī…œī„›:100
ī„Žī…‹ī„Ĩ: ī„ģī„ĩī…‡ī„› ī„˛ī…‚ī„ ī…ƒī„¸ī…žī„ ī…Œ ī„žī…žī„Ĩī…‹-ī„žī…žī„Ĩ ī„“ī…žī„› ī„¯ī…Ĩī„šī„Ž ī„ ī……ī„Š ī„›ī…‡, ī„›ī……ī…žī„ļ ī„¯ī…œī„ĸ ī„¯ī…Ĩī„šī„Žī…‚ī„ ī„Ŧī…„ ī„›ī…‹ī…ƒī„šī„š ī„›ī„ĩī…‹îĨĄ ī„¯ī…œī„ĸī„¸ī…‚ī„ ī„¯ī…Ĩī„šī„Ž
ī…ƒī„›ī„ēī…‡ ī„¸ī…„ ī„˛ī…‚ī„ ī…ƒī„¸ī…žī„ ī…Œ ī„›ī…„ī„Ēī…‚ ī„ĸī…‚ ī„ēī„›ī„Ŧī…‚ ī„ģī…‰îĨĄ ī„ēī…‚ī„ĩī…‡ ī„¯ī…Ĩī„šī„Žī…‚ī„ ī„Ŧī…‡ ī„‘ī…œī„› ī„ąī„ĩī…‚ī„ąī„ĩ ī„ģī„ŽîĨĄ
ī„˛ī…‚ī„-ī„—
1. (1) ī„ģī…žī„ļ ī„›ī„ĩī…‹
îēžîēŸ
îēžī„žîēŸ
īŒ
īƒš
īƒĸ

îēžîē 
îēžī„žîē 
īŒ
īƒš
īƒšīƒš
îē‡îē”îēŠî˜ƒ
îē îēž
îē ī„žîēž
īŒ
īƒš
īƒšīƒ™
(ii) ī„ģī…žī„ļ ī„›ī„ĩī…‹:
îĻž
îēž ī…
īƒĄ
îĻž
îēž
īŒ īƒŸ
(iii) ī„ŗī…œī„ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„›ī…Ĩī„ŗī„¸ī…‚ī„ĩ p2 + q2 = 20 ī„‘ī„Ēī…‡ 2p + q = 8 ī„Ŧī…‡ ī„ĩī…†ī„¯ ī…ƒī„¸ī…žī„  ī…ƒī„Ŧī…žī„Ēī…‡ ī„ī„— ī„ģī„Ž ī…ƒī„ĸī…žī„Ģī…‡ p ī„›ī…„ī„ŗī„Ē
ī„ģī…‰ ī„‘ī„Ēī…‡ q ī„ŗī…‚ī„Ēī„ĩī…‚ ī„ģī…‰îĨĄ ī„ēī…œī„Ēī……ī„ļī„Ž ī„›ī…„ī„ŗī„Ē ī„‘ī„Ēī…‡ ī„ŗī…‚ī„Ēī„ĩī…‚ ī„¯ī„Ēī…‚ ī„›ī„ĩī…‹îĨĄ
ī„•ī…žī„Ēī„ĩ:ķˇ…ļķˇ…ąķˇ…ēķˇ…ˇķˇ…¸ķˇ…šī„“ī…žī„› ī„›ī…‹ī„ŗī„ļ ī„šī……ī„ĩī…†ī„’ī„Ē...
ī„›ī„ļī„¯ī„Žī…‚ ī„›ī„ĩī…‹ ī…ƒī„› ī„“ī…žī„› ī„šī…‚ī„ī„Ē ī…ƒī„¯ī…œī„§ ī…ƒ ī…žī„  ī„“ī…žī„› ī„ēī……ī„ģī…‚ ī„Šī…„ ī„ē ī…‡ī„ĩ, ī…ƒī„Ēī…œī„Ž ī„Ŧī…‹ī„ēī„Ē - ī„’ī„ĩī…„ī„‘ī„Ž, ī„ąī…‡ī„ļī…‚ ī„‘ī„Ēī…‡ ī„žī…‹ī„“ī„’ -
ī„“ī…žī„› ī…žī„§ī…‡ ī„ąī…‹ī„ģī„ŋ ī„Ŧī…‡ ī„Ŧī„ĩī…žī„œī„Ē ī„ģī…‡ī„Ļ ī„’ī„¯ī„Šī…‡ ī„ēī„Ģī…‚ī„Žī„› ī„ī…ƒī„Šī„Ē ī„‘ī…ƒī„­ī„’ī„¯ī„› ī„›ī…‹ī„ļ ī„ī„—îĨĄ ī„•ī„ģ ī„ģī……ī„Šī…‡ ī„ēī„›ī…†ī„ļ ī„Ēī…Œ ī…‚ī„¯ī„ē
ī„’ī„— ī„ēī„Ž, ī„‘ī„Ēī…‡ ī„•ī„Žī…§ī…‚ī„ ī„Ŧī…‡ ī…ƒī„ēī„ĩ ī„ēī…œī…ƒī„œī„’ ī…‚ī„, ī…ƒī„ ī…œī„Žī…§ī…‚ī„ ī„‘ī„Ēī…‡ ī„ēī„ŗī…„ī„›ī„ĩī„Žī…‚ī„ ī„Žī…‚ī„ļ ī„žī……ī…œī„ŗ ī„ĩī„ģī…‡ ī„ēī„ŽîĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„’ī„ĩī…„ī„‘ī„Ž ī„ēī„˛ ī„Ēī…Œ ī…žī„­ ī„•ī„ļī„Ŗī„Š ī…ƒ ī…žī„  ī„ēī…„îĨĄ "ī„ēī„ĩ," ī„•ī„ēī„Žī…‡ ī…ƒī„›ī„ģī…‚, "ī„ŗī…‰ī„Žī…†ī…œ ī„“ī„ģ ī„ē ī…‚ī„ļ ī„ēī…‚ī„§ī…‡ ī„‘ī„ēī…‚ī„”ī„Žī„ŗī…Šī„Ĩ ī…ƒ ī…žī„ 
ī…ƒī„ŗī…ƒī„ļī„’ ī„ģī…‰îĨĄ ī„“ī„ģ ī„“ī…žī„›ī…‹ ī„ēī„ŗī…ˆ ī…ƒī„Ēī…œī„Ž ī„¯ī„ģī…‡ī„ļī…„ī„’ī„ ī„ģī…žī„ļ ī„›ī„ĩī„Ž ī„ĩī„ī…‚ ī„ģī…‰!"
ī„•ī„Žī…§ī…‚ī„ ī„Ŧī…‡ ī„•ī„ēī„Ēī…‚ī„Ŧ, ī…ƒī„ēī„’ī„Šī…‡ ī„‘ī„Ēī…‡ ī…ƒī„Ŧī„’ī„ļī…†, ī„ŗī……ī„ēī„›ī„ĩī…‚ī„— ī„‘ī„Ēī…‡ ī…ƒī„›ī„ģī…‚, "ī„’ī„™ ī„“ī„›ī…žī„Ļī…‡ ī„“ī„ģī„Žī…‚ī„ ī„¯ī„ģī…‡ī„ļī…„ī„’ī„ ī„Žī…†ī…œ ī„“ī…žī„›-ī„“ī…žī„›
ī„›ī„ĩī„›ī…‡ ī„ģī…žī„ļ ī„›ī„ĩī…„ī„—îĨĄ ī„´ī…‚ī„Ŧ ī„ĩī…žī„œī…‹, ī„ī…ƒī„Šī„Ē ī„“ī…žī„› ī„ī…œī„¨ ī„œī…‹ī„ļī…§ī„Š ī…‚ī„ī„ ī„ģī…‰ - ī„“ī…žī„› ī„ēī„ŗī…ˆ ī…ƒ ī…žī„  ī„“ī…žī„› ī„Ąī…‹ī„Ĩī…‚ ī…ƒī„ĸī„ģī…‚ ī„­ī…‚ī„ī…‚îĨĄ"
ī„‘ī„Ēī…‡ ī„“ī„ē ī„Ēī„ĩī…§ī…‚ī„, ī…ƒī„Ēī…œī„Ž ī…žī„œ- ī…žī„œ ī…ƒī„›ī„ēī„ŗī…‚ī„ ī„Ŧī…„ī„’ī„ ī„ī…ƒī„Šī…ƒī„Ēī„› ī„ēī„ŗī…žī…ƒī„ēī„’ ī…‚ī„ ī„Žī…†ī…œ ī„ģī…žī„ļ ī„›ī„ĩī„Ž ī„ļī„” ī„•ī„Žī…§ī…‚ī„ ī„Ŧī…‚ ī„›ī„Ŧī„ŗ-ī„Ŧī„ĩ-
ī„›ī„Ŧī„ŗ ī„ēī…‚ī„ģī„ē ī„šī……ī„ĩī…† ī„ģī…‹ī„“ī„’îĨĄ ī„’ī„™ ī„•ī„Žī…§ī…‚ī„ ī„Ŧī…‡ ī„Žī…‚ī„ļ ī„ ī…žī„ļī…„ī„— ī„‘ī„Ēī…‡ ī„•ī„ēī…‡ ī„Ēī„ĩī…§ī…‚ī„ ī…ƒī„ēī…žī„œī…„ī„— - ī„ģī…ī„ļī…„, ī„ēī„¯ī…žī„šī„Ĩ ī„‘ī„Ēī…‡ ī„•ī„Ŧī…‡ī„š
ī„Žī…‚ī„ļîĨĄ
ķŧŠ•ķŧŠ–ķŧŠ—ķŧŠ˜ķŧŠ™ķŧŠš(i) ī„ēī„ŗī…„ī„›ī„ĩī„Žī…‚ī„ ī„Ŧī…„ ī„¯ī…Ĩī„Šī…‚ī„ļī…„ ī„Žī…†ī…œ ī„ģī…žī„ļ ī„›ī„ĩī„Žī…‚:
ī„ēī…‚ī„Žī…†ī…œ ī„ģī…‡ī„Ļ ī…ƒī„ļī„œī…‡ ī…ƒī„Ŧī…žī„Ēī…‡ ī„ī„— ī„ģī„Ž:
ī„’ī„™ ī„ģī„ĩī…‡ī„› ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„Žī…†ī…œ ī„ļī…Šī„Ŧī…‡ ī„ģī…‚ī„ ī„‘ī„Ēī…‡ ī„Ŧī…‹ ī…‚ī„ ī„¯ī…‚ī…ƒī„ēī„’ī„ ī„Žī…†ī…œ ī„•ī„ļī„Ĩī…‚ ī„›ī…‡ ī„ ī…„ī„žī…‚ī„ ī„Žī…†ī…œ ī„’ī„ēī…‚ī„Ž ī„ąī„Šī…‚ī„•
ī„
ī„Ŧī…‡ ī„ģī…‚ī„îĨĄ ī„“ī„ģ ī„ ī…‚ī„ļ
ī„‘ī„›ī„ēī„ĩ ī„•ī„Ŧī…Œ ī„•ī„¯ī„´ī…‹ī„ī…„ ī„ģī……ī…œī„Ŧī…„ ī„ģī…‰ ī„ĸī„Ŧī…Œ ī…‡ī„ĩī…„ī„—ī„ąī„ļ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„˛ī…‚ī„ĸ ī…ƒ ī…žī„  ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„›ī„Ŧī„ŗ 1: ī„¯ī„Ąī…‚ī„Š ī„Ŧī…„ ī„¸ī„ĩī„Ēī…Œ ī„›ī„ĩī…‹
ī„“ī…žī„› ī„ŗī„Ŧī„Ŧī„ī…‚ī„ĩ ī„¯ī„Ąī…‚ī„Š ī„´ī…‚ī„Ŧ ī„›ī„ĩī…‹:
ī„’ī„™ ī„“ī„ē ī„Ēī„ĩī„› ī„Žī…†ī…œ ī„ēī„ŗī…„ī„›ī„ĩī„Žī…‚ī„ ī„ļī„” ī„ĩī„Ēī…„ī„—îĨĄ
ī„“ī„ē ī„ļī„”,
ī„’ī„™ ī„Žī…‚ī„ŗ ī„Ŧī…‡ī„”ī„—:
â€ĸ ī„— = 1/x
â€ĸ ī„ąī…„ = 1/y
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
â€ĸ ī„ēī…„ = 1/ī„žī…‰ī„§
ī„ģī……ī„Š ī„ąī„Ŧī„ļī…‹:
â€ĸ A + B = 9 â€Ļ (i)
â€ĸ A + C = 11 â€Ļ (ii)
â€ĸ C + A = 10 â€Ļ (iii)
ī„›ī„Ŧī„ŗ 2: ī„ēī„ŗī…„ī„›ī„ĩī„Žī…‚ī„ ī„Žī…†ī…œ ī„žī„Ĩī…‚ī„™
ī„’ī„™ ī„žī„Ĩī…‚ī„™ (ii) – (iii):
ī„™ī„ģ! ī„“ī„ģ ī„•ī„ļī„Ŗī„Š ī…‚ī„ļī…‚ ī„ļī…žī„ī„Ŧī…‚ ī„ģī…‰ - ī„Ŧī…‹ ī…ˆ (ii) ī„‘ī„Ēī…‡ (iii) A + C ī„ąī…‚ī„ĩī…‡ ī„ģī„Ž, ī„¯ī„ĩ ī„•ī„ģ ī…žī„œ- ī…žī„œ ī„Žī„Ēī…„ī„ĸī…‡ ī…ƒī„Ŧī…œī„Ŧī…‡ ī„ģī„Ž:
11 ī„‘ī„Ēī…‡ 10îĨĄ ī„“ī„ēī„Ŧī…‚ ī„ŗī„Ēī„ļī„ą ī„ģī…‰ ī…ƒī„› ī„“ī„ģī„Žī…‚ī„ ī…ƒ ī…žī„ ī…Œ ī„“ī…žī„› ī…žī„œī„ĩī…‚ ī„ģī…‹ī„Šī…‚ ī„ ī…‚ī„ģī…„ī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„’ī„™ ī„Ŧī……ī„ąī…‚ī„ĩī…‚ ī„ĸī…‚ī„ī„  ī„›ī„ĩī…„ī„—:
ī„ŗī…†ī„ļ ī„Ēī…Œ:
ī„ĩī……ī„›ī…‹! ī„¯ī„ĩ (2) ī„‘ī„Ēī…‡ (3) ī„Ŧī…‹ ī…ˆ ī„ŗī……ī…žī„ļ ī…ƒ ī…žī„  ī„“ī…žī„›ī…‹ ī…ƒī„ĸī„ģī…‡ ī„ģī„Ž, ī„ąī„ē ī…žī„œī„ĩī…‡ ī„¨ī…œī„ ī„Žī…‚ī„ļ ī…ƒī„ļī„œī…‡ ī„ī„— ī„ģī„ŽîĨĄ
ī„Ēī…‚ī„ ī„•ī„ģ ī…ƒī„› ī…ˆ ī…žī„œī„ĩī…‡ ī„ģī…‹ ī„ēī„›ī„Ŧī…‡ ī„ģī„Ž? ī„“ī„ģ ī„ēī…œī„˛ ī„Žī„ģī…„
ī„
ī„ģī…‰ ī„ĸī„Ŧī…Œ ī„Ēī…žī„› ī„ē ī…‚ī„ļ ī„ĸī…‚ī„ ī„ŗī……ī…žī„ļ ī…ƒ ī…žī„  ī„›ī…‹ī„” ī„ī„ļī„Ēī…„ ī„Žī…‚ ī„ģī…‹ ī…‡îĨĄ
ī„ ī„ļī…‹ ī„“ī„ēī„Žī…†ī…œ ī„Ļī…„ī„› ī„›ī„ĩī„Ŧī…‡ ī„ģī…‚ī„îĨĄ ī„ŗī…œī„Ž ī„ļī„™:
ī„’ī„™ ī„ļī…Šī„Ŧī…‡ ī„ģī…‚ī„:
â€ĸ ī„— = 1/x
â€ĸ ī„ąī…„ = 1/y
â€ĸ ī„ēī…„ = 1/ī„žī…‰ī„§
ī„¸ī…žī„ļī…Œ:
â€ĸ ī„— + ī„ąī…„ = 9 → (1)
â€ĸ ī„— + ī„ēī…„ = 11 → (2)
â€ĸ ī„ēī…„ + ī„— = 10 → (3)
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„ģī……ī„Š ī„ēī„¯ī…žī„šī„Ĩ ī„Ēī…ī„ĩ 'ī„Ēī…‡, (2) ī„‘ī„Ēī…‡ (3) ī„›ī„ģī…‹:
â€ĸ ī„— + ī„ēī…„ = 11
â€ĸ ī„— + ī„ēī…„ = 10
ī…ƒī„°ī„ĩ ī…ƒī„¸ī„ĩī…‹ī„­ī…‚ī„˛ī…‚ī„ēîĨĄ
ī„Ēī…‚ī„ ī„ģī…‹ ī„ēī„›ī„Ŧī…‚ ī„ģī…‰ ī…ƒī„› ī„Ēī…„ī„ĸī…‚ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„›ī……ī„Ŗ ī„ģī…‹ī„ĩ ī„ģī…‹ ī…‡îĨĄ ī„¯ī„ĩ ī„‘ī…ƒī„ĸī„ģī…‡ ī„¯ī…Ĩī„šī„Žī…‚ī„ ī…ƒ ī…žī„ , ī„Ŧī…‹ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī…žī„œī„ĩī…‡ ī„ģī…‹ī„Šī…‡ ī„ ī…‚ī„ģī…„ī„Ŧī…‡
ī„ģī„ŽîĨĄ ī„ŗī…œī„Ž ī„ļī„™ ī…ƒī„› ī„›ī…‹ī„” ī„Ĩī…‚ī„”ī„¯ī…‹ ī„ī„ļī„Ēī…„ ī„ģī…‰, ī„‘ī„Ēī…‡ ī…ƒī„›ī„ēī…‡ ī…„ ī„Ŧī…‹ ī„Žī…‚ī„ļ ī„‘ī…žī„ī…‡ ī„­ī„Ŧī…‡ ī„ģī…‚ī„:
ī„¸ī…žī„ļī…Œ:
â€ĸ ī„— + ī„ąī…„ = 9 (1)
â€ĸ ī„— + ī„ēī…„ = 11 (2)
ī„ģī……ī„Š (2) – (1) ī„žī„Ĩī…‚ī„™:
(A + C) – (A + B) = 11 – 9C – B = 2ī„ąī…„ = ī„ēī…„–2
ī„ģī……ī„Š (1) ī„Ēī…Œ:
ī„— + ī„ąī…„ = 9ī„— + (ī„ēī…„)–2) = 9A + C = 11 ī„“ī„ģ ī„ēī„ŗī…„ī„›ī„ĩī„Ž (2) ī„Žī…‚ī„ļ ī„ŗī…‡ī„ļ ī„œī…‚ī„ī„Ŧī…‚ ī„ģī…‰, ī„ąī„ģī……ī„Ē ī„­ī…„ī„’!
ī„’ī„™ ī„“ī…žī„› ī„ŗī……ī…žī„ļ ī…ƒī„Žī„ĩī„­ī…‚ī„ĩī„Ē ī„›ī„ĩī…„ī„—:
ī„ŗī…œī„Ž ī„ļī„™ A = 4 (ī…ƒī„ēī„ĩī…€ ī„‘ī…œī„Ŧī…‚ī„žī…‚ ī„ļī„ī…‚ ī…ƒī„ĩī„ģī…‚ ī„ģī…‚ī„), ī…ƒī„°ī„ĩ C = 7 (A + C = 11 ī„Ēī…Œ), ī…ƒī„°ī„ĩ B = C – 2 = 5
ī„Ēī…‚ī„ ī„ēī…‚ī„§ī…‡ ī„›ī…‹ī„ļ ī„ģī…‰:
A = 1/x = 4 → x = 1/4B = 1/y = 5 → y = 1/5C = 1/z = 7 → z = 1/7
ķˇƒ†ķŧŊĸī„ģī……ī„Š ī„ŗī…†ī„ļ ī„ēī„ŗī…„ī„›ī„ĩī„Žī…‚ī„ ī…ƒī„¸ī…žī„  ī„¸ī…‚ī„¯ī„ē ī„¯ī„ļī…žī„ ī„“ī„Ž ī„›ī„ĩī…‹ ī„‘ī„Ēī…‡ ī„¯ī……ī„šī„Ĩī…„ ī„›ī„ĩī…‹:
â€ĸ xy / (x + y) = (1/4 * 1/5) / (1/4 + 1/5) = (1/20) / (9/20) = 1/9ķˇƒ†ķŧŊĸ
â€ĸ xz / (x + z) = (1/4 * 1/7) / (1/4 + 1/7) = (1/28) / (11/28) = 1/11ķˇƒ†ķŧŊĸ
â€ĸ zx / (z + x) = (1/7 * 1/4) / (1/7 + 1/4) = (1/28) / (11/28) = 1/11ķŊ…‚
ī„’ī„ģ! ī„“ī…žī„› ī…ƒ ī„ĩī…‹ī„­ī…‚ī„˛ī…‚ī„ē ī„ģī…‰îĨĄ ī„‘ī…œī…ƒī„Ēī„ŗ ī„ŗī……ī…žī„ļ ī…ƒī„Ēī…œī„Žī…‚ī„ ī„Žī…‚ī„ļ ī„ŗī…‡ī„ļ ī„Žī„ģī…„
ī„
ī„œī…‚ī„ī„Ŧī…‡îĨĄ
ķˇĩģķˇĩŧķˇĩŊķˇĩžī„“ī„ē ī„ļī„” ī…ƒī„ēī„ĩī…€ ī„Ŧī…‹ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„“ī„›ī„ēī…‚ī„ĩ ī„ģī„ŽîĨĄ ī„“ī„ē ī„ļī„”, ī„Ēī……ī„ēī…„
ī„
2 ī…‡ī„ĩī…„ī„—ī„ąī„ļī…‚ī„ ī„ļī„” ī„›ī…‹ī„” ī…„ ī„Ŧī…‹ ī„ēī„ŗī…„ī„›ī„ĩī„Ž
ī„ģī…žī„ļ ī„›ī„ĩī„Ŧī…‡ ī„ģī…‹îĨĄ ī„ĸī…‡ī„›ī„ĩ ī„Ēī…„ī„ĸī…‚ ī…ƒ ī„ĩī…‹ī„­ ī„›ī„ĩī„Ŧī…‚ ī„ģī…‰, ī„Ēī…‚ī„ ī„“ī„ģ ī„“ī…žī„› ī„ī„ļī„Ēī…„ ī„ģī…‹ ī„ēī„›ī„Ŧī…„ ī„ģī…‰îĨĄ
ķŧ¨ģķŧ¨ŧ(ii) ī„ģī…žī„ļ ī„›ī„ĩī…‹:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„’ī„™ ī„“ī„ēī„Žī…†ī…œ ī„ēī„ĩī„ļ ī„ąī„Šī…‚ī„•ī„Š ī„ļī„” ī„“ī…žī„› ī„ąī„ģī……ī„Ē ī„ģī…„ ī„­ī…„ī„’ ī„ąī„Ŧī„ļ ī„ļī…Šī„Ŧī…‡ ī„ģī…‚ī„îĨĄ
ī„›ī„Ŧī„ŗ 1: ī„ŗī…œī„Ž ī„ļī„™ √x = a
ī…ƒī„°ī„ĩ:
ī„‘ī…œī„š ī„Žī…†ī…œ ī„ģī„Ĩī…‚ī„•ī„Š ī„ļī„” ī„Ŧī…‹ ī…‚ī„ ī„¯ī…‚ī…ƒī„ēī„’ī„ ī„Žī…†ī…œ a ī„Žī…‚ī„ļ ī„ī……ī„Šī…‚ ī„›ī„ĩī…‹:
ī„›ī„Ŧī„ŗ 2: ī„ ī„Ēī……ī„ĩī„˛ī……ī„ĸ ī„ģī…žī„ļ ī„›ī„ĩī…‹
ī„“ī„ē ī„ļī„”:
√x = 4x = 16
√x = 2x = 4
ķˇƒ†ķŧŊĸī„‘ī…œī…ƒī„Ēī„ŗ ī„•ī…žī„Ēī„ĩ: x = 4 ī„ĸī…‚ī„ x = 16
ķšŗ¨ķšŗ¤ķšŗŠķšŗĒķšŗĢ(iii) ī„ŗī…œī„ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„ģī…žī„ļ ī„›ī„ĩī…‹:
ī„ēī…‚ī„Žī…†ī…œ ī…ƒī„Ŧī…žī„Ēī…‡ ī„ī„— ī„ģī„Ž:
ī„ēī…‚ī„Žī…†ī…œ ī„ēī…œī„Ēī……ī„ļī„Ž ī„›ī…„ī„ŗī„Ē (p) ī„‘ī„Ēī…‡ ī„ŗī…‚ī„Ēī„ĩī…‚ (q) ī„ļī…žī„˛ī„Š ī„Ŧī…„ ī„ļī…‹ī„ŋ ī„ģī…‰îĨĄ
ī„›ī„Ŧī„ŗ 1: ī„ēī„ŗī…„ī„›ī„ĩī„Ž (2) ī„Ēī…Œ:
2p+q=8ī…ƒī„›ī„–=8-2p(3)2p
ī„ģī……ī„Š ī„“ī„ēī„Žī…†ī…œ ī„ēī„ŗī…„ī„›ī„ĩī„Ž (1) ī…ƒī„¸ī…žī„  ī„ļī„ī…‚ī„™:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
p2+(8−2p)2=20p2 + (8 - 2p)2 = 20p2+(8−2p)2=20
(8 – 2p)² ī„Ŧī…‚ ī…ƒī„¸ī„ēī„Ēī…‚ī„ĩ ī„›ī„ĩī…‹:
ī„›ī„Ŧī„ŗ 2: ī„ ī„Ēī……ī„ĩī„˛ī……ī„ĸ ī„ģī…žī„ļ ī„›ī„ĩī…‹
ī„Ŧī…‹ī„ģī„ĩī…„ ī„°ī…‚ī„ĩī„ŗī…†ī„ļī…‚ ī„¸ī„ĩī„Ēī…‹:
ī„“ī„ē ī„ļī„”:
ī„ģī……ī„Š ī„ēī…œī„ąī…œī…ƒī„­ī„Ē q ī„ļī…žī„˛ī…‹:
1. ī„ĸī…‡ī„›ī„ĩ p = 4.4:
q = 8 – 2(4.4) = 8 – 8.8 = –0.8ķŊ…‚(ī„ē ī…„ī„›ī…‚ī„ĩī„´ī…‹ī„ ī„Žī„ģī…„
ī„
, ī„ŗī…‚ī„Ēī„ĩī…‚ ī„ģī…‹ ī„ēī„›ī„Ŧī…„ ī„ģī…‰'ī„Žī„›ī…‚ī„ĩī…‚ī„Ēī„ŗī„› ī„Žī„ģī…„
ī„
ī„ģī…‹ī„Šī…‚ ī„ ī…‚ī„ģī…„ī„Ŧī…‚)
2. ī„ĸī…‡ī„›ī„ĩ p = 2:
q = 8 – 2(2) = 8 – 4 = 4ķˇƒ†ķŧŊĸ
ķˇƒ†ķŧŊĸī„‘ī…œī…ƒī„Ēī„ŗ ī„ēī…œī„Ēī……ī„ļī„Ž:
â€ĸ ī„›ī…„ī„ŗī„Ē (p) = 2
â€ĸ ī„ŗī…‚ī„Ēī„ĩī…‚ (q) = 4
ķˇ‡´ķˇ‡ĩķˇ‡ļķˇ‡ˇķˇ‡¸ķˇ‡šī„´ī…‚ī„Ŧ ī„ĩī…žī„œī„Š ī„¸ī…‚ī„ļī…„ ī„“ī…žī„› ī„›ī„ģī…‚ī„Šī…„ - ī„ŗī…‚ī„ĩī„›ī…„ī„Ĩ ī„°ī…‡ī„‘ī„ĩ ī„ēī„ŗī…‚ī„Žī„Ēī…‚:
ī…ƒī„¯ī…œī„§ ī„Ŧī…‡ ī…žī„§ī…‡ ī„ąī…‚ī„žī…‚ī„ĩ ī„ŗī…‡ī„ļī…‡ ī…ƒ ī…žī„ , ī…‡ī„ ī„Š ī…‚ī„ļī…‚ ī„“ī…žī„› ī„›ī…„ī„ŗī„Ē p ī…ƒī„Žī„ĩī„­ī…‚ī„ĩī„Ē ī„›ī„ĩī„Ŧī…‚ ī„ģī…‰, ī„‘ī„Ēī…‡ ī„ļī…‹ī„› ī„•ī„ē ī„›ī…„ī„ŗī„Ē 'ī„Ēī…‡ q
ī„ŗī…‚ī„Ēī„ĩī…‚ ī„Ŧī…„ ī„ŗī…œī„ ī„›ī„ĩī„Ŧī…‡ ī„ģī„ŽîĨĄ ī„¯ī„ĩ ī„“ī…žī„› ī„ī……ī„¯ī„Ē ī…ƒī„Žī„´ī„ŗ ī„ģī…‰: ī„›ī…„ī„ŗī„Ē ī„‘ī„Ēī…‡ ī„ŗī…‚ī„Ēī„ĩī…‚ ī„Žī…†ī…œ ī„Ŧī…‹ ī„ąī…žī…ƒī„ ī„’ī„ ī…‚ī„ī„ ī„ēī…œī„Ēī……ī…ƒī„ļī„Ē
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„›ī„ĩī„Žī…‚ ī„ ī…‚ī„ģī…„ī„Ŧī…‚ ī„ģī…‰îĨĄ ī„ĸī…‡ī„›ī„ĩ ī„›ī…„ī„ŗī„Ē ī„ąī„ģī……ī„Ē ī…ƒī„žī„’ī„Ŧī…‚ ī„ģī…‰, ī„Ēī…‚ī„ ī„›ī…‹ī„” ī…„ ī…ƒī„žī„’ī„Ŧī…‚ ī„Žī„ģī…„
ī„
ī„œī„ĩī…„ī„Ŧī„Ŧī…‚; ī„ĸī…‡ī„›ī„ĩ ī„ŗī…‚ī„Ēī„ĩī…‚ ī„ąī„ģī……ī„Ē
ī…ƒī„žī„’ī„Ŧī…‚ ī„ģī…‰, ī„Ēī…‚ī„ ī…‡ī„ ī„Š ī…‚ī„ļī…‚ ī„¯ī…‰ī„ēī…‡ ī„ī……ī„’ ī…ƒī„Ŧī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„’ī„ĩī…„ī„‘ī„Ž ī„Žī…†ī…œ ī„“ī„ģ ī…ƒī„Žī„´ī„ŗ ī„´ī…‚ī„Ŧ ī„’ī„“ī„’ ī„‘ī„Ēī…‡ ī„•ī„ēī„Žī…‡ ī…ƒī„Ŧī…žī„Ēī…‡ ī„ī„— ī„Ŧī…‹ ī„ēī„ŗī…„ī„›ī„ĩī„Žī…‚ī„ ī„Ŧī…„ ī„ĩī„Ēī…Œ ī„›ī„ĩī„›ī…‡ ī„œī……ī„Ŧ ī„ēī…œī„Ēī……ī„ļī„Ž
ī„‘ī„žī„ŗī…‚ī„•ī„Š ī„Ŧī…‚ ī„°ī…‰ī„ēī„ļī…‚ ī„›ī…„ī„Ēī…‚îĨĄ ī„•ī„ēī„Žī…‡ ī„ēī„ŗī…„ī„›ī„ĩī„Žī…‚ī„ ī„Žī…‚ī„ļ ī„Ąī…‡ī„ŋī„Ąī…‚ī„ŋ ī„›ī…„ī„Ēī…„, ī…žī„œ- ī…žī„œ ī„ŗī……ī…žī„ļī…‚ī„ ī„Ŧī…„ ī„›ī…‹ī…ƒī„šī„š ī„›ī…„ī„Ēī…„, ī„‘ī„Ēī…‡
ī„‘ī…œī„Ē ī…ƒ ī…žī„ , ī„ĸī„Ŧī…Œ ī„•ī„ģ p = 2 ī„‘ī„Ēī…‡ q = 4 'ī„Ēī…‡ ī„¯ī„ģī……ī…œī…ƒī„ ī„’, ī„Ēī…‚ī„ ī„Ŧī…‹ ī…ˆ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„“ī…žī„› ī…ƒī„ąī„ļī„›ī……ī„ļ ī„ąī„ĩī…‚ī„ąī„ĩ ī„’ī„ĩī…‡ ī…‚ī„ī„
ī„ēī…œī„Ēī……ī…ƒī„ļī„Ē ī„ģī…‹ ī„ī„—îĨĄ ī„•ī„Ŧī…Œ ī„ģī…„ ī„•ī„ģ ī„ŗī……ī„ēī„›ī„ĩī…‚ī„“ī„’îĨĄ "ī„ēī…œī„Ēī……ī„ļī„Ž ī„¯ī…Ĩī…‚ī„¯ī„Ē ī„ģī…‹ī„“ī„’," ī„•ī„ēī„Žī…‡ ī„°ī……ī„ēī„°ī……ī„ēī…‚ī„“ī„’îĨĄ
ķŊ„ĄķŊ„ĸķŊ„ŖķŊ„¤ķŊ„ĨķŊ„Ļī„‘ī…œī…ƒī„Ēī„ŗ ī„•ī…žī„Ēī„ĩī…‚ī„ ī„Ŧī…‚ ī„ēī…‚ī„ĩ:
ķš°¤ķš°Ĩķš°Ļķš°§ķš°¨ī„ī…ƒī„Šī„Ē ī„Ŧī…„ ī„›ī„ģī…‚ī„Šī…„ ī„Ŧī…‚ ī„Žī…‰ī…ƒī„Ēī„› ī…ƒī„ēī„­ī…‚ī„ī„Ē:
ī„ģī„ĩ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„“ī…žī„› ī„ąī……ī„Ŗī…‚ī„ĩī„Ē ī„Ŧī…‡ ī„Ĩī……ī„›ī„ŋī…‡ ī…‚ī„ī„ ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰îĨĄ ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī„•ī„ģī„Žī…‚ī„ ī„ēī…‚ī…ƒī„ĩī„’ī„ ī„Žī…†ī…œ ī„“ī…žī„›ī…‹ ī…‚ī„ĩ ī…ƒ ī…žī„  ī…ƒī„°ī…žī„Ĩ ī„›ī„ĩī„Ž
ī„ļī„” ī„ĸī„ļī„Ŧī„ąī…‚ī„žī…„ ī„›ī„ĩī„Ž ī„Ŧī…„ ī„ļī…‹ī„ŋ ī„Žī„ģī…„
ī„
ī„ģī…‰îĨĄ ī„“ī…žī„› ī„§ī…†ī…œī„žī…‚ ī„ēī…‚ī„ģ ī„ļī„™, ī„“ī…žī„› ī„ēī„ŗī…ˆ ī…ƒ ī…žī„  ī„“ī…žī„› ī„›ī„Ŧī„ŗ ī„ ī……ī…žī„›ī„Š ī„Ŧī…„ ī„›ī…‹ī…ƒī„šī„š
ī„›ī„ĩī…‹, ī„‘ī„Ēī…‡ ī„ēī„˛ ī„Ēī…Œ ī„ŗī„ģī…žī„Ē ī„¯ī…†ī„ĩī„Ž, ī„“ī„ēī„Žī…†ī…œ ī„ģī…žī„ļ ī„›ī„ĩī„Ž ī„Ŧī…‡ ī„ēī…€ī„ĩ ī„Ŧī…‚ ī„’ī„Žī…œ ī„Ŧ ī„ŗī…‚ī„Šī…‹îĨĄ ī„˛ī…‚ ī…ˆ ī„“ī„ģ ī„‘ī„ļī„ĸī„ąī„ĩī…‚ ī„Žī…†ī…œ ī„ģī…žī„ļ
ī„›ī„ĩī„Žī…‚ ī„ģī…‹ ī…‡, ī„ ī„Ēī……ī„ĩī„˛ī……ī„ĸ ī„›ī…‹ī„§ī…‚ī„ ī„Žī…†ī…œ ī„Ēī…‹ī„ŋī„Žī…‚ ī„ģī…‹ ī…‡, ī„ĸī…‚ī„ ī„ąī…‚ī„žī…‚ī„ĩ ī„Ŧī…„ī„’ī„ ī„ŗī…œī„ī…‚ī„ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Šī…‚ ī„ģī…‹ ī…‡ - ī„ĸī„Ŧī…Œ ī„Ēī……ī„ēī…„
ī„
ī„“ī„ēī„Žī…†ī…œ ī„’ī„¯ī„Šī…‚
ī„Ŧī…‹ī„ēī„Ē ī„ąī„Šī…‚ī„•
ī„
ī„Ŧī…‡ ī„ģī…‹ ī„Ēī…‚ī„ ī„ī…ƒī„Šī„Ē ī„ŗī„žī…‡ī„Ŧī…‚ī„ĩ ī„ąī„Š ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„ĸī…‡ī„›ī„ĩ ī„’ī„ĩī…„ī„‘ī„Ž, ī„ąī…‡ī„ļī…‚ ī„‘ī„Ēī…‡ ī„žī…‹ī„“ī„’ ī„“ī„ģ ī„ąī…‹ī„ģī„ŋ ī„Ŧī…‡ ī„Ŧī„ĩī…žī„œī„Ē ī„ģī…‡ī„Ļ ī„›ī„ĩ ī„ēī„›ī„Ŧī…‡ ī„ģī„Ž, ī„Ēī…‚ī„ ī„Ēī……ī„ēī…„
ī„
ī…„ ī„›ī„ĩ ī„ēī„›ī„Ŧī…‡ ī„ģī…‹ -
ī„Ģī…‹ī„ŋī…§ī…„ ī…ƒī„ĸī„ģī…„ ī„•ī„Ēī„ēī……ī„›ī„Ēī…‚ ī„‘ī„Ēī…‡ ī„šī…‚ī„ī„Ēī…„ ī„Žī…‚ī„ļîĨĄ
2. (i) 3 ī„‘ī„Ēī…‡ 24 ī„Ŧī…‡ ī…ƒī„¸ī„ ī„›ī…‚ī„ĩ 6 ī„‘ī…œī„›ī„ī…ƒī„Šī„Ē ī„Ŧī…‡ ī„ēī…‚ī„­ī„Ž ī„¯ī…‚ī„™îĨĄ
(ii) ī„ļī„ŋī…„ ī„Ŧī…‚ ī…ƒī„›ī„ģī„ŋī…‚ ī„¯ī„Ŧ 1 + 1/2 + 1/4 + 1/8 1/512
(iii) ī„ļī„ŋī…„ ī„Ŧī…‡ n ī„¯ī„Ŧī…‚ī„ ī„Ēī…žī„› ī„Ŧī…‚ ī„ĸī…‹ī„ŋ ī„¯ī„Ēī…‚ ī„›ī„ĩī…‹îĨĄ
ī„•ī…žī„Ēī„ĩ:ķˇ‡´ķˇ‡ĩķˇ‡ļķˇ‡ˇķˇ‡¸ķˇ‡šī„ĸī…‹ī„ŋ ī„šī……ī„ĩī…† ī„ģī…‹ī„Š ī„Ēī…Œ ī„¯ī…ƒī„ģī„ļī…‚ī„ ī„Ŧī…„ ī„“ī…žī„› ī„›ī„ģī…‚ī„Šī…„...
