Question:`a = sqrt(13) + 2sqrt(3)` হলে, ক. `1/a` নির্ণয় কর। খ. ` (13a)/(a^2 - sqrt(13a) + 1) = sqrt(13)` প্রমাণ কর। গ. দেখাও যে, `a^4 = 2498 - 1/a^4`
Answer ক. দেওয়া আছে, `a = sqrt(13) + 2sqrt(3)` `:. 1/a = 1/(sqrt(13) + 2sqrt(3)` `= (sqrt(13) - 2sqrt(3))/((sqrt(13) + 2sqrt(3)) (sqrt(13) - 2sqrt(3))` [লব ও হরকে `sqrt(13) - 2sqrt(3)` দ্বারা গুণ করে] `= (sqrt(13) - 2sqrt(3))/((sqrt(13)^2 - (2sqrt(3))^2` `= (sqrt(13) - 2sqrt(3))/(13 - 12)` `= sqrt(13) - 2sqrt(3)` `:. 1/a = sqrt(13) - 2sqrt(3)` (Ans) খ. ‘ক’ হতে ` a = sqrt(13) + 2sqrt(3)` এবং` 1/a = sqrt(13) - 2sqrt(3)` এখন, `(13a)/(a^2 - sqrt(13a) + 1` `= (13a)/(a(a - sqrt(13) + 1/a)` `= (13)/(a + 1/a - sqrt(13)` `= (13)/(sqrt(13) + 2sqrt(3) + sqrt(13) - 2sqrt(3) - sqrt(13)` `= (13)/(sqrt(13) = sqrt(13)` `:. (13a)/(a^2 - sqrt(13a) + 1) = sqrt(13)` (প্রমাণিত) গ. আমরা জনি, `(a + 1/a)^2 = a^2 + 2.a.1/a + (1/a)^2` `(sqrt(13) + 2sqrt(3) + sqrt(13) - 2sqrt(3))^2 = a^2 + 1/a^2 + 2` `[:. a = sqrt(13) + 2sqrt(3) = 1/a = sqrt(13) - 2sqrt(3)]` বা, ` (2sqrt(13))^2 = a^2 + 1/a^2 + 2` বা, `52 = a^2 + 1/a^2 + 2` বা, `a^2 + 1/a^2 = 50` বা, `(a^2 + 1/a^2)^2 = (50)^2` [উভয়পক্ষকে বর্গ করে] বা, `(a^2)^2 + 2.a^2.1/a^2 + (1/a^2)^2 = 2500` বা, `a^4 + 1/a^4 + 2 = 2500` বা, `a^4 + 1/a^4 = 2500 - 2` বা, `a^4 + 1/a^4 = 2498` `:. a^4 = 2498 - 1/a^4` (দেখানো হলো)
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`a = sqrt(13) + 2sqrt(3)` hole, ka. `1/a` nirony karo. kh. ` (13a)/(a^2 - sqrt(13a) + 1) = sqrt(13)` proman karo. ga. dekhao je, `a^4 = 2498 - 1/a^4`