Question:`x^2 = 5 + 2sqrt(6)` ক. x এর মান নির্ণয় কর। খ. `x^2 + 1/x^2` এর মান কত? গ. `(x^6 + 1)/x^3 = 18sqrt(3)` প্রমাণ কর।
Answer ক. দেওয়া আছে, `x^2 = 5 + 2sqrt(6)` `= 3 + 2sqrt(3) xx 2 + 2` `= (sqrt(3))^2 + 2.sqrt(3).sqrt(2) + (sqrt(2))^2` `= (sqrt(3) + sqrt(2))^2` `:. x = sqrt(3) + sqrt(2)` খ. ’ক’ হতে পাই, `x = sqrt(3) + sqrt(2)` `:. 1/x = 1/(sqrt(3) + sqrt(2)` `= (1 (sqrt(3) - sqrt(2)))/((sqrt(3) + sqrt(2)) (sqrt(3) - sqrt(2))` `= (sqrt(3) - sqrt(2))/(sqrt(3))^2 - (sqrt(2))^2` `= (sqrt(3) - sqrt(2))/(3 - 2)` `= sqrt(3) - sqrt(2)` `:. x + 1/x = sqrt(3) + sqrt(2) + sqrt(3) - sqrt(2) = 2sqrt(3)` প্রদত্ত রাশি `= x^2 + 1/x^2` `= (x + 1/x)^2 - 2.x.1/x` `= (2sqrt(3)^2 - 2` `= 4.3 - 2` = 12 - 2 = 10 (Ans) গ. ‘খ’ থেকে পাই, বামপক্ষ, `x + 1/x = 2sqrt(3)` `= (x^6 + 1)/x^3` `= x^6/x^3 + 1/x^3` `= x^3 + 1/x^3` `= (x + 1/x)^3 - 3.x.1/x (x + 1/x)` `= (2sqrt(3))^3 - 3.2sqrt(3)` `= 8.3sqrt(3) - 6sqrt(3)` `= 24sqrt(3) - 6sqrt(3)` `= 18sqrt(3)` = ডানপক্ষ `:. (x^6 + 1)/x^3 = 18sqrt(3)`
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`x^2 = 5 + 2sqrt(6)` ka. x ar man nirony karo. kh. `x^2 + 1/x^2` ar man koto? ga. `(x^6 + 1)/x^3 = 18sqrt(3)` proman karo.