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Haya, twende na maswali haya! 13. Fanya K kuwa mada ya fomula K = (ab)/(b-a). K tayari ni mada ya fomula hii. 14. Tafuta mlinganyo wa quadratic ambao mizizi yake ni 2.5 + sqrt(5) na 2.5 - sqrt(5), ukiueleza katika fomu ax^2 + bx + c = 0 ambapo a, b na c ni namba kamili. Step 1: Tafuta jumla ya mizizi ( + ). Acha mizizi iwe = 2.5 + sqrt(5) na = 2.5 - sqrt(5). + = (2.5 + sqrt(5)) + (2.5 - sqrt(5)) + = 2.5 + 2.5 + = 5 Step 2: Tafuta zao la mizizi ( ). = (2.5 + sqrt(5))(2.5 - sqrt(5)) Hii ni tofauti ya miraba (A+B)(A-B) = A^2 - B^2. = (2.5)^2 - (sqrt(5))^2 = 6.25 - 5 = 1.25 Step 3: Unda mlinganyo wa quadratic. Mlinganyo wa quadratic ni x^2 - ( + )x + = 0. x^2 - 5x + 1.25 = 0 Step 4: Badilisha namba za desimali kuwa namba kamili. Zidisha mlinganyo mzima kwa 4 ili kuondoa desimali. 4(x^2 - 5x + 1.25) = 4(0) 4x^2 - 20x + 5 = 0 Mlinganyo wa quadratic ni 4x^2 - 20x + 5 = 0. 15. Kiasi Z kinatofautiana moja kwa moja na mraba wa x na kinyume na mzizi wa mraba wa y. Ikiwa x inaongezeka kwa 20% na y inapungua kwa 36%, tafuta asilimia ya mabadiliko katika Z. Step 1: Andika fomula ya utofauti. Z = k (x^2)/(sqrt(y)) Step 2: Eleza x mpya (x') na y mpya (y'). x inaongezeka kwa 20%, hivyo x' = x + 0.20x = 1.2x. y inapungua kwa 36%, hivyo y' = y - 0.36y = 0.64y. Step 3: Tafuta Z mpya (Z'). Z' = k ((1.2x)^2)/(sqrt(0.64y)) Z' = k (1.44x^2)/(0.8sqrt(y)) Z' = (1.44)/(0.8) (k (x^2)/(sqrt(y))) Z' = 1.8 Z Step 4: Kokotoa asilimia ya mabadiliko katika Z. Mabadiliko katika Z ni Z' - Z = 1.8Z - Z = 0.8Z. Asilimia ya mabadiliko ni (0.8Z)/(Z) × 100\% = 80\%. Asilimia ya ongezeko katika Z ni 80%. 16. Neno la nne la G.P ni 48 na neno la saba ni 384. Tafuta uwiano wa kawaida na kisha kokotoa jumla ya maneno sita ya kwanza. Step 1: Andika milinganyo kwa kutumia fomula ya neno la G.P (a_n = ar^n-1). Neno la nne: ar^3 = 48 (Equation 1) Neno la saba: ar^6 = 384 (Equation 2) Step 2: Tafuta uwiano wa kawaida (r). Gawanya Equation 2 kwa Equation 1: (ar^6)/(ar^3) = (384)/(48) r^3 = 8 r = [3]8 r = 2 Step 3: Tafuta neno la kwanza (a). Badilisha r=2 kwenye Equation 1: a(2)^3 = 48 8a = 48 a = (48)/(8) a = 6 Step 4: Kokotoa jumla ya maneno sita ya kwanza (S_6). Fomula ya jumla ya G.P ni S_n = (a(r^n - 1))/(r-1). S_6 = (6(2^6 - 1))/(2-1) S_6 = (6(64 - 1))/(1) S_6 = 6(63) S_6 = 378 17. Kiasi P kinatofautiana moja kwa moja na mraba wa Q na kinyume na kiasi R. Ikiwa P=2 wakati Q=4 na R=6, tafuta P wakati Q=8 na R=4. Step 1: Andika fomula ya utofauti. P = k (Q^2)/(R) Step 2: Tumia maadili ya kwanza kupata kigezo k. 2 = k (4^2)/(6) 2 = k (16)/(6) 2 = k (8)/(3) k = 2 × (3)/(8) k = (6)/(8) k = (3)/(4) Hivyo, fomula kamili ni P = (3)/(4) (Q^2)/(R). Step 3: Tafuta P kwa kutumia maadili mapya. Badilisha Q=8 na R=4 kwenye fomula. P = (3)/(4) (8^2)/(4) P = (3)/(4) (64)/(4) P = (3)/(4) × 16 P = 3 × 4 P = 12 18. B