ī„“ī…žī„› ī…‚ī„ĩ ī„Ŧī…„ ī„ī…žī„ļ ī„ģī…‰, ī„ŗī…‰ī„Ģī…‡ī„ŗī…‰ī…ƒī„ĸī„›ī…‚ ī„Žī…‚ī„ŗī„› ī„“ī…žī„› ī…ƒī„¯ī…œī„§ ī…ƒ ī…žī„ , ī„’ī„ĩ ī„Žī…‚ī„ŗ ī„Ŧī…‚ ī„“ī…žī„› ī„•ī„Ēī„ēī……ī„› ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ ī„ĩī…ƒī„ģī…œī„Ŧī…‚
ī„ēī…„îĨĄ ī„•ī„ēī„Žī…†ī…œ ī„ĩī„ģī…žī„ēī…‚ī„ ī„Žī…†ī…œ ī„ēī……ī„ĩī…‚ī„ī…‚ī„ ī„Žī…‚ī„ļ ī„Žī„ģī…„
ī„
, ī„ēī„ī…Œ ī„‘ī…œī„›ī…ƒī„ŋī„’ī„ ī„Žī…‚ī„ļ ī„ģī…žī„ļ ī„›ī„ĩī„Žī…‚ ī„ąī„ģī……ī„Ē ī„¯ī„ēī…œī„Ŧ ī„ēī…„îĨĄ ī„“ī…žī„› ī…ƒī„Ŧī„Ž, ī„•ī„ēī„Ŧī…‡
ī„‘ī…ƒī„­ī„’ī„¯ī„› ī„Žī…‡ ī„•ī„ēī„Žī…†ī…œ ī…ƒī„Ēī…œī„Ž ī„¯ī„ģī…‡ī„ļī…„ī„’ī„ ī…ƒī„ļī„œī…„ ī„ģī…‹ī„” ī„“ī…žī„› ī„ēī„›ī…Ĩī…ī„ļ ī…ƒī„Ŧī…žī„Ēī…„îĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
"ī„ĸī…‡ ī„Ēī……ī„ēī…„
ī„
ī„“ī„ģī„Žī…‚ī„ ī„Žī…†ī…œ ī„ģī…žī„ļ ī„›ī„ĩ ī„ļī…Šī„Ŧī…‡ ī„ģī…‹," ī„‘ī…ƒī„­ī„’ī„¯ī„› ī„ŗī……ī„ēī„›ī„ĩī…‚ī„“ī„’, "ī„Ēī……ī„ēī…„
ī„
ī„‘ī…œī„›ī„ī…ƒī„Šī„Ē ī„‘ī„Ēī…‡ ī…ƒī„ĸī„™ī„ŗī…‰ī…ƒī„Ĩī…Ĩī„› ī„ļī„ŋī…„ ī„Ŧī…‡
ī„ŗī…‚ī„ēī„Ĩī„ĩ ī„ąī„Š ī„ĸī…‚ī„™ī„ī…‡!"
ī„•ī„Ēī„šī…‚ī…ƒī„ģī„Ē ī„ģī…‹ ī„›ī…‡, ī„’ī„ĩ ī„Žī…‡ ī„ēī„›ī…Ĩī…ī„ļ ī„œī…‹ī…ƒī„ļī…§ī„’îĨĄ ī„’ī„™ ī„•ī„ēī„Ŧī…‡ ī„Žī…‚ī„ļ ī„ ī…žī„ļī…„ī„— ī…ƒī„›ī„•
ī„
ī…ƒī„› ī„•ī„ģ ī„ģī„ĩ ī„ēī„ŗī…žī…ƒī„ēī„’ ī„Žī…†ī…œ ī„›ī„Ŧī„ŗ-
ī„Ŧī„ĩ-ī„›ī„Ŧī„ŗ ī„ģī…žī„ļ ī„›ī„ĩī„Ŧī…‚ ī„ģī…‰, ī„“ī„ēī„Žī…†ī…œ ī„“ī…žī„› ī„›ī…œī„ŗ ī„Ŧī…„ ī„ąī„ĸī…‚ī„— ī„“ī…žī„› ī„´ī…‚ī„Ēī„ĩī…‚ ī…‚ī„ī„ ī„ŗī…ƒī„ģī„ēī…†ī„ē ī„›ī„ĩ ī…‚ī„•
ī„
ī„Ŧī…‚ ī„ģī…‰îĨĄ
ķš´Ąķš´ĩķš´Ŗķš´¤ī„ąī……ī„Ŗī…‚ī„ĩī„Ē 1: 3 ī„‘ī„Ēī…‡ 24 ī„Ŧī…‡ ī…ƒī„¸ī„ ī„›ī…‚ī„ĩ 6 ī„‘ī…œī„›ī„ī…ƒī„Šī„Ē ī„Ŧī…‡ ī„ēī…‚ī„­ī„Ž ī„¯ī…‚ī„™îĨĄ
ķš¸¯ķš¸­ķš¸Žī„ēī„¸ī…‚ī„ļ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Šī…‚
ī„ēī…‚ī„Žī…†ī…œ ī„Ŧī…žī…ƒī„ēī„’ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰:
â€ĸ ī„¯ī…ƒī„ģī„ļī…‚ ī„¯ī„Ŧ (a) = 3
â€ĸ ī„’ī„œī„ĩī…„ ī„Ĩī„ĩī„ŗ = 24
â€ĸ ī„ēī…‚ī„Žī…†ī…œ 6 ī„ī…ƒī„Šī„Ē ī„Ŧī…‡ ī„ēī…‚ī„­ī„Ž, ī„˛ī…‚ , 3 ī„‘ī„Ēī…‡ 24 ī„Ŧī…‡ ī…ƒ ī„ ī„›ī…‚ī„ĩ ī„Ąī…‡ ī„ēī…œī…ƒī„œī„’ ī…‚ī„ ī„¯ī…‚ī„•ī„Š ī„Ŧī…„ ī„ļī…‹ī„ŋ ī„ģī…‰ ī„Ēī…‚ī„ ī„ĸī…‹ ī„¯ī…†ī„ĩī…‚
ī„›ī…Ĩī„ŗ ī„‘ī…œī„›ī„ī…ƒī„Šī„Ē ī„¯ī…Ĩī„ī„Ēī…„ (AP) ī…ƒ ī…žī„  ī„ģī…‹ ī…‡îĨĄ
ī„“ī„ē ī„ļī„”, ī„›ī……ī…žī„ļ ī„¯ī„Ŧī…‚ī„ ī„Ŧī…„ ī…ƒī„ī„Šī„Ēī…„ ī„ģī…‹ī„¸ī…‡ī„ī…„:
ķˇĩģķˇĩŧķˇĩŊķˇĩž3 (ī„šī……ī„ĩī…†), ī…ƒī„°ī„ĩ 6 ī„Ŧī…‚ ī„ŗī„Ēī„ļī„ą ī„ģī…‰, ī…ƒī„°ī„ĩ 24 (ī„‘ī…œī„Ē) ī„“ī„ģ AP ī…ƒ ī…žī„  ī„›ī……ī…žī„ļ 8 ī„šī„ąī„Ŧ ī„ģī„ŽîĨĄ
ķš¸Ŋī„‘ī…œī„›ī„ī…ƒī„Šī„Ē ī„¯ī…Ĩī„ī„Ēī…„ ī„›ī…„ ī„ģī…‰?
AP ī…ƒ ī…žī„ , ī„ģī„ĩī…‡ī„› ī„ēī…œī…ƒī„œī„’ ī„“ī…žī„› ī…ƒī„Žī„šī…ƒī„ ī„Ē ī„ŗī……ī…žī„ļ ī„Žī…‚ī„ļ ī„­ī„Ŧī…„ ī„ģī…‰ ī…ƒī„ĸī„ēī„Žī…†ī…œ ī„’ī„ŗ ī„‘ī…œī„Ēī„ĩ (d) ī…ƒī„›ī„ģī…‚ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ AP ī„Ŧī…‚
ī„’ī„ŗ ī„ĩī…†ī„¯:
a, a + d, a + 2d, a + 3d, ..., a + (n - 1)d
ķŧ¨ķŧ¨‘ķŧ¨’ī„ąī……ī„Ŗī…‚ī„ĩī„Ē ī„ģī…žī„ļ ī„›ī„ĩī„Žī…‚
ī„‘ī„ēī…„
ī„
ī„ĸī…‚ī„Šī„Ŧī…‡ ī„ģī…‚ī„:
â€ĸ ī„¯ī…ƒī„ģī„ļī…‚ ī„¯ī„Ŧ, a = 3
â€ĸ ī„¯ī„Ŧī…‚ī„ ī„Ŧī…„ ī…ƒī„ī„Šī„Ēī…„, n = 8
â€ĸ ī„’ī„œī„ĩī…„ ī„Ĩī„ĩī„ŗ (8 ī…‚ī„ ī„Ĩī„ĩī„ŗ) = 24
ī„‘ī„ēī…„
ī„
ī„“ī…žī„› AP ī„Ŧī…‡ ī„Žī… ī…ˆ ī„¯ī„Ŧ ī„ļī„” ī„°ī…‚ī„ĩī„ŗī…†ī„ļī…‚ ī„ĩī„Ēī„Ŧī…‡ ī„ģī…‚ī„:
ī„ŗī……ī…žī„ļ ī„ļī„ī…‚ī„•ī„Šī…‚:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„ģī……ī„Š ī„‘ī„ēī…„
ī„
ī…ƒī„¸ī„ ī„›ī…‚ī„ĩī„ļī…‡ 6 ī„šī„ąī„Ŧ ī„ļī…žī„˛ ī„ēī„›ī„Ŧī…‡ ī„ģī…‚ī„:
â€ĸ ī„Ŧī…†ī„ĸī…‚ ī„¯ī„Ŧ: a+d=3+3=6
â€ĸ ī„Ēī…„ī„ĸī…‚ ī„Ĩī„ĩī„ŗ: 3+23=9
â€ĸ ī„ ī…ī„Ģī…‚ ī„Ĩī„ĩī„ŗ: 3+33=12
â€ĸ 5 ī…‚ī„ ī„Ĩī„ĩī„ŗ: 3+43=15
â€ĸ ī„Ąī…‡ ī…‚ī„ ī„Ĩī„ĩī„ŗ: 3+53=18
â€ĸ 7 ī…‚ī„ ī„Ĩī„ĩī„ŗ: 3+63=21
ķˇƒ†ķŧŊĸī„“ī„ē ī„ļī„”, ī„¯ī…†ī„ĩī…‚ ī„›ī…Ĩī„ŗ ī„“ī„ģ ī„ąī„Šī„Ŧī…‚ ī„ģī…‰:
3, 6, 9, 12, 15, 18, 21, 24
ķˇ“ ķˇ“Ąķˇ“ĸķˇ“Ŗķˇ“¤ķˇ“Ĩķˇ“¨ķˇ“Šķˇ“Ēķˇ“Ģķˇ“Ļķˇ“§ķˇ“Ŧī„¯ī„ģī…‡ī„ļī…„ 1 ī„ģī…žī„ļ ī„ģī…‹ ī„ī„”!
ķš´Ąķš´ĩķš´Ŗķš´¤ī„ąī……ī„Ŗī…‚ī„ĩī„Ē 2: ī„ļī„ŋī…„ ī„Ŧī…‚ ī…ƒī„›ī„ģī„ŋī…‚ ī„šī„ąī„Ŧ
1 + 1/2 + 1/4 + 1/8 + ... = 1/512
ī„“ī„ģ ī„“ī…žī„› ī…ƒī„ĸī„™ī„ŗī…‰ī…ƒī„Ĩī…Ĩī„› ī„¯ī…Ĩī…‹ī„ī„ĩī…‰ī„ēī„Ž (GP) ī„ģī…‰, ī„ī…ƒī„Šī„Ē ī„Žī„ģī…„
ī„
îĨĄ
ķš¸¯ķš¸­ķš¸Žī„ēī„¸ī…‚ī„ļ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Šī…‚
ī…ƒī„Ŧī…žī„Ēī…‚ ī…ƒī„ī„’:
â€ĸ ī„¯ī…ƒī„ģī„ļī…‚ ī„¯ī„Ŧ (a) = 1
â€ĸ ī„’ī„ŗ ī„‘ī„Žī……ī„¯ī…‚ī„Ē (r) = 1/2
â€ĸ ī„ēī…‚ī„Žī…†ī…œ ī„Ŧī…žī…ƒī„ēī„’ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰ ī…ƒī„› ī„Žī… ī…‚ī„ ī„¯ī„Ŧ 1/512 ī„ģī…‰, ī„‘ī„Ēī…‡ ī„ēī…‚ī„Žī…†ī…œ ī„¯ī……ī…žī…ƒī„Ąī„’ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰ ī…ƒī„› ī„“ī„ģ ī…ƒī„›ī„ģī„ŋī…‚ ī„¯ī„Ŧ ī„ģī…‰?
ķŧ¨ķŧ¨‘ķŧ¨’ī„ąī……ī„Ŗī…‚ī„ĩī„Ē ī„ģī…žī„ļ ī„›ī„ĩī„Žī…‚
GP ī…ƒī„¸ī…žī„ , nī„¸ī…‚ī„ ī„¯ī„Ŧ ī„ģī…‰:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„ŗī……ī…žī„ļ ī„ļī„ī…‚ī„•ī„Šī…‚:
ī„Ēī…‚ī„ ī„ēī…‚ī„§ī…‡ ī„›ī…‹ī„ļ ī„ģī…‰:
ī„ģī……ī„Š 512 ī„Žī…†ī…œ 2 ī„Ŧī…„ ī„žī…‚ī„Ē ī„¸ī„ĸī…Œ ī…ƒī„ļī„œī…‹:
ī„“ī„ē ī„ļī„”,
ī„“ī„ē ī„ļī„”,
n−1=9n=10
ķˇƒ†ķŧŊĸī„“ī„ē ī„ļī„”, 1/512 GP ī„Ŧī…‚ 10 ī…‚ī„ ī„›ī…‚ī„ĩī„ĸī„›ī…‚ī„ļ ī„ģī…‰îĨĄ
ķˇ“ ķˇ“Ąķˇ“ĸķˇ“Ŗķˇ“¤ķˇ“Ĩķˇ“¨ķˇ“Šķˇ“Ēķˇ“Ģķˇ“Ļķˇ“§ķˇ“Ŧī„ąī……ī„Ŗī…‚ī„ĩī„Ē 2 ī„ģī…žī„ļ ī„ģī…‹ ī„ī„”!
ķš´Ąķš´ĩķš´Ŗķš´¤ī„ąī……ī„Ŗī…‚ī„ĩī„Ē 3: ī„ļī„ŋī…„ ī„Ŧī…‡ n ī„Ēī…žī„› ī„Ŧī…‡ ī„šī„ąī„Ŧī…‚ī„ ī„Ŧī…‚ ī„ĸī…‹ī„ŋ ī„ļī…žī„˛ī…‹îĨĄ
ī„ŗī…œī„Ž ī„ļī„™ ī…ƒī„› ī„ē ī…‚ī„ļ ī„¯ī…ƒī„ģī„ļī…‚ī„ ī…‚ī„ī„ ī„ģī…„ ī…ƒī„ĸī„™ī„ŗī…‰ī…ƒī„Ĩī…Ĩī„› ī„ļī„ŋī…„ ī„Ŧī…‡ ī„ĸī…‹ī„ŋ ī„ļī„” ī„¯ī……ī…žī„Ą ī…ƒī„ĩī„ģī…‚ ī„ģī…‰:
ķš¸¯ķš¸­ķš¸Žī„ēī…œī„›ī„ļī„¯ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Šī…‚
ī„“ī„ģ ī…ƒī„°ī„ĩ ī„“ī…žī„› ī„ĸī…„ī„¯ī…„ ī„ģī…‰, ī…ƒī„ĸī„ēī„Ŧī…‡ ī„Žī…‚ī„ļ:
â€ĸ ī„¯ī…ƒī„ģī„ļī…‚ ī„¯ī„Ŧ, a=1a = 1a=1
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
â€ĸ ī„’ī„ŗ ī„‘ī„Žī……ī„¯ī…‚ī„Ē, r=
ī„ĩ
ī„ļ
ī„“ī…žī„› GP ī„Ŧī…‡ ī„¯ī…ƒī„ģī„ļī…‡ nnn ī„¯ī„Ŧī…‚ī„ ī„Ŧī…‚ ī„ĸī…‹ī„ŋ ī„“ī„ģ ī„ģī…‰:
ķŧ¨ķŧ¨‘ķŧ¨’ī„ąī……ī„Ŗī…‚ī„ĩī„Ē ī„ģī…žī„ļ ī„›ī„ĩī„Žī…‚
ī„“ī„ē ī„ļī„”, ī„ļī„ŋī…„ ī„Ŧī…‡ ī„¯ī…ƒī„ģī„ļī…‡ nnn ī„¯ī„Ŧī…‚ī„ ī„Ŧī…‚ ī„ĸī…‹ī„ŋ ī„“ī„ģ ī„ģī…‰:
ī„’ī„™ ī„“ī„ēī„Žī…†ī…œ n ī„Ŧī…‡ ī„›ī……ī„Ŗ ī„ŗī……ī…žī„ļī…‚ī„ ī„Žī…‚ī„ļ ī„‘ī„žī„ŗī…‚ī„”ī„—:
â€ĸ n = 1 ī„ļī„”:
S1=2(1−1/2)=2(1/2)=1
â€ĸ n = 2 ī„ļī„”:
S2=2(1−1/4)=2(3/4)=1.5
â€ĸ n = 3 ī„ļī„”:
S3=2(1−1/8)=2(7/8)=1.75
ķˇƒ†ķŧŊĸī„°ī…‚ī„ĩī„ŗī…†ī„ļī…‚ ī„¯ī…Ĩī„ŗī…‚ī…ƒī„Šī„Ē!ķˇ“ ķˇ“Ąķˇ“ĸķˇ“Ŗķˇ“¤ķˇ“Ĩķˇ“¨ķˇ“Šķˇ“Ēķˇ“Ģķˇ“Ļķˇ“§ķˇ“Ŧ
ķŧ¨ģķŧ¨ŧī„’ī„ĩī„¸ ī„Ŧī…‚ ī„‘ī…œī…ƒī„Ēī„ŗ ī…ƒī„¸ī„ ī…‚ī„ĩ...
ī…ƒī„ĸ ī…ˆ ī„ģī…„ ī„’ī„ĩ ī„Žī…‡ ī„ēī„›ī…Ĩī…ī„ļ ī„ąī…œī„Ŧ ī„›ī…„ī„Ēī…‚, ī„•ī„ģ ī„ŗī……ī„ēī„›ī„ĩī…‚ī„“ī„’îĨĄ "ī„“ī„ģ ī„¯ī„ģī…‡ī„ļī…„ī„’ī„ ī…ƒī„ēī„ĩī…€ ī„°ī…‚ī„ĩī„ŗī…†ī…ƒī„ļī„’ī„ ī„ąī…‚ī„ĩī…‡ ī„Žī„ģī…„
ī„
ī„ēī„Ž," ī„•ī„ēī„Žī…‡ ī„°ī……ī„ēī„°ī……ī„ēī…‚ī„“ī„’, "ī„“ī„ģ ī„¯ī…‰ī„Ĩī„ĩī„Žī…‚ī„, ī„Ēī„ĩī„› ī„‘ī„Ēī…‡ ī„•ī„ģī„Žī…‚ī„ ī„Žī…†ī…œ ī„Ēī…‹ī„ŋī„Ž ī„Ŧī…„ ī„ŗī„ēī„Ēī…„ ī„ąī…‚ī„ĩī…‡ ī„ēī„Ž!"
ī„‘ī„Ēī…‡ ī…ƒī„¯ī„’ī„ĩī…‡ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„™, ī„“ī„ģ ī…„ ī„ĩī…‚ī„ž ī„ģī…‰â€”ī„ī…ƒī„Šī„Ē ī„°ī…‚ī„ĩī„ŗī…†ī…ƒī„ļī„’ī„ ī„Žī…†ī…œ ī„´ī…‚ī„Ŧ ī„›ī„ĩī„Ž ī„ąī…‚ī„ĩī…‡ ī„Žī„ģī…„
ī„
ī„ģī…‰îĨĄ ī„“ī„ģ ī„¯ī…‰ī„Ĩī„ĩī„Ž
ī„Žī…†ī…œ ī„¯ī„Ąī…‚ī„Šī„Ž, ī„›ī„ģī…‚ī„Šī…„ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Š ī„‘ī„Ēī…‡ ī„“ī„ēī„Žī…†ī…œ ī„ģī…žī„ļ ī„›ī„ĩī„Ž ī„Ŧī…‡ ī„ēī…€ī„ĩ ī„Ŧī…‚ ī„’ī„Žī…œ ī„Ŧ ī„ļī…‰ī„Š ī„ąī…‚ī„ĩī…‡ ī„ģī…‰îĨĄ
ķˇƒ†ķŧŊĸī„‘ī…œī…ƒī„Ēī„ŗ ī„ĸī„¸ī…‚ī„ą:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
1. 3 ī„‘ī„Ēī…‡ 24 ī„Ŧī…‡ ī…ƒī„¸ī„ ī„›ī…‚ī„ĩ ī„¯ī…‚ī„— ī„ī„— ī„‘ī…œī„›ī„ī…ƒī„Šī„Ē ī„Ŧī…‡ ī„ēī…‚ī„­ī„Ž:
6, 9, 12, 15, 18, 21
2. GP ī…ƒī„¸ī…žī„  ī…ƒī„›ī„ģī„ŋī…‚ ī„¯ī„Ŧ 1/512 ī„ģī…‰:
10 ī…‚ī„ ī„Ĩī„ĩī„ŗ
ī„˛ī…‚ī„-ī„ąī…„
3. (i) ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ ī…ƒī„› y = 5x - 7 ī„‘ī„Ēī…‡ 2y = 10x + 5 ī„ēī„ŗī…‚ī„ī„Ēī„ĩ ī„ģī„ŽîĨĄ
(ii) ī„•ī„ē ī„ēī…ˆī„Ĩ ī„ĩī…‡ī„œī…‚ ī„Ŧī…‚ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„ļī…žī„˛ī…‹ ī„ĸī…‹ (1, 3, 5) ī…ƒī„¸ī…žī„ ī…Œ ī„ļī…œī„žī„Ŧī…„ ī„ģī…‰ ī„‘ī„Ēī…‡ ī…ƒī„Žī„ĩī„Ŧī…‡ī„šī…‚ī„ī„› ī„­ī……ī…ƒī„ĩī„’ī„ 'ī„Ēī…‡ ī„“ī„ēī„Ŧī…‡
ī„‘ī…œī„Ēī„ĩī…‚ī„ļī…‚ī„ ī„Ŧī…‚ ī„ĸī…‹ī„ŋ 9 ī„ģī…‰îĨĄ
(ii) ī„Ŧī……ī…žī„­ ī„Ŧī…„ ī„ŗī…œī„ ī„“ī„ē ī„Ēī„ĩī…§ī…‚ī„ ī…ƒī„Ŧī…žī„Ēī…„ ī„ĸī…‚ī„ī„Ŧī…„ ī„ģī…‰:
ī„›ī…„ī„ŗī„Ē ī„ĩī……ī„¯ī„—/ī„ļī…„ī„Ĩī„ĩ
ī„ļī…„ī„Ĩī„ĩī…‚ī„ ī…ƒī„¸ī…žī„  ī„ŗī…œī„
1
100
2
50
3
0
ī„ĩī…‡ī…ƒī„œī„› ī„ŗī…œī„ ī„°ī…œī„›ī„šī„Ž ī„ļī…žī„˛ī…‹îĨĄ
ī„•ī…žī„Ēī„ĩ:ķˇ‡´ķˇ‡ĩķˇ‡ļķˇ‡ˇķˇ‡¸ķˇ‡šī„“ī…žī„› ī„Žī„¸ī…„
ī„
ī„šī……ī„ĩī…†ī„’ī„Ē: ī„Ŧī……ī…žī„­ ī…ƒī„¸ī„›ī„ĩī…‡ī„Ēī…‚ ī„Ŧī…‡ ī„ī…ƒī„Šī„Ē ī„Ŧī…„ī„’ī„ ī„ēī„ŗī…žī…ƒī„ēī„’ī„¸ī…‚ī„ ī„Ŧī…‚ ī„•ī„Ēī„ēī……ī„› ī„ŗī…‚ī„ŗī„ļī…‚
ī„“ī…žī„› ī„Ąī…‹ī„Ĩī…‡ ī…ƒī„ĸī„ģī…‡ ī„šī…ƒī„ģī„ĩ ī…ƒ ī…žī„ , ī„ĩī…‚ī„ĸī…† ī„Žī…‚ī„ŗ ī„Ŧī…‚ ī„“ī…žī„› ī„ ī„ļī…‚ī„› ī„Ŧī……ī…žī„­ ī…‡ī„ ī„Š ī…‚ī„ļī…‚ ī„ĩī…ƒī„ģī…œī„Ŧī…‚ ī„ēī…„îĨĄ ī„ĸī„Ŧī…Œ ī…ƒī„› ī„Ŧī…†ī„ēī„ĩī…‡ ī„Ŧī……ī…žī„­ ī…‡ī„ ī„Ŧī…‡
ī„ēī„Ž ī„‘ī„Ēī…‡ ī„ŗī…‹ī„Ĩī…‡ ī„Ēī…ī„ĩ 'ī„Ēī…‡ ī„›ī…„ī„ŗī„Ēī…‚ī„ ī„Ŧī…‚ ī…ƒī„ģī„ēī…‚ī„ą ī„ļī„ī…‚ī„•
ī„
ī„Ŧī…‡ ī„ēī„Ž, ī„ĩī…‚ī„ĸī…† ī„Žī…†ī…œ ī„‘ī…œī„› ī„‘ī„Ēī…‡ ī…ƒī„ĸī„™ī„ŗī…‰ī„Ĩī„ĩī…„ ī„ąī„ģī……ī„Ē ī„¯ī„ēī…œī„Ŧ ī„ēī…„îĨĄ ī„“ī…žī„›
ī…ƒī„Ŧī„Ž, ī„•ī„ēī„Ŧī…‡ ī„ĩī…‹ī„žī…‚ī„Žī…‚ ī„Ŧī…‡ ī„›ī…œī„ŗ ī„Žī…‚ī„ļ ī„ēī„ąī…œī„­ī„Ē ī…ƒī„Ēī…œī„Ž ī„ī…ƒī„Šī„Ē ī„Ŧī…‡ ī„ē ī…‚ī„ļ ī„•ī„ēī„Ŧī…‡ ī„ēī…‚ī„ģī„ŗī„Šī…‡ ī„’ī„—îĨĄ ī„•ī„Ēī„ēī……ī„› ī„‘ī„Ēī…‡
ī„•ī„Ēī„šī…‚ī…ƒī„ģī„Ē, ī„•ī„ēī„Žī…‡ ī„•ī„ģī„Žī…‚ī„ ī„Žī…†ī…œ ī„›ī„Ŧī„ŗ-ī„Ŧī„ĩ-ī„›ī„Ŧī„ŗ ī„ģī…žī„ļ ī„›ī„ĩī„Ž ī„Ŧī…‚ ī„°ī…‰ī„ēī„ļī…‚ ī„›ī…„ī„Ēī…‚îĨĄ ī„’ī„™ ī„ĩī…‚ī„ĸī…† ī„Ŧī…‡ ī„ēī…€ī„ĩ 'ī„Ēī…‡ ī„ ī…žī„ļī…„ī„—îĨĄ
ī„˛ī…‚ī„ (i): ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ ī…ƒī„› y = 5x - 7 ī„‘ī„Ēī…‡ 2y = 10x + 5 ī„ēī„ŗī…‚ī„ī„Ēī„ĩ ī„ģī„ŽîĨĄ
ī„ĩī…‚ī„ĸī…† ī„Žī…†ī…œ ī„´ī…‚ī„Ŧ ī„’ī„“ī„’ ī…ƒī„› ī„•ī„ēī„Ŧī…‡ ī„ēī„›ī…†ī„ļ ī„Ŧī…‡ ī„‘ī…ƒī„­ī„’ī„¯ī„› ī„Žī…‡ ī„“ī…žī„› ī„¸ī…‚ī„ĩ ī„•ī„ēī„Žī…†ī…œ ī…ƒī„›ī„ģī…‚ ī„ēī…„:
"ī„ĸī…‡ī„›ī„ĩ ī„Ŧī…‹ ī„ļī…‚ī„”ī„Žī…‚ī„ ī„Ŧī…„ ī„¨ī„ļī…‚ī„Š ī„“ī…žī„›ī…‹ ī…ƒī„ĸī„ģī…„ ī„ģī…‰, ī„Ēī…‚ī„ ī„•ī„ģ ī„ēī„ŗī…‚ī„Žī…‚ī„ī„Ēī„ĩ ī„ģī„ŽîĨĄ ī„•ī„ģ ī„›ī„Ŧī…‡ ī„Žī„ģī…„
ī„
ī…ƒī„ŗī„ļī„Šī„ī…„ī„’ī„, ī…ƒī„ĸ ī…ˆ ī„Ŧī…‹
ī„ĩī…‡ī„ļ ī„¯ī„Ĩī„ŋī…„ī„’ī„ ī„Žī…‚ī„ļ-ī„Žī…‚ī„ļ ī„ąī…‡ī„‘ī…œī„Ē ī„ ī…žī„ļī„Ŧī…„ī„’ī„ ī„ģī„ŽîĨĄ"
ī„’ī„™ ī„¯ī…ƒī„ģī„ļī…„ ī„ļī…‚ī„”ī„Ž ī„Žī…†ī…œ ī„ēī„ŗī„Ŗ ī„›ī…‡ ī„šī……ī„ĩī…†ī„’ī„Ē ī„›ī„ĩī…„ī„—:
ī„ļī…‚ī„”ī„Ž 1:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī…ƒī„Ŧī…žī„Ēī…‚ ī…ƒī„ī„’ ī„ēī„ŗī…„ī„›ī„ĩī„Ž:
y=5x−7y = 5x - 7y=5x−7
ī„“ī„ģ ī„¯ī…ƒī„ģī„ļī…‚ī„ ī„ģī…„ ī„¨ī„ļī…‚ī„Š-ī„ĩī…‹ī„›ī„Ģī…‚ī„ŗ ī„ĩī…†ī„¯ ī…ƒ ī…žī„  ī„ģī…‰:
y=mx+c
ī„“ī…žī„Ģī…‡, m=5m = 5m=5 ī„¨ī„ļī…‚ī„Ž ī„ģī…‰îĨĄ
ī„ļī…‚ī„”ī„Ž 2:
ī…ƒī„Ŧī…žī„Ēī…‚ ī…ƒī„ī„’ ī„ēī„ŗī…„ī„›ī„ĩī„Ž:
2y=10x+5
ī„¨ī„ļī…‚ī„Š ī„Ŧī…„ ī„Ēī……ī„ļī„Žī…‚ ī„›ī„ĩī„Ž ī„ļī„”, ī„ēī…‚ī„Žī…†ī…œ ī„“ī„ēī„Žī…†ī…œ ī„ēī„Ĩī…Šī„§ī„ĩī„§ ī„¨ī„ļī…‚ī„Š-ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„¯ī„Ĩ ī„ĩī…†ī„¯ ī…ƒ ī…žī„  ī„ąī„Ŧī„ļī„Š ī„Ŧī…„ ī„ļī…‹ī„ŋ ī„ģī…‰îĨĄ
ī„›ī„Ŧī„ŗ 1: ī„Ŧī…‹ī„¸ī…ˆ ī„¯ī…‚ī…ƒī„ēī„’ī„ ī„Žī…†ī…œ 2 ī„Žī…‚ī„ļ ī„¸ī…œī„§ī…‹:
y=5x+52
ī„ģī……ī„Š, ī„“ī„ģ ī„ēī„¯ī…žī„šī„Ĩ ī„Ēī…ī„ĩ 'ī„Ēī…‡ y=mx+c ī„ĩī…†ī„¯ ī…ƒ ī…žī„  ī„ģī…‰îĨĄ ī„“ī…žī„Ģī…‡ ī…ƒī„°ī„ĩ, ī„¨ī„ļī…‚ī„Ž (m) 5 ī„ģī…‰îĨĄ
ķˇƒ†ķŧŊĸī…ƒī„ēī…žī„Ĩī…‚:
ī…ƒī„›ī„•
ī„
ī…ƒī„› ī„Ŧī…‹ ī…ˆ ī„ĩī…‡ī„œī…‚ ī…‚ī„ ī„Ŧī…„ ī„¨ī„ļī…‚ī„Š ī„“ī…žī„›ī…‹ ī…ƒī„ĸī„ģī…„ ī„ģī…‰ (5) ī„‘ī„Ēī…‡ ī…žī„œ- ī…žī„œ ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„¯ī„Ĩ (−7 ī„‘ī„Ēī…‡ 2.5), ī„“ī„ģ ī„ēī„ŗī…‚ī„Žī…‚ī„ī„Ēī„ĩ
ī„ģī„ŽîĨĄ
ī„ĩī…‚ī„ĸī…† ī„Žī…‡ ī„ŗī……ī„ēī„›ī„ĩī…‚ī„“ī„’ ī„‘ī„Ēī…‡ ī…ƒī„›ī„ģī…‚, "ī„“ī…žī„›ī…‹ ī„ĩī„ēī„Ēī…‡ 'ī„Ēī…‡ ī„Ŧī…‹ ī…ƒī„§ī„ļī…„ ī„ĩī…„ ī„ŗī……ī…œī…ƒī„§ī„’ī„ ī…‚ī„ī„ ī„¯ī„ĩ ī…žī„œī…‹- ī…žī„œī„ĩī…‡ ī„ēī„ŗī…ˆ ī„Ŧī…‡ ī„ēī„ļī…‚ī„Ĩ -
ī„•ī„ģ ī„“ī…žī„›ī…‹ ī„ĩī„ēī„Ēī…‡ 'ī„Ēī…‡ ī„ ī…žī„ļī„Ŧī…‡ ī„ģī„Ž ī„¯ī„ĩ ī„›ī„Ŧī…‡ ī…„ ī„“ī…žī„› ī„Ŧī…†ī„ĸī…‡ ī„Žī…‚ī„ļ ī„Žī„ģī…„
ī„
ī„Ĩī„›ī„ĩī…‚ī„•
ī„
ī„Ŧī…‡!"
