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On (\mathbb{R}), (a*b=ka+b). For which (k) will the operation be associative?(\mathbb{R}) पर (a*b=ka+b) है। किस (k) के लिए संक्रिया साहचर्य होगी?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: (k=1)(k=1)
Step 1: ((a*b)*c=k(ka+b)+c=k^2a+kb+c). Step 2: (a*(b*c)=ka+(kb+c)=ka+kb+c). Step 3: Equality needs (k^2a=ka) for all (a), so (k^2=k); among the options, (k=1) works.चरण 1: ((a*b)*c=k(ka+b)+c=k^2a+kb+c)। चरण 2: (a*(b*c)=ka+(kb+c)=ka+kb+c)। चरण 3: बराबरी के लिए (k^2a=ka) सभी (a) के लिए चाहिए, अतः (k^2=k); दिए विकल्पों में (k=1) साहचर्य देता है।
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