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Functions question

2021 · 31 Aug · Shift 2 · Q25
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  5. /2021 · 31 Aug · Shift 2 · Q25

Functions question

2021 · 31 Aug · Shift 2 · Q25

JEE MainMathematicsFunctionsMCQ+4 / −1
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 :
  1. A
    6
  2. B
    54
  3. C
    18
  4. D
    36
View written solutionFree

Correct answer: B

  1. We are given that

\mathbb{N}\to\mathbb{N}satisfies satisfies satisfiesf(m+n)=f(m)+f(n)\quad \text{for all } m,n\in \mathbb{N}.$$

  1. This is the additive functional equation on natural numbers. For such functions, we can determine values using repeated addition.

  2. Since 6=3+3,6=3+3,6=3+3, we get f(6)=f(3)+f(3)=2f(3).f(6)=f(3)+f(3)=2f(3).f(6)=f(3)+f(3)=2f(3). Given f(6)=18f(6)=18f(6)=18, we have 2f(3)=18  ⟹  f(3)=9.2f(3)=18 \implies f(3)=9.2f(3)=18⟹f(3)=9.

  3. Also, 6=2+2+2,6=2+2+2,6=2+2+2, so f(6)=f(2)+f(2)+f(2)=3f(2).f(6)=f(2)+f(2)+f(2)=3f(2).f(6)=f(2)+f(2)+f(2)=3f(2). Hence, 3f(2)=18  ⟹  f(2)=6.3f(2)=18 \implies f(2)=6.3f(2)=18⟹f(2)=6.

  4. Therefore, f(2)⋅f(3)=6⋅9=54.f(2)\cdot f(3)=6\cdot 9=54.f(2)⋅f(3)=6⋅9=54.

  5. Checking options:

    • A: 666 ❌
    • B: 545454 ✅
    • C: 181818 ❌
    • D: 363636 ❌

So the correct answer is Option B.

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