ī„˛ī…‚ī„ (ii): (1, 3, 5) ī„ĩī…‚ī„ģī…„
ī„
ī„“ī…žī„› ī…ƒī„ēī…žī„­ī…„ ī„ĩī…‡ī„œī…‚ ī„Ŧī…„ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„ļī…žī„˛ī…‹, ī„Ēī…‚ī„ ī„ĸī…‹ ī„“ī„ēī„Ŧī…‡ ī„‘ī…œī„Ēī„ĩī…‚ī„ļī…‚ī„ ī„Ŧī…‚ ī„ĸī…‹ī„ŋ 9 ī„ģī…‹ī„¸ī…‡îĨĄ
ī„ģī……ī„Š ī„ĩī…‚ī„ĸī…† ī„Ŧī…‡ ī…žī„§ī…‡ ī„ ī„ ī…‡ī„ĩī…‡ ī„˛ī„ĩī…‚, ī„ĸī…‹ ī„›ī…‚ī„ļī„ĸ ī…ƒ ī…žī„  ī„¯ī„ŋī…§ ī…ƒī„ĩī„ģī…‚ ī„ēī…„, ī„Žī…‡ ī„•ī„ēī„Žī…†ī…œ ī„¯ī……ī…žī…ƒī„Ąī„’:
"ī„›ī…„ ī„Ēī……ī„ēī…„
ī„
3D ī…ƒī„ĸī„™ī„ŗī…‰ī„Ĩī„ĩī…„ ī„Ŧī…„ ī„ēī„ŗī…žī…ƒī„ēī„’ ī…ƒ ī…žī„  ī„ŗī…‡ī„ĩī…„ ī„ŗī„Ŧī„Ŧ ī„›ī„ĩ ī„ēī„›ī„Ŧī…‡ ī„ģī…‹?"
ī„ĩī…‚ī„ĸī…† ī„Žī…†ī…œ ī„“ī…žī„› ī„ ī……ī„Šī…ī„Ēī…„ ī„ąī„ģī……ī„Ē ī„¯ī„ēī…œī„Ŧ ī„’ī„”!
ī„ēī…‚ī„Žī…†ī…œ ī…ƒī„Ŧī…žī„Ēī…‡ ī„ī„— ī„ģī„Ž:
â€ĸ ī„“ī…žī„› ī„ĩī…‡ī„œī…‚ ī…ƒī„ąī…œī„Ŧī…† ī…ƒ ī…žī„ ī…Œ ī„ļī…œī„žī„Ŧī…„ ī„ģī…‰ (1, 3, 5)
â€ĸ ī„›ī…‹ī„’ī„ĩī„§ī…„ī„Žī…‡ī„Ĩ ī„­ī……ī…ƒī„ĩī„’ī„ 'ī„Ēī…‡ ī„“ī„ēī„Ŧī…‡ ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„¯ī„Ĩī…‚ī„ ī„Ŧī…‚ ī„ĸī…‹ī„ŋ 9 ī„ģī…‰ ī„˛ī…‚ ,
a+b+c=9
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„’ī„™ ī„“ī„ēī„Žī…†ī…œ ī„›ī„Ŧī„ŗ ī„Ŧī„ĩ ī„›ī„Ŧī„ŗ ī„ģī…žī„ļ ī„›ī„ĩī…„ī„—îĨĄ
ķŧŠŽķŧŠķŧŠķŧŠ‘ķŧŠ’ķŧŠ“ķŧŠ”ī„“ī…žī„› ī„ēī„ŗī„Ēī„ļ (ī„ĸī…‚ī„ ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„¯ī„Ĩ ī„¸ī…‚ī„ļī…„ ī„ĩī…‡ī„œī…‚) ī„Ŧī…‡ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„Ŧī…‡ ī„ĩī…†ī„¯ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Šī…‚
ī„“ī…žī„› ī„ĩī…‡ī„œī…‚ ī„Ŧī…‚ ī„’ī„ŗ ī„ĩī…†ī„¯ ī„ĸī…‹ ī…ƒī„Žī„ĩī„Ŧī…‡ī„šī…‚ī„ī„› ī„­ī……ī…ƒī„ĩī„’ī„ ī„Žī…†ī…œ ī„ĩī…‹ī„›ī„Ŧī…‚ ī„ģī…‰:
ī…ƒī„›ī…žī„Ģī…‡:
â€ĸ a x-ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„¯ī„Ĩ ī„ģī…‰,
â€ĸ b, y-ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„¯ī„Ĩ ī„ģī…‰,
â€ĸ c z-ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„¯ī„Ĩ ī„ģī…‰
ī„ēī…‚ī„Žī…†ī…œ ī„Ŧī…žī…ƒī„ēī„’ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰:
â€ĸ ī„ĩī…‡ī„œī…‚ (1, 3, 5) ī…ƒ ī…žī„ ī…Œ ī„ļī…œī„žī„Ŧī…„ ī„ģī…‰îĨĄ
â€ĸ a+b+c=9
ķŧ¨ķŧ¨‘ķŧ¨’ī„›ī„Ŧī„ŗ-ī„Ŧī„ĩ-ī„›ī„Ŧī„ŗ ī„ąī„Ŧī„ļ
ī„ģī……ī„Š ī…ƒī„ąī…œī„Ŧī…† (1, 3, 5) ī„Žī…†ī…œ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī…ƒī„¸ī…žī„  ī„ąī„Ŧī„ļī…‹:
ī„Žī…‚ī„ļ ī„ģī…„, ī…ƒī„Ŧī…žī„Ēī…‡ ī„ī„— ī„Ēī…Œ:
ī„ģī……ī„Š ī„šī„œī…‚ ī…ƒī„ģī…žī„ēī…‚ ī„’ī„•
ī„
ī„Ŧī…‚ ī„ģī…‰: ī„“ī„ģī„Žī…‚ī„ ī„Ŧī…‹ ī„ēī„ŗī…„ī„›ī„ĩī„Žī…‚ī„ ī„Žī…†ī…œ ī…ƒī„Ēī…œī„Ž ī„‘ī„Šī„ĸī…‚ī„Šī…‚ī„ ī„Žī…‚ī„ļ ī„ģī…žī„ļ ī„›ī„ĩī„Žī…‚ ī…ƒī„ēī…žī„­ī…‡ ī„Ēī…ī„ĩ 'ī„Ēī…‡ ī„ēī…œī„˛
ī„Žī„ģī…„
ī„
ī„ģī…‰ ī„ĸī„Ŧī…Œ ī„Ēī…žī„› ī„‘ī„ēī…„
ī„
ī„œī…‚ī„ē ī„ģī…žī„ļ ī„ļī…žī„˛ī„Š ī„ļī„” ī„“ī…žī„› ī…‡ī„ĩī…„ī„—ī„ąī„ļ ī„Žī…†ī…œ ī„Žī„ģī…„
ī„
ī„ŗī…œī„Žī„Ŧī…‡îĨĄ
ķšēŠī„ŗī…œī„Ž ī„ļī„™ a=3a = 3a=3
ī„“ī„ē ī„ļī„”, ī„ēī„ŗī…„ī„›ī„ĩī„Ž 2 ī„Ŧī…„ ī„¸ī„ĩī„Ēī…Œ ī„›ī„ĩī„Ŧī…‡ ī„ģī…‹ī„—:
ī„ģī……ī„Š ī„“ī„ģī„Žī…‚ī„ ī„ŗī……ī…žī„ļī…‚ī„ ī„Žī…†ī…œ ī„ēī„ŗī…„ī„›ī„ĩī„Ž 1 ī…ƒ ī…žī„  ī„ąī„Ŧī„ļī…‹:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„“ī„ģ ī„“ī…žī„› ī„Ēī„ĩī„›ī„šī…„ī„ļ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„ģī…‰îĨĄ ī„’ī„™ ī„“ī„ēī„Žī…†ī…œ ī„ģī…žī„ļ ī„›ī„ĩī…„ī„—îĨĄ
ī„Ŧī…‹ ī…‚ī„ ī„¯ī…‚ī…ƒī„ēī„’ī„ ī„Žī…†ī…œ 3 ī„Žī…‚ī„ļ ī„ī……ī„Šī…‚ ī„›ī„ĩī…‹:
ī„ģī……ī„Š ī„Ŧī…‹ ī…‚ī„ ī„¯ī…‚ī…ƒī„ēī„’ī„ ī„Žī…†ī…œ b(6−b) ī„Žī…‚ī„ļ ī„ī……ī„Šī…‚ ī„›ī„ĩī…‹:
ī„ēī„˛ ī„›ī……ī„Ŗ ī„“ī…žī„› ī„¯ī…‚ī„ēī…‡ ī„›ī„ĩ ī…ƒī„Ŧī„™:
ī„“ī„ē ī„Ŧī…„ī„’ī„ ī„›ī…‹ī„” ī„‘ī„ēī„ļ ī„ĸī„ŋī…§ī…‚ī„ ī„Žī„ģī…„
ī„
ī„ģī„ŽîĨĄ ī„“ī„ē ī„ļī„” ī„ēī…‚ī„§ī…„ ī„­ī…‚ī„ĩī„Žī…‚ a=3a = 3a=3 ī„“ī…žī„› ī„ēī…‚ī…€ ī„ģī…žī„ļ ī„Žī„ģī…„
ī„
ī…ƒī„Ŧī…œī„Ŧī…„îĨĄ
🛠ī„“ī…žī„› ī„ģī…‹ī„ĩ ī„­ī…‚ī„ĩī„Žī…‚ ī„‘ī„žī„ŗī…‚ī„™: ī„ŗī…œī„Ž ī„ļī„™ a=6a = 6a=6
ī…ƒī„°ī„ĩ, ī„ēī„ŗī…„ī„›ī„ĩī„Ž 2 ī„Ēī…Œ:
ī„ēī„ŗī…„ī„›ī„ĩī„Ž 1 ī„ąī„Š ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰:
ī„Ŧī…‹ ī…‚ī„ ī„¯ī…‚ī…ƒī„ēī„’ī„ ī„Žī…†ī…œ 6 ī„Žī…‚ī„ļ ī„ī……ī„Šī…‚ ī„›ī„ĩī…‹:
ī„Ŧī…‹ī„ģī…‚ī„ ī„¯ī…‚ī…ƒī„ēī„’ī„ ī„Žī…†ī…œ b(3−b) ī„Žī…‚ī„ļ ī„ī……ī„Šī…‚ ī„›ī„ĩī…‹:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„“ī„ēī„Žī…†ī…œ ī„ģī…žī„ļ ī„›ī„ĩī„Ž ī„Ŧī…„ ī„›ī…‹ī…ƒī„šī„š ī„›ī„ĩī…‹:
ī„Ŧī……ī„ąī…‚ī„ĩī…‚ ī…ƒī„°ī„ĩ, ī„›ī…‹ī„” ī„‘ī„ēī„ļī…„ ī„ŗī…†ī„ļ ī„Žī„ģī…„
ī„
îĨĄ ī„“ī„ēī„Ŧī…‚ ī„ŗī„Ēī„ļī„ą ī„ģī…‰ ī…ƒī„› ī„‘ī„ēī…„
ī„
ī„ ī…œī„ī…‡ ī„Žī…œ ī„ąī„ĩ ī„¯ī…Ĩī…‚ī„¯ī„Ē ī„Žī„ģī…„
ī„
ī„›ī„ĩ ī„ēī„›ī„Ŧī…‡ ī„ĸī„Ŧī…Œ ī„Ēī…žī„›
ī„ģī…‹ī„ĩ ī„§ī…‡ī„Ĩī…‚ ī„Žī„ģī…„
ī„
ī…ƒī„Ŧī…žī„Ēī…‚ ī„ĸī…‚ī„ī„Ŧī…‚îĨĄ ī„“ī„ē ī„ļī„”, ī„‘ī„›ī„ēī„ĩ ī„‘ī…ƒī„ĸī„ģī…‡ ī„¯ī…Ĩī„šī„Žī…‚ī„ ī…ƒ ī…žī„ , ī„•ī„ģ ī„°ī…‚ī„ĩī„ŗ ī„Ŧī…„ ī„ĩī„Ēī…Œ ī„›ī„ĩī„Ŧī…‡ ī„ģī…‹ī„— ī„“ī…žī„›
ī„’ī„ŗ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„Ŧī…„ ī„•ī„ŗī…„ī„Ŧ ī„›ī„ĩī„Ŧī…‡ ī„ģī„Ž:
ī…ƒī„ĸī…žī„Ģī…‡ ī„Ēī……ī„ēī…„
ī„
a ī„‘ī„Ēī…‡ b ī„Ŧī…‡ ī„›ī…‹ī„” ī…„ ī„Ŧī…‹ ī„ŗī……ī…žī„ļ ī„ ī……ī„Š ī„ēī„›ī„Ŧī…‡ ī„ģī…‹ ī„‘ī„Ēī…‡ c ī„Žī…†ī…œ 9−a−b9 - a - b9−a−b ī„Ŧī…‡ ī„ĩī…†ī„¯ ī…ƒ ī…žī„  ī…ƒī„ī„Š ī„ēī„›ī„Ŧī…‡
ī„ģī…‹îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž ī„‘ī…œī…ƒī„Ēī„ŗ ī„•ī…žī„Ēī„ĩ: ī„ŗī…œī„Ž ī„ļī„™:
ī„Ēī…‚ī„ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„“ī„ģ ī„ģī…‰:
ī„‘ī„Ēī…‡ ī„“ī„ģ ī„ĸī„ģī…‚ī„ž ī„¯ī……ī„ļī…‚ī„ŋ ī„Ŧī…‡ ī…ƒī„›ī„ēī…‡ ī…ƒī„ąī…œī„Ŧī…† ī…ƒ ī…žī„ ī…Œ ī„ļī…œī„žī„Ŧī…‚ ī„ģī…‰îĨĄ ī„“ī„ģ ī„´ī„›ī…„ī„Žī…„ ī„ąī„Šī…‚ī„•ī„Š ī„ļī„” ī…ƒī„› ī„“ī„ģ (1, 3, 5) ī…ƒ ī…žī„ ī…Œ
ī„ļī…œī„žī„Ŧī…‚ ī„ģī…‰, ī„“ī„ēī„Žī…†ī…œ ī„“ī…žī„› ī…‚ī„ĩ ī…ƒī„°ī„ĩ ī„ĸī…‚ī„ī„ ī…‹îĨĄ
ī„˛ī…‚ī„ (iii): ī„ĩī…‡ī…ƒī„œī„› ī„ŗī…œī„ ī„°ī…œī„›ī„šī„Ž (ī„›ī…„ī„ŗī„Ē ī„ąī„Žī…‚ī„ŗ ī„Ŧī……ī…žī„­ ī„Ŧī…„ ī„ŗī…œī„)
ī„ĩī…‚ī„ĸī…† ī„‘ī…œī„Ē ī…ƒ ī…žī„  ī„’ī„¯ī„Šī…„ ī„Ŧī……ī„›ī…‚ī„Ž ī…ƒ ī…žī„  ī„ąī…‰ī„Ļ ī…ƒī„ī„’, ī„“ī„ģ ī„Ŧī…‡ī„œī„Ŧī…‚ ī…ƒī„ĩī„ģī…‚ ī…ƒī„› ī„Ŧī……ī…žī„­ ī…ƒī„› ī…ˆ ī…žī„œ- ī…žī„œ ī„›ī…„ī„ŗī„Ēī…‚ī„ 'ī„Ēī…‡ ī…ƒ ī„›ī„Ŧī…‚
ī„ģī…‰îĨĄ
ī„›ī…„ī„ŗī„Ē (ī„ĩī……ī„¯ī„—/ī„ļī…„ī„Ĩī„ĩ)
ī„ŗī…œī„ (ī„ļī…„ī„Ĩī„ĩ)
1
100
2
50
3
0
ī„•ī„ēī„Žī…‡ ī„Ŧī…‡ī…ƒī„œī„’ ī…ƒī„› ī„ĸī„Ŧī…Œ ī„›ī…„ī„ŗī„Ē ī„­ī„Ŧī…„ ī„ģī…‰, ī„Ēī…‚ī„ ī„ŗī…œī„ ī„žī…žī„Ĩ ī„ĸī…‚ī„ī„Ŧī…„ ī„ģī…‰îĨĄ ī„“ī„ģ ī„“ī…žī„› ī„ĩī…‡ī…ƒī„œī„› ī„ŗī…œī„ ī„°ī…œī„›ī„šī„Ž ī„ļī„” ī„“ī…žī„›
ī„ēī…œī„¯ī…†ī„ĩī„Ž ī„›ī…‡ī„ē ī„ģī…‰, ī…ƒī„ĸī„ēī„Ŧī…‚ ī„‘ī„ĩī„Ģ ī„ģī…‰ ī…ƒī„› ī„ŗī…œī„ ī„‘ī„Ēī…‡ ī„›ī…„ī„ŗī„Ē ī„“ī…žī„› ī…ƒī„ēī…žī„­ī…„-ī„ĩī…‡ī„œī…‚ ī„ēī„ąī…œī„­ ī„Ŧī…„ ī„¯ī…‚ī„ļī„Šī…‚ ī„›ī„ĩī„Ŧī…‡ ī„ģī„ŽîĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„›ī„Ŧī„ŗ 1: ī„’ī„™ ī„°ī…‚ī„ĩī„ŗ ī…ƒī„¸ī…žī„  ī„ĩī…‡ī…ƒī„œī„› ī„°ī…œī„›ī„šī„Ž ī„ļī…žī„˛ī…„ī„—:
ī„§ī…„=ī„˜ī„ŗī„¯ī…„+ī„ēī…„
ī…ƒī„›ī…žī„Ģī…‡:
â€ĸ ī„§ī…„ ī„ŗī…œī„ ī„ģī…‰îĨĄ
â€ĸ ī„¯ī…„ ī„›ī…„ī„ŗī„Ē ī„ģī…‰
â€ĸ m ī„¨ī„ļī…‚ī„Ž ī„ģī…‰
â€ĸ c, y-ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„¯ī„Ĩ ī„ģī…‰îĨĄ
ī„›ī„Ŧī„ŗ 2: ī„Ŧī…‹ ī…ƒī„ąī…œī„Ŧī…†ī„’ī„ ī„Ŧī…„ ī„¸ī„ĩī„Ēī…Œ ī„›ī„ĩī…‹
ī„’ī„™ ī„Ŧī…‹ ī„Žī……ī„›ī„Ēī…‡ ī„ļī…Šī„Ŧī…‡ ī„ģī…‚ī„:
â€ĸ (1,100)(1, 100)(1,100)
â€ĸ (2,50)(2, 50)(2,50)
ī„ģī……ī„Š ī„¨ī„ļī…‚ī„Š ī„Ŧī…„ ī„ī„Šī„Žī…‚ ī„›ī„ĩī…‹:
ī„ģī……ī„Š ī…ƒī„ąī…œī„Ŧī…†-ī„¨ī„ļī…‚ī„Ž ī„°ī…‚ī„ĩī„ŗ ī„Ŧī…„ ī„¸ī„ĩī„Ēī…Œ ī„›ī„ĩī…‹:
ķˇƒ†ķŧŊĸī„‘ī…œī…ƒī„Ēī„ŗ ī„ĩī…‡ī…ƒī„œī„› ī„ŗī…œī„ ī„°ī…œī„›ī„šī„Ž:
ī„’ī„™ ī„“ī„ēī„Žī…†ī…œ ī„Ēī…„ī„ĸī…‡ ī„Žī……ī„›ī„Ēī…‡ ī„Žī…‚ī„ļ ī„¯ī…Ĩī„ŗī…‚ī…ƒī„Šī„Ē ī„›ī„ĩī…„ī„—:
ī„ģī…‚ī„, ī„“ī„ģ ī„ŗī…‡ī„ļ ī„œī…‚ī„ī„Ŧī…‚ ī„ģī…‰!
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ķšŗ´ķšŗĩķšŗļķšŗˇī„‘ī…œī…ƒī„Ēī„ŗ ī„ēī…‚ī„ĩ
1. ī„ēī„ŗī…‚ī„Žī…‚ī„ī„Ēī„ĩ ī„ĩī…‡ī„œī…‚ī„¸ī…‚ī„:
ī„ĩī…‡ī„œī…‚ ī…‚ī„ y=5x−7y = 5x - 7y=5x−7 ī„‘ī„Ēī…‡ 2y=10x+52y = 10x + 52y=10x+5 ī„ēī„ŗī…‚ī„Žī…‚ī„ī„Ēī„ĩ ī„ģī„Ž
ī…ƒī„›ī„•
ī„
ī…ƒī„› ī„Ŧī…‹ ī…‚ī„ ī„Ŧī…„ ī„¨ī„ļī…‚ī„Š ī„“ī…žī„›ī…‹ ī…ƒī„ĸī„ģī…„ ī„ģī…‰ = 5îĨĄ
2. 3D ī„ēī„¯ī…‡ī„ē ī…ƒ ī…žī„  ī„ļī…‚ī„”ī„Ž ī…ƒī„ĸī„ē ī…ƒ ī…žī„  ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„¯ī„Ĩī„ē 9 ī„Ēī…žī„› ī„ĸī…‹ī„ŋī„Ŧī…‡ ī„ģī„Ž: ī„’ī„ŗ ī„ĩī…†ī„¯:
ī„•ī„Ŧī…‚ī„ģī„ĩī„Š ī„ĸī…Œ, ī„ĸī…‡ī„›ī„ĩ a=3,b=2,c=4a = 3,b = 2,c = 4a=3,b=2,c=4, ī„Ēī…‚ī„ ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„“ī„ģ ī„ģī…‰:
3. ī„Ŧī……ī…žī„­ ī„ļī„” ī„ĩī…‡ī…ƒī„œī„› ī„ŗī…œī„ ī„°ī…œī„›ī„šī„Ž:
ķŊ„ģķŊ„ŧķŊ„Ŋī„‘ī…œī…ƒī„Ēī„ŗ ī…ƒī„¸ī„ ī…‚ī„ĩ
ī„ī…ƒī„Šī„Ē, ī…ƒī„žī…œī„Ŧī„ī…„ ī…‚ī„ī„ ī„ģī…„, ī„•ī„Ŧī…Œ ī„’ī„ēī…‚ī„Ž ī„ģī…‹ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰ ī„ĸī„Ŧī…Œ ī„‘ī„ēī…„
ī„
ī„“ī„ēī„Žī…†ī…œ ī„ēī„­ī…‚ī„ĩī„Ž ī…ƒī„Žī„ĩī…„ī„œī„Šī…‚ī„ ī„Žī…‚ī„ļ ī„ĸī…‹ī„ŋī„Ŧī…‡ ī„ģī…‚ī„îĨĄ ī„ĩī…‚ī„ĸī…†
ī„Žī…‡ ī…ƒī„ēī„ĩī…€ ī„“ī„ģī„Žī…‚ī„ ī„ēī„ŗī…žī…ƒī„ēī„’ ī…‚ī„ ī„Žī…†ī…œ ī„ģī…žī„ļ ī„Žī„ģī…„
ī„
ī„›ī…„ī„Ēī…‚; ī„•ī„ēī„Žī…‡ ī„Ŧī……ī…žī„­ ī„Ŧī…„ī„’ī„ ī„ąī…‹ī„Ēī„ļī…‚ī„, ī…ƒī„§ī„ļī…„ ī„ĩī…„ ī„ĩī…†ī„Ĩī…‚ī„ ī„‘ī„Ēī…‡ ī„šī…ƒī„ģī„ĩ ī„Ŧī…„ī„’ī„
ī„ī„ļī…„ī„’ī„ ī…ƒ ī…žī„  ī„‘ī…œī„›ī…‚ī„ ī„Ŧī…„ ī„ēī……ī…œī„Ŧī„ĩī„Ēī…‚ ī„Ŧī…‡ī„œī…„îĨĄ ī„ģī„ĩ ī„°ī…‚ī„ĩī„ŗī…†ī„ļī…‚, ī„ģī„ĩ ī„ēī„ŗī…„ī„›ī„ĩī„Ž, ī„“ī…žī„› ī„›ī„ģī…‚ī„Šī…„ ī„Ŧī…žī„ēī„Ŧī…‚ ī„ģī…‰ — ī„‘ī„Ēī…‡ ī„“ī…žī„›
ī…‚ī„ĩ ī„ĸī„Ŧī…Œ ī„Ēī……ī„ēī…„
ī„
ī„ēī……ī„Šī„Žī…‚ ī…ƒī„ēī…žī„œ ī„ļī…Šī„Ŧī…‡ ī„ģī…‹, ī„Ēī…‚ī„ ī„ī…ƒī„Šī„Ē ī„Ēī……ī„ģī…‚ī„§ī…‚ ī„ēī„˛ ī„Ēī…Œ ī„­ī…„ī„’ ī„Ŧī…‹ī„ēī„Ē ī„ąī„Š ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ
4. (i) ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Žī…†ī…œ ī„¯ī…ƒī„ĩī„˛ī…‚ī…ƒī„šī„Ē ī„›ī„ĩī…‹îĨĄ ī„¸ī…žī„œ-ī„¸ī…žī„œ ī…ƒī„›ī„ēī„ŗī…‚ī„ ī„Ŧī…‡ ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…„ ī…ƒī„¸ī„’ī…ƒī„œī„’ ī„›ī„ĩī…‹îĨĄ
(ii) ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…‡ ī„ēī…œī„ž, ī„ļī…‚ī„ī„žī…‡, ī„‘ī…œī„Ēī„ĩ ī„‘ī„Ēī…‡ ī„ēī„ŗī…ƒī„ŗī„Ēī…„ ī„‘ī…œī„Ēī„ĩ ī„Ŧī…„ ī…ƒī„¸ī„’ī…ƒī„œī„’ ī„›ī„ĩī…‹îĨĄ
(iii) 70 ī…ƒī„¸ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Ŧī…„ ī„“ī…žī„› ī„›ī„ļī…‚ī„ē, ī…ƒī„ĸī„Žī…§ī…‚ī„ ī…ƒī„¸ī…žī„ ī…Œ 30 ī„Žī…‡ ī„ī…ƒī„Šī„Ē ī„Ŧī…‚ ī…ƒī„¸ī„šī…‚ ī…ƒī„ļī„’ ī„ģī…‰ ī„‘ī„Ēī…‡ 20 ī„Žī…‡
ī„ī…ƒī„Šī„Ē ī„Ŧī…‚ ī…ƒī„¸ī„šī…‚ ī…ƒī„ļī„’ ī„ģī…‰ ī„¯ī„ĩ ī„‘ī…œī„›ī„ŋī…‚ ī„Žī„ģī…„
ī„
ī…ƒī„ļī„’îĨĄ ī„ī…ƒī„Šī„Ē ī„‘ī„Ēī…‡ ī„‘ī…œī„›ī„ŋī…‚ ī…ƒī„¸ī…ƒī„ī„’ī„Ž ī„ļī…‰ī„Š ī„¸ī…‚ī„ļī…‡
ī…ƒī„¸ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Ŧī…„ ī…ƒī„ī„Šī„Ēī…„ ī„ļī…žī„˛ī…‹ ī„‘ī„Ēī…‡ ī…ƒī„ĸī„Žī…§ī…‚ī„ ī„Žī…‡ ī„‘ī…œī„›ī„ŋī…‚ ī…ƒī„¸ī…ƒī„ī„’ī„Ž ī…ƒī„ļī„’ ī„ģī…‰ ī„¯ī„ĩ ī„ī…ƒī„Šī„Ē ī„Žī„ģī…„
ī„
ī…ƒī„ļī„’
ī„ģī…‰îĨĄ
ī„•ī…žī„Ēī„ĩ: ķˇ‡´ķˇ‡ĩķˇ‡ļķˇ‡ˇķˇ‡¸ķˇ‡šī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…„ ī„Ŧī……ī„Žī…„ī„’ ī…ƒ ī…žī„ ī…Œ ī„“ī…žī„› ī„´ī…‚ī„Ēī„ĩī…‚
ī„’ī„™ ī„’ī„¯ī„Šī…„ ī…ƒī„ēī…žī„œī„Š ī„Ŧī…„ ī„´ī…‚ī„Ēī„ĩī…‚ ī…ƒī„›ī„ēī…‡ ī„›ī„ļī…‚ī„ēī„ĩī…†ī„ŗ ī…ƒ ī…žī„  ī„Žī„ģī…„
ī„
, ī„ēī„ī…Œ ī…ƒī„ī„’ī„Ž ī„Ŧī…„ ī„“ī…žī„› ī„ĸī…‚ī„Ŧī…†ī„” ī„ļī…‚ī„“ī„ąī…Ĩī…‡ī„ĩī…„ ī…ƒ ī…žī„ 
ī„šī……ī„ĩī…† ī„›ī„ĩī…„ī„—, ī…ƒī„ĸī…žī„Ģī…‡ ī„ģī„ĩ ī…ƒī„›ī„Ēī…‚ī„ą ī„“ī…žī„› ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ ī„Žī…†ī…œ ī„Ŧī„ĩī„ēī…‚ī„•
ī„
ī„Ŧī…„ ī„ģī…‰, ī„‘ī„Ēī…‡ ī„ģī„ĩ ī…ƒī„›ī„Ēī…‚ī„ąī…‚ī„ ī„Ŧī…„ ī„šī…‰ī„ļī„° ī„“ī…žī„› ī„ēī…‰ī…žī„Ĩ ī„Žī…†ī…œ
ī„Ŧī„ĩī„ēī…‚ī„•
ī„
ī„Ŧī…„ ī„ģī…‰îĨĄ ī„›ī„ļī„¯ī„Žī…‚ ī„›ī„ĩī…‹ ī…ƒī„› ī„“ī„ē ī„ļī…‚ī„“ī„ąī…Ĩī…‡ī„ĩī…„ ī„Ŧī…‚ ī„¯ī…Ĩī„ąī…œī„­ī„Ž ī„’ī„ĩī…„ī„‘ī„Ž ī„Žī…‚ī„ŗ ī„Ŧī…‡ ī„“ī…žī„› ī„•ī„Ēī„ēī……ī„› ī„ŗī……ī…œī„§ī…‡ ī„Ŧī……ī„’ī„ĩī…‚
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„›ī…„ī„Ēī…‚ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰, ī„ĸī…‹ ī„¯ī…‰ī„Ĩī„ĩī„Žī…‚ī„ ī„Žī…†ī…œ ī„Ŧī…‡ī„œī„Šī…‚, ī„ĸī…‚ī„Šī„›ī…‚ī„ĩī…„ ī„Žī…†ī…œ ī„ēī…œī„ī…ƒī„Ļī„Ē ī„›ī„ĩī„Žī…‚ ī„‘ī„Ēī…‡ ī„¯ī„ģī…‡ī„ļī…„ī„’ī„ ī„Žī…†ī…œ ī„ģī…žī„ļ ī„›ī„ĩī„Žī…‚ ī„¯ī„ēī…œī„Ŧ
ī„›ī„ĩī„Ŧī…‚ ī„ģī…‰îĨĄ ī„“ī…žī„› ī…ƒī„Ŧī„Ž, ī„ĸī„Ŧī…Œ ī„•ī„ģ ī„’ī„¯ī„Šī…‡ ī„‘ī…ƒī„­ī„’ī„¯ī„› ī„Žī…†ī…œ ī„ī…ƒī„Šī„Ē ī„‘ī„Ēī…‡ ī„‘ī…œī„›ī„ŋī…‚ ī…ƒ ī…ƒī„ī„’ī„Ž ī„Ŧī…„ ī„ ī…‹ī„Š ī„›ī„ĩī„Ž ī…‚ī„ļī…‡
ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„ąī…‚ī„ĩī…‡ ī„°ī…‚ī„”ī„ļī…‚ī„ ī„Ŧī…‚ ī„¯ī…Ĩī„ąī…œī„­ ī„›ī„ĩī„Ž ī…ƒ ī…žī„  ī„ŗī„Ŧī„Ŧ ī„›ī„ĩī„Ŧī…‚ ī„ēī…„, ī„Ēī…‚ī„ ī„•ī„ēī„Žī…‡ ī„›ī……ī„Ŗ ī„šī…‚ī„Žī„Ŧī…‚ī„ĩ ī„œī…‹ī„ĸ ī„›ī…„ī„Ēī…„ - ī„ēī…‰ī…žī„Ĩ
ī…ƒī„Ģī„–ī„ĩī…„!