inatofautiana kwa sehemu kama mraba wa M na kwa sehemu kama kinyume cha N. B, M na N ni hivi kwamba wakati M=2, N=1/2, B=96 wakati M=3, N=2, B=46. Andika usemi wa B katika maneno ya M na N. Step 1: Andika fomula ya utofauti. B = k_1 M^2 + k_2 (1)/(N) Step 2: Unda milinganyo kwa kutumia data uliyopewa. Kwa M=2, N=1/2, B=96: 96 = k_1 (2)^2 + k_2 (1)/(1/2) 96 = 4k_1 + 2k_2 (Equation 1) Kwa M=3, N=2, B=46: 46 = k_1 (3)^2 + k_2 (1)/(2) 46 = 9k_1 + (1)/(2)k_2 (Equation 2) Step 3: Tatua milinganyo ili kupata k_1 na k_2. Zidisha Equation 2 kwa 4 ili kuondoa sehemu na kurahisisha: 4(46) = 4(9k_1) + 4((1)/(2)k_2) 184 = 36k_1 + 2k_2 (Equation 3) Toa Equation 1 kutoka Equation 3: (184 - 96) = (36k_1 - 4k_1) + (2k_2 - 2k_2) 88 = 32k_1 k_1 = (88)/(32) k_1 = (11)/(4) Badilisha k_1 = (11)/(4) kwenye Equation 1: 96 = 4((11)/(4)) + 2k_2 96 = 11 + 2k_2 2k_2 = 96 - 11 2k_2 = 85 k_2 = (85)/(2) Step 4: Andika usemi kamili wa B. B = (11)/(4) M^2 + (85)/(2N) 19. Tatua kwa x na y. (3x)/(y-1) = 1 (2x+2) : (y-5) = 1 : 2 Step 1: Rahisisha mlinganyo wa kwanza. (3x)/(y-1) = 1 3x = y-1 y = 3x+1 (Equation 1) Step 2: Rahisisha mlinganyo wa pili. (2x+2)/(y-5) = (1)/(2) 2(2x+2) = 1(y-5) 4x+4 = y-5 y = 4x+9 (Equation 2) Step 3: Tatua milinganyo kwa kutumia njia ya kubadilisha. Linganisha Equation 1 na Equation 2 (kwa sababu zote mbili zinalingana na y): 3x+1 = 4x+9 1-9 = 4x-3x -8 = x x = -8 Step 4: Tafuta thamani ya y. Badilisha x=-8 kwenye Equation 1: y = 3(-8)+1 y = -24+1 y = -23 20. Fanya x kuwa mada ya fomula P = ((x-1)/(x+2)). Step 1: Zidisha pande zote mbili na denominata. P(x+2) = x-1 Step 2: Panua na panga upya maneno ili kukusanya maneno ya x. Px + 2P = x-1 2P+1 = x - Px 2P+1 = x(1-P) Step 3: Gawanya ili kutenga x. x = (2P+1)/(1-P) 21. Fanya d kuwa mada ya fomula a^2 = sqrt((1+d^2+b)/(b^2)). Step 1: Piga pande zote mbili mraba. (a^2)^2 = (sqrt((1+d^2+b)/(b^2)))^2 a^4 = (1+d^2+b)/(b^2) Step 2: Zidisha pande zote mbili na b^2. a^4 b^2 = 1+d^2+b Step 3: Tenga d^2. d^2 = a^4 b^2 - 1 - b Step 4: Chukua mzizi wa mraba wa pande zote mbili. d = sqrt(a^4 b^2 - 1 - b) 22. Z inatofautiana kwa pamoja kama mraba wa x na kinyume na mraba wa y. Wakati x=10 na y=4 basi z=15. Step 1: Andika fomula ya utofauti. Z = k (x^2)/(y^2) Step 2: Tumia maadili uliyopewa kupata kigezo k. 15 = k (10^2)/(4^2) 15 = k (100)/(16) 15 = k (25)/(4) k = 15 × (4)/(25) k = (3 × 5 × 4)/(5 × 5) k = (12)/(5) a) Tafuta z katika maneno ya x na y. Z = (12x^2)/(5y^2) b) Tafuta thamani ya x wakati z=8 na y=12. Step 1: Badilisha maadili ya Z na y kwenye fomula. 8 = (12 x^2)/(5 (12)^2) 8 = (12 x^2)/(5 × 144) 8 = (12 x^2)/(720) Step 2: Tatua kwa x^2. 