This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Haya hapa masuluhisho ya maswali yako: 2. b) Kutafuta thamani ya 45: Step 1: Tumia sheria za logariti kueleza 45 kwa kutumia 3 na 5. Tunajua kuwa 45 = 9 × 5 = 3^2 × 5. Hivyo, 45 = (3^2 × 5). Step 2: Tumia sheria ya kuzidisha ya logariti ((ab) = a + b) na sheria ya nguvu ((a^n) = n a). 45 = (3^2) + 5 45 = 2 3 + 5 Step 3: Badilisha thamani ulizopewa za 3 na 5. 45 = 2(0.4771) + 0.6990 Step 4: Fanya hesabu. 45 = 0.9542 + 0.6990 45 = 1.6532 Thamani ya 45 ni 1.6532. 3. a) Swali hili halijasomeka kikamilifu kutokana na picha iliyokatika. Tafadhali tuma picha kamili ya swali hili. 3. b) Kutafuta seti A' na (A B)': Kwanza, orodhesha vipengele vya kila seti: Seti ya ulimwengu U = \1, 2, 3, ..., 25\ Seti A = \nambari asilia chini ya 20\ = \1, 2, 3, ..., 19\ Seti B = \nambari shufwa chini ya 10\ = \2, 4, 6, 8\ i) A' (Kikamilisho cha A): Step 1: A' inajumuisha vipengele vyote vilivyo katika U lakini havipo katika A. A' = U - A A' = \1, 2, ..., 25\ - \1, 2, ..., 19\ Step 2: Orodhesha vipengele vilivyobaki. A' = \20, 21, 22, 23, 24, 25\ Seti A' ni \20, 21, 22, 23, 24, 25\. ii) (A B)' (Kikamilisho cha makutano ya A na B): Step 1: Kwanza, tafuta makutano ya A na B, yaani A B. Hivi ni vipengele vilivyo katika A na B. A B = \1, 2, ..., 19\ \2, 4, 6, 8\ A B = \2, 4, 6, 8\ Step 2: Kisha, tafuta kikamilisho cha A B, yaani (A B)'. Hivi ni vipengele vyote vilivyo katika U lakini havipo katika A B. (A B)' = U - (A B) (A B)' = \1, 2, ..., 25\ - \2, 4, 6, 8\ Step 3: Orodhesha vipengele vilivyobaki. (A B)' = \1, 3, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25\ Seti (A B)' ni \1, 3, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25\. 4. a) Kutafuta mlinganyo wa mstari ulionyooka unaopita kwenye pointi P(-4, 5) na Q(-2, 1): Step 1: Tafuta mteremko (m) wa mstari kwa kutumia formula m = (y_2 - y_1)/(x_2 - x_1). Kwa pointi P(-4, 5) na Q(-2, 1): m = (1 - 5)/(-2 - (-4)) m = (-4)/(-2 + 4) m = (-4)/(2) m = -2 Step 2: Tumia mlinganyo wa mstari y - y_1 = m(x - x_1) na pointi P(-4, 5) na mteremko m = -2. y - 5 = -2(x - (-4)) y - 5 = -2(x + 4) Step 3: Rahisisha mlinganyo katika mfumo wa y = mx + c. y - 5 = -2x - 8 y = -2x - 8 + 5 y = -2x - 3 Mlinganyo wa mstari ni y = -2x - 3. 4. b) Mstari unaopita kwenye pointi A(2, -k) na B(4, 3) una mteremko wa 2: i) Thamani ya k: Step 1: Tumia formula ya mteremko m = (y_2 - y_1)/(x_2 - x_1) na mteremko uliotolewa m=2. Kwa pointi A(2, -k) na B(4, 3): 2 = (3 - (-k))/(4 - 2) 2 = (3 + k)/(2) Step 2: Tatua kwa k. 2 × 2 = 3 + k 4 = 3 + k k = 4 - 3 k = 1 Thamani ya k ni 1. ii) Mlinganyo wa mstari AB: Step 1: Sasa tunajua k=1, hivyo pointi A ni (2, -1). Tuna pointi A(2, -1) na B(4, 3), na mteremko m=2. Tumia mlinganyo wa mstari y - y_1 = m(x - x_1) na pointi A(2, -1) na mteremko m=2. y - (-1) = 2(x - 2) y + 1 = 2x - 4 Step 2: Rahisisha mlinganyo katika mfumo wa y = mx + c. y = 2x - 4 - 1 y = 2x - 5 Mlinganyo wa mstari AB ni y = 2x - 5. That's 2 down. 3 left today — send the next one.