ī„’ī„™ ī„’ī„ĩī…„ī„‘ī„Ž ī„Ŧī…‡ ī„Žī…‚ī„ļ ī„ ī…žī„ļī…„ī„— ī…ƒī„ĸ ī…ˆ ī…ƒī„› ī„•ī„ģ ī„œī…‹ī„ĸī„Ŧī…‚ ī„ģī…‰ ī…ƒī„› ī„ēī…‰ī…žī„Ĩ ī„›ī…„ ī„ģī„Ž, ī…žī„œ- ī…žī„œ ī…ƒī„›ī„ēī„ŗī…‚ī„ ī„Ŧī…‡ ī„ēī…‰ī…žī„Ĩ, ī„‘ī„Ēī…‡ ī„‘ī„ēī…„
ī„
ī„´ī…†ī„Žī…„ī„‘ī„Ž, ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„›ī„šī„Ž, ī„°ī„ĩī„›, ī„‘ī„Ēī…‡ ī„ēī„ŗī…ƒī„ŗī„Ēī…„ ī„‘ī…œī„Ēī„ĩ ī„ĩī„ī…‡ ī„›ī…‚ī„ĩī„ĸ ī…ƒī„› ī…ˆ ī„›ī„ĩī„Ŧī…‡ ī„ģī…‚ī„îĨĄ ī„‘ī…œī„Ē ī…ƒ ī…žī„ , ī„‘ī„ēī…„
ī„
ī„“ī„ē
ī…ƒī„ī„’ī„Ž ī„Ŧī…„ ī„ĩī„Ēī…Œ ī„›ī„ĩī„›ī…‡ ī„’ī„ĩī…„ī„‘ī„Ž ī„Žī…†ī…œ ī„‘ī„ēī„ļ-ī„ĸī…„ ī„Ž ī„Ŧī…„ ī„ēī„ŗī…žī…ƒī„ēī„’ ī„Žī…†ī…œ ī„ģī…žī„ļ ī„›ī„ĩī„Ž ī…ƒ ī…žī„  ī…„ ī„ŗī„Ŧī„Ŧ ī„›ī„ĩī…‚ī„ī„ī…‡îĨĄ ī„›ī…„
ī„Ēī……ī„ēī…„
ī„
ī…ƒī„Ēī„’ī„ĩ ī„ģī…‹? ī„ ī„ļī…‹ ī„ ī…žī„ļī…„ī„—!
ķšģ€(i) ī„ēī…‰ī…žī„Ĩ ī„›ī…„ ī„ģī…‰?
ī„“ī…žī„› ī„ēī…‰ī…žī„Ĩ ī…ƒī„ēī„ĩī…€ ī„ ī…œī„ī…„ ī„Ēī„ĩī…§ī…‚ī„ ī„¯ī…ƒī„ĩī„˛ī…‚ī…ƒī„šī„Ē ī„‘ī„Ēī…‡ ī…žī„œī„ĩī…„ī„’ī„ ī„ēī„Ēī…†ī„’ī„ ī„Ŧī…‚ ī„ēī…œī„ī…Ĩī…ƒī„ģ ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰îĨĄ ī„“ī„ģī„Žī…‚ī„ ī„ēī„Ēī…†ī„’ī„ ī„Žī…†ī…œ
ī„˜ī„ļī…„ī„ŗī…Šī„Ĩī„ē ī„ĸī…‚ī„ ī„ēī…‰ī…žī„Ĩ ī„Ŧī…‡ ī„ŗī…Šī„ąī„ĩ ī…ƒī„›ī„ģī…‚ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ ī„ēī„˛ ī„Ēī…Œ ī„­ī…„ī„’ ī…ƒī„ģī…žī„ēī…‚? ī„Ēī……ī„ēī…„
ī„
ī„›ī„ĩī„ļī…„ ī„ąī„ĩī…‰ī„›ī„Ĩ { } ī„Ŧī…„ ī„ĩī„Ēī…Œ ī„›ī„ĩī„›ī…‡
ī„“ī…žī„› ī„ēī…‰ī…žī„Ĩ ī…ƒī„ļī„œ ī„ēī„›ī„Ŧī…‡ ī„ģī…‹îĨĄ
ķŊ„ģķŊ„ŧķŊ„Ŋī„¯ī…ƒī„ĩī„˛ī…‚ī„šī…‚:
ī„“ī…žī„› ī„ēī…‰ī…žī„Ĩ ī„ēī„Ēī…†ī„’ī„ ī„Ŧī…‚ ī„“ī…žī„› ī„ ī…œī„ī…„ ī„Ēī„ĩī…§ī…‚ī„ ī„¯ī…ƒī„ĩī„˛ī…‚ī…ƒī„šī„Ē ī„ēī…œī„ī…Ĩī…ƒī„ģ ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰îĨĄ ī„“ī…žī„› ī„ēī…‰ī…žī„Ĩ ī„Ŧī…‡ ī„˜ī„ļī…„ī„ŗī…Šī„Ĩī„ē ī„›ī„ĩī„ļī…„ ī„ąī„ĩī…‰ī„›ī„Ĩī„ē
ī„Ŧī…‡ ī„‘ī…œī„Ŧī„ĩ ī…ƒī„ļī„œī…‡ ī„ĸī…‚ī„ī„Ŧī…‡ ī„ģī„Ž, ī„›ī…‚ī…ƒī„ŗī„’ī„ ī„Žī…‚ī„ļ ī…žī„œ ī„›ī…„ī„Ēī…‡ ī„ĸī…‚ī„ī„Ŧī…‡ ī„ģī„ŽîĨĄ
ķŊ„ģķŊ„ŧķŊ„Ŋī„•ī„Ŧī…‚ī„ģī„ĩī„Žī…‚ī„:
â€ĸ ī„‘ī…œī„ī„ĩī…‡ī„žī…„ ī„¸ī„ĩī„Šī„ŗī…‚ī„ļī…‚ ī…ƒī„¸ī…žī„  ī„ēī„¸ī„ĩī…‚ī„ ī„Ŧī…‚ ī„“ī…žī„› ī„ēī„ŗī…†ī„ģ:
ī„— = {ī„—, ī„”, ī„’ī„”, ī„™, ī„´ī…†}
â€ĸ ī„¯ī…ƒī„ģī„ļī…‡ ī„¯ī…œī„ĸ ī„›ī……ī„Ŧī„ĩī„Ēī…„ ī„ēī…œī…ƒī„œī„’ī„¸ī…‚ī„ ī„Ŧī…‚ ī„ēī…‰ī…žī„Ĩ:
ī„ąī…„ = {1, 2, 3, 4, 5}
â€ĸ ī„ēī„Ēī„ĩī…œī„ī…„ ī„¯ī…„
ī„
ī„ž ī…ƒ ī…žī„  ī„ĩī…œī„ī…‚ī„ ī„Ŧī…‚ ī„ēī„ŗī…†ī„ģ: C = {ī„ļī…‚ī„ļ, ī„ēī…œī„Ēī„ĩī…„, ī„¯ī…„ī„ļī…‚, ī„ģī„ĩī…‚, ī„Žī…„ī„ļī…‚, ī„Žī…„ī„ļī…‚, ī„ĸī…‚ī„ŗī„Žī…„}
ķšģ€ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…„ī„’ī„ ī…ƒī„›ī„ēī„ŗī…‚ī„
ī„’ī„ĩī…„ī„‘ī„Ž ī…ƒī„›ī„Ēī…‚ī„ąī…‚ī„ (ī„ēī…‰ī…žī„Ĩī…‚ī„) ī„Žī…†ī…œ ī„šī…Ĩī…‡ī„Šī…„ī„ąī…žī„­ ī„›ī„ĩī„Žī…‚ ī„ ī…‚ī„ģī……ī…œī„Ŧī…‚ ī„ēī…„, ī„“ī„ē ī„ļī„” ī„•ī„ēī„Ŧī…‡ ī„‘ī…ƒī„­ī„’ī„¯ī„› ī„Žī…‡ ī„•ī„ēī„Žī…†ī…œ ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…„ī„’ī„
ī…ƒī„›ī„ēī„ŗī…‚ī„ ī„Žī…‚ī„ļ ī„ĸī…‚ī„Šī…† ī„›ī„ĩ ī…‚ī„“ī„’îĨĄ ī„“ī…žī„Ģī…‡ ī„•ī„ēī„Žī…†ī…œ ī„›ī…„ ī…ƒī„ŗī…ƒī„ļī„’:
1. ī„œī…‚ī„ļī…„ ī„ēī…‰ī…žī„Ĩ (ī„Žī„ļ ī„ēī…‰ī…žī„Ĩ):
ī„“ī…žī„› ī„ēī…‰ī…žī„Ĩ ī…ƒī„ĸī„ē ī…ƒ ī…žī„  ī„›ī…‹ī„” ī„Ēī…žī„Ē ī„Žī„ģī…„
ī„
ī„ģī……ī…œī„Ŧī…‚, ī„•ī„ēī„Žī…†ī…œ ī„œī…‚ī„ļī…„ ī„ēī…‰ī…žī„Ĩ ī„ĸī…‚ī„ ī„Žī„ļ ī„ēī…‰ī…žī„Ĩ ī…ƒī„›ī„ģī…‚ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„“ī„ēī„Žī…†ī…œ ÎĻ ī„ĸī…‚ī„ { } ī„Žī…‚ī„ļ ī„Ŧī„ĩī„ēī…‚ī„“ī„’ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
1 → ÎĻ ī„Ēī…Œ ī„žī…žī„Ĩ ī„›ī……ī„Ŧī„ĩī„Ēī…„ ī„ēī…œī…ƒī„œī„’ ī…‚ī„ ī„Ŧī…‚ ī„ēī…‰ī…žī„Ĩ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
2. ī…ƒī„ēī…œī„ī„ļī„Ĩī„Ž ī„ēī…‰ī…žī„Ĩ:
ī„“ī…žī„› ī„ēī…‰ī…žī„Ĩ ī…ƒī„ĸī„ē ī…ƒ ī…žī„  ī…ƒī„ēī„ĩī…€ ī„“ī…žī„› ī„Ēī…žī„Ē ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰, ī„Žī…†ī…œ ī…ƒī„ēī…œī„ī„ļī„Ĩī„Ž ī„ēī…‰ī…žī„Ĩ ī…ƒī„›ī„ģī…‚ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
3 ī„Ēī…Œ ī„žī…žī„Ĩ ī„‘ī„ĸī…„ī„ą ī„‘ī„˛ī…‚ī„ĸ ī„ēī…œī…ƒī„œī„’ ī…‚ī„ ī„Ŧī…‚ ī„ēī…‰ī…žī„Ĩ → {2}
3. ī„ēī…„ī„ŗī„Ē ī„ēī…‰ī…žī„Ĩ:
ī„“ī…žī„› ī„ēī…‰ī…žī„Ĩ ī…ƒī„ĸī„ē ī…ƒ ī…žī„  ī„Ēī…žī„Ēī…‚ī„ ī„Ŧī…„ ī…ƒī„ī„Šī„Ēī…„ ī„›ī„ĩī„Ž ī„´ī…‹ī„ ī…ƒī„ī„Šī„Ēī…„ ī„ģī…‹ ī…‡îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž: ī„ģī…€ī„Ēī…‡ ī…ƒ ī…žī„  ī…ƒī„Ŧī„Žī…‚ī„ ī„Ŧī…‚ ī„ēī…‰ī…žī„Ĩ → {ī„ēī…‹ī„ŗ ī…‚ī„ĩ, ī„ŗī…œī„ī„ļ ī…‚ī„ĩ, ..., ī„˜ī„Ē ī…‚ī„ĩ}
4. ī„‘ī„Žī…œ ī„Ē ī„ēī…‰ī…žī„Ĩ:
ī„‘ī„Šī…ƒī„ī„Šī„Ē/ī„‘ī„Žī…œ ī„Ē ī„Ēī…žī„Ēī…‚ī„ ī…‚ī„ļī…‚ ī„ēī…‰ī…žī„ĨîĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
ī„ēī…‚ī„ĩī…„ī„’ī„ ī„›ī……ī„Ŧī„ĩī„Ēī…„ ī„ēī…œī…ƒī„œī„’ ī…‚ī„ ī„Ŧī…‚ ī„ēī…‰ī…žī„Ĩ → {1, 2, 3, 4, ...}
5. ī„ąī„ĩī…‚ī„ąī„ĩ ī„ēī…‰ī…žī„Ĩ:
ī„Ŧī…‹ ī„ēī…‰ī…žī„Ĩ ī„ąī„ĩī…‚ī„ąī„ĩ ī„ģī……ī…œī„Ŧī…‡ ī„ģī„Ž ī„ĸī…‡ī„›ī„ĩ ī„•ī„ģī„Žī…‚ī„ ī…ƒ ī…žī„  ī…ƒī„ąī„ļī„›ī……ī„ļ ī„“ī…žī„›ī…‹ ī…ƒī„ĸī„ģī…‡ ī„Ēī…žī„Ē ī„ģī…‹ī„Š, ī„˛ī…‚ ī…ˆ ī„›ī…Ĩī„ŗ ī„›ī…‹ī„” ī…„ ī„ģī…‹ ī…‡îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Š: A = {1, 2, 3}, B = {3, 2, 1}ī„— = ī„ąī…„
6. ī„ąī„ĩī…‚ī„ąī„ĩ ī„ēī…‰ī…žī„Ĩ:
ī„Ŧī…‹ ī„ēī…‰ī…žī„Ĩ ī…ƒī„ĸī„Žī…§ī…‚ī„ ī…ƒ ī…žī„  ī„“ī…žī„›ī…‹ ī…ƒī„ĸī„ģī…‡ ī„Ēī…žī„Ē ī„ģī…‹ī„Š, ī„¯ī„ĩ ī„žī„ĩī…†ī„ĩī…„ ī„Žī„ģī…„
ī„
ī…ƒī„› ī„•ī„ģī…„ ī„Ēī…žī„Ē ī„ģī…‹ī„ŠîĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
ī„— = {ī„—, ī„ąī…„, ī„ēī…„}, ī„ąī…„ = {1, 2, 3}ī„— ≈ ī„ąī…„
7. ī„ēī„ąī„ēī…‰ī…žī„Ĩ:
ī„ĸī…‡ī„›ī„ĩ ī„ēī…‰ī…žī„Ĩ A ī„Ŧī…‚ ī„ģī„ĩ ī„Ēī…žī„Ē ī„ēī…‰ī…žī„Ĩ B ī…ƒ ī…žī„  ī…„ ī„ģī…‰, ī„Ēī…‚ī„ A B ī„Ŧī…‚ ī„“ī…žī„› ī„•ī„¯ ī„ēī„ŗī…†ī„ģ ī„ģī…‰îĨĄ ī…ƒī„ ī…œī„Žī…§: Aī„ąī…„
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
A = {1, 2}, B = {1, 2, 3}ī„—îąƒī„ąī…„
8. ī„ēī„ģī…„ ī„ēī„ąī„ēī…‰ī…žī„Ĩ:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
A, B ī„Ŧī…‚ ī„“ī…žī„› ī„ēī„ģī…„ ī„•ī„¯-ī„ēī…‰ī…žī„Ĩ ī„ģī…‰ ī„ĸī…‡ī„›ī„ĩ AB, ī„‘ī„Ēī…‡ A ≠ B. ī…ƒī„ ī…œī„Žī…§: Aî°ŋī„ąī…„
9. ī„´ī…†ī„Žī…„ī„¸ī„ĩī„ēī„ļ ī„ēī…‰ī…žī„Ĩ:
ī„“ī…žī„› ī„ēī…‰ī…žī„Ĩ ī…ƒī„ĸī„ē ī…ƒ ī…žī„  ī„ēī…‚ī„ĩī…‡ ī…ƒ ī„ ī…‚ī„ĩ ī„‘ī„­ī…„ī„Ž ī„Ēī…žī„Ē ī„šī…‚ī„ŗī„ļ ī„ģī„ŽîĨĄ U ī„Ŧī……ī„’ī„ĩī…‚ ī„Ŧī„ĩī„ēī…‚ī„“ī„’ ī…ƒī„ī„’îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
ī„ĸī…‡ī„›ī„ĩ U = ī„ēī…‚ī„ĩī…„ī„’ī„ ī„›ī……ī„Ŧī„ĩī„Ēī…„ ī„ēī…œī…ƒī„œī„’ ī…‚ī„ ≤ 10, ī„Ēī…‚ī„ U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
10. ī„¯ī…‚ī„¸ī„ĩ ī„ēī…‰ī…žī„Ĩ:
ī…ƒī„›ī„ēī…‡ ī„ēī…‰ī…žī„Ĩ ī„Ŧī…‡ ī„ēī…‚ī„ĩī…‡ ī„ēī„ąī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…‡ ī„ēī…‰ī…žī„Ĩ ī„Žī…†ī…œ ī„“ī„ēī„Ŧī…‚ ī„¯ī…‚ ī„ĩ ī„ēī…‰ī…žī„Ĩ ī…ƒī„›ī„ģī…‚ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ ī„ĸī…‡ī„›ī„ĩ A = {1, 2}, ī„Ēī…‚ī„ P(A) = {ÎĻ, {1},
{2}, {1, 2}}
ķšģ€(ii) ī„ēī…‰ī…žī„Ĩī…‚ī„ 'ī„Ēī…‡ ī„›ī…‚ī„ĩī„¸ī…‚ī„”ī„’ī„
ī…ƒī„ĸ ī…ˆ ī„ģī…„ ī„’ī„ĩī…„ī„‘ī„Ž ī…ƒī„›ī„Ēī…‚ī„ąī…‚ī„ ī„Žī…†ī…œ ī…ƒ ī„ēī…ƒī„Ģī„Ē ī„›ī„ĩ ī…ƒī„ĩī„ģī…‚ ī„ēī…„, ī„•ī„ēī„Žī…†ī…œ ī„‘ī…ƒī„ģī„ēī…‚ī„ē ī„ģī…‹ī„“ī„’ ī…ƒī„› ī„›ī……ī„Ŗ ī…ƒī„›ī„Ēī…‚ī„ąī…‚ī„ ī„“ī…žī„› ī„Ēī…Œ ī…žī„­
ī„šī…‰ī„ļī„°ī…‚ī„ ī„Ŧī…„ī„’ī„ ī„ēī„ŽîĨĄ ī„“ī„ģ ī„•ī„Ŧī…Œ ī„ģī…‹ī„“ī„’ ī„ĸī„Ŧī…Œ ī„•ī„ēī„Ŧī…‡ ī„‘ī…ƒī„­ī„’ī„¯ī„› ī„Žī…‡ ī„•ī„ēī„Žī…†ī…œ ī„ ī…‚ī„ĩ ī„šī„›ī„Ēī…„ī„šī…‚ī„ļī…„ ī…ƒī„›ī…ƒī„ĩī„’ ī…‚ī„
ī…ƒī„ēī„œī…‚ī„”ī„’ī„:
ķˇƒ†ķŊ„ž1. ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…‚ ī„ēī…œī„ž (Aî¯Ģī„ąī…„)
ī„´ī…†ī„Žī…„ī„‘ī„Ž ī„Ŧī…‚ ī„‘ī„ĩī„Ģ ī„ģī…‰ ī„Ŧī…‹ ī…‚ī„ ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…‡ ī„ēī…‚ī„ĩī…‡ ī„Ēī…žī„Ēī…‚ī„ ī„Žī…†ī…œ ī„ĸī…‹ī„ŋī„Žī…‚îĨĄ ī„ĸī…‡ī„›ī„ĩ ī„›ī……ī„Ŗ ī„Ēī…žī„Ēī…‚ī„ ī„Žī…†ī…œ ī„Ŧī……ī„ģī„ĩī…‚ī„“ī„’ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰, ī„Ēī…‚ī„ ī„‘ī„ēī…„
ī„
ī„•ī„ģī„Žī…‚ī„ ī„Žī…†ī…œ ī…ƒī„ēī„ĩī…€ ī„“ī…žī„› ī…‚ī„ĩ ī„šī…‚ī„ŗī„ļ ī„›ī„ĩī„Ŧī…‡ ī„ģī…‚ī„îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
A = {1, 2, 3}, B = {3, 4, 5}Aî¯Ģī„ąī…„ = {1, 2, 3, 4, 5}
ķˇƒ†ķŊ„ž2. ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…‚ ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„›ī„šī„Ž (A ∊ B)
ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„›ī„šī„Ž ī„Ŧī…‹ ī„ēī…‰ī…žī„Ĩī…‚ī„ ī…ƒ ī„ ī„›ī…‚ī„ĩ ī„ēī…‚ī„ī„Ŗī…‡ ī„Ēī…žī„Ē ī…ƒī„Ŧī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
A = {1, 2, 3}, B = {2, 3, 4}A ∊ B = {2, 3}
ķˇƒ†ķŊ„ž3. ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…‚ ī„‘ī…œī„Ēī„ĩ (A - B)
ī„“ī„ģ ī…ƒī„ēī„ĩī…€ A ī…ƒ ī…žī„  ī„Ēī…žī„Ē ī…ƒī„Ŧī…œī„Ŧī…‚ ī„ģī…‰ ī„¯ī„ĩ B ī…ƒ ī…žī„  ī„Žī„ģī…„
ī„
îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Š: A = {1, 2, 3}, B = {2, 3, 4}A – B = {1}
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ķˇƒ†ķŊ„ž4. ī„ēī„ŗī…ƒī„ŗī„Ēī…„ ī„‘ī…œī„Ēī„ĩ (A Δ B)
ī„ēī„ŗī…ƒī„ŗī„Ēī…„ ī„‘ī…œī„Ēī„ĩ ī„‘ī…ƒī„ĸī„ģī…‡ ī„Ēī…žī„Ē ī…ƒī„Ŧī…œī„Ŧī…‚ ī„ģī…‰ ī„ĸī…‹ A ī„ĸī…‚ī„ B ī…ƒ ī…žī„  ī„ģī„Ž, ī„¯ī„ĩ ī„Ŧī…‹ ī…‚ī„ ī…ƒ ī…žī„  ī„Žī„ģī…„
ī„
îĨĄ ī„¯ī…Ĩī„Ēī…„ī„›: A Δ B = (A –
B)î¯Ģ(ī„ąī…„–ī„—)
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
A = {1, 2, 3}, B = {3, 4, 5}A Δ B = {1, 2, 4, 5}
ī„’ī„™ ī„“ī„ēī„Žī…†ī…œ ī„ģī…‹ī„ĩ ī…„ ī„ēī„¯ī…žī„šī„Ĩ ī„›ī„ĩī„Ž ī„ļī„” ī„“ī…žī„Ģī…‡ ī„“ī…žī„› ī„Ąī…‹ī„Ĩī…„ ī…ƒī„ĸī„ģī…„ ī„›ī„ģī…‚ī„Šī…„ ī„ĸī…‹ī„ŋī„Ŧī…‡ ī„ģī…‚ī„îĨĄ
ķš´Ąķš´ĩķš´Ŗķš´¤ī…ƒī„ŗī…œī„Žī…„ ī„›ī„ģī…‚ī„Šī…„: ī…ƒī„›ī…Ĩī„›ī„Ĩ ī„‘ī„Ēī…‡ ī„°ī……ī…žī„Ĩī„ąī…‚ī„ļ ī„›ī„ļī…žī„ą
ī„“ī…žī„› ī„›ī…‚ī„ļī„ĸ ī…ƒ ī…žī„ , ī„›ī……ī„Ŗ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ ī…ƒī„›ī…Ĩī„›ī„Ĩ (ī„ēī…‰ī…žī„Ĩ ī„—) ī„œī…‡ī„§ī„Ŧī…‡ ī„ģī„Ž ī„‘ī„Ēī…‡ ī„›ī……ī„Ŗ ī„°ī……ī…žī„Ĩī„ąī…‚ī„ļ (ī„ēī…‰ī…žī„Ĩ ī„ąī…„) ī„œī…‡ī„§ī„Ŧī…‡ ī„ģī„ŽîĨĄ ī„›ī……ī„Ŗ
ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ ī„Ŧī…‹ ī…ˆ ī„œī…‡ī„§ī…‚ī„ ī„œī…‡ī„§ī„Ŧī…‡ ī„ģī„ŽîĨĄ
ī„’ī„™:
â€ĸ ī„‘ = {ī„ĩ ī…„, ī„ŗī…‹ī„ģī„Ž, ī„ēī…„ī„Ēī…‚}
â€ĸ ī„ą = {ī„ēī…„ī„Ēī…‚, ī„›ī„ĩī„Ž, ī„ŗī…„ī„Žī…‚}
ī„ģī……ī„Š:
â€ĸ ī„´ī…†ī„Žī…„ī„‘ī„Ž (ī„—)î¯Ģī„‘) = ī„•ī„ģ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ ī„ĸī…‹ ī…ƒī„›ī…Ĩī„›ī„Ĩ ī„ĸī…‚ī„ ī„°ī……ī…žī„Ĩī„ąī…‚ī„ļ ī„ĸī…‚ī„ ī„Ŧī…‹ ī…ˆ ī„œī…‡ī„§ī„Ŧī…‡ ī„ģī„Ž = {ī„ĩ ī…„, ī„ŗī…‹ī„ģī„Ž,
ī„ēī…„ī„Ēī…‚, ī„›ī„ĩī„Ž, ī„ŗī…„ī„Žī…‚}
â€ĸ ī„“ī…œī„Ĩī„ĩī„ēī…‰ī„›ī„šī„Ž (A ∊ B) = ī„Ŧī…‹ ī…ˆ ī„œī…‡ī„§ī„Š ī…‚ī„ļī…‡ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ = {ī„ēī…„ī„Ēī…‚}
â€ĸ A – B = ī…ƒī„ēī„ĩī…€ ī…ƒī„›ī…Ĩī„›ī„Ĩ = {ī„ĩ ī…„, ī„ŗī…‹ī„ģī„Ž}
â€ĸ B – A = ī…ƒī„ēī„ĩī…€ ī„°ī……ī…žī„Ĩī„ąī…‚ī„ļ = {ī„›ī„ĩī„Ž, ī„ŗī…„ī„Žī…‚}
â€ĸ A Δ B = ī…ƒī„›ī…Ĩī„›ī„Ĩ ī„ĸī…‚ī„ ī„°ī……ī…žī„Ĩī„ąī…‚ī„ļ, ī„¯ī„ĩ ī„Ŧī…‹ ī…ˆ ī„Žī„ģī…„
ī„
= {ī„ĩ ī…„, ī„ŗī…‹ī„ģī„Ž, ī„›ī„ĩī„Ž, ī„ŗī…„ī„Žī…‚}
ī„ģī…‰ ī„Žī…‚ ī„ŗī„žī…‡ī„Ŧī…‚ī„ĩ? ī„Ļī…„ī„› ī„“ī„ēī…‡ ī„Ēī„ĩī…§ī…‚ī„, ī„’ī„ĩī…„ī„‘ī„Ž ī„Žī…†ī…œ ī„ģī……ī„Š ī„ī…ƒī„Šī„Ē ī„›ī…‚ī…€ī…„ ī…ƒī„Ŧī„ļī„ ī„ēī„¯ ī„ļī…žī…ƒī„ī„’îĨĄ
ķšģ€(iii) ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…„ ī„¸ī„ĩī„Ēī…Œ ī„›ī„ĩī„›ī…‡ ī„ēī„ŗī…žī…ƒī„ēī„’ ī„ģī…žī„ļ ī„›ī„ĩī„Žī…‚
ī„ģī……ī„Š, ī…ƒī„Ŧī…žī„Ēī…„ ī„ī„” ī„ēī„ŗī…žī…ƒī„ēī„’ ī„Žī…†ī…œ ī„›ī„Ŧī„ŗ-ī„Ŧī„ĩ-ī„›ī„Ŧī„ŗ ī„ģī…žī„ļ ī„›ī„ĩī…„ī„—îĨĄ
ķŧŠ•ķŧŠ–ķŧŠ—ķŧŠ˜ķŧŠ™ķŧŠšī„ēī„¸ī…‚ī„ļ:
70 ī…ƒī„¸ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Ŧī…„ ī„“ī…žī„› ī„›ī„ļī…‚ī„ē ī…ƒī„¸ī…žī„ :
â€ĸ 30 ī„Žī…‡ ī„ī…ƒī„Šī„Ē ī…ƒī„ļī„’ ī„ģī…‰îĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
â€ĸ 20 ī„Žī…‡ ī„ī…ƒī„Šī„Ē ī…ƒī„ļī„’ ī„ģī…‰ ī„¯ī„ĩ ī„‘ī…œī„›ī„ŋī…‚ ī„Žī„ģī…„
ī„
îĨĄ
ī„ļī…žī„˛ī…‹:
1. ī„ī…ƒī„Šī„Ē ī„‘ī„Ēī…‡ ī„‘ī…œī„›ī„ŋī…‚ ī„Ŧī…‹ ī…ˆ ī„ļī…‰ī„Š ī…‚ī„ļī…‡ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Ŧī…„ ī…ƒī„ī„Šī„Ēī…„
2. ī…ƒī„ēī„ĩī…€ ī„‘ī…œī„›ī„ŋī…‚ ī„¯ī…Ĩī…„ī…ƒī„œī„’ ī„Ŧī…‡ī„Š ī…‚ī„ļī…‡ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Ŧī…„ ī…ƒī„ī„Šī„Ēī…„
ķˇƒ†ķŧŊĸī„›ī„Ŧī„ŗ 1: ī„’ī„™ ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Žī…†ī…œ ī„ŗī…œī„Ž ī„ļī„”ī„—
ī„’ī„™:
â€ĸ M = ī„ī…ƒī„Šī„Ē ī„ļī…‰ī„Š ī…‚ī„ļī…‡ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Ŧī…‚ ī„ēī„ŗī…†ī„ģ
â€ĸ S = ī„‘ī…œī„›ī„ŋī…‚ ī„¯ī…Ĩī…„ī…ƒī„œī„’ ī„Ŧī…‡ī„Š ī…‚ī„ļī…‡ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Ŧī…‚ ī„ēī„ŗī…†ī„ģ
â€ĸ n(M) = ī„ī…ƒī„Šī„Ē ī„ļī…‰ī„Š ī…‚ī„ļī…‡ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Ŧī…„ ī…ƒī„ī„Šī„Ēī…„ = 30
â€ĸ n(M – S) = ī„•ī„ģ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ ī…ƒī„ĸī„Žī…§ī…‚ī„ ī„Žī…‡ ī„ī…ƒī„Šī„Ē ī…ƒī„ļī„’ ī„¯ī„ĩ ī„‘ī…œī„›ī„ŋī…‚ ī„Žī„ģī…„
ī„
ī…ƒī„ļī„’ = 20
â€ĸ ī„›ī……ī…žī„ļ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ = 70
ķˇƒ†ķŧŊĸī„›ī„Ŧī„ŗ 2: ī„•ī„ģī„Žī…‚ī„ ī…ƒī„¸ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Žī…†ī…œ ī„ļī…žī„˛ī…‹ ī…ƒī„ĸī„Žī…§ī…‚ī„ ī„Žī…‡ ī„ī…ƒī„Šī„Ē ī„‘ī„Ēī…‡ ī„‘ī…œī„›ī„ŋī…‚ ī„Ŧī…‹ī„¸ī…ˆ ī„ļī„— ī„ģī„ŽîĨĄ
ī„ēī…‚ī„Žī…†ī…œ ī…ƒī„Ŧī…žī„Ēī…‡ ī„ī„— ī„ģī„Ž:
n(M – S) = 20 ī„‘ī„Ēī…‡ ī„›ī……ī…žī„ļ ī„ī…ƒī„Šī„Ē ī…ƒī„›ī„ēī„Žī…‡ ī…ƒī„ļī„’ (n(M)) = 30
ī„“ī„ē ī„ļī„”:
n(M ∩ S) = n(M) – n(M – S)= 30 – 20 = 10
ī„“ī„ē ī„ļī„”, 10 ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Žī…‡ ī„ī…ƒī„Šī„Ē ī„‘ī„Ēī…‡ ī„‘ī…œī„›ī„ŋī…‚ ī„Ŧī…‹ ī…ˆ ī„ļī„—îĨĄ
ķˇƒ†ķŧŊĸī„›ī„Ŧī„ŗ 3: ī„•ī„ģī„Žī…‚ī„ ī…ƒī„¸ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Žī…†ī…œ ī„ļī…žī„˛ī…‹ ī…ƒī„ĸī„Žī…§ī…‚ī„ ī„Žī…‡ ī…ƒī„ēī„ĩī…€ ī„‘ī…œī„›ī„ŋī…‡ ī„ļī„— ī„ģī„ŽîĨĄ
ī„’ī„™ ī„ļī…žī„˛ī…„ī„—:
n(S – M) = ī…ƒī„ēī„ĩī…€ ī„‘ī…œī„›ī„ŋī…‚ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„
ī„ēī…‚ī„Žī…†ī…œ ī„Žī„ģī…„
ī„
ī„¯ī„Ēī…‚ ī…ƒī„› ī„Ĩī…‹ī„Ĩī„ļ ī…ƒī„›ī„ēī„Žī…‡ ī…ƒī„ēī…žī„­ī…‡ ī„Ēī…ī„ĩ 'ī„Ēī…‡ ī„ēī„Ĩī…‰ī…ƒī„Ĩī„ēī…ƒī„Ĩī„›ī„ē ī„ļī„—, ī„¯ī„ĩ ī„‘ī„ēī…„
ī„
ī„“ī„ēī„Žī…†ī…œ ī„ļī…žī„˛ ī„ēī„›ī„Ŧī…‡ ī„ģī…‚ī„îĨĄ
ī„’ī„™:
â€ĸ ī„›ī……ī…žī„ļ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ = 70
â€ĸ n(M)î¯Ģī„ē) = ī„•ī„ģ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ ī…ƒī„ĸī„Žī…§ī…‚ī„ ī„Žī…‡ ī„ī…ƒī„Šī„Ē ī„ĸī…‚ī„ ī„‘ī…œī„›ī„ŋī…‚ ī„ĸī…‚ī„ ī„Ŧī…‹ ī…ˆ ī„ļī„— ī„ēī„ŽîĨĄ
ī„¯ī„ĩ ī„ēī…‰ī…žī„Ĩ ī…ƒī„Ģī„–ī„ĩī…„ ī…ƒī„¸ī…žī„ ,
n(M)î¯Ģī„ē) = ī„Ž(ī„˜ī„ŗ) + ī„Ž(ī„ē)–n(M)∊ī„ē)
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„’ī„™ n(S) ī„ļī…žī„˛ī…„ī„—:
ī„ŗī…œī„Ž ī„ļī„™ n(S) = x
ī„“ī„ē ī„ļī„”:
n(M)î¯Ģī„ē) = 70 = 30 + x–10 => 70 = 20 + x => x = 50
ī„“ī„ē ī„ļī„”, ī„‘ī…œī„›ī„ŋī…‚ ī„¯ī…Ĩī…„ī…ƒī„œī„’ ī„Ŧī…‡ī„Š ī…‚ī„ļī…‡ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Ŧī…„ ī…ƒī„ī„Šī„Ēī…„ = 50
ī„ģī……ī„Š: ī…ƒī„ēī„ĩī…€ ī„‘ī…œī„›ī„ŋī…‡ = n(S – M) = n(S) – n(M ∊ S)= 50 – 10 = 40
ķˇƒ†ķŧŊĸī„‘ī…œī…ƒī„Ēī„ŗ ī„ĸī„¸ī…‚ī„ą:
1. ī„ī…ƒī„Šī„Ē ī„‘ī„Ēī…‡ ī„‘ī…œī„›ī„ŋī…‚ ī„Ŧī…‹ ī…ˆ ī…ƒ ī„šī…‡ ī„ļī…‰ī„Š ī…‚ī„ļī…‡ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ = 10
2. ī…ƒī„ĸī„Žī…§ī…‚ī„ ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ī„’ī„ ī„Žī…‡ ī…ƒī„ēī„ĩī…€ ī„‘ī…œī„›ī„ŋī…‚ ī„¯ī…Ĩī…„ī…ƒī„œī„’ ī…ƒī„Ŧī…žī„Ēī…„ ī„ģī…‰ = 40
ķˇ—­ķˇ—¨ķˇ—Šķˇ—Ēķˇ—Ģķˇ—Ŧī…ƒī„ēī…žī„Ĩī…‚
ī„’ī„ĩī…„ī„‘ī„Ž ī„ąī„ģī……ī„Ē ī„œī……ī„š ī„ēī…„îĨĄ ī…ƒī„ēī„ĩī…€ ī„ēī„­ī…‚ī„ĩī„Ž ī„ēī…‰ī…žī„Ĩ ī„™ī„¯ī„ĩī…‡ī„šī„Žī…‚ī„ ī„Ŧī…‡ ī…ƒī„ī„’ī„Ž ī„‘ī„Ēī…‡ ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Ŧī…„ī„’ī„ ī…ƒī„›ī„ēī„ŗī…‚ī„ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Š
ī„Žī…‚ī„ļ, ī„•ī„ģ ī„“ī…žī„› ī„‘ī„ēī„ļ-ī„ēī…œī„ēī…‚ī„ĩ ī„ēī„ŗī…žī…ƒī„ēī„’ ī„Žī…†ī…œ ī„ģī…žī„ļ ī„›ī„ĩī„Ž ī„Ŧī…‡ ī„´ī…‹ī„ ī„ēī…„! ī…ƒī„ąī„ļī„›ī……ī„ļ ī„’ī„ĩī…„ī„‘ī„Ž ī…‚ī„ī„, ī„“ī…žī„› ī…‚ī„ĩ ī„ĸī„Ŧī…Œ
ī„‘ī„ēī…„
ī„
ī„ŗī…†ī„ļ ī„ī…žī„ļī…‚ī„ ī„ēī„ŗī„Ŗ ī„ļī…Šī„Ŧī…‡ ī„ģī…‚ī„:
â€ĸ ī„ēī…‰ī…žī„Ĩ ī„›ī…„ ī„ģī„Ž?