8 × 720 = 12 x^2 5760 = 12 x^2 x^2 = (5760)/(12) x^2 = 480 Step 3: Tafuta x. x = sqrt(480) x = sqrt(16 × 30) x = 4sqrt(30) x ≈ 21.91 23. Kiasi R kinatofautiana kwa sehemu kama n na kwa sehemu kama mzizi wa mraba wa n. Wakati n=9 R=45 na wakati n=25 R=100. Tafuta R wakati n=16. Step 1: Andika fomula ya utofauti. R = k_1 n + k_2 sqrt(n) Step 2: Unda milinganyo kwa kutumia data uliyopewa. Kwa n=9, R=45: 45 = k_1 (9) + k_2 sqrt(9) 45 = 9k_1 + 3k_2 (Equation 1) Kwa n=25, R=100: 100 = k_1 (25) + k_2 sqrt(25) 100 = 25k_1 + 5k_2 (Equation 2) Step 3: Tatua milinganyo ili kupata k_1 na k_2. Gawanya Equation 1 kwa 3: 15 = 3k_1 + k_2 (Equation 3) Gawanya Equation 2 kwa 5: 20 = 5k_1 + k_2 (Equation 4) Toa Equation 3 kutoka Equation 4: (20 - 15) = (5k_1 - 3k_1) + (k_2 - k_2) 5 = 2k_1 k_1 = (5)/(2) Badilisha k_1 = (5)/(2) kwenye Equation 3: 15 = 3((5)/(2)) + k_2 15 = (15)/(2) + k_2 k_2 = 15 - (15)/(2) k_2 = (30-15)/(2) k_2 = (15)/(2) Step 4: Andika fomula kamili ya R. R = (5)/(2) n + (15)/(2) sqrt(n) Step 5: Tafuta R wakati n=16. R = (5)/(2) (16) + (15)/(2) sqrt(16) R = 5(8) + (15)/(2) (4) R = 40 + 15(2) R = 40 + 30 R = 70 24. Fanya b kuwa mada ya fomula a = (bd)/(sqrt(b^2+d)). Step 1: Zidisha pande zote mbili na denominata. asqrt(b^2+d) = bd Step 2: Piga pande zote mbili mraba. (asqrt(b^2+d))^2 = (bd)^2 a^2(b^2+d) = b^2d^2 Step 3: Panua na panga upya maneno ili kukusanya maneno ya b^2. a^2 b^2 + a^2 d = b^2 d^2 a^2 d = b^2 d^2 - a^2 b^2 a^2 d = b^2 (d^2 - a^2) Step 4: Gawanya ili kutenga b^2 na kisha b. b^2 = (a^2 d)/(d^2 - a^2) b = sqrt((a^2 d)/(d^2 - a^2)) 25. P inatofautiana kwa sehemu kama Q na kwa sehemu kama mzizi wa mraba wa Q. Wakati Q=4, P=22 na wakati Q=9, P=42. Tafuta thamani ya P wakati Q=25. Step 1: Andika fomula ya utofauti. P = k_1 Q + k_2 sqrt(Q) Step 2: Unda milinganyo kwa kutumia data uliyopewa. Kwa Q=4, P=22: 22 = k_1 (4) + k_2 sqrt(4) 22 = 4k_1 + 2k_2 (Equation 1) Kwa Q=9, P=42: 42 = k_1 (9) + k_2 sqrt(9) 42 = 9k_1 + 3k_2 (Equation 2) Step 3: Tatua milinganyo ili kupata k_1 na k_2. Gawanya Equation 1 kwa 2: 11 = 2k_1 + k_2 (Equation 3) Gawanya Equation 2 kwa 3: 14 = 3k_1 + k_2 (Equation 4) Toa Equation 3 kutoka Equation 4: (14 - 11) = (3k_1 - 2k_1) + (k_2 - k_2) 3 = k_1 Badilisha k_1 = 3 kwenye Equation 3: 11 = 2(3) + k_2 11 = 6 + k_2 k_2 = 11 - 6 k_2 = 5 Step 4: Andika fomula kamili ya P. P = 3Q + 5sqrt(Q) Step 5: Tafuta P wakati Q=25. P = 3(25) + 5sqrt(25) P = 75 + 5(5) P = 75 + 25 P = 100 26. Fanya C kuwa mada ya fomula b = sqrt((k-aC)/(k-a)). Kisha tafuta thamani ya C wakati K=1, a=4 na b=2. Step 1: Piga pande zote mbili mraba. b^2 = (k-aC)/(k-a) Step 2: Zidisha pande zote mbili na denominata. b^2(k-a) = k-aC Step 3: Panua na panga upya maneno ili kutenga C. b^2 k - b^2 a = k - aC aC = k - b^2 k + b^2 a aC = k(1-b^2) + ab^2 Step 4: Gawanya ili kutenga C. C = (k(1-b^2) + ab^2)/(a) Step 5: Tafuta thamani ya C wakati K=1, a=4 na b=2. Badilisha maadili haya kwenye fomula ya C. C = (1(1-2^2) + 4(2^2))/(4) C = (1(1-4) + 4(4))/(4) C = (1(-3) + 16)/(4) C = (-3 + 16)/(4) C = (13)/(4) C = 3.25 3 done, 2 left today. You're making progress.