\( \left\{ \begin{array} { l } { 1.1 ) = \frac { 101 } { 100 } + \frac { 100 } { 101 } \quad b = \frac { 101 } { 100 } - \frac { 100 } { 101 } } \\ { a ^ { 2 } - b ^ { 2 } + 1 = ? } \\ { 2 x + y + z = 0 \times y z = - 18 \quad x ^ { 3 } + y ^ { 3 } + z ^ { 3 } = ? } \end{array} \right. \)

**1.2) Hisoblash:** Berilgan: \[ a = \frac{101}{100} + \frac{100}{101} \] \[ b = \frac{101}{100} - \frac{100}{101} \] Topshiriq: \[ a^2 - b^2 + 1 = ? \] **YeChim:** \[ a^2 - b^2 = (a + b)(a - b) \] Dastlab \( a + b \) va \( a - b \) ni hisoblaymiz: \[ a + b = \left( \frac{101}{100} + \frac{100}{101} \right) + \left( \frac{101}{100} - \frac{100}{101} \right) = 2 \times \frac{101}{100} = \frac{202}{100} = \frac{101}{50} \] \[ a - b = \left( \frac{101}{100} + \frac{100}{101} \right) - \left( \frac{101}{100} - \frac{100}{101} \right) = 2 \times \frac{100}{101} = \frac{200}{101} \] Endi \( a^2 - b^2 \) ni hisoblaymiz: \[ a^2 - b^2 = \left( \frac{101}{50} \right) \times \left( \frac{200}{101} \right) = \frac{101 \times 200}{50 \times 101} = \frac{200}{50} = 4 \] Shunday qilib: \[ a^2 - b^2 + 1 = 4 + 1 = 5 \] **Javob:** \( 5 \) --- **2. Hisoblash:** Berilgan: \[ 2x + y + z = 0 \] \[ y \times z = -18 \] Topshiriq: \[ x^3 + y^3 + z^3 = ? \] **YeChim:** Birinchi tenglamani \( x \) bo'yicha ifodalaymiz: \[ y + z = -2x \] Qabul qilaylik: \[ S = x + y + z = x + (-2x) = -x \] \[ P = y \times z = -18 \] Kub yig'indisi formulasi: \[ x^3 + y^3 + z^3 = (x + y + z)^3 - 3(x + y + z)(xy + yz + zx) + 3xyz \] Ammo qo'shimcha ma'lumot kerak. Alternativ yondashuv: Agar \( y + z = -2x \) va \( yz = -18 \), demak \( y \) va \( z \) kvadrat tenglamasining ildizlari: \[ t^2 + 2x t - 18 = 0 \] Ilgari berilgan formulani ishlatamiz: \[ y^3 + z^3 = (y + z)^3 - 3yz(y + z) = (-2x)^3 - 3(-18)(-2x) = -8x^3 - 108x \] Shunday qilib: \[ x^3 + y^3 + z^3 = x^3 + (-8x^3 - 108x) = -7x^3 - 108x \] Agar qo'shimcha ma'lumot bo'lmasa, javobni shu shaklda ifodalash mumkin. Ammo agar \( x + y + z = 0 \) bo'lsa, unda: \[ x^3 + y^3 + z^3 = 3xyz \] Bu holda: \[ 3x \times (-18) = -54x \] Ammo bu faqat \( x + y + z = 0 \) uchun to'g'ri. **Javob:** Qo'shimcha ma'lumot yo'qligi sababli, umumiy yechim \( x^3 + y^3 + z^3 = -7x^3 - 108x \)