â€ĸ ī„•ī„ģ ī…ƒī„›ī„ē ī…ƒī„›ī„ēī„ŗ ī„Ŧī…‡ ī„ģī„Ž?
â€ĸ ī„‘ī„ēī…„
ī„
ī„•ī„Žī…§ī…‚ī„ 'ī„Ēī…‡ ī„™ī„¯ī„ĩī…‡ī„šī„Ž ī…ƒī„› ī…ˆ ī„›ī„ĩī„Ŧī…‡ ī„ģī…‚ī„
â€ĸ ī„‘ī„Ēī…‡ ī„‘ī„ēī…„
ī„
ī…‡ī„Ž ī„§ī…‚ī„“ī„ī…Ĩī…‚ī„ŗ ī„ĸī…‚ī„ ī„ēī„­ī…‚ī„ĩī„Ž ī„ēī„ŗī…„ī„›ī„ĩī„Žī…‚ī„ ī„Ŧī…„ ī„ĩī„Ēī…Œ ī„›ī„ĩī„›ī…‡ ī…ƒ ī„ģī…‚ī„ĩī„› ī„ēī„ŗī…žī…ƒī„ēī„’ ī…‚ī„ ī„Žī…†ī…œ ī…ƒī„› ī…ˆ ī„ģī…žī„ļ
ī„›ī„ĩī„Ŧī…‡ ī„ģī…‚ī„
...ī„ī…ƒī„Šī„Ē ī…ƒī„ēī„ĩī…€ ī„’ī„ēī…‚ī„Ž ī„ģī…„ ī„Žī„ģī…„
ī„
ī„ēī„ī…Œ ī„ŗī„žī…‡ī„Ŧī…‚ī„ĩ ī…„ ī„ąī„Š ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰! ī„ēī…‰ī…žī„Ĩ ī…ƒī„Ģī„–ī„ĩī…„ ī„Ŧī…„ ī„ēī……ī…œī„Ŧī„ĩī„Ēī…‚ ī„“ī„ēī„Ŧī…„ ī„ēī„¯ī…žī„šī„Ĩī„Ēī…‚
ī„‘ī„Ēī…‡ ī„ēī…œī„ī„Ļī„Ž ī…ƒ ī…žī„  ī„ģī…‰îĨĄ ī„“ī„ģ ī„›ī…œī…ƒī„¯ī„–ī„Ĩī„ĩ ī…ƒ ī…ƒī„ī„’ī„Ž, ī„Ēī„ĩī„›, ī„‘ī…œī„›ī„ŋī…‚ ī„‘ī„Ēī…‡ ī„ĩī…‹ī„žī…‚ī„Žī…‚ ī„ĸī…„ ī„Ž ī„Ŧī…„ ī„´ī…‹ī„ĸī„Žī…‚ī„ąī…œī„Ŧī…„ ī„ĩī„ī…‡
ī…žī„œ- ī…žī„œ ī„œī…‡ī„Ēī„ĩī…‚ī„ ī…ƒ ī…žī„  ī„ēī…‚ī„§ī…„ ī„ŗī„Ŧī„Ŧ ī„›ī„ĩī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„“ī„ē ī„ļī„”, ī„‘ī„ī„ļī…„ ī…‚ī„ĩ ī„ĸī„Ŧī…Œ ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī„¯ī……ī…žī…ƒī„Ąī„’ ī„ĸī…‚ ī…‡ ī…ƒī„› ī„›ī…‹ī„” ī…ƒī„›ī„ēī…‡ ī„ēī„ŗī…†ī„ģ ī„Ŧī…‚ ī…ƒī„ģī…žī„ēī…‚ ī„ģī…‰ ī„ĸī…‚ī„ ī„Žī„ģī…„
ī„
- ī„´ī…‚ī„Ŧ ī„ĩī…žī„œī…‹, ī„Ēī……ī„ēī…„
ī„
ī„ēī…‰ī…žī„Ĩī…‚ī„ ī„Žī…‚ī„ļ ī„Žī…ƒī„ĸī…žī„Ļ ī„ĩī„ģī…‡ ī„ģī…‹!
ī„˛ī…‚ī„-ī„ēī…„
5. (1) ī„°ī…œī„›ī„šī„Ž ī„Ŧī…„ ī„­ī…‚ī„ĩī„Žī…‚ ī„‘ī„Ēī…‡ ī„¸ī…žī„œ-ī„¸ī…žī„œ ī…ƒī„›ī„ēī„ŗī…‚ī„ ī„Ŧī…‡ ī„°ī…œī„›ī„šī„Žī…‚ī„ ī„Ŧī…„ ī…ƒī„¸ī„’ī…ƒī„œī„’ ī„›ī„ĩī…‹îĨĄ
(ii) ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ ī…ƒī„›
îē’îēîē“îēîēš
îēžî­œīƒš

îēž
īƒ›
ī„ŋīƒœîēžī„žīƒ›
îēžī„ŋīƒš
īŒ ī†īƒš
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„•ī…žī„Ēī„ĩ:ķˇ‡´ķˇ‡ĩķˇ‡ļķˇ‡ˇķˇ‡¸ķˇ‡šī„“ī…žī„› ī„Žī„¸ī…„
ī„
ī„šī……ī„ĩī…†ī„’ī„Ē: ī„ĩī…„ī„’ ī„‘ī„Ēī…‡ ī„ĸī…‚ī„Ŧī…†ī„” ī„ŗī„šī…„ī„Ž ī„Ŧī…„ ī„›ī„ģī…‚ī„Šī…„
ī„“ī…žī„› ī…‚ī„ĩ ī„Ŧī…„ ī„ī…žī„ļ ī„ģī…‰, ī„Žī„ŗī…ƒ ī„ļ ī„Žī…‚ī„ŗ ī„Ŧī…‡ ī„“ī…žī„› ī„Ąī…‹ī„Ĩī…‡ ī…ƒī„ĸī„ģī…‡ ī„›ī„ēī„ąī…‡ ī…ƒ ī…žī„ , ī„ĩī…„ī„’ ī„Žī…‚ī„ŗ ī„Ŧī…„ ī„“ī…žī„› ī„•ī„Ēī„ēī……ī„› ī„›ī……ī„ŋī…„ ī„ĩī…ƒī„ģī…œī„Ŧī…„
ī„ēī…„ ī…ƒī„ĸī„ēī„Žī…†ī…œ ī„¯ī„ģī…‡ī„ļī…„ī„’ī„ ī„ģī…žī„ļ ī„›ī„ĩī„Žī…‚ ī„ąī„ģī……ī„Ē ī„¯ī„ēī…œī„Ŧ ī„ēī…„îĨĄ ī„“ī…žī„› ī…ƒī„Ŧī„Ž, ī„•ī„ēī„Žī…†ī…œ ī„’ī„¯ī„Šī…‡ ī„Ŧī…‚ī„Ŧī…‚ ī„ĸī…„ ī„Ŧī…‡ ī„‘ī„Ĩī…‚ī„ĩī…„ ī…ƒ ī…žī„  ī„“ī…žī„›
ī„‘ī„ĸī…„ī„ą ī„ŗī„šī…„ī„Ž ī…ƒī„ŗī„ļī…„îĨĄ ī„“ī„ē ī„•ī…žī„Ēī…‡ ī„“ī…žī„› ī„ļī…‡ī„ąī„ļ ī„ēī…„: "ī„“ī…žī„› ī„Žī…œ ī„ąī„ĩ ī„“ī„Žī„¯ī……ī„Ĩ ī„›ī„ĩī…‹, ī„‘ī„Ēī…‡ ī„“ī…žī„› ī…ƒ ī„ļī…žī„œī„Š ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ
ī„¯ī…Ĩī…‚ī„¯ī„Ē ī„›ī„ĩī…‹îĨĄ" ī„ŗī…‹ī…ƒī„ģī„Ē ī„ģī…‹ ī„›ī…‡, ī„•ī„ēī„Žī…‡ ī„“ī„ē ī…ƒ ī…žī„  ī„“ī…žī„› ī„Žī…œ ī„ąī„ĩ ī„›ī…‚ī„ĩī„§ ī„ĩī…žī…ƒī„œī„’—5, ī„‘ī„Ēī…‡ ī„“ī…žī„› ī„ģī…‹ī„ĩ ī„›ī…‚ī„ĩī„§ ī…ƒī„Žī„›ī…ƒī„ļī„’
ī…ƒī„ĸī„ē ī„•ī…žī„Ēī…‡ 13 ī…ƒī„ļī…ƒī„œī„’ ī„ģī…‹ī„“ī„’ ī„ēī…„îĨĄ ī…ƒī„°ī„ĩ ī„•ī„ēī„Žī…‡ 6 ī„Ŧī…„ ī„›ī…‹ī…ƒī„šī„š ī„›ī…„ī„Ēī…„ ī„‘ī„Ēī…‡ 15 ī„¯ī…Ĩī…‚ī„¯ī„Ē ī„›ī…„ī„Ēī…‡, ī„‘ī„Ēī…‡ ī„“ī„ē ī„Ēī„ĩī…§ī…‚ī„ ī„ģī…„îĨĄ
ī„•ī„ēī„Ŧī…‡ ī„Ŧī…‚ī„Ŧī…‚ ī„ĸī…„ ī„Žī…‡ ī„ŗī……ī„ēī„›ī„ĩī…‚ī„•
ī„
ī„Ŧī…‡ ī„ģī…‹ī„— ī…ƒī„›ī„ģī…‚, "ī„ĩī…„ī„’, ī„ĸī…‹ ī„Ēī……ī„ēī…„
ī„
ī„ģī……ī„Šī…‡ ī„ĩī…ƒī„Ēī„’ ī„ģī…‰ ī„•ī„ģ ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī„Ŧī…„ ī„‘ī„ēī„ļ
ī…ƒī„žī…œī„Ŧī„ī…„ ī„Ŧī…„ ī„•ī„Ŧī…‚ī„ģī„ĩī„Š ī„Ēī…Œ ī„“ī„ļī…‚ ī…‚ ī„›ī……ī„Ŗ ī„Žī„ģī…„
ī„
ī„ģī…‰!"
ī„“ī„ē ī„Žī…‚ī„ļ ī„•ī„ēī„Ŧī…„ ī„•ī„Ēī„ēī……ī„›ī„Ēī…‚ ī„­ ī„ī„” - ī„‘ī„Ēī…‡ ī„šī…‚ī„“ī„Ŧ ī„Ēī……ī„ģī…‚ī„§ī…„ ī…„îĨĄ
ī„’ī„™ ī„°ī…œī„›ī„šī„Žī…‚ī„ ī„Ŧī…„ ī„“ī„ē ī„šī…‚ī„Žī„Ŧī…‚ī„ĩ ī„´ī…‚ī„Ēī„ĩī…‚ ī„Ŧī…„ ī„šī……ī„ĩī…†ī„’ī„Ē ī„›ī„ĩī…„ī„—, ī„•ī„ģī„Žī…‚ī„ ī„Žī…†ī…œ ī„§ī…†ī…œī„žī…‚ī„” ī„Žī…‚ī„ļ ī„ēī„ŗī„Ŗī…„ī„—, ī„‘ī„Ēī…‡ ī…ƒī„°ī„ĩ
ī„ēī…„ī„ŗī…‚ ī„ēī„ŗī…žī…ƒī„ēī„’ ī„Ŧī…‚ ī„ŗī……ī„ļī…‚ī„ī„›ī„Š ī„›ī„ĩī„Ž ī…žī„ļ ī„­ī…„ī„— ī„ĸī…‹ ī…ƒī„› ī„ŗī……ī„šī„›ī„ļ ī„ĸī…‚ī„¯ī„Ŧī…„ ī„ģī…‰ ī„¯ī„ĩ ī„ēī„ģī…„ ī„ŗī…‚ī„ĩī„ī„Ŧī„ĩī„šī„Ž ī„Žī…‚ī„ļ ī„ąī„ģī……ī„Ē
ī„ēī„ĩī„ļ ī„ģī…‹ ī„ĸī…‚ī„ī„Ŧī…„ ī„ģī…‰îĨĄ
ķš´Ąķš´ĩķš´Ŗķš´¤ī„˛ī…‚ī„ (i): ī„°ī…œī„›ī„šī„Ž ī„Ŧī…„ ī„­ī…‚ī„ĩī„Žī…‚ ī„‘ī„Ēī…‡ ī„°ī…œī„›ī„šī„Žī…‚ī„ ī„Ŧī…„ī„’ī„ ī„›ī„” ī…ƒī„›ī„ēī„ŗī…‚ī„
ķš¸¯ķš¸­ķš¸Ž1. ī„°ī…œī„›ī„šī„Ž ī„›ī…„ ī„ģī…‰?
ī„ēī„ĩī„ļ ī„šī„ąī„Ŧī…‚ī„ ī…ƒī„¸ī…žī„ :
ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī„“ī…žī„› ī„œī…‚ī„ē ī…ƒī„ĩī„šī„Ēī…‚ ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰ ī…ƒī„ĸī…žī„Ģī…‡ ī„ģī„ĩī…‡ī„› ī„“ī„Žī„¯ī……ī…žī„Ĩ ī„Ŧī…‚ ī„“ī…žī„› ī„ģī…„ ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„ĸī…‚ī„, ī…ƒī„ĸ ī…ˆ ī…ƒī„› ī„ĩī…„ī„’ ī„Ŧī…‡ ī„Ŧī…‚ī„Ŧī…‚ ī„ĸī…„ ī„Žī…‡ ī…ƒī„›ī„ģī…‚ ī„ēī…„, "ī„ĸī…‡ ī„Ēī……ī„ēī…„
ī„
ī„ŗī„šī…„ī„Ž ī„Žī…†ī…œ ī„“ī…žī„› ī„Žī…œ ī„ąī„ĩ ī…ƒī„Ŧī…œī„Ŧī…‡ ī„ģī…‹, ī„Ēī…‚ī„ ī„“ī„ģ ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī…ƒī„ēī„ĩī…€ ī„“ī…žī„›
ī„Žī…œ ī„ąī„ĩ ī…‚ī„¯ī„ē ī„Ŧī…‡ ī…‡ī„ī…‚îĨĄ ī„›ī…‹ī„” ī„•ī„ļī„Ŗī„Š ī„Žī„ģī…„
ī„
, ī„›ī…‹ī„” ī„Ŧī…‹ī„ģī„ĩī…‡ ī„ĸ ī…‚ī„ą ī„Žī„ģī…„
ī„
îĨĄ"
ī„ī…ƒī„Šī…ƒī„Ēī„› ī„¯ī…ƒī„ĩī„˛ī…‚ī„šī…‚:
ī„“ī…žī„› ī„ēī…‰ī…žī„Ĩ A ī„Ēī…Œ ī„“ī…žī„› ī„ēī…‰ī…žī„Ĩ B ī„Ēī…žī„› ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī„“ī…žī„› ī…ƒī„Žī„´ī„ŗ ī„ģī…‰ ī„ĸī…‹ ī„ēī…‰ī…žī„Ĩ A ī…ƒ ī…žī„  ī„ģī„ĩī…‡ī„› ī„˜ī„ļī…„ī„ŗī…Šī„Ĩ x ī„Žī…†ī…œ ī…ƒī„Žī„ĩī„­ī…‚ī„ĩī„Ē
ī„›ī„ĩī„Ŧī…‚ ī„ģī…‰, ī„ēī…‰ī…žī„Ĩ B ī…ƒ ī…žī„  ī…ƒī„ąī„ļī„›ī……ī„ļ ī„“ī…žī„› ī„˜ī„ļī…„ī„ŗī…Šī„Ĩ f(x)îĨĄ
ī„‘ī„ēī…„
ī„
ī„“ī„ēī„Žī…†ī…œ ī„“ī„ē ī„Ēī„ĩī…§ī…‚ī„ ī…ƒī„ļī„œī„Ŧī…‡ ī„ģī…‚ī„:
f:A→B
ī…ƒī„›ī…žī„Ģī…‡:
â€ĸ A ī„Žī…†ī…œ ī„§ī…‹ī„ŗī…‡ī„Ž (ī„“ī„Žī„¯ī……ī„Ĩī„ē ī„Ŧī…‚ ī„ēī…‰ī…žī„Ĩ) ī…ƒī„›ī„ģī…‚ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ
â€ĸ B ī„Žī…†ī…œ ī„›ī…‹ī„§ī…‹ī„ŗī…‡ī„Ž (ī„ēī…œī„˛ī…‚ī…ƒ ī„Ē ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ ī„Ŧī…‚ ī„ēī…‰ī…žī„Ĩ) ī…ƒī„›ī„ģī…‚ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ
â€ĸ f(x) ī„Žī…†ī…œ x ī„Žī…‚ī„ļ ī„ēī…œī„ąī…œī…ƒī„­ī„Ē ī…ƒī„ ī…žī„Ēī„ĩ ī„ĸī…‚ī„ ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ ī…ƒī„›ī„ģī…‚ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ķš´‚ķš´ƒķš´„ķš´…ķš´‰ķš´Šķš´†ķš´‹ķš´‡ķš´ˆī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
ī„’ī„™ ī„›ī…ƒī„ģī…œī„Ŧī…‡ ī„ģī…‚ī„:
f(x)=2x+3f
ī„ĸī…‡ī„›ī„ĩ ī„Ēī……ī„ēī…„
ī„
x = 1 ī„“ī„Žī„¯ī……ī…žī„Ĩ ī„›ī„ĩī„Ŧī…‡ ī„ģī…‹, ī„Ēī…‚ī„ ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī…ƒī„ŗī„ļī„Ŧī…‚ ī„ģī…‰:
f(1)=2(1)+3=5
ī„“ī„Žī„¯ī……ī…žī„Ĩ x = 2, ī„‘ī„Ēī…‡:
f(2)=2(2)+3=7f
ī„“ī„ģ ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī„ģī…‰ ī…ƒī„›ī„•
ī„
ī…ƒī„› ī„ģī„ĩī…‡ī„› ī„“ī„Žī„¯ī……ī„Ĩ ī…ƒī„ēī„ĩī…€ ī„“ī…žī„› ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ ī…ƒī„Ŧī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
ķŧ¨ķŧ¨‘ķŧ¨’2. ī„°ī…œī„›ī„šī„Ž ī„Žī…‚ī„ļ ī„ēī„ąī…œī„­ī„Ē ī„ŗī„ģī…žī„Ēī„¸ī„¯ī…†ī„ĩī„Ž ī„šī„ąī„Ŧ
1. ī„§ī…‹ī„ŗī…‡ī„Ž: ī„ēī…‚ī„ĩī…‡ ī„“ī„Žī„¯ī……ī…žī„Ĩ ī„ŗī……ī…žī„ļī…‚ī„ (x-ī„ŗī……ī…žī„ļ) ī„Ŧī…‚ ī„ēī…‰ī…žī„ĨîĨĄ
2. ī„ēī…ƒī„ģ-ī„§ī…‹ī„ŗī…‡ī„Ž:ī„•ī„ģ ī„ēī…‰ī…žī„Ĩ ī…ƒī„ĸī„ē ī…ƒ ī…žī„  ī„ēī…‚ī„ĩī…‡ ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ ī„’ī„•
ī„
ī„Ŧī…‡ ī„ģī„Ž (ī„ģī„ŗī…‡ī„šī…‚ ī„ēī…‚ī„ĩī…‡ ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ ī„Žī„ģī…„
ī„
ī„ĩī„Ēī…‡
ī„ĸī…‚ī„ī„Ŧī…‡)îĨĄ
3. ī„ēī…„ī„ŗī…‚:ī„‘ī„ēī„ļ ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ ī„ŗī……ī…žī„ļī…‚ī„ ī„Ŧī…‚ ī„ēī…‰ī…žī„Ĩ ī„ĸī…‹ ī„°ī…œī„›ī„šī„Ž ī…ƒī„Ŧī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
ķŧ¨ģķŧ¨ŧ3. ī…ƒī„¸ī„žī…†ī„‘ī„ļī…‚ī„”ī…ƒī„žī…œī„ ī„°ī…œī„›ī„šī„Ž
ī„“ī…žī„› ī„¸ī…Šī…ƒī„§ī…œī„ ī„ŗī„šī…„ī„Ž ī„ąī…‚ī„ĩī…‡ ī„ēī…‹ī„ ī…‹:
â€ĸ ī„Ēī……ī„ēī…„
ī„
ī„ąī„Ĩī„Ž A1 (ī„“ī„Žī„¯ī……ī„Ĩ) ī„Ŧī„ąī…‚ī„™îĨĄ
â€ĸ ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī„ģī„ŗī…‡ī„šī…‚ ī„ļī…‡ī„‘ī„ž ī…ƒī„ ī„¯ī„ē (ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ) ī…ƒī„ŗī„ļī„Ŧī…„ ī„ģī…‰îĨĄ
â€ĸ ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī„›ī„Ŧī…‡ ī…„ ī„“ī…žī„›ī…‹ ī„ąī„Ĩī„Ž ī„ļī„” ī„ļī…‡ī„‘ī„ž ī„‘ī„Ēī…‡ ī„›ī…‹ī„› ī„Ŧī…‹ ī…ˆ ī„Žī„ģī…„
ī„
ī…ƒī„ŗī„ļī„Ŧī…‡îĨĄ
ī„“ī„ģ ī…Šī…ƒī„§ī…œī„ ī„ŗī„šī…„ī„Ž ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī„Žī…†ī…œ ī„Ŧī„ĩī„ēī…‚ī„•
ī„
ī„Ŧī…„ ī„ģī…‰ ī…ƒī„›ī„•
ī„
ī…ƒī„› ī„ģī„ĩ ī„ąī„Ĩī„Ž ī…ƒī„ąī„ļī„›ī……ī„ļ ī„“ī…žī„› ī„ ī…„ī„ž ī…ƒī„Ŧī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
ķš´ˇķš´ēķš´¸ķš´šķš´ģķš´ŧķš´Ŋķš´žķš´ŋķšĩ€ķšĩķšĩ‚4. ī„¸ī…žī„œ-ī„¸ī…žī„œ ī…ƒī„›ī„ēī„ŗī…‚ī„ ī„Ŧī…‡ ī„°ī…œī„›ī„šī„Ž (ī„•ī„Ŧī…‚ī„ģī„ĩī„Šī…‚ī„ ī„Ŧī…‡ ī„Žī…‚ī„ļ)
ī„’ī„™ ī„ģī……ī„Š ī„°ī…œī„›ī„šī„Žī…‚ī„ ī„Ŧī…„ ī„ēī……ī…œī„Ŧī„ĩ ī…ƒ ī…ƒī„˛ī…œī„Žī„Ēī…‚ ī„Ŧī…„ ī„¯ī„ŋī„ ī…‹ī„ļ ī„›ī„ĩī…„ī„—, ī„ģī„ĩī…‡ī„› ī„Ŧī…„ ī„ī…ƒī„Šī„Ē ī…ƒ ī…žī„  ī„’ī„¯ī„Šī…„ ī„˛ī…†ī…ƒī„ŗī„›ī…‚ ī„ģī…‰îĨĄ
ķšģ€(i) ī„“ī…žī„›-ī„Ēī…Œ-ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž (ī„“ī…œī„ĸī…‰ī„›ī…ƒī„Ĩī„¸)
ī„§ī…‹ī„ŗī…‡ī„Ž ī„Ŧī…‚ ī„ģī„ĩī…‡ī„› ī„Ēī…žī„Ē ī„›ī…‹ī„§ī…‹ī„ŗī…‡ī„Ž ī…ƒ ī…žī„  ī„“ī…žī„› ī…ƒ ī„ļī…žī„œī„Š ī„Ēī…žī„Ē ī„Žī…‚ī„ļ ī„ŗī…‡ī„ļ ī„œī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
f(x)=x+5f
ī„ēī…‚ī„ĩī…‡ x ī„ļī„”, ī„ŗī……ī…žī„ļ ī…žī„œī„ĩī…‚ ī„‘ī„Ēī…‡ ī…ƒ ī„ļī…žī„œī„Š ī„ģī…‰îĨĄ
ķšģ€(ii) ī„°ī…œī„›ī„šī„Ž ī„¸ī…žī„ļ (ī„ēī„ĩī„ĸī…‰ī„›ī…ƒī„Ĩī„¸)
ī„›ī…‹ī„§ī…‹ī„ŗī…‡ī„Ž ī…ƒ ī…žī„  ī„ģī„ĩī…‡ī„› ī„Ēī…žī„Ē ī„§ī…‹ī„ŗī…‡ī„Ž ī„Ēī…Œ ī„žī…žī„Ĩī…‹-ī„žī…žī„Ĩ ī„“ī…žī„› ī„Ēī…žī„Ē ī„Ŧī…‚ ī…ƒī„ ī…žī„Ēī„ĩ ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
f(x)=x
ī„ēī…‚ī„ĩī…‡ ī„‘ī„ēī„ļ ī„Žī…œ ī„ąī„ĩ ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ (codomain = ī„ĩī…ˆī„ĸ) ī„Ŧī…‡ ī„ĩī…†ī„¯ ī…ƒ ī…žī„  ī„› ī„ĩ ī„›ī…„ī„Ēī…‡ ī„ī„— ī„ģī„ŽîĨĄ
ķšģ€(iii) ī„“ī…žī„›-ī„Ēī…Œ-ī„“ī…žī„› ī„‘ī„Ēī…‡ 'ī„Ēī…‡ (ī„Ŧī…‹ī„˛ī…‚ī„šī…„ ī„°ī…œī„›ī„šī„Ž)
ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī„ĸī…‹ ī„“ī…žī„›-ī„Ēī…Œ-ī„“ī…žī„› ī„‘ī„Ēī…‡ ī„šī„Ž ī„Ŧī…‹ ī…ˆ ī„ģī…‰îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
f(x)=x
ī„“ī„ģ ī„¯ī„Ąī…‚ī„Š ī„°ī…œī„›ī„šī„Ž ī„ģī…‰â€”ī„ģī„ĩī…‡ī„› ī„ŗī……ī…žī„ļ ī„’ī„¯ī„Šī…‡ ī„’ī„¯ ī…ƒ ī…žī„  ī„ŗī…‰ī„¯ ī„›ī„ĩī„Ŧī…‚ ī„ģī…‰îĨĄ
ķšģ€(iv) ī„ēī…ƒī„Ģī„ĩ ī„°ī…œī„›ī„šī„Ž
ī„ģī„ĩī…‡ī„› ī„“ī„Žī„¯ī……ī„Ĩ ī„ļī„” ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ ī„“ī…žī„›ī…‹ ī…ƒī„ĸī„ģī…‚ ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
ī„ēī…‚ī„ĩī…‡ x ī„ļī„” f(x)=7
ī„Ēī……ī„ēī…„
ī„
ī„ĸī…‹ ī…„ x ī„ ī……ī„Šī„Ŧī…‡ ī„ģī…‹, ī„Žī„Ēī…„ī„ĸī…‚ ī„ģī„ŗī…‡ī„šī…‚ 7 ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
ķšģ€(v) ī„¯ī„Ąī…‚ī„Š ī„°ī…œī„›ī„šī„Ž
f(x)=x
ī„Ŧī…‡ī„œī„Š ī„Žī…†ī…œ ī„Ēī…‚ī„ ī„“ī„ģ ī„ąī„ģī……ī„Ē ī„ģī…„ ī„ēī…‚ī„Ŧī…‚ ī„ļī…žī„ī„Ŧī…‚ ī„ģī…‰! ī„“ī„Žī„¯ī……ī…žī„Ĩ ī„‘ī„Ēī…‡ ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ ī„“ī…žī„›ī…‹ ī…ƒī„ĸī„ģī…‡ ī„ģī„ŽîĨĄ
ķšģ€(vi) ī„ąī„ģī……ī„¯ī„Ŧ ī„°ī…œī„›ī„šī„Ž
ī„°ī…œī„›ī„šī„Ž ī…ƒī„ĸī„Žī…§ī…‚ī„ ī…ƒī„¸ī…žī„  ī„šī„ąī„Ŧ ī„šī…‚ī„ŗī„ļ ī„ģī……ī…œī„Ŧī…‡ ī„ģī„Ž ī…ƒī„ĸī„¸ī…ˆ ī…ƒī„›:
f(x)=4x2+3x+7
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ķšģ€(vii) ī„Ēī„ĩī„›ī„šī…„ī„ļ ī„›ī…‚ī„ĩī„ĸ
ī„°ī…‚ī„ĩī„ŗ ī„Ŧī…‚ ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž:
ī…ƒī„ĸī…žī„Ģī…‡ P(x) ī„‘ī„Ēī…‡ Q(x) ī„ąī„ģī……ī„¯ī„Ŧ ī„ģī„Ž ī„‘ī„Ēī…‡ Q(x) ≠ 0 ī„ģī…‰îĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
ķšģ€(viii) ī„ēī„ŗ ī„‘ī„Ēī…‡ ī…ƒī„¸ī„šī„ŗ ī„°ī…œī„›ī„šī„Ž
â€ĸ ī„ēī„ŗ ī„°ī…œī„›ī„šī„Ž: f(−x)=f(x)
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:f(x)=x2
â€ĸ ī„šī„§ ī„°ī…œī„›ī„šī„Ž: f(−x)=−f(x)
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:f(x)=x3
ķšģ€(ix) ī„¯ī…„ī„ēī„¸ī…‚ī„”ī„ž ī„°ī…œī„›ī„šī„Ž
x ī„Ŧī…‡ ī…žī„œ- ī…žī„œ ī„‘ī…œī„Ēī„ĩī…‚ī„ļī…‚ī„ ī„ļī„” ī…žī„œ- ī…žī„œ ī„ēī„ŗī…„ī„›ī„ĩī„Žī…‚ī„ ī„Ŧī……ī„’ī„ĩī…‚ ī„¯ī…ƒī„ĩī„˛ī…‚ī…ƒī„šī„ĒîĨĄ
ī„•ī„Ŧī…‚ī„ģī„ĩī„Ž:
ķšģ€(x) ī„žī…‚ī„Ēī„› ī„‘ī„Ēī…‡ ī„ļī„žī…†ī„ī„Šī„› ī„°ī…œī„›ī„šī„Ž
ķšģ€(xi) ī„ŗī…‚ī…ƒī„§ī„–ī„ļī„ē ī„°ī…œī„›ī„šī„Ž
f(x)=ī„˜ī„›ī„ē
ī„ģī„ŗī…‡ī„šī…‚ ī„ī…‰ī„ĩ-ī„Žī…‰ī„ī…‡ī…ƒī„Ĩ ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ ī…ƒī„Ŧī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ķšģ€(xii) ī…ƒī„ēī„ī„Žī„ŗ ī„°ī…œī„›ī„šī„Ž
ķŧŠ•ķŧŠ–ķŧŠ—ķŧŠ˜ķŧŠ™ķŧŠšī„˛ī…‚ī„ (ii): ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ ī…ƒī„›
ķš¸¯ķš¸­ķš¸Žī„›ī„Ŧī„ŗ-ī„Ŧī„ĩ-ī„›ī„Ŧī„ŗ ī„›ī„ģī…‚ī„Šī…„: ī„ēī…„ī„ŗī…‚ī„¸ī…‚ī„ ī„Ŧī…‡ ī„ĩī„ģī…žī„ē ī„Žī…†ī…œ ī„Ēī…‹ī„ŋī„Žī…‚
ī„’ī„™ ī„ĩī…„ī„’ ī…žī„ļ ī…‚ī„¯ī„ē ī„ ī…žī„ļī…„ī„—îĨĄ ī„“ī…žī„› ī…ƒī„Ŧī„Ž, ī„•ī„ēī„Žī…†ī…œ ī„“ī…žī„› ī„•ī„ļī„Ŗī„Š ī…‚ī„ļī…‚ ī„¯ī…Ĩī„ī„Ĩī…‚ ī…‚ ī…ƒī„ŗī…ƒī„ļī„’:
"ī„“ī„ē ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„Ŧī…‚ ī„›ī…„ ī„ģī…‹ ī…‡ī„ī…‚ ī„ĸī„Ŧī…Œ x ī„ēī…žī„ ī„ŗī……ī…žī„ , ī„ēī…žī„ ī„ŗī……ī…žī„  1 ī„Ŧī…‡ ī„Žī…‡ī„ŋī…‡ ī„’ ī„ĸī…‚ ī…‡ī„ī…‚?"
