JEE MAIN - Mathematics (2021 - 31st August Evening Shift - No. 4)
Let f : N $$\to$$ N be a function such that f(m + n) = f(m) + f(n) for every m, n$$\in$$N. If f(6) = 18, then f(2) . f(3) is equal to :
6
54
18
36
Explanation
f(m + n) = f(m) + f(n)
Put m = 1, n = 1
f(2) = 2f(1)
Put m = 2, n = 1
f(3) = f(2) + f(1) = 3f(1)
Put m = 3, n = 3
f(6) = 2f(3) $$\Rightarrow$$ f(3) = 9
$$\Rightarrow$$ f(1) = 3, f(2) = 6
f(2) . f(3) = 6 $$\times$$ 9 = 54
Put m = 1, n = 1
f(2) = 2f(1)
Put m = 2, n = 1
f(3) = f(2) + f(1) = 3f(1)
Put m = 3, n = 3
f(6) = 2f(3) $$\Rightarrow$$ f(3) = 9
$$\Rightarrow$$ f(1) = 3, f(2) = 6
f(2) . f(3) = 6 $$\times$$ 9 = 54
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