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On real numbers, (a*b=ka+kb). For which (k) will this operation be associative?वास्तविक संख्याओं पर (a*b=ka+kb) है। किस (k) के लिए यह संक्रिया साहचर्य होगी?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: (k=0) or (k=1)(k=0) या (k=1)
Step 1: ((a*b)*c=k(ka+kb)+kc=k^2a+k^2b+kc). Step 2: (a*(b*c)=ka+k(kb+kc)=ka+k^2b+k^2c). Equality for all (a,c) requires (k^2=k). Step 3: (k(k-1)=0), so (k=0) or (k=1).चरण 1: ((a*b)*c=k(ka+kb)+kc=k^2a+k^2b+kc)। चरण 2: (a*(b*c)=ka+k(kb+kc)=ka+k^2b+k^2c)। तुलना से (k^2=k) चाहिए। चरण 3: (k^2=k\Rightarrow k(k-1)=0), इसलिए (k=0) या (k=1)।
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