ī„•ī„ēī„Žī…‡ ī„Ŧī…‡ī…ƒī„œī„’:
ī„ĸī„Ŧī…Œ ī„•ī„ēī„Žī…‡ x = 1 ī„Žī…†ī…œ ī…ƒī„ēī…žī„­ī…‚ ī„¯ī„ļī…žī„ ī„“ī„Ž ī„›ī…„ī„Ēī…‚:
ī„“ī„ēī„Žī…†ī…œ ī„‘ī…ƒī„Žī„šī…ƒī„ ī„Ē ī„ĩī…†ī„¯ ī…ƒī„›ī„ģī…‚ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ ī…ƒī„ĸ ī…ˆ ī„ĸī„Ŧī…Œ ī„Ŧī…‹ ī„ąī……ī„Ŗī…‚ī„ĩī„Ē ī„Ŧī…‡ ī„Ĩī……ī„›ī„ŋī…‡ ī„“ī…žī„›ī…‹ ī…ƒī„ĸī„ģī…‡ ī…ƒī„Ŧī„œī…‚ī„” ī…ƒī„Ŧī…œī„Ŧī…‡ ī„ģī„Ž ī„¯ī„ĩ ī…ƒī„°ī…žī„Ĩ
ī„Žī„ģī…„
ī„
ī„ąī…‰ī„Ļī„Ŧī…‡ - ī„ēī…‚ī„Žī…†ī…œ ī„ēī„ĩī„ļ ī„ąī„Šī…‚ī„•ī„Š ī„Ŧī…„ ī„ļī…‹ī„ŋ ī„ģī…‰!
ķŧŦ°ķŧŦŽķŧŦ¯ī„›ī„Ŧī„ŗ 1: ī„‘ī…œī„›ī…‚ī„ ī„Žī…†ī…œ ī„ī……ī„Šī„Žī„œī…œī…ƒī„§ī„Ē ī„›ī„ĩī…‹
ī„¯ī…Ĩī„ī„Ĩī…‚ī„¸ī…‚ ī„“ī„ģ ī„ģī…‰:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„’ī„™ ī„‘ī…œī„š ī„Ŧī…‚ ī„ī……ī„Šī„Žī„œī…œī„§ ī„›ī„ĩī…„ī„—:
ī„Ēī…‚ī„ ī„ģī……ī„Š ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„ąī„Š ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰:
ķˇƒ†ķŧŊĸī„‘ī…œī…ƒī„Ēī„ŗ ī„ĸī„¸ī…‚ī„ą:
ķˇ—­ķˇ—¨ķˇ—Šķˇ—Ēķˇ—Ģķˇ—Ŧī„“ī„ģ ī…ƒī„›ī„•
ī„
ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰:
ī„ĸī„Ŧī…Œ ī„Ēī……ī„ēī…„
ī„
ī„“ī…žī„› ī„ēī…„ī„ŗī…‚ ī„Ŧī…‚ ī„ēī…‚ī„ģī„ŗī„Šī…‚ ī„›ī„ĩī„Ŧī…‡ ī„ģī…‹ ī„ĸī…‹ 0/0 ī…ƒī„Ŧī…œī„Ŧī…„ ī„ģī…‰, ī„Ēī…‚ī„ ī„“ī„ēī„Ŧī…‚ ī„’ī„ŗ ī„Ēī…ī„ĩ 'ī„Ēī…‡ ī„ŗī„Ēī„ļī„ą ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰:
â€ĸ ī„‘ī…œī„š ī„‘ī„Ēī…‡ ī„˛ī…‚ī„ĸ ī„Ŧī…‚ ī„“ī…žī„› ī„ēī…‚ī„ī„Ŗī…‚ ī„ī……ī„Šī„Žī„œī…œī„§ ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
â€ĸ ī„’ī„ŗ ī„ī……ī„Šī„› ī„Žī…†ī…œ ī„ģī„Ĩī…‚ī„™îĨĄ
â€ĸ ī…ƒī„°ī„ĩ x ī„Ŧī…‡ ī„ŗī……ī…žī„ļ ī„Žī…†ī…œ ī…ƒī„ēī…žī„­ī…‚ ī„ąī„Ŧī„ļ ī…ƒī„Ŧī„™îĨĄ
ķˇ•˜ķˇ•™ķˇ•šī…ƒī„ēī…žī„Ĩī…‚: ī„›ī…‚ī„ĩī„ĸī…‚ī„ ī„‘ī„Ēī…‡ ī„ēī…„ī„ŗī…‚ī„¸ī…‚ī„ ī…ƒī„¯ī…žī„Ąī…‡ ī…ƒī„ēī„’ī„Šī„¯
ī„“ī„ē ī„´ī…‚ī„Ēī„ĩī…‚ ī…ƒī„¸ī…žī„ , ī„‘ī„ēī…„
ī„
ī…ƒī„ēī…žī…ƒī„œī„’ ī„ģī…‰:
â€ĸ ī„°ī…œī„›ī„šī„Ž ī„˛ī„ĩī…‹ī„ēī…‡ī„´ī…‹ī„ ī„ŗī„šī…„ī„Žī…‚ī„ ī…‚ī„ī„ ī„ģī„Ž: ī„“ī…žī„› ī„“ī„Žī„¯ī……ī…žī„Ĩ = ī„“ī…žī„› ī„’ī„•ī„Ĩī„¯ī……ī…žī„ĨîĨĄ
â€ĸ ī„“ī„ģ ī„›ī„” ī„’ī„›ī…‚ī„ĩī…‚ī„ ī„‘ī„Ēī…‡ ī„˛ī…†ī…ƒī„ŗī„›ī…‚ ī…‚ī„ ī…ƒ ī…žī„  ī„’ī„•
ī„
ī„Ŧī…‡ ī„ģī„Žâ€”ī„ĩī…‡ī…ƒī„œī„› ī„Ēī…Œ ī„žī…‚ī„Ē ī„‘ī…œī„›ī…„ ī„‘ī„Ēī…‡ ī„Ĩī……ī„›ī„ŋī…‡ ī„‘ī„Žī……ī„ēī…‚ī„ĩîĨĄ
â€ĸ ī„‘ī„Ēī…‡ ī„ēī…„ī„ŗī…‚ ī…‚ī„ ī„ēī…‚ī„Žī…†ī…œ ī„“ī„ģ ī„ēī„ŗī„Ŗī„Š ī…ƒ ī…žī„  ī„ŗī„Ŧī„Ŧ ī„›ī„ĩī„Ŧī…„ī„’ī„ ī„ģī„Ž ī…ƒī„› ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī„“ī…žī„› ī…ƒī„ąī…œī„Ŧī…† ī„Ŧī……ī„’ī„ļī…‡ ī„›ī…„
ī„›ī„ĩ ī…ƒī„ĩī„ģī…‚ ī„ģī…‰, ī„œī…‚ī„ē ī„›ī„ĩī„›ī…‡ ī„ĸī„Ŧī…Œ ī…ƒī„ēī…žī„­ī…‚ ī„ąī„Ŧī„ļ ī„•ī„ļī„Ŗī„Š ī…‚ī„ļī…‡ ī„ĩī…†ī„¯ ī…ƒī„Ŧī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„ĩī…„ī„’, ī„•ī„Ēī„ēī……ī„› ī„›ī……ī„ŋī…„, ī„Žī…‡ ī„’ī„œī„ĩī„›ī…‚ī„ĩ ī„’ī„¯ī„Šī…‡ ī„Ŧī…‚ī„Ŧī…‚ ī„ĸī…„ ī„Ŧī…„ ī„ŗī„Ŧī„Ŧ ī„Žī…‚ī„ļ ī„’ī„¯ī„Šī…„ "ī„°ī…œī„›ī„šī„Ž ī„ŗī„šī…„ī„Ž" ī„ąī„Šī…‚ī„”îĨĄ ī„‘ī„Ēī…‡
ī„ģī„ĩ ī…‚ī„ĩ ī„ĸī„Ŧī…Œ ī„•ī„ģ ī…ƒī„›ī„ēī…‡ ī„ąī……ī„Ŗī…‚ī„ĩī„Ē ī„Ŧī…‚ ī„ēī…‚ī„ģī„ŗī„Šī…‚ ī„›ī„ĩī„Ŧī…„ ī„ēī…„, ī„•ī„ēī„Žī…†ī…œ ī„´ī…‚ī„Ŧ ī„’ī„•
ī„
ī„Ŧī…‚ ī„ēī…„:
"ī„ĸī…‡ī„›ī„ĩ ī„ĸ ī…‚ī„ą ī„¯ī…ƒī„ģī„ļī…‚ī„ ī„Ēī…‚ī„ ī„•ī„ļī„Ŗī„Š ī…‚ī„ļī…‚ ī„ļī…žī„ī„Ŧī…‚ ī„ģī…‰ ī„Ēī…‚ī„ ī„žī„ąī„ĩī…‚ī„™ ī„Žī…‚îĨĄ ī„ąī„ē ī„“ī„ēī„Žī…†ī…œ ī„°ī…‰ī„›ī„Ĩī„ĩ ī„›ī„ĩī…‹, ī„“ī„ēī„Žī…†ī…œ ī„ēī„ĩī„ļ
ī„ąī„Šī…‚ī„™, ī„‘ī„Ēī…‡ ī„Ēī„ĩī„› ī„Žī…†ī…œ ī„’ī„¯ī„Šī…‚ ī„ŗī…‚ī„ĩī„ī„Ŧī„ĩī„šī„› ī„ąī„Šī„Ž ī…ƒī„Ŧī„™îĨĄ"
ķŊ„ģķŊ„ŧķŊ„Ŋī„›ī„ģī…‚ī„Šī…„ ī„Ŧī…‚ ī„Žī…‰ī…ƒī„Ēī„› ī…ƒī„ēī„­ī…‚ī„ī„Ē
ī„ī…ƒī„Šī„Ē ī…ƒī„ēī„ĩī…€ ī„‘ī…œī„›ī…‚ī„ ī„Žī…†ī…œ ī„ģī…žī„ļ ī„›ī„ĩī„Ž ī„ąī…‚ī„ĩī…‡ ī„Žī„ģī…„
ī„
ī„ģī…‰ - ī„“ī„ģ ī„¯ī…‰ī„Ĩī„ĩī„Žī…‚ī„ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Š, ī„ēī„ąī…œī„­ ī„ąī„Šī…‚ī„•ī„Š ī„‘ī„Ēī…‡ ī„¯ī„ĩī„Ŧī…‡ ī…ƒī„¯ī…žī„Ąī…‡ ī„Ŧī…‡
ī„ĸī…‚ī„Ŧī…† ī„Žī…†ī…œ ī„•ī„ĸī…‚ī„ī„ĩ ī„›ī„ĩī„Ž ī„ąī…‚ī„ĩī…‡ ī„ģī…‰îĨĄ
6. (i) ī„“ī…žī„› ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„°ī…œī„›ī„šī„Ž ī„‘ī„Ēī…‡ ī„‘ī„ēī…œī„Ēī……ī„šī„Ĩ ī„°ī…œī„›ī„šī„Ž ī…ƒī„¸ī…žī„  ī„‘ī…œī„Ēī„ĩ ī„Ŧī…žī„ēī…‹îĨĄ
(ii) ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ ī…ƒī„› x = 2 ī„Ēī…‡ ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„ģī…‰
îēž
īƒ›
ī„žīƒš
îēž
īƒ›
ī„ŋīƒš

ī„•ī…žī„Ēī„ĩ:ī„›ī„ļī„¯ī„Žī…‚ ī„›ī„ĩī…‹ ī…ƒī„› ī„Ēī……ī„ēī…„
ī„
ī„“ī…žī„› ī…ƒī„Žī„ĩī…ƒ ī„žī„Ž ī„¯ī„ģī…‚ī„ŋī…„ ī„ĩī„ēī„Ēī…‡ 'ī„Ēī…‡ ī„ ī…žī„ļ ī„ĩī„ģī…‡ ī„ģī…‹îĨĄ ī„ĩī„ēī„Ēī…‚ ī„ēī…‚ī…€ ī„ģī…‰, ī„‘ī„Ēī…‡ ī„Ēī……ī„ēī…„
ī„
ī…ƒī„ąī„Žī…‚ī„
ī„ĩī……ī„›ī…‡ ī„ĸī…‚ī„ ī„Ąī…‚ī„ļ ī„ŗī…‚ī„ĩī…‡ ī„Ēī……ī„ĩī„Ŧī…‡ ī„ĩī…ƒī„ģ ī„ēī„›ī„Ŧī…‡ ī„ģī…‹îĨĄ ī„‘ī„ ī…‚ī„Žī„›, ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī„’ī„¯ī„Šī…‡ ī„ĩī„ēī„Ēī…‡ ī…ƒ ī…žī„  ī„“ī…žī„› ī„Ĩī……ī…žī…ƒī„Ĩī„’ ī„ģī…‹ī„“ī„’ ī„¯ī……ī„ļ
ī…ƒī„ŗī„ļī„Ŧī…‚ ī„ģī…‰ - ī„ģī……ī„Š ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī„ĸī…‚ī„ ī„Ēī…‚ī„ ī„œī…žī„§ ī„•ī…žī„Ēī…‡ ī„Ąī…‚ī„ļ ī„ŗī…‚ī„ĩī„Žī…„ ī„ ī…‚ī„ģī…„ī„Ŧī…„ ī„ģī…‰ ī„ĸī…‚ī„ ī„ŗī……ī„ŋī„Žī…‚ ī„ ī…‚ī„ģī…„ī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„ī…ƒī„Šī„Ē ī„Ŧī…„ ī„Ŧī……ī„Žī…„ī„’ ī…ƒ ī…žī„ , ī„°ī…œī„›ī„šī„Ž ī„•ī„ēī…‡ ī„ĩī„ēī„Ēī…‡ ī…‚ī„ī„ ī„ģī…„ ī…ƒ ī„ģī…‚ī„ĩ ī„›ī„ĩī„Ŧī…‡ ī„ģī„ŽîĨĄ ī„›ī……ī„Ŗ ī…ƒī„Žī„ĩī…ƒ ī„žī„Ž ī„‘ī„Ēī…‡ ī…ƒī„Žī„ĩī…ƒ ī„žī„Ž
(ī…ƒī„Žī„ĩī…œī„Ēī„ĩ) ī„ģī……ī…œī„Ŧī…‡ ī„ģī„Ž, ī„ĸī„Ŧī…Œ ī…ƒī„› ī„Ŧī…†ī…ƒī„ĸī„’ī„ ī…ƒ ī…žī„  ī„ąī…Ĩī…‡ī„› ī„ĸī…‚ī„ ī„ĸī…œī„¯ (ī…ƒī„Žī„ĩī…œī„Ēī„ĩ) ī„ģī……ī…œī„Ŧī…‡ ī„ģī„ŽîĨĄ ī„“ī„ē ī„¯ī…‚ī„Ļ ī…ƒ ī…žī„ , ī„’ī„™ ī„¯ī„ŋī„ ī…‹ī„ļ
ī„›ī„ĩī…„ī„— ī…ƒī„› ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī„Ŧī…‚ ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„ĸī…‚ī„ ī…ƒī„Žī„ĩī…ƒ ī„žī„Ž ī„ģī…‹ī„Šī…‚ ī„›ī…„ ī„‘ī„ĩī„Ģ ī„ĩī…žī„œī„Ŧī…‚ ī„ģī…‰, ī„‘ī„Ēī…‡ ī…ƒī„°ī„ĩ ī„‘ī„ēī…„
ī„
ī„›ī…‹ī„ŗī„ļ ī„Ēī„ĩī„›
ī„Žī…‚ī„ļ ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‚ī„ī„ī…‡ ī…ƒī„› ī„“ī…žī„› ī„œī…‚ī„ē ī„°ī…œī„›ī„šī„Ž ī„“ī…žī„› ī„œī…‚ī„ē ī…ƒī„ąī…œī„Ŧī…† 'ī„Ēī…‡ ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī…ƒ ī„ģī…‚ī„ĩ ī„›ī„ĩī„Ŧī…‚ ī„ģī…‰îĨĄ ī„‘ī„ēī…„
ī„
ī„“ī„ē ī„ēī…œī„›ī„ļī„¯
ī„Žī…†ī…œ ī„ĸī…„ ī…œī„Ē ī„›ī„ĩī„Ž ī„ļī„” ī„“ī…žī„› ī„Ąī…‹ī„Ĩī…„ ī„›ī„ģī…‚ī„Šī…„ ī…ƒ ī…žī„  ī…„ ī„§ī……ī…žī„ąī…‚ī„ī„ī…‡îĨĄ ī„’ī„™ ī„šī……ī„ĩī…† ī„›ī„ĩī…„ī„—!
ķŧ¨ķŧ¨‘ķŧ¨’ī„˛ī…‚ī„ (i): ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„‘ī„Ēī…‡ ī„‘ī„ēī…œī„Ēī……ī„šī„Ĩ ī„›ī…‚ī„ĩī„ĸī…‚ī„ ī…ƒī„¸ī…žī„  ī„‘ī…œī„Ēī„ĩ ī„›ī„ĩī„Žī…‚
ķŧ¨ģķŧ¨ŧī„’ī„™ ī„ŗī……ī…žī„¨ī„ļī…„ī„’ī„ ī„ī…žī„ļī…‚ī„ ī„Žī…‚ī„ļ ī„šī……ī„ĩī…†ī„’ī„Ē ī„›ī„ĩī…„ī„—
ī„¯ī…ƒī„ĩī„˛ī…‚ī„šī…‚ ī…‚ī„ ī…ƒ ī…žī„  ī„§ī……ī…žī„ąī„Š ī„Ēī…Œ ī„¯ī…ƒī„ģī„ļī…‚ī„, ī„’ī„™ ī„’ī„¯ī„Šī…‡ ī„’ī„¯ ī„Ēī…Œ ī„¯ī……ī…žī„Ąī…„ī„— - ī„ĩī…‹ī„žī…‚ī„Žī…‚ ī„ĸī…„ ī„Ž ī…ƒ ī…žī„  "ī…ƒī„Žī„ĩī…œī„Ēī„ĩī„Ēī…‚" ī„Ēī…Œ ī„ēī…‚ī„§ī…‚
ī„›ī…„ ī„˛ī…‚ ī„ģī…‰?
ī…ƒī„Žī„ĩī…œī„Ēī„ĩī„Ēī…‚ ī„Ŧī…‚ ī„‘ī„ĩī„Ģ ī„ģī…‰ "ī„›ī…‹ī„” ī„ĩī……ī„›ī…‚ ī„Ĩ ī„Žī„ģī…„
ī„
" ī„ĸī…‚ī„ "ī„›ī…‹ī„” ī…ƒ ī„ĩī…‚ī„ŗ ī„Žī„ģī…„
ī„
îĨĄ"
ī„ēī…‹ī„ ī…‹:
â€ĸ ī„“ī…žī„› ī…ƒī…€ī„ļī„ŗ ī„ĸī…‹ ī„ąī„°ī…ƒī„ĩī…œī„ ī„Ēī…Œ ī…ƒī„ąī„Žī…‚ī„ ī„ ī…žī„ļī„Ŧī…„ ī„ģī…‰ — ī…ƒī„Žī„ĩī…ƒ ī„žī„Ž ī„‘ī„Ēī…‡ ī…ƒī„Žī„ĩī…œī„Ēī„ĩîĨĄ
â€ĸ ī„“ī…žī„› ī„ēī„ŋī„› ī…ƒī„ĸī„ē ī…ƒ ī…žī„  ī„›ī…‹ī„” ī„Ĩī…‹ī„— ī„Žī„ģī…„
ī„
ī„ģī„Ž - ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„ēī„ŋī„›îĨĄ
â€ĸ ī„“ī…žī„› ī„žī„ŋī…„ ī„ĸī…‹ ī„ģī„ĩ ī„ēī…ƒī„›ī…œī„Ĩ ī…ƒī„ąī„Žī…‚ī„ ī„ĩī……ī„›ī…‡ ī…ƒī„Ĩī„› ī…ƒī„Ĩī„› ī„›ī„ĩī„Ŧī…„ ī„ģī…‰ - ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„ī„Ēī…„îĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„ī…ƒī„Šī„Ē ī…ƒ ī…žī„ , ī„‘ī„ēī…„
ī„
ī„›ī…ƒī„ģī…œī„Ŧī…‡ ī„ģī…‚ī„ ī…ƒī„› ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī„“ī…žī„› ī…ƒī„ąī…œī„Ŧī…† 'ī„Ēī…‡ ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰ ī„ĸī…‡ī„›ī„ĩ ī„•ī„ēī„Ŧī…‚ ī„ŗī……ī…žī„ļ ī„•ī„ē ī…ƒī„ąī…œī„Ŧī…† 'ī„Ēī…‡
ī„‘ī„ ī…‚ī„Žī„› ī„Žī„ģī…„
ī„
ī„­ī„Ŧī…‚ ī„ĸī…‚ī„ ī„Ĩī……ī…žī„Ĩī„Ŧī…‚ ī„Žī„ģī…„
ī„
ī„ģī…‰îĨĄ
ķˇ•˜ķˇ•™ķˇ•šī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„°ī…œī„›ī„šī„Ž ī„Ŧī…„ ī„ĩī„ēī„ŗī…„ ī„¯ī…ƒī„ĩī„˛ī…‚ī„šī…‚:
ķŧŠ•ķŧŠ–ķŧŠ—ķŧŠ˜ķŧŠ™ķŧŠšī„ģī……ī„Š, ī„“ī…žī„› ī…ƒī„§ī„ēī„›ī…œī„Ĩī…„ī…ƒī„Žī„–ī„ē ī„°ī…œī„›ī„šī„Ž ī„›ī…„ ī„ģī…‰?
ī„ĸī…‡ī„›ī„ĩ ī„•ī„¯ī„ĩī…‹ī„›ī„Ē ī…ƒī„Ēī…œī„Žī…‚ī„ ī…ƒ ī…žī„ ī…Œ ī„›ī…‹ī„” ī„“ī…žī„› ī„šī„ĩī„Ē ī„‘ī„ēī„°ī„ļ ī„ģī…‹ ī„ĸī…‚ī„ī„Ŧī…„ ī„ģī…‰ ī„Ēī…‚ī„ ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī„Žī…†ī…œ ī„“ī…žī„› ī…ƒī„ąī…œī„Ŧī…† 'ī„Ēī…‡ ī„‘ī„ēī…œī„Ēī……ī„šī„Ĩ
ī…ƒī„›ī„ģī…‚ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ ī„´ī…‚ī„Žī…„:
â€ĸ ī„ĸī…‚ī„ ī„Ēī…‚ī„ ī„°ī…œī„›ī„šī„Ž ī„•ī„ē ī…ƒī„ąī…œī„Ŧī…† 'ī„Ēī…‡ ī„¯ī…ƒī„ĩī„˛ī…‚ī…ƒī„šī„Ē ī„Žī„ģī…„
ī„
ī„ģī…‰,
â€ĸ ī„ĸī…‚ī„ ī„ēī…„ī„ŗī…‚ ī„ŗī…ī„ĸī…†ī„Ŧ ī„Žī„ģī…„
ī„
ī„ģī…‰,
â€ĸ ī„ĸī…‚ī„ ī„ēī…„ī„ŗī…‚ ī„ŗī…ī„ĸī…†ī„Ŧ ī„ģī…‰ ī„¯ī„ĩ ī„•ī„ē ī…ƒī„ąī…œī„Ŧī…† 'ī„Ēī…‡ ī„°ī…œī„›ī„šī„Ž ī„Ŧī…‡ ī„ŗī……ī…žī„ļ ī„Žī…‚ī„ļ ī„ŗī…‡ī„ļ ī„Žī„ģī…„
ī„
ī„œī…‚ī„ī„Ŧī…„îĨĄ
ī„“ī„ģ ī„ąī…Ĩī…‡ī„› ī„ĸī…‚ī„ ī„¯ī…‚ī„ŋī…‚ ī„•ī„ē ī„ēī„Ģī…‚ī„Ž 'ī„Ēī…‡ ī„°ī…œī„›ī„šī„Ž ī„Žī…†ī…œ ī„‘ī„ēī…œī„Ēī……ī„šī„Ĩ ī„ąī„Šī…‚ī„•
ī„
ī„Ŧī…‚ ī„ģī…‰îĨĄ
ķē‚Ÿķē‚ ķē‚§ķē‚Ąķē‚ĸķē‚Ŗķē‚¤ķē‚Ĩķē‚Ļķē‚¨ī„ēī„­ī…‚ī„ĩī„Ž ī„ī…Ĩī…‚ī…ƒī„°ī„›ī„ļ ī„ēī„ŗī„Ŗ:
â€ĸ ī„“ī…žī„› ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„°ī…œī„›ī„šī„Ž ī„“ī…žī„› ī…ƒī„Žī„ĩī…ƒ ī„žī„Ž, ī„‘ī„Ĩī……ī…žī„Ĩ ī„›ī„ĩ ī…‚ī„ī„ ī…ƒī„Ŧī„œī…‚ī„” ī…ƒī„Ŧī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
â€ĸ ī„“ī…žī„› ī…ƒī„§ī„ēī„›ī…œī„Ĩī…„ī…ƒī„Žī„–ī„ē ī„°ī…œī„›ī„šī„Ž ī…ƒ ī…žī„  ī„ī…Ĩī…‚ī…€ ī…ƒ ī…žī„  ī„¯ī…‚ī„ŋī…‡, ī„ĸī…œī„¯, ī„ĸī…‚ī„ ī„Ąī…‡ī„› ī„ģī……ī…œī„Ŧī…‡ ī„ģī„ŽîĨĄ
ķˇ—›ķˇ—œī…ƒī„Žī„ĩī…œī„Ēī„ĩī„Ēī…‚ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Š ī„ļī„” ī„“ī…žī„› ī„›ī„ģī…‚ī„Šī…„
ī„’ī„™ ī„ĩ ī…„ ī„Žī…†ī…œ ī…ƒī„ŗī„ļī„Ŧī…‡ ī„ģī…‚ī„, ī„“ī…žī„› ī…ƒī„§ī„ļī…„ ī„ĩī…„ ī„ąī……ī„’ī„—îĨĄ
ī„ģī„ĩ ī„ĩī…‹ī„ž, ī„ĩ ī…„ ī„’ī„¯ī„Šī…‡ ī„žī„ĩ ī„Ēī…Œ ī„§ī…‚ī„›ī„žī„ĩ ī„Ēī…žī„› ī„ēī…‚ī„”ī„›ī„ļ 'ī„Ēī…‡ ī„•ī„ēī…‡ ī„ĩī„ēī„Ēī…‡ 'ī„Ēī…‡ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ ī„“ī…žī„› ī…ƒī„Ŧī„Ž, ī„ĩī„ēī„Ēī…‡ ī…ƒ ī…žī„  ī„“ī…žī„›
ī„¯ī……ī„ļ ī„Ĩī……ī…žī„Ĩ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ ī„ģī……ī„Š ī„•ī„ēī„Žī…†ī…œ ī„ĩī……ī„›ī„Šī…‚ ī„¯ ī…‡ī„ī…‚, ī„“ī…žī„› ī„ŗī…‹ī„ŋ ī„ļī…‰ī„Šī…‚ ī„¯ ī…‡ī„ī…‚, ī„ĸī…‚ī„ ī„¯ī…‚ī„ĩ ī„Ąī…‚ī„ļ ī„ŗī…‚ī„ĩī„Žī…„ ī„¯ ī…‡ī„ī…„îĨĄ ī„“ī„ģ
ī„•ī„ēī„Ŧī…„ ī„´ī…‚ī„Ēī„ĩī…‚ ī…ƒ ī…žī„  ī„“ī…žī„› ī…ƒ ī„žī„Ž ī„ģī…‰îĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„¯ī……ī„ļ ī„Ēī…Œ ī„¯ī…ƒī„ģī„ļī…‚ī„, ī„•ī„ēī„Ŧī…‚ ī„ĩī„ēī„Ēī…‚ ī…ƒī„Žī„ĩī…ƒ ī„žī„Ž ī„ēī…„ (ī„“ī…žī„› ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„°ī…œī„›ī„šī„Ž ī…‚ī„ī„)îĨĄ ī„¯ī„ĩ ī„•ī„ē ī„ī……ī…œī„ŗ ī„ģī…‹ī„— ī„Ĩī……ī„›ī„ŋī…‡ ī„Ŧī…‡
ī„›ī…‚ī„ĩī„Ž, ī„“ī„ģ ī„“ī…žī„› ī„¯ī…‚ī„ŋī…‡ ī…‚ī„ļī…‡ ī„°ī…œī„›ī„šī„Ž ī…‚ī„ī„ ī„ąī„Š ī…ƒī„ī„’îĨĄ
ī…ƒī„ĸ ī…ˆ ī„ĩ ī…„ ī„Žī…†ī…œ ī„ēī„ŗī…ˆ ī…ƒī„ēī„ĩ ī„§ī…‚ī„› ī„¯ī„ģī……ī…œī„ ī…‚ī„•ī„Š ī„ļī„” ī„“ī…žī„› ī„‘ī„Ĩī……ī…žī„Ĩ ī„ĩī„ēī„Ēī…‡ ī„Ŧī…„ ī„ļī…‹ī„ŋ ī„ģī……ī…œī„Ŧī…„ ī„ģī…‰, ī„•ī„ēī…‡ ī„Ēī„ĩī…§ī…‚ī„ ī„ēī…‚ī„Žī…†ī…œ ī„ī…ƒī„Šī„Ē ī…ƒ ī…žī„ 
ī„ēī……ī„ ī…‚ī„ĩī…† ī„ī„Šī„Žī…‚ ī…‚ī„ ī„ļī„” ī„‘ī„Ĩī……ī…žī„Ĩ (ī…ƒī„Žī„ĩī…œī„Ēī„ĩ) ī„°ī…œī„›ī„šī„Žī…‚ī„ ī„Ŧī…„ ī„ļī…‹ī„ŋ ī„ģī……ī…œī„Ŧī…„ ī„ģī…‰îĨĄ
ķŧ¨ķŧ¨‘ķŧ¨’ī…ƒī„§ī„ēī„›ī…œī„Ĩī…„ī…ƒī„Žī„–ī„Ĩī…„ ī„Ŧī…„ī„’ī„ ī…ƒī„›ī„ēī„ŗī…‚ī„ (ī„§ī…†ī…œī„žī…„ ī„ēī„ŗī„Ŗ ī„ļī„”):
1. ī…ƒī„ąī…œī„Ŧī…† ī…ƒī„§ī„ēī„›ī…œī„Ĩī…„ī…ƒī„Žī„–ī„Ĩī…„ (ī„ģī„Ĩī…‚ī„•ī„Šī„´ī…‹ī„):ī„ī…Ĩī…‚ī…€ ī…ƒ ī…žī„  ī„“ī…žī„› ī„Ąī…‹ī„Ĩī…‚ ī…ƒī„ĸī„ģī…‚ ī„Ąī…‡ī„›îĨĄ
2. ī„Ąī…‚ī„ļ ī„Ŧī…„ ī„‘ī„ēī…œī„Ēī……ī„šī„Ĩī…„:ī„°ī…œī„›ī„šī„Ž ī„“ī…žī„› ī„ŗī……ī…žī„ļ ī„Ēī…Œ ī„Ŧī…†ī„ĸī…‡ ī„ŗī……ī…žī„ļ 'ī„Ēī…‡ ī„Ąī…‚ī„ļ ī„ŗī…‚ī„ĩī„Ŧī…‚ ī„ģī…‰îĨĄ
3. ī„‘ī„Žī…œ ī„Ē ī…ƒī„¸ī„žī„Ž:ī„°ī…œī„›ī„šī„Ž ī„‘ī„Žī…œ ī„Ēī„Ēī…‚ ī„Ēī…žī„› ī„­ī„Ŧī…‚ ī„ģī…‰ (ī„“ī…žī„› ī„ļī…œī„ąī„›ī…‚ī„ĩī…„ ī„‘ī…ƒī„ēī…œī„¯ī…‹ī„Ĩ ī…‚ī„ī„)îĨĄ
ķˇƒ†ķŧŊĸī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„›ī…‚ī„ĩī„ĸī…‚ī„ ī„Ŧī…„ī„’ī„ ī„•ī„Ŧī…‚ī„ģī„ĩī„Šī…‚ī„:
â€ĸ ī„ąī„ģī……ī„¯ī„Ŧ ī„‘ī„Ēī…‡ ī„¯ī…ƒī„ĩī„ŗī…‡ī„´ ī„°ī…œī„›ī„šī„Ž (ī…ƒī„ĸī…žī„Ģī…‡ ī„ģī„ĩ ≠ 0)
ķŊ…‚ī„‘ī„ēī…œī„Ēī……ī…ƒī„ļī„Ē ī„›ī…‚ī„ĩī„ĸī…‚ī„ ī„Ŧī…„ī„’ī„ ī„•ī„Ŧī…‚ī„ģī„ĩī„Šī…‚ī„:
ķŧ¨Ŋķŧ¨žķŧ¨ŋķŧŠķŧŠ€ī„˛ī…‚ī„ (ii): ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ ī…ƒī„›
x=2x = 2x=2 ī„Ēī…‡ ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„ģī…‰
ķšŗ¸ķšŗēķšŗšī„›ī„Ŧī„ŗ 1: ī„°ī…œī„›ī„šī„Ž ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī…‹
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī…ƒī„Ŧī…žī„Ēī…‚ ī…ƒī„ī„’ ī„°ī…œī„›ī„šī„Ž:
ī„’ī„™ ī„˛ī…‚ī„ĸ ī„Žī…†ī…œ ī„ēī„ĩī„ļ ī„ąī„Šī…‚ī„”ī„—:
ī„“ī„ē ī„ļī„”, ī„°ī…œī„›ī„šī„Ž ī„Žī…†ī…œ x=1 ī„‘ī„Ēī…‡ x=−1 ī„Žī…†ī…œ ī„Ąī…žī„§ ī„›ī…‡ ī„ēī…‚ī„ĩī…‡ ī„‘ī„ēī„ļī…„ x ī„ļī„” ī„¯ī…ƒī„ĩī„˛ī…‚ī…ƒī„šī„Ē ī„›ī…„ī„Ēī…‚ ī…ƒī„ī„’ ī„ģī…‰, ī…ƒī„›ī„•
ī„
ī…ƒī„› ī„“ī„ģ
ī„ŗī……ī…žī„ļ ī„ģī„ĩ ī„Žī…†ī…œ ī„žī…„ī„ĩī…‹ ī„ąī„Šī…‚ī„•
ī„
ī„Ŧī…‡ ī„ģī„Ž (ī„žī…„ī„ĩī…‹ ī„Žī…‚ī„ļ ī„˛ī…‚ī„ ī„¯ī…ƒī„ĩī„˛ī…‚ī…ƒī„šī„Ē ī„Žī„ģī…„
ī„
ī„ģī…‰)îĨĄ
ī„“ī„ē ī„ļī„”, ī„°ī…œī„›ī„šī„Ž ī„Žī…†ī…œ x=2 'ī„Ēī…‡ ī„¯ī…ƒī„ĩī„˛ī…‚ī…ƒī„šī„Ē ī„›ī…„ī„Ēī…‚ ī…ƒī„ī„’ ī„ģī…‰îĨĄ
ķˇƒ†ķŧŊĸī„›ī„Ŧī„ŗ 2: x=2 'ī„Ēī…‡ ī…ƒī„Žī„ĩī…œī„Ēī„ĩī„Ēī…‚ ī„Ŧī…„ ī„ĸī…‚ī„ī„  ī„›ī„ĩī…‹
ī„‘ī„ēī…„
ī„
ī„¯ī…ƒī„ģī„ļī…‚ī„ ī„Ŧī…žī„ēī…‡ ī„ī„— ī…ƒī„Ēī…œī„Ž-ī„¯ī„ŋī…‚ī„‘ ī„¸ī…‚ī„ļī…‡ ī„Ĩī…‰ī„ēī„Ĩ ī„Ŧī…„ ī„¸ī„ĩī„Ēī…Œ ī„›ī„ĩī…‚ī„ī„ī…‡:
ī„›ī„Ŧī„ŗ 1: ī„›ī…„ f(2) ī„¯ī…ƒī„ĩī„˛ī…‚ī…ƒī„šī„Ē ī„ģī…‰?
