1. Masala: Soat 7:30 da soat va minut milari orasidagi burchakni toping. Soat milining har bir son uchun aylanma 360°/12 = 30° qadam bilan yurishini hisoblaymiz. Minut mili esa 60 daqiqada 360° aylana to'liq aylanadi. 2. Hisoblashlar: - Soat mili 7 soatda 30 daqiqada joylashadi, ya'ni $7 \times 30° + \frac{30}{60} \times 30° = 210° + 15° = 225°$. - Minut mili esa 30 daqiqada $30 \times 6° = 180°$. - Ikki mil orasidagi burchak katta yoki kichik burchak bo'lishi mumkin: $$|225° - 180°| = 45°$$ Javob: 45° yani C. --- 2. Masala: Go'zal va Dononing yoshlarining hozirgi qiymatini toping. Shart: 2 yil oldingi farq 7, hozirgi yoshlar yig'indisi 9, Go'zal Donodan katta. Tenglamalar: Let $D$ = Dononing hozirgi yoshi, $G$ = Go'z alning hozirgi yoshi. $ (G - 2) - (D - 4) = 7 $ $ G + D = 9 $ Birinchi tenglamani ochamiz: $ G - 2 - D + 4 = 7 \Rightarrow G - D + 2 = 7 \Rightarrow G - D = 5 $ Ikkinchi tenglamadan $ G = 9 - D $ Substitutsiya qilib: $ 9 - D - D = 5 \Rightarrow 9 - 2D = 5 \Rightarrow 2D = 4 \Rightarrow D = 2 $ Shuning uchun $G = 9 - 2 = 7$ Dono hozir 2 yoshda. Javob: 2 yani D. --- 3. Masala: Otaning hozirgi yoshini topish. Shartlar:
5eerta 1: Ota yoshi uchta farzand yoshlari yig'indisidan 3 marta katta.
5eerta 2: 4 yil o'tgach, ota yoshi farzandlar yig'indisidan 2 marta katta bo'ladi. Let $O$ = otaning hozirgi yoshi, $F$ = farzandlarning joriy yoshi yig'indisi. Tenglamalar: $ O = 3F $ $ O + 4 = 2(F + 12) $ (farzandlar 4 yildan keyin jami $F + 12$, chunki 3 farzand x 4 yil) Birinchi tenglamadan $O = 3F$ o'rniga qo'yamiz ikkinchisiga: $ 3F + 4 = 2(F + 12) $ $ 3F + 4 = 2F + 24 $ $ 3F - 2F = 24 - 4 $ $ F = 20 $ Shunday qilib, $ O = 3 imes 20 = 60 $ Javob: 60 yani D. --- 11. Masala: Fermadagi tovuqlar sonini toping. Qo'ylar soni $x$, tovuqlar soni $2x$. Qo'ylarning oyoqlari soni $4x$, tovuqlarning oyoqlari soni $2x imes 2 = 4x$ (tovuqda 2 oyoq bor) Jami oyoqlar: $4x + 4x = 8x = 600$ $8x = 600 \Rightarrow x = 75$ Demak, tovuqlar soni $2x = 150$ Javob: 150 yani C. --- 12. Masala: Aka va ukaning yoshlari yig'indisini toping. Let $A$ = aka yoshi, $U$ = uka yoshi. $ A - U = 10 $ 3 yil o'tgach: $ A + 3 = 1.5(U + 3) $ Birinchi tenglamadan $ A = U + 10 $ Ikkinchisiga: $ U + 10 + 3 = 1.5(U + 3) $ $ U + 13 = 1.5U + 4.5 $ $ 13 - 4.5 = 1.5U - U $ $ 8.5 = 0.5U $ $ U = 17$ Demak, $A = 27$ Yig'indisi: $ 17 + 27 = 44 $ Javob: 44 yani C. --- 13. Masala: Otaning hozirgi yoshini toping. Otasi 32 yoshda o'g'li tug'ilganida. Hozir otaning yoshi o'g'lidan 3 marta katta. Let $x$ = o'g'lining hozirgi yoshi. Ota yoshi hozir: $x + 32$ Shart: $x + 32 = 3x$ $32 = 2x \Rightarrow x = 16$ Ota yoshi: $16 + 32 = 48$ Javob: 48 yani A. --- 20. Masala: Otaning va o'g'lining yoshini toping. $O$ = ota yoshi, $S$= o'g'lining yoshi. $O = 4S$ 6 yil keyin: $O + 6 = 2 (S + 6)$ $4S + 6 = 2S + 12$ $4S - 2S = 12 - 6$ $2S = 6 \Rightarrow S=3$ Demak, $O=4 imes 3=12$. Javob: Bu variantlardan javob yo'q, lekin eng yaqin 6 yani D. --- 21. Masala: 14 ta bola o'rtasidagi umumiy salomlashishlar sonini toping. Har bir juftlik faqat 1 marta salomlashadi. Formula: $$ \frac{n(n-1)}{2} = \frac{14 \times 13}{2} = 91 $$ Javob: 91 yani A. --- 22. Masala: Onaning hozirgi yoshini toping. Let $Q$ = qizning yoshi, $O$ = onaning yoshi. Onaning yoshi qiznikidan 4 marta katta: $O = 4Q$ Qiz tug'ilganda ona 24 yosh edi, ya'ni $O - Q = 24$ Tenglamalar: $O - Q = 24$ $O = 4Q$ Substitutsiya $4Q - Q = 24$ $3Q = 24 \Rightarrow Q = 8$ $O = 4 imes 8 = 32$ Javob: 32 yani B. --- 23. Masala: Ota va bolaning yoshi farqi 28 ga teng. Besh yildan keyin ota bolaning yoshidan 4,5 marta katta bo'ladi. Let $O$ ota yoshi, $B$ bola yoshi. $O - B = 28$ Besh yildan keyin: $O + 5 = 4.5(B + 5)$ Tenglamalar: $O = B + 28$ Substitutsiya: $B + 28 + 5 = 4.5(B + 5)$ $B + 33 = 4.5B + 22.5$ $33 - 22.5 = 4.5B - B$ $10.5 = 3.5B$ $B = \frac{10.5}{3.5} = 3$ Ota yoshi: $O = 3 + 28 = 31$ So'ralgan tenglama: $$\begin{cases} O - B = 28 \\\ O + 5 = 4.5(B + 5) \end{cases}$$