ī„ģī…‚ī„îĨĄ
ķˇƒ†ķŧŊĸī„“ī„ē ī„ļī„”, f(2) ī„Žī…†ī…œ ī„¯ī…ƒī„ĩī„˛ī…‚ī…ƒī„šī„Ē ī„›ī…„ī„Ēī…‚ ī…ƒī„ī„’ ī„ģī…‰îĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ķˇ“ ķˇ“Ąķˇ“ĸķˇ“Ŗķˇ“¤ķˇ“Ĩķˇ“¨ķˇ“Šķˇ“Ēķˇ“Ģķˇ“Ļķˇ“§ķˇ“Ŧī…ƒī„ēī…žī„Ĩī…‚:
ī…ƒī„Ēī…œī„Žī…Œ ī„šī„ĩī„Ēī…‚ī„ ī„¯ī…†ī„ĩī…„ī„’ī„ ī„ģī……ī…œī„Ŧī…„ī„’ī„ ī„ģī„ŽîĨĄ
ī„“ī„ē ī„ļī„”,
ķˇƒ†ķš¸ƒķš¸„ī…ƒī„¸ī…ƒī„Ŧī„’ī„ĩī„Ģī…„-ī„‘ī„Žī……ī„›ī…†ī„ļ ī„¨ī…œī„ ī„Žī…‚ī„ļ ī„ēī…œī„œī…‡ī„¯:
ī„‘ī„ēī…„
ī„
ī„›ī…„ ī„›ī…„ī„Ēī…‚:
1. ī„‘ī„ēī…„
ī„
ī„ēī„ŗī„Ŗ ī„ī„— ī…ƒī„› ī…ƒī„Žī„ĩī…œī„Ēī„ĩī„Ēī…‚ ī„Ŧī…‚ ī„›ī…„ ī„‘ī„ĩī„Ģ ī„ģī…‰ - ī„“ī…žī„› ī…ƒī„Žī„ĩī…ƒ ī„žī„Ž ī„ēī„ŋī„› ī…‚ī„ī„îĨĄ
2. ī„‘ī„ēī…„
ī„
ī„Ŧī…‡ī…ƒī„œī„’ ī…ƒī„› ī„ĸī…‡ī„›ī„ĩ ī„›ī…‹ī„” ī„Ąī…‡ī„›, ī„Ąī…‚ī„ļ, ī„ĸī…‚ī„ ī„ąī…Ĩī…‡ī„› ī„Žī„ģī…„
ī„
ī„ģī…‰, ī„Ēī…‚ī„ ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰îĨĄ
3. ī„‘ī„ēī…„
ī„
x=2x = 2x=2 'ī„Ēī…‡ ī„°ī…œī„›ī„šī„Ž ī„Ŧī…‡ ī„ŗī……ī…žī„ļ ī„‘ī„Ēī…‡ ī„ēī…„ī„ŗī…‚ ī„Ŧī…„ ī„ĸī…‚ī„ī„  ī„›ī…„ī„Ēī…„îĨĄ
4. ī…ƒī„›ī„•
ī„
ī…ƒī„› ī„•ī„ē ī„ēī„ŗī…ˆ ī„°ī…œī„›ī„šī„Ž ī„­ī…„ī„’ ī…ƒ ī„ģī…‚ī„ĩ ī„›ī„ĩī„Ŧī…‚ ī„ģī…‰, ī„‘ī„ēī…„
ī„
ī„ēī…‚ī„ąī„Ē ī„›ī…„ī„Ēī…‚ ī…ƒī„› ī„“ī„ģ ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„ģī…‰îĨĄ
ķš¸¯ķš¸­ķš¸Žī„‘ī„ēī„ļ-ī„ĸī…„ī„¸ī„Ž ī„ēī„ŗī…‚ī„Žī„Ēī…‚ (ī„Ŧī…†ī„ĸī…„ ī„Ąī…‹ī„Ĩī…„ ī„›ī„ģī…‚ī„Šī…„)
ī„¯ī…‚ī„Šī…„ ī„Ŧī…„ ī„¯ī…‚ī„”ī„¯ ī„ąī…‚ī„ĩī…‡ ī„ēī…‹ī„ ī…‹îĨĄ ī„ĸī…‡ī„›ī„ĩ ī„¯ī…‚ī„Šī…„ ī„“ī„ē ī…ƒ ī…žī„ ī…Œ ī„ēī……ī„ ī…‚ī„ĩī…† ī„¨ī…œī„ ī„Žī…‚ī„ļ ī…ƒī„ģī…œī„Ŧī…‚ ī„ģī…‰, ī„Ēī…‚ī„ ī„“ī„ģ ī…ƒī„Žī„ĩī…œī„Ēī„ĩī„Ēī…‚ ī„ģī…‰îĨĄ ī„ĸī…‡ī„›ī„ĩ
ī„›ī…‹ī„” ī„ĩī……ī„›ī…‚ ī„Ĩ ī„ĸī…‚ī„ ī„Ąī…‡ī„› ī„ģī…‰, ī„Ēī…‚ī„ ī„“ī„ģ ī„“ī…žī„› ī…ƒ ī„žī„Ž ī„ģī…‰îĨĄ
ī„ŗī…œī„Ž ī„ļī„™ ī…ƒī„› ī„“ī…žī„› ī„ŗī…‚ī„ļī…„ ī„¯ī…‚ī„”ī„¯ ī„ĩī…‚ī„ģī…„
ī„
ī„¯ī…ī…ƒī„Ŧī„’ī„ ī„Žī…†ī…œ ī„¯ī…‚ī„Šī…„ ī„Ŧī…‡ ī…ƒī„ĩī„ģī…‚ ī„ģī…‰îĨĄ ī…ƒī„žī„’ī„Ŧī…‚ī„Ēī„ĩ ī„Ģī…‚ ī…‚ī„ 'ī„Ēī…‡, ī„¯ī…‚ī„Šī…„ ī…ƒī„ąī„Žī…‚ī„ ī…ƒī„›ī„ēī…‡
ī„ēī„ŗī…žī…ƒī„ēī„’ ī„Ŧī…‡ ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ ī„¯ī„ĩ ī„“ī…žī„› ī„Ģī…‚ī„ 'ī„Ēī…‡, ī„“ī…žī„› ī„Ąī…‹ī„Ĩī…‚ ī…ƒī„ĸī„ģī…‚ ī…ƒ ī„ĩī…‚ī„ŗ ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰, ī„‘ī„Ēī…‡ ī„¯ī…‚ī„Šī…„ ī„Ŧī…‡ ī…ƒī„Ąī…žī„Ĩī…‡ ī…ƒī„Žī„›ī„ļī„Ŧī…‡ ī„ģī„ŽîĨĄ
ī„“ī„ģ ī…ƒī„ąī„ļī„›ī……ī„ļ ī„“ī…žī„› ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„°ī…œī„›ī„šī„Ž ī…‚ī„ī„ ī„ģī…‰ - ī„ēī…‚ī„ĩī…‡ ī„“ī„Žī„¯ī……ī„Ĩ ī…ƒī„Žī„ĩī…ƒ ī„žī„Ž ī„’ī„•ī„Ĩī„¯ī……ī…žī„Ĩ ī„Žī„ģī…„
ī„
ī…ƒī„Ŧī…œī„Ŧī…‡ ī„ģī„ŽîĨĄ
ī„¯ī…‚ī„”ī„¯ ī„Ŧī…‡ ī„›ī…‚ī„ĩī„ĸī„šī…„ī„ļ ī„Ēī…ī„ĩ 'ī„Ēī…‡ ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„ĩī…ƒī„ģī„Š ī„ļī„”, ī„¯ī…‚ī„Šī…„ ī„Žī…†ī…œ ī„ģī„ĩ ī…ƒī„ģī…žī„ēī…‡ ī…ƒ ī…žī„ ī…Œ ī„ēī……ī„ ī…‚ī„ĩī…† ī„¨ī…œī„ ī„Žī…‚ī„ļ ī…ƒī„ģī„Šī…‚ ī„ ī…‚ī„ģī…„ī„Ŧī…‚
ī„ģī…‰, ī…ƒī„ąī„ļī„›ī……ī„ļ ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī…‚ī„ī„ ī„ĸī…‹ ī„’ī„¯ī„Šī…‡ ī„œī…‡ī„Ēī„ĩ ī„Ŧī…‡ ī„ēī…‚ī„ĩī…‡ ī…ƒī„ąī…œī„Ŧī…†ī„’ī„ 'ī„Ēī…‡ ī„­ī…„ī„’ ī…ƒ ī„ģī…‚ī„ĩ ī„›ī„ĩī„Ŧī…‚ ī„ģī…‰îĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ķš´Ąķš´ĩķš´Ŗķš´¤ī„‘ī…œī…ƒī„Ēī„ŗ ī…ƒī„¸ī„ ī…‚ī„ĩ:
ī„ī…ƒī„Šī„Ē ī…ƒ ī…žī„ , ī„œī…‚ī„ē ī„›ī„ĩī„›ī…‡ ī„›ī…‰ī„ļī„›ī…†ī„ļī„ē ī…ƒ ī…žī„ , ī…ƒī„Žī„ĩī…œī„Ēī„ĩī„Ēī…‚ ī„°ī…œī„›ī„šī„Žī…‚ī„, ī„ī…Ĩī…‚ī„°ī…‚ī„ ī„‘ī„Ēī…‡ ī„Ēī„ąī„Ŧī…„ī„ļī…„ī„’ī„ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Š ī„Ŧī…„ ī„Žī…„
ī„
ī„ģ
ī„ģī…‰îĨĄ ī„Ēī……ī„ēī…„
ī„
ī„ēī…„ī„ŗī…‚ ī…‚ī„, ī„§ī…‰ī„ĩī…„ ī…‡ī…ƒī„Ĩ ī„ž, ī„ĸī…‚ī„ ī„“ī…œī„Ĩī…„ī„ī„ĩī„ļ ī„ąī…‚ī„ĩī…‡ ī„ī…žī„ļ ī„Žī„ģī…„
ī„
ī„›ī„ĩ ī„ēī„›ī„Ŧī…‡ ī„ĸī„Ŧī…Œ ī„Ēī…žī„› ī„“ī„ģ ī„´ī„›ī…„ī„Žī…„ ī„Žī…‚
ī„ąī„Šī…‚ī„“ī„’ ī„ĸī…‚ ī…‡ ī…ƒī„› ī„Ēī……ī„ģī…‚ī„§ī…‚ ī„°ī…œī„›ī„šī„Ž ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„ģī…‰îĨĄ
ī„“ī„ē ī„ļī„”, ī„ēī…œī„œī…‡ī„¯ ī…ƒī„¸ī…žī„ :
â€ĸ ī„“ī…žī„› ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„›ī…‚ī„ĩī„ĸ ī„“ī…žī„› ī…ƒī„Žī„ĩī…ƒ ī„žī„Ž ī„´ī…‚ī„Ēī„ĩī…‚ ī…‚ī„ī„ ī„ģī…‰îĨĄ
â€ĸ ī„“ī…žī„› ī…ƒī„Žī„ĩī…œī„Ēī„ĩ ī„Žī…‚ ī„ģī…‹ī„Š ī…‚ī„ļī…‚ ī„›ī…‚ī„ĩī„ĸ ī„“ī…žī„› ī„ēī„ŋī„› ī…‚ī„ī„ ī„ģī……ī…œī„Ŧī…‚ ī„ģī…‰ ī…ƒī„ĸī„ē ī…ƒ ī…žī„  ī„ąī„ĩī…‡ī„› ī„ģī……ī…œī„Ŧī…„ ī„ģī…‰îĨĄ
ī„ēī…‰ī„›ī„šī„Ž-ī„§ī…„
7. (i) (7x - 8)4 (5x - 1)3 ī„¯ī„Ŧī…‚ī„ ī„Žī…†ī…œ ī„¸ī…žī„œ ī„›ī„ĩī…‹
(ii) wrtx ī„Ŧī…‚ ī„§ī…‰ī„ĩī…„ī„¸ī…‡ī…ƒī„Ĩī„¸ ī„ļī…žī„˛ī…‹îĨĄ
îēžī„žīƒ›
īƒœī„žîē’îē•îēî˜ƒîēžî˜ƒ

(iii) wr.tx ī„Žī…†ī…œ ī„¸ī…žī„œī„ĩī…‚ ī„›ī„ĩī…‹īƒž
īŽĨ
īƒšī„žîē‹îēž
īƒ›
ī„•ī…žī„Ēī„ĩ: ī„ŗī…Š ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī„“ī…žī„› ī„›ī„ļī…‚ī„ēī„ĩī…†ī„ŗ ī…ƒ ī…žī„  ī„ļī…‰ ī„ĸī…‚ī„ī„Ŧī…„ ī„ģī…‚ī„, ī…ƒī„ĸī…žī„Ģī…‡ ī„ĩī…„ī„’ ī„Žī…‚ī„ŗ ī„Ŧī…„ ī„“ī…žī„› ī„•ī„Ēī„ēī……ī„› ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī„Š ī„Žī…‡
ī„’ī„¯ī„Šī…‚ ī„ģī…žī„Ģ ī„•ī…žī„ ī…‚ ī„›ī…„ī„Ēī…‚ ī„‘ī„Ēī…‡ ī„¯ī……ī…žī…ƒī„Ąī„’, "ī„ŗī…‰ī„§ī„ŗ, ī…ƒī„§ī„°ī„ĩī…Šī„šī…„ī„—ī„šī„Ž ī„ģī„ŗī…‡ī„šī…‚ ī„ŗī…‰ī„Žī…†ī…œ ī„“ī…žī„› ī„ąī……ī„Ŗī…‚ī„ĩī„Ē ī…ƒī„›ī„•
ī„
ī„ļī…žī„ī„Ŧī…‚ ī„ģī…‰? ī„ŗī…‰ī„Žī…†ī…œ
ī„°ī…‚ī„ĩī„ŗī…†ī„ļī…‡ ī„¯ī„Ēī…‚ ī„ģī„Ž ī„¯ī„ĩ ī„ŗī…‰ī„Žī…†ī…œ ī„“ī„ģ ī„Žī„ģī…„
ī„
ī„¯ī„Ēī…‚ ī…ƒī„› ī„•ī„ģī„Žī…‚ī„ ī„Žī…†ī…œ ī„›ī„Ŧī…Œ ī„‘ī„Ēī…‡ ī…ƒī„› ī…ˆ ī„ĩī„Ēī„Šī…‚ ī„ģī…‰!"
ī„‘ī…ƒī„­ī„’ī„¯ī„› ī„Žī…‡ ī„ī„ĩī„ŗī„ĸī…‹ī„šī…„ ī„Žī…‚ī„ļ ī„ŗī……ī„ēī„›ī„ĩī…‚ī„•
ī„
ī„Ŧī…‡ ī„ģī…‹ī„— ī„ĸ ī…‚ī„ą ī…ƒī„Ŧī…žī„Ēī…‚, "ī„ąī„ģī……ī„Ē ī„­ī…„ī„’ ī„ē ī…‚ī„ļ, ī„ĩī…„ī„’! ī„“ī„ģ ī„“ī„ē
ī„ļī„” ī„ģī…‰ ī…ƒī„›ī„•
ī„
ī…ƒī„› ī„ī…ƒī„Šī„Ē ī…ƒī„ēī„ĩī…€ ī„ģī…žī„ļ ī„›ī„ĩī„Ž ī„ąī…‚ī„ĩī…‡ ī„Žī„ģī…„
ī„
ī„ģī…‰ - ī„“ī„ģ ī„¯ī…‰ī„Ĩī„ĩī„Žī…‚ī„ ī„Žī…†ī…œ ī„Ŧī…‡ī„œī„Š ī„‘ī„Ēī…‡ ī„ēī„ģī…„ ī„šī„žī…‚ī„ĩī…‚ī„ ī„Ŧī…„ ī„ ī…‹ī„Š
ī„›ī„ĩī„Ž ī„ąī…‚ī„ĩī…‡ ī„ģī…‰îĨĄ ī„’ī„™ ī„‘ī…žī„ĸ ī…ƒī„Ēī…œī„Ž ī„ēī„ŗī…žī…ƒī„ēī„’ ī…‚ī„ ī„ģī…žī„ļ ī„›ī„ĩī…„ī„—, ī„ģī„ĩ ī„“ī…žī„› ī…žī„œī„ĩī…„ ī…ƒī„›ī„ēī„ŗ ī„Ŧī…‡ ī„Ēī…‚ī„ļī…‡ ī…‚ī„ī„ ī…ƒī„ĸī„ē ī„ļī„” ī„“ī…žī„›
ī…žī„œī„ĩī…„ ī„ ī…‚ī„ąī…„ ī„Ŧī…„ ī„ļī…‹ī„ŋ ī„ģī……ī…œī„Ŧī…„ ī„ģī…‰!"
ī„‘ī„Ēī…‡ ī„“ī„ē ī„Ēī„ĩī…§ī…‚ī„ ī„ēī…‚ī„§ī…„ ī„ī…ƒī„Šī„Ē ī„Ŧī…„ ī„›ī„ģī…‚ī„Šī…„ ī„šī……ī„ĩī…† ī„ģī……ī…œī„Ŧī…„ ī„ģī…‰ — ī…ƒī„Ēī…œī„Ž ī„ēī„ŗī…„ī„›ī„ĩī„Žī…‚ī„ ī„Žī…‚ī„ļ, ī„ģī„ĩī…‡ī„› ī„Žī…†ī…œ x ī„Ŧī…‡ ī„ēī…œī„ąī…œī„­ ī…ƒ ī…žī„ 
ī…žī„œī„ĩī…‚ ī„›ī„ĩī„Ž ī„ļī„” ī„“ī…žī„› ī…žī„œī„ĩī…‡ ī…ƒī„Ŧī…Ĩī„šī„Ĩī…„ī„›ī…‹ī„Š ī„Ŧī…„ ī„ļī…‹ī„ŋ ī„ģī……ī…œī„Ŧī…„ ī„ģī…‰îĨĄ
ķŊ„ĄķŊ„ĸķŊ„ŖķŊ„¤ķŊ„ĨķŊ„Ļī„¯ī…Ĩī„šī„Ž 7 (i): (7x - 8)⁴(5x - 1)Âŗ wrt x ī„Žī…†ī…œ ī„¸ī…žī„œ ī„›ī„ĩī…‹
ķš¸¯ķš¸­ķš¸Žī„›ī„Ŧī„ŗ 1: ī„°ī…œī„›ī„šī„Ž ī„Ŧī…„ ī…ƒī„›ī„ēī„ŗ ī„Ŧī…„ ī„¯ī„Ąī…‚ī„Š ī„›ī„ĩī…‹
ī„ēī…‚ī„Žī…†ī…œ ī…ƒī„Ŧī…žī„Ēī…‡ ī„ī„— ī„ģī„Ž:
y=(7x−8)4(5x−1)3
ī„“ī„ģ ī„Ŧī…‹ ī„°ī…œī„›ī„šī„Žī…‚ī„ ī„Ŧī…‚ ī„•ī„Ēī„¯ī…‚ī„Ŧ ī„ģī…‰, ī„“ī„ē ī„ļī„” ī„ēī…‚ī„Žī…†ī…œ ī„•ī„Ēī„¯ī…‚ī„Ŧ ī…ƒī„Žī„´ī„ŗ ī„Ŧī…„ ī„ĩī„Ēī…Œ ī„›ī„ĩī„Žī…„ ī„ ī…‚ī„ģī…„ī„Ŧī…„ ī„ģī…‰îĨĄ
ķŧ¨ķŧ¨‘ķŧ¨’ī„•ī„Ēī„¯ī…‚ī„Ŧ ī…ƒī„Žī„´ī„ŗ ī„ĩī…„ī„ŗī…‚ī„”
ī„
ī„§ī„ĩ:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„ĸī…‡
ī„’ī„™ ī…ƒī„Žī„ĩī„­ī…‚ī„ĩī„Ē ī„›ī„ĩī…„ī„—:
â€ĸ u=(7x−8)4
â€ĸ v=(5x−1)3
ķŊ„ŦķŊ„­ķŊ„ŽķŊ„¯ķŊ„°ī„›ī„Ŧī„ŗ 2: ī„ģī„ĩī…‡ī„› ī…ƒī„ģī…žī„ēī…‡ ī„Žī…†ī…œ ī„¸ī…žī„œī„ĩī…‚ ī„›ī„ĩī…‹
ķˇĩģķˇĩŧķˇĩŊķˇĩžu = (7x - 8)4 ī…ƒ ī…žī„  ī„‘ī…œī„Ēī„ĩ ī„›ī„ĩī…‹îĨĄ
ī„ ī…‡ī„Ž ī…ƒī„Žī„´ī„ŗ ī„Ŧī…„ ī„¸ī„ĩī„Ēī…Œ ī„›ī„ĩī…‹:
ķˇĩģķˇĩŧķˇĩŊķˇĩžv = (5x - 1)3 ī„Ŧī…‚ ī„‘ī…œī„Ēī„ĩ ī„›ī„ĩī…‹
ķšš‹ķššŒī„›ī„Ŧī„ŗ 3: ī„•ī„Ēī„¯ī…‚ī„Ŧ ī…ƒī„Žī„´ī„ŗ ī„ļī…‚ī„ī…† ī„›ī„ĩī…‹
ī„ģī……ī„Š,
ī„ąī„Ŧī„ļ ī„ŗī……ī…žī„ļ:
ī„“ī„ģ ī„Ēī……ī„ģī…‚ī„§ī…‚ ī„’ī„œī„ĩī…„ ī…ƒ ī…ƒī„˛ī…œī„Ž ī„ĩī…†ī„¯ ī„ģī…‰îĨĄ ī„Ēī……ī„ēī…„
ī„
ī„“ī„ēī„Žī…†ī…œ ī„ēī„ĩī„ļī„Ēī…‚ ī„ļī„” ī„°ī…‰ī„›ī„Ĩī„ĩ ī„ĩī…žī„œ ī„ēī„›ī„Ŧī…‡ ī„ģī…‹îĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„ģī……ī„Š ī„’ī„™ ī„‘ī„ĩī„ĸī……ī„Ž ī„Žī…‚ī„ŗ ī„Ŧī…‡ ī„“ī…žī„› ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ ī„Žī…†ī…œ ī…ƒī„ŗī„ļī„Ŧī…‡ ī„ģī…‚ī„, ī…ƒī„ĸī„ēī„Žī…†ī…œ ī…ƒī„˛ī…œī„Žī…‚ī„ ī„Žī…†ī…œ ī„ģī…žī„ļ ī„›ī„ĩī„Žī…‚ ī„ąī„ģī……ī„Ē ī„¯ī„ēī…œī„Ŧ ī„ģī…‰ ī„¯ī„ĩ
ī„ĸī„Ŧī…Œ ī„“ī…žī„› ī„ļī„žī…†ī„ī„Šī„› ī…ƒī„Ŧī„œī…‚ī„” ī…ƒī„Ŧī…œī„Ŧī…‚ ī„ģī…‰ ī„Ēī…‚ī„ ī„žī„ąī„ĩī…‚ ī„ĸī…‚ī„ī„Ŧī…‚ ī„ģī…‰îĨĄ ī„‘ī…ƒī„­ī„’ī„¯ī„› ī„•ī„ēī„Žī…†ī…œ ī„›ī…ƒī„ģī…œī„Ŧī…‚ ī„ģī…‰, "ī„ļī…žī„ ī„Ēī…Œ ī„Žī…‚ ī„§ī„ĩī…‹ -
ī„“ī„ēī„Žī…†ī…œ ī„“ī…žī„› ī„ģī…‹ī„ĩ ī„°ī…œī„›ī„šī„Ž ī…‚ī„ī„ ī„ēī„ŗī„Ŗī…‹!"
ķŧ¨ķŧ¨‘ķŧ¨’ī„›ī„Ŧī„ŗ 1: ī„˛ī…‚ī„ī„°ī„ļ ī…ƒī„Žī„´ī„ŗ ī„Žī…†ī…œ ī„¯ī„Ąī…‚ī„Šī…‹
ī„“ī„ģ ī„“ī…žī„› ī„˛ī…‚ī„ī„°ī„ļ ī„ģī…‰ — ī„“ī…žī„› ī„°ī…œī„›ī„šī„Ž ī„Ŧī…†ī„ĸī…‡ ī„Žī…‚ī„ļ ī…œī…ƒī„§ī„’ ī„ģī…‹ī„“ī„’ ī„ģī…‰îĨĄ ī„’ī„™ ī…ƒī„ļī„œī„Ŧī…‡ ī„ģī…‚ī„:
ī„’ī„™:
â€ĸ u=x+2
â€ĸ v=3+ī„ļī…ī„ī„˜ī„›ī„ē
ķŊ„ŦķŊ„­ķŊ„ŽķŊ„¯ķŊ„°ī„›ī„Ŧī„ŗ 2: ī„˛ī…‚ī„ī„°ī„ļ ī…ƒī„Žī„´ī„ŗ ī„Ŧī…„ ī„¸ī„ĩī„Ēī…Œ ī„›ī„ĩī…‹
ī„˛ī…‚ī„ī„°ī„ļ ī…ƒī„Žī„´ī„ŗ:
ī„ģī……ī„Š ī„§ī…‰ī„ĩī…„ ī…‡ī…ƒī„Ĩ ī„ļī…žī„˛ī…‹:
ķšš‹ķššŒī„›ī„Ŧī„ŗ 3: ī„˛ī…‚ī„ī„°ī„ļ ī…ƒī„Žī„´ī„ŗ ī…ƒī„¸ī…žī„  ī„¯ī„ļī…žī„ ī„“ī„Ž ī„›ī„ĩī…‹
ī„ģī……ī„Š ī„‘ī…œī„š ī„Žī…†ī…œ ī„ēī„ĩī„ļ ī„ąī„Šī…‚ī„™:
ī„Ēī……ī„ēī…„
ī„
ī„ĸ ī…‚ī„ą ī„“ī„ē ī„Ēī„ĩī…§ī…‚ī„ ī„Ąī…žī„§ ī„ēī„›ī„Ŧī…‡ ī„ģī…‹:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„ĸī…‚ī„, ī„ĸī…‡ ī„ļī…‹ī„ŋ ī„ģī…‹ ī…‡, ī„Ēī…‚ī„ ī„ģī…‹ī„ĩ ī„ēī„ĩī„ļ ī„ąī„Šī…‚ī„™îĨĄ
ī„“ī„ģ ī„ģī…‰ ī„ŗī„žī…‡ī„Ŧī…‚ī„ĩ! ī„“ī„ē ī„°ī…œī„›ī„šī„Ž ī„Žī…†ī…œ ī„“ī…žī„› ī„ļī…‡ī„‘ī„ĩī„§ ī„›ī…‡ī„› ī„Ŧī…‡ ī„ĩī…†ī„¯ ī…ƒ ī…žī„  ī„›ī„ļī„¯ī„Žī…‚ ī„›ī„ĩī…‹ — ī„“ī…žī„› ī„žī…‚ī„Ē ī„‘ī…œī„› ī„Ŧī…‡
ī„ĩī„ī„ŗī…†ī„ļ ī„Ŧī…„ ī„“ī…žī„› ī„šī„›ī„Ēī…„îĨĄ ī„ļī„žī…†ī„ī„Šī„› ī…ƒ ī…ƒī„˛ī…œī„Žī„Ēī…‚ ī„‘ī„Ēī…‡ ī„ ī…‡ī„Ž ī…ƒī„Žī„´ī„ŗ ī„Ŧī…‹ ī…‚ī„ ī„Ŧī…„ ī„ĩī„Ēī…Œ ī„›ī„ĩī„Ž ī„Ŧī…‚ ī„ēī„ŗī…‚ī„îĨĄ
ī„’ī„™ ī„›ī…ƒī„ģī…œī„Ŧī…‡ ī„ģī…‚ī„:
ķš¸¯ķš¸­ķš¸Žī„›ī„Ŧī„ŗ 1: ī„¯ī„Ąī…‚ī„Š ī„Ŧī…„ ī„¸ī„ĩī„Ēī…Œ ī„›ī„ĩī…‹
ī„´ī…‚ī„Ŧ ī„ĩī…žī„œī…‹:
ī„“ī„Ģī…‡,
â€ĸ a=5a
ķŊ„ŦķŊ„­ķŊ„ŽķŊ„¯ķŊ„°ī„›ī„Ŧī„ŗ 2: f(x) ī„Žī…†ī…œ ī„¸ī…žī„œī„ĩī…‚ ī„›ī„ĩī…‹
ī„ēī…‚ī„Žī…†ī…œ ī„“ī„ģ ī„ļī…žī„˛ī„Š ī„Ŧī…„ ī„ļī…‹ī„ŋ ī„ģī…‰:
ī„ģī……ī„Š ex2 ī„Žī…†ī…œ ī…žī„œī„ĩī…‚ ī„›ī„ĩī…‹:
ī„Ēī…‚ī„ ī„‘ī…œī„Ē ī…ƒ ī…žī„ :
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ķšš‹ķššŒī„›ī„Ŧī„ŗ 3: ī„¯ī…†ī„ĩī…‚ ī„°ī…‚ī„ĩī„ŗī…†ī„ļī…‚ ī„ļī…‚ī„ī…† ī„›ī„ĩī…‹
ī„ģī……ī„Š ī„ļī…‚ī„ī…† ī„›ī„ĩī…‹:
ī„ēī„ĩī„ļ ī„ąī„Šī…‚ī„™:
ķŧ¨ķŧ¨‘ķŧ¨’ī„ĩī…„ī„’ ī„‘ī„Ēī…‡ ī„‘ī„ĩī„ĸī……ī„Ž ī„ļī„” ī„‘ī…œī…ƒī„Ēī„ŗ ī…ƒī„¸ī„ ī…‚ī„ĩ...
ī„‘ī…ƒī„­ī„’ī„¯ī„› ī„Žī…‡ ī„•ī„Žī…§ī…‚ī„ ī„Žī…†ī…œ ī…ƒī„›ī„ģī…‚,
"ī…ƒ ī…ƒī„˛ī…œī„Žī„Ēī…‚ ī„“ī…žī„› ī„ĸī…‚ī„ēī…†ī„ē ī„ģī…‹ī„Š ī…‚ī„ī„ ī„ģī…‰ - ī„Ēī……ī„ģī…‚ī„§ī…‚ ī„›ī…œī„ŗ ī„°ī…œī„›ī„šī„Ž ī„Ŧī…„ ī„ąī„Šī„Ēī„ĩ ī„Žī…†ī…œ ī…‡ī„œī„Šī…‚ ī„‘ī„Ēī…‡ ī„“ī„ģ ī„¯ī„Ēī…‚ ī„ļī„ī…‚ī„•ī„Šī…‚ ī„ģī…‰
ī…ƒī„› ī…ƒī„›ī„ģī„ŋī…‚ ī…ƒī„Žī„´ī„ŗ ī„ēī„˛ ī„Ēī…Œ ī„­ī…„ī„’ ī…ƒī„°ī…žī„Ĩ ī„ąī…‰ī„Ļī„Ŧī…‚ ī„ģī…‰îĨĄ ī„“ī…žī„› ī…‚ī„ĩ ī„ĸī„Ŧī…Œ ī„Ēī……ī„ēī…„
ī„
ī„’ī„¯ī„Šī…‡ ī„šī„žī…‚ī„ĩī…‚ī„ ī„Žī…†ī…œ ī„ĸī…‚ī„Š ī„ļī…Šī„Ŧī…‡ ī„ģī…‹ -
ī„•ī„Ēī„¯ī…‚ī„Ŧ ī…ƒī„Žī„´ī„ŗ, ī„˛ī…‚ī„ī„°ī„ļ ī…ƒī„Žī„´ī„ŗ, ī„ ī…‡ī„Ž ī…ƒī„Žī„´ī„ŗ, ī„ļī„žī…†ī„ī„Šī„› ī„‘ī…œī„Ēī„ĩ - ī„Ēī…‚ī„ ī„Ēī……ī„ēī…„
ī„
ī„Ŧī……ī„ąī…‚ī„ĩī…‚ ī„›ī„Ŧī…‡ ī„Žī„ģī…„
ī„
ī„§ī„ĩī…‹ī„ī…‡îĨĄ"
8. (i) ī„ŗī…œī„ ī„°ī…œī„›ī„šī„Ž p = 50 - 3q ī…ƒī„Ŧī…žī„Ēī…‡ ī„ĸī…‚ī„Š 'ī„Ēī…‡ ī„ŗī…œī„ p = 5 ī„Ŧī…„ ī„ļī„ ī„›ī„Ēī…‚ ī„¯ī„Ēī…‚ ī„›ī„ĩī…‹îĨĄ
(ii) ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ ī…ƒī„› ī„ŗī…œī„ ī„°ī…œī„›ī„šī„Ž ī„Ŧī…„ ī„ļī„ ī„›ī„Ēī…‚ = ī„ģī…‰ ī…ƒī„ĸī…žī„Ģī…‡ ī„ŗī…œī„ p = 50 - 3x ī„‘ī„Ēī…‡ p = 5 ī„ģī…‰îĨĄ
îš­îšž
îš­îšžī„ŋîššîšž
(iii) ī„›ī……ī…žī„ļ ī„ļī…‚ī„ī„Ē ī„°ī…œī„›ī„šī„Ž C = 60 - 12q + 2q2 ī…ƒī„Ŧī…žī„Ēī…‡ ī„ĸī…‚ī„Š 'ī„Ēī…‡ AC, MC ī„ļī…žī„˛ī…‹ ī„‘ī„Ēī…‡ AC ī„Ŧī…„ ī„¨ī„ļī…‚ī„Š = (MC -
AC) ī…ƒī„Ŧī„œī…‚ī„™îĨĄ
īƒš
îē—
ī„•ī…žī„Ēī„ĩ:ķˇ‡´ķˇ‡ĩķˇ‡ļķˇ‡ˇķˇ‡¸ķˇ‡š ī„ĸī…‚ī„Š-ī„¯ī„Ąī…‚ī„Š: ī„‘ī„ĩī„Ģī„šī…‚ī„ēī„Ēī„ĩ ī„Ŧī…„ ī„Ŧī……ī„Žī…„ī„’ ī…ƒ ī…žī„  ī„“ī…žī„› ī„ēī…‰ī„ĩ
ī„›ī„ļī„¯ī„Žī…‚ ī„›ī„ĩī…‹ ī…ƒī„› ī„‘ī„ĩī„ĸī……ī„Ž ī„Žī…‚ī„ŗ ī„Ŧī…‚ ī„“ī…žī„› ī„Žī…ī„ĸ ī…‚ī„Ž ī„•ī…žī„Ŧī„ŗī…„ ī„ģī…‰ ī„ĸī…‹ ī„ģī…žī„Ģ ī„Žī…‚ī„ļ ī„ąī„Šī…„ī„’ī„ ī„Žī…‹ī„Ĩī„ąī……ī…žī„›ī…‚ī„ ī…‡ī„ ī„Š ī„Ŧī…‚ ī„“ī…žī„›
ī„Ąī…‹ī„Ĩī…‚ ī…ƒī„ĸī„ģī…‚ ī„šī„Žī„ļī…‚ī„”ī„Ž ī„›ī…‚ī„ĩī…‹ī„ąī…‚ī„ĩ ī„ ī„ļī…‚ī„•
ī„
ī„Ŧī…‚ ī„ģī…‰îĨĄ ī„•ī„ģ ī„“ī„ģ ī„ēī„ŗī„Ŗī„Š ī„Ŧī…„ ī„›ī…‹ī…ƒī„šī„š ī„›ī„ĩ ī…ƒī„ĩī„ģī…‚ ī„ģī…‰ ī…ƒī„› ī„›ī…„ī„ŗī„Ē ī…ƒ ī…žī„ 
ī„ąī„Ŧī„ļī…‚ī„‘ ī„ŗī…œī„ ī„Žī…†ī…œ ī…ƒī„› ī…ˆ ī„¯ī…Ĩī„˛ī…‚ī…ƒ ī„Ē ī„›ī„ĩī„Ŧī…‡ ī„ģī„Ž, ī„•ī„ģ ī…ƒī„›ī…œī„Žī…‚ ī„ŗī…‚ī„ļī…„ī„’ ī„›ī„ŗī…‚ī„•
ī„
ī„Ŧī…‚ ī„ģī…‰, ī„‘ī„Ēī…‡ ī„’ī„¯ī„Šī…‡ ī„•ī„Ēī„¯ī…‚ī„Ŧī…‚ī„ ī„Žī…†ī…œ
ī…ƒī„Ēī„’ī„ĩ ī„›ī„ĩī„Ž ī„ļī„” ī…ƒī„›ī…œī„Žī…‚ ī„œī„ĩī„ ī…‚ ī„’ī„•
ī„
ī„Ŧī…‚ ī„ģī…‰îĨĄ ī„“ī…žī„› ī…ƒī„Ŧī„Ž, ī„•ī„ģ ī„’ī„¯ī„Šī…„ ī„¯ī……ī„ĩī…‚ī„Šī…„ ī„ēī„›ī…†ī„ļ ī„‘ī…ƒī„­ī„’ī„¯ī„›ī…‚, ī„šī…Ĩī…„ī„ŗī„Ēī…„
ī„ŗī…ƒī„ģī„ĩī…‚ ī„Žī…†ī…œ ī…ƒī„ŗī„ļī„Ŧī…‚ ī„ģī…‰, ī„ĸī…‹ ī„“ī…žī„› ī„ģī……ī…ƒī„šī„’ī„ĩ ī„‘ī„ĩī„Ģī„šī…‚ī„ēī„Ēī„ĩī…„ ī„ģī…‰, ī„ĸī…‹ ī„•ī„ēī„Žī…†ī…œ ī„ēī„­ī…‚ī„ĩī„Ž ī„ī…ƒī„Šī„Ē ī„‘ī„Ēī…‡ ī„Ēī„ĩī„› ī„Ŧī…„ ī„ĩī„Ēī…Œ
ī„›ī„ĩī„›ī…‡ ī„“ī„ģī„Žī…‚ī„ ī„ēī…‚ī„ĩī…„ī„’ī„ ī„­ī…‚ī„ĩī„Žī…‚ ī…‚ī„ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Š ī…ƒ ī…žī„  ī„ŗī„Ŧī„Ŧ ī„›ī„ĩī„Ž ī„ļī„” ī„ēī…ƒī„ģī„ŗī„Ē ī„ģī……ī…œī„Ŧī…„ ī„ģī…‰îĨĄ ī…ƒī„ĸ ī…ˆ ī„ģī…„ ī„•ī„ģ ī„•ī„ēī„Ŧī…„
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ī„’ī„ĩī…‚ī„ŗī„Ŧī…‚ī„“ī„› ī„Ąī…‹ī„Ĩī…„ ī„ĩī„›ī„šī…‚ī„¯ ī…ƒ ī…žī„  ī„“ī„›ī…žī„Ļī…‡ ī„ąī…‰ī„Ļī„Ŧī…‡ ī„ģī„Ž, ī„•ī„ģ ī„’ī„¯ī„Šī…„ ī„ī…žī„ļī„ąī…‚ī„Ē ī„šī……ī„ĩī…† ī„›ī„ĩī„Ŧī…‡ ī„ģī„ŽîĨĄ ī„Ēī……ī„ēī…„
ī„
, ī„“ī…žī„›
ī„•ī„Ēī„ēī……ī„› ī…ƒ ī…ƒī„Ŧī„’ī„ĩī„Ģī…„ ī„Ŧī…‡ ī„ĩī…†ī„¯ ī…ƒ ī…žī„ , ī„ ī……ī…žī„¯ī„ ī…‚ī„¯ ī„Ŧī…‡ī„œ ī„ĩī„ģī…‡ ī„ģī…‹ ī„‘ī„Ēī…‡ ī„ēī……ī„Š ī„ĩī„ģī…‡ ī„ģī…‹îĨĄ
ī„’ī„™ ī„‘ī„ĩī„ĸī……ī„Ž ī„‘ī„Ēī…‡ ī„šī…Ĩī…„ī„ŗī„Ēī…„ ī„ŗī…ƒī„ģī„ĩī…‚ ī„Ŧī…‡ ī„Žī…‚ī„ļ ī„ ī…žī„ļī…„ī„— ī…ƒī„ĸ ī…ˆ ī…ƒī„› ī„‘ī„ēī…„
ī„
ī„ēī…†ī„œī„ŗ ī„‘ī„ĩī„Ģī„šī…‚ī„ēī„Ēī„ĩ ī„Ŧī…‡ ī…ƒī„Ēī…œī„Ž ī„ŗī„ģī…žī„Ē ī„¯ī…†ī„ĩī„Ž
ī…ƒī„ģī…žī…ƒī„ēī„’ī„ ī„Ŧī…„ ī„¯ī„ŋī„ ī…‹ī„ļ ī„›ī„ĩī„Ŧī…‡ ī„ģī…‚ī„: ī„ŗī…œī„ ī„Ŧī…„ ī„ļī„ ī„›ī„Ēī…‚, ī„ļī„ ī„›ī„Ēī…‚ ī„‘ī„Ēī…‡ ī„ŗī…‚ī„ļī…„ī„— ī…ƒ ī„ ī„›ī…‚ī„ĩ ī„ēī„ąī…œī„­, ī„‘ī„Ēī…‡ ī„ļī…‚ī„ī„Ē
ī…ƒ ī„šī„ļī…‡ī„šī„Š - ī„›ī„Ŧī„ŗ ī„Ŧī„ĩ ī„›ī„Ŧī„ŗîĨĄ
ķŧŠ•ķŧŠ–ķŧŠ—ķŧŠ˜ķŧŠ™ķŧŠš ī„˛ī…‚ī„ (i): ī„ŗī…œī„ ī„Ŧī…„ ī„ļī„ ī„›ī„Ēī…‚ ī„ĸī„Ŧī…Œ p = 5 ī„‘ī„Ēī…‡ p = 50  3q
ī„šī…Ĩī…„ī„ŗī„Ēī…„ ī„ŗī…ƒī„ģī„ĩī…‚ ī„šī……ī„ĩī…† ī„›ī„ĩī„Ŧī…„ī„’ī„ ī„ģī„Ž:
"ī„‘ī„ĩī„ĸī……ī„Ž, ī„ŗī…œī„ ī„Ŧī…„ ī„ļī„ ī„›ī„Ēī…‚ ī„ēī…‚ī„Žī…†ī…œ ī„Ŧī…žī„ēī„Ŧī…„ ī„ģī…‰ ī…ƒī„› ī„œī„¯ī„Ēī„›ī…‚ī„ĩ ī„›ī…„ī„ŗī„Ēī…‚ī„ ī…ƒ ī…žī„  ī„Ēī„ąī„Ŧī…„ī„ļī…„ī„’ī„ ī„¯ī…Ĩī„Ēī…„ ī…ƒī„›ī…œī„Žī…‡ ī„ēī…œ ī…‡ī„Ŧī„Žī„šī…„ī„ļ
ī„ģī„ŽîĨĄ ī„“ī„ģ ī„¯ī……ī…žī„Ąī„Š ī„ĩī„ī…‚ ī„ģī…‰ - ī„ĸī…‡ ī„ŗī…Š ī„“ī…žī„› ī„Žī…‹ī„Ĩī„ąī……ī…žī„› ī„Ŧī…„ ī„›ī…„ī„ŗī„Ē ī„žī„Ĩī…‚ ī…ƒī„Ŧī…œī„Ŧī…‚ ī„ģī…‚ī„, ī„Ēī…‚ī„ ī„ļī…‹ī„› ī„ģī…‹ī„ĩ ī…ƒī„›ī…œī„Žī…‚ ī„œī„ĩī…„ī„Ŧī„Šī…‚
ī„ ī…‚ī„ģī……ī„Šī„ī…‡?"
ī„ŗī…œī„ ī„°ī…œī„›ī„šī„Ž ī„“ī„ģ ī„ģī…‰:
p=50−3qp = 50 - 3qp=50−3q
ī„¯ī„ĩ ī„ļī„ ī„›ī„Ēī…‚ ī„Ŧī…„ ī„ī„Šī„Žī…‚ ī„›ī„ĩī„Ž ī„ļī„”, ī„ēī…‚ī„Žī…†ī…œ p ī„Ŧī…‡ ī„°ī…œī„›ī„šī„Ž ī„ĸī…Œ q ī„Ŧī…„ ī„ļī…‹ī„ŋ ī„ģī…‰îĨĄ ī„“ī„ē ī„ļī„”, ī„ēī„ŗī…„ī„›ī„ĩī„Ž ī„Žī…†ī…œ ī„ŗī……ī„ŋ
ī…ƒ ī„ēī…ƒī„Ģī„Ē ī„›ī„ĩī…‹:
ī„ģī……ī„Š ī„ŗī…œī„ ī„Ŧī…„ ī„›ī…„ī„ŗī„Ē ī„ļī„ ī„›ī„Ēī…‚ (Ep) ī„ļī„” ī„°ī…‚ī„ĩī„ŗī…†ī„ļī…‚ ī„´ī…‚ī„Ŧ ī„›ī„ĩī…‹:
ī„’ī„™ ī„“ī„ēī„Žī…†ī…œ ī„›ī„Ŧī„ŗ-ī„Ŧī„ĩ-ī„›ī„Ŧī„ŗ ī„Ēī…‹ī„ŋī…„ī„—:
p ī„Ŧī…‡ ī„ēī…œī„ąī…œī„­ ī…ƒ ī…žī„  ī„‘ī…œī„Ēī„ĩ ī„›ī„ĩī…‹:
ķŧ¨ģķŧ¨ŧ ī„›ī„Ŧī„ŗ 2: ī„ļī„ ī„›ī…„ī„ļī…‡ī„¯ī„Š ī„°ī…‚ī„ĩī„ŗī…†ī„ļī…‡ ī…ƒī„¸ī…žī„  ī„¯ī„ļī…žī„ ī„“ī„Ž ī„›ī„ĩī…‹
ī„ēī…‚ī„§ī…‡ ī„›ī…‹ī„ļ ī„¯ī…ƒī„ģī„ļī…‚ī„ ī„ģī…„ ī„ģī…‰:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ķˇƒ†ķŧŊĸ ī„‘ī…œī…ƒī„Ēī„ŗ ī„•ī…žī„Ēī„ĩ:
p=5 'ī„Ēī…‡ ī„ŗī…œī„ ī„Ŧī…„ ī„›ī…„ī„ŗī„Ē ī„ļī„ ī„›ī„Ēī…‚ ī„“ī„ģ ī„ģī…‰:
ī„“ī„ēī„Ŧī…‚ ī„ŗī„Ēī„ļī„ą ī„ģī…‰ ī…ƒī„› ī„“ī„ē ī„›ī…„ī„ŗī„Ē ī…ƒī„ąī…œī„Ŧī…† 'ī„Ēī…‡ ī„ŗī…œī„ ī„‘ī„ēī…ƒī„Ģī„ĩ ī„ģī…‰ - ī„ī…‚ī„ģī„› ī„›ī…„ī„ŗī„Ēī…‚ī„ ī…ƒ ī…žī„  ī„Ēī„ąī„Ŧī…„ī„ļī…„ī„’ī„ 'ī„Ēī…‡ ī„ąī„ģī……ī„Ēī…‚
ī„¯ī…Ĩī„Ēī…„ī…ƒī„›ī…ƒī„ĩī„’ ī„Žī„ģī…„
ī„
ī„›ī„ĩī„Ŧī…‡îĨĄ
ķŧŠ•ķŧŠ–ķŧŠ—ķŧŠ˜ķŧŠ™ķŧŠš ī„˛ī…‚ī„ (ii): ī„ļī„ ī„›ī„Ēī…‚ ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī„Žī…‚ = AR / (AR  MR)
ī„šī…Ĩī…„ī„ŗī„Ēī…„ ī„ŗī…ƒī„ģī„ĩī…‚ ī„Žī…‡ ī„ŗī……ī„ēī„›ī„ĩī…‚ī„•
ī„
ī„Ŧī…‡ ī„ģī…‹ī„— ī…ƒī„›ī„ģī…‚, "ī„ģī……ī„Š, ī„‘ī„ĩī„ĸī……ī„Ž, ī„ŗī…Š ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī„“ī…žī„› ī„ĸī…‚ī„Ŧī…†ī„” ī„°ī…‚ī„ĩī„ŗī…†ī„ļī…‚ ī…ƒī„Ŧī„œī…‚ī„•
ī„
ī„Ŧī…‚ ī„ģī…‚ī„ ī„ĸī…‹
ī„ŗī…œī„, ī„ŗī…‚ī„ļī…„ī„’ ī„‘ī„Ēī…‡ ī„ļī„ ī„›ī„Ēī…‚ ī„Žī…†ī…œ ī„ĸī…‹ī„ŋī„Ŧī…‚ ī„ģī…‰îĨĄ"
ķˇ•˜ķˇ•™ķˇ•š ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī„Ž ī„ļī„” ī„ŗī……ī…žī„œ ī„°ī…‚ī„ĩī„ŗī…†ī„ļī…‚:
ī„’ī„™ ī„“ī„ēī„Žī…†ī…œ ī„›ī„Ŧī„ŗī…‚ī„ ī…ƒī„¸ī…žī„  ī„¸ī…œī„§ī…„ī„—:
ķŧ¨ģķŧ¨ŧ ī„›ī„Ŧī„ŗ 1: ī„šī„ĩī„Ēī…‚ī„ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī…‹
â€ĸ AR (ī„šī„ēī„Ē ī„’ī„ŗī„Ŧī„Ž) = ī„›ī…„ī„ŗī„Ē (p)
â€ĸ TR (ī„›ī……ī…žī„ļ ī„ŗī…‚ī„ļī…„ī„’) = p × q
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
ķŧŠ•ķŧŠ–ķŧŠ—ķŧŠ˜ķŧŠ™ķŧŠš ī„˛ī…‚ī„ (iii): ī…ƒī„Ŧī…žī„Ēī…‡ ī„ī„— ī„ļī…‚ī„ī„Ē ī„°ī…œī„›ī„šī„Ž, AC, MC ī„ļī…žī„˛ī…‹ ī„‘ī„Ēī…‡ AC ī„Ŧī…„ ī„¨ī„ļī…‚ī„Š ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ =
(1/q)(MC  AC)
ī„šī…Ĩī…„ī„ŗī„Ēī…„ ī„ŗī…ƒī„ģī„ĩī…‚ ī„Žī…‡ ī„‘ī…žī„ī…‡ ī…ƒī„›ī„ģī…‚, "ī„ģī……ī„Š, ī„’ī„™ ī„ļī…‚ī„ī„Ē ī„Ŧī…„ ī„Ŧī……ī„Žī…„ī„’ ī…ƒ ī…žī„  ī„§ī…†ī„ąī„Ŧī…‡ ī„ģī…‹ī„”ī„—îĨĄ ī„‘ī„ĩī„ĸī……ī„Ž, ī„“ī„ģ ī„Ēī……ī„ģī…‚ī„§ī…„
ī„Žī…‹ī„Ĩī„ąī……ī…žī„› ī„•ī„Ēī„¯ī…‚ī„Ŧī„Ž ī„ļī„” ī„›ī……ī…žī„ļ ī„ļī…‚ī„ī„Ē ī„°ī…œī„›ī„šī„Ž ī„ģī…‰îĨĄ"
C=60−12q+2q2
ī„ēī…‚ī„Žī…†ī…œ ī„“ī„ģ ī„›ī„ĩī„Žī…‚ ī„ģī…‰:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
1. AC (ī„šī„ēī„Ē ī„ļī…‚ī„ī„Ē) = C/q ī„ļī…žī„˛ī…‹îĨĄ
2. MC (ī„ēī…„ī„ŗī…‚ī„ī„Ē ī„ļī…‚ī„ī„Ē) = dC/dq ī„ļī…žī„˛ī…‹
3. ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹:
ķŊ„ĄķŊ„ĸķŊ„ŖķŊ„¤ķŊ„ĨķŊ„Ļ ī„›ī„Ŧī„ŗ 1: ī„šī„ēī„Ē ī„ļī…‚ī„ī„Ē (AC)
ķŊ„ĄķŊ„ĸķŊ„ŖķŊ„¤ķŊ„ĨķŊ„Ļ ī„›ī„Ŧī„ŗ 2: ī„ēī…„ī„ŗī…‚ī„ī„Ē ī„ļī…‚ī„ī„Ē (MC)
ķŊ„ĄķŊ„ĸķŊ„ŖķŊ„¤ķŊ„ĨķŊ„Ļ ī„›ī„Ŧī„ŗ 3: ī„ēī…‚ī„ąī„Ē ī„›ī„ĩī…‹ ī…ƒī„›:
AC ī„Žī…†ī…œ q ī„Ŧī…‡ ī„ēī…œī„ąī…œī„­ ī…ƒī„¸ī…žī„  ī„¸ī…žī„œī„ĩī…‚ ī„›ī„ĩī…‹:
ī„ģī……ī„Š RHS ī„Ŧī…„ ī„ī„Šī„Žī…‚ ī„›ī„ĩī…„ī„—:
ī„ēī…‚ī„§ī…‡ ī„›ī…‹ī„ļ ī„¯ī…ƒī„ģī„ļī…‚ī„ ī„ģī…„ ī„ģī…‰:
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
â€ĸ ī„˜ī„ŗī„ēī…„ = -12 + 4 ī…ƒī„›ī„–
ī„“ī„ē ī„ļī„”:
ī„ēī„ĩī„ļ ī„ąī„Šī…‚ī„™:
ī„ģī……ī„Š:
ī„“ī„ē ī„Žī…‚ī„ļ ī„Ēī……ī„ļī„Žī…‚ ī„›ī„ĩī…‹:
ī„Ŧī…‹ī„¸ī…ˆ ī„¯ī…Ĩī„ī„Ĩī…‚ī„¸ī…‡ ī„“ī…žī„›ī…‹ ī…ƒī„ĸī„ģī…‡ ī„ģī„Ž:
ķˇƒ†ķŧŊĸ ī„“ī„ē ī„ļī„” ī„ēī…‚ī„ąī„Ē ī„ģī…‹ī„“ī„’!
ķš´ˇķš´ēķš´¸ķš´šķš´ģķš´ŧķš´Ŋķš´žķš´ŋķšĩ€ķšĩķšĩ‚ ī„‘ī„ĩī„ĸī……ī„Ž ī„Ŧī…‡ ī„šī„ąī„Ŧī…‚ī„ ī…ƒī„¸ī…žī„  ī„ēī…œī„œī…‡ī„¯
ī„‘ī„ĩī„ĸī……ī„Ž ī…ƒī„¯ī…žī„Ąī…‡ ī„Ŗī……ī…ƒī„›ī„’ ī„‘ī„Ēī…‡ ī…ƒī„›ī„ģī…‚, " ī…‚ī„ģ! ī„Ēī…‚ī„ ī„“ī„ģ ī„ģī…‰ ī„ĸī…‹ ī„ŗī…Š ī„‘ī…žī„ĸ ī…ƒī„ēī…žī…ƒī„œī„’ ī„ģī…‰:
â€ĸ ī„ļī„ ī„›ī„Ēī…‚ ī„ŗī…‰ī„Žī…†ī…œ ī„Ŧī…žī„ēī„Ŧī…„ ī„ģī…‰ ī…ƒī„› ī„ī…‚ī„ģī„› ī„›ī…„ī„ŗī„Ē ī…ƒ ī…žī„  ī„Ēī„ąī„Ŧī…„ī„ļī…„ī„’ī„ ī„¯ī…Ĩī„Ēī…„ ī…ƒī„› ī…ˆ ī„¯ī…Ĩī„Ēī…„ī…ƒī„›ī…ƒī„ĩī„’ ī„›ī„ĩī„Ŧī…‡ ī„ģī„ŽîĨĄ
â€ĸ ī„ĸī…‡ī„›ī„ĩ ī„ļī„ ī„›ī„Ēī…‚ ī„žī…žī„Ĩ ī„ģī…‰, ī„Ēī…‚ī„ ī„›ī…„ī„ŗī„Ēī…‚ī„ ī„žī„Ĩī…‚ī„•ī„Š ī„Žī…‚ī„ļ ī„ŗī…‡ī„ĩī…„ ī…ƒ ī„›ī„ĩī…„ ī…ƒī„žī„’ī„Ŧī…‚ ī„Žī„ģī…„
ī„
ī„­ī…‡ī„ī…„îĨĄ
â€ĸ AR, MR, ī„‘ī„Ēī…‡ ī„ļī„ ī„›ī„Ēī…‚ ī„ēī…‚ī„ĩī…‡ ī„“ī…žī„› ī„ēī……ī…œī„Ŧī„ĩ ī„°ī…‚ī„ĩī„ŗī…†ī„ļī…‡ ī…ƒ ī…žī„  ī„ĸī……ī„ŋī…‡ ī„ģī…‹ī„— ī„ģī„ŽîĨĄ
â€ĸ ī„ļī…‚ī„ī„Ē ī„°ī…œī„›ī„šī„Ž ī„ŗī…‰ī„Žī…†ī…œ ī„“ī„ģ ī„¯ī„Ēī…‚ ī„ļī„ī…‚ī„•ī„Š ī…ƒ ī…žī„  ī„ŗī„Ŧī„Ŧ ī„›ī„ĩ ī„ēī„›ī„Ŧī…‡ ī„ģī„Ž ī…ƒī„› ī„ŗī…Š ī„›ī……ī„šī„ļ ī„ģī…‚ī„ ī„ĸī…‚ī„ ī„Žī„ģī…„
ī„
îĨĄ
ī„”ī„žī…„2ī…ƒī„ēī„›ī„šī…‚.ī„›ī…‚ī…žī„ŗ
â€ĸ ī„‘ī„Ēī…‡ ī„ēī„˛ ī„Ēī…Œ ī„ģī…‰ī„ĩī…‚ī„Žī…„ī„ĸī„Žī„› ī„ī…žī„ļ ī„“ī„ģ ī„ģī…‰ ī…ƒī„› ī„ŗī…‡ī„ĩī…„ ī„šī„ēī„Ē ī„ļī…‚ī„ī„Ē ī„›ī„ĩ ī„Ŧī…„ ī„¨ī„ļī…‚ī„Š ī…„ ī„“ī…žī„› ī„ī…ƒī„Šī…ƒī„Ēī„›
ī…ƒī„Žī„´ī„ŗ ī„Ŧī…„ ī„¯ī…‚ī„ļī„Šī…‚ ī„›ī„ĩī„Ŧī…„ ī„ģī…‰!”
ī„šī…Ĩī…„ī„ŗī„Ēī…„ ī„ŗī…ƒī„ģī„ĩī…‚ ī„ŗī……ī„ēī„›ī„ĩī…‚ī„”, "ī…ƒī„ąī„ļī„›ī……ī„ļ, ī„‘ī„ĩī„ĸī……ī„ŽîĨĄ ī„‘ī„ĩī„Ģī„šī…‚ī„ēī„Ēī„ĩ ī…ƒī„ēī„ĩī…€ ī„‘ī…œī„›ī…ƒī„ŋī„’ī„ ī„ąī…‚ī„ĩī…‡ ī„Žī„ģī…„
ī„
ī„ģī…‰ - ī„“ī„ģ ī„ŗī„Žī……ī…ž ī„œī…„
ī…ƒ ī„ģī…‚ī„ĩ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Š ī„‘ī„Ēī…‡ ī„ēī„ŗī„Ŗī„Ŧī…‚ī„ĩī…„ ī„Žī…‚ī„ļ ī„°ī…‰ī„ēī„ļī…‡ ī„ļī…‰ī„Š ī„ąī…‚ī„ĩī…‡ ī„ģī…‰îĨĄ"
ķˇ™Žķˇ™ķˇ™ ī…ƒī„ēī…žī„Ĩī…‚: ī„ēī„­ī…‚ī„ĩī„Ž ī„’ī„ĩī…ƒī„Ģī„› ī„ī…ƒī„Šī„Ē ī„Ŧī…„ ī„ēī……ī…œī„Ŧī„ĩī„Ēī…‚
ī„‘ī„ĩī„ĸī……ī„Ž ī…‚ī„ī„, ī„ēī…‚ī„§ī…‡ ī…ƒ ī…žī„ ī…Œ ī„ąī„ģī……ī„Ē ī„ēī…‚ī„ĩī…‡ ī„ļī…‹ī„›ī…‚ī„ ī„Žī…†ī…œ ī„“ī„ģ ī„ēī„ŗī„Ŗī„Š ī…ƒ ī…žī„  ī„ŗī……ī„šī„›ī„ļ ī„’ī„•
ī„
ī„Ŧī…„ ī„ģī…‰ ī…ƒī„› ī„›ī…„ī„ŗī„Ē, ī„’ī„ŗī„Ŧī„Ž ī„‘ī„Ēī…‡
ī„ļī…‚ī„ī„Ē ī„Ŧī…‡ ī„Ĩī……ī„›ī„ŋī…‡ ī…ƒī„› ī…ˆ ī„“ī„›ī…žī„Ļī…‡ ī…ƒī„°ī…žī„Ĩ ī„ģī……ī…œī„Ŧī…‡ ī„ģī„ŽîĨĄ ī„¯ī„ĩ ī„“ī…žī„› ī…‚ī„ĩ ī„ĸī„Ŧī…Œ ī„Ēī……ī„ēī…„
ī„
ī„“ī„ēī„Žī…†ī…œ ī„ēī„­ī…‚ī„ĩī„Ž ī„›ī„Ŧī„ŗī…‚ī„ ī„‘ī„Ēī…‡ ī„‘ī„ēī„ļ-
ī„ĸī…„ ī„Ž ī„Ŧī…‡ ī„Ēī„ĩī„› ī„Ŧī…„ ī„ĩī„Ēī…Œ ī„›ī„ĩī„›ī…‡ ī„Ēī…‹ī„ŋ ī…ƒī„Ŧī…œī„Ŧī…‡ ī„ģī…‹, ī„Ēī…‚ī„ ī„ģī„ĩ ī„ ī…„ī„ž ī„ēī„ŗī„Ŗ ī„’ī„•ī„Š ī„ļī…žī„ ī„¯ī…Šī„Ŧī…„ ī„ģī…‰îĨĄ ī„ģī„ŗī…‡ī„šī…‚ ī„´ī…‚ī„Ŧ ī„ĩī…žī„œī…‹:
â€ĸ ī„ļī„ ī„›ī…„ī„ļī…‚ī„¯ī„Š ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī„ī…‚ī„ģī„›ī…‚ī„ ī„Ŧī…„ī„’ī„ ī„¯ī…Ĩī„Ēī…„ī…ƒī„›ī…ƒī„ĩī„’ ī…‚ī„ ī„Žī…†ī…œ ī„ēī„ŗī„Ŗī„Š ī…ƒ ī…žī„  ī„ŗī„Ŧī„Ŧ ī„›ī„ĩī„Ŧī…‚ ī„ģī…‰îĨĄ
â€ĸ AR, MR, ī„‘ī„Ēī…‡ ī„ļī„ ī„›ī„Ēī…‚ ī„¯ī…ƒī„ĩ ī…‚ī„ĩ ī…‚ī„ī„ ī„ĸī……ī„ŋī…‡ ī„ģī…‹ī„— ī„ģī„ŽîĨĄ
â€ĸ ī„ļī…‚ī„ī„Ē ī„›ī„ĩ ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī„Ŧī…žī„ēī„Ŧī…‡ ī„ģī„Ž ī…ƒī„› ī„Ēī……ī„ēī…„
ī„
ī…ƒī„›ī…œī„Žī…„ ī„›ī……ī„šī„ļī„Ēī…‚ ī„Žī…‚ī„ļ ī„•ī„Ēī„¯ī…‚ī„Ŧī„Ž ī„›ī„ĩ ī„ĩī„ģī…‡ ī„ģī…‹îĨĄ
ī„“ī„ē ī„ļī„”, ī„°ī…‚ī„ĩī„ŗī…†ī…ƒī„ļī„’ī„ ī„Ēī…Œ ī„Žī…‚ ī„§ī„ĩī…‹â€”ī„‘ī„ĩī„ĸī……ī„Ž ī…‚ī„ī„ ī„•ī„Ēī„ēī……ī„›ī„Ēī…‚ ī„Žī…‚ī„ļ ī„•ī„Žī…§ī…‚ī„ ī„Žī…†ī…œ ī„‘ī„¯ī„Šī…‚ī„™îĨĄ ī„˛ī…‚ ī…ˆ ī„Ēī……ī„ēī…„
ī„
ī„›ī…‹ī„”
ī„›ī…‚ī„ĩī…‹ī„ąī…‚ī„ĩ ī„ ī„ļī…‚ī„•
ī„
ī„Ŧī…‡ ī„ģī…‹ ī„ĸī…‚ī„ ī„¯ī…Ĩī…„ī…ƒī„œī„’ ī„Ŧī…‡ ī„¯ī…Ĩī„šī„Ž ī„ģī…žī„ļ ī„›ī„ĩī„Ŧī…‡ ī„ģī…‹, ī„“ī„ģ ī„ēī…œī„›ī„ļī„¯ ī„ģī„ŗī…‡ī„šī…‚ ī„Ēī……ī„ģī…‚ī„§ī…‚ ī„ŗī…‚ī„ĩī„ī„Ŧī„ĩī„šī„Ž
ī„›ī„ĩī„Žī„ī…‡îĨĄ
"ī„“ī„ģ ī„¯ī…‡ī„¯ī„ĩ ī…ƒī„¸ī…ƒī„Ŧī„‘ī„› ī„•ī„Ŧī…‡ī„šī…‚ī„ ī„ļī„” ī…ƒī„­ī„’ī„Ž ī„Žī…‚ī„ļ ī…ƒī„Ēī„’ī„ĩ ī„›ī…„ī„Ēī…‚ ī…ƒī„ī„’ ī„ģī…‰îĨĄ ī„ĸī…‡ī„›ī„ĩ ī„Ēī……ī„ģī…‚ī„Žī…†ī…œ ī„›ī…‹ī„” ī„ī„ļī„Ēī…„ ī„Žī„žī„ĩ
ī„’ī„•
ī„
ī„Ŧī…„ ī„ģī…‰ ī„ĸī…‚ī„ ī„Ēī……ī„ģī…‚ī„§ī…‡ ī„›ī…‹ī„ļ ī„›ī…‹ī„” ī„ēī……ī„Ŗī…‚ī„‘ ī„ģī„Ž, ī„Ēī…‚ī„ ī„’ī„¯ī„Šī…„ ī„°ī…„ī„§ī„ąī…‰ī„› ī„ēī…‚ī„ī„Ŗī…„ ī„›ī„ĩī„Ž ī„ļī„” ī„ąī…‡ī…ƒī„Ŗī„ĸī„› ī„ŗī…ƒī„ģī„ēī…†ī„ē ī„›ī„ĩī…‹îĨĄ"