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JEE Main 2023 · 25 Jan · Shift 2

JEE Main 2023 Mathematics question on Sets Relations and Functions

Let f(x)=2xn+λf(x) = 2x^n + \lambda, λ∈R\lambda \in \mathbb{R}, n∈Nn \in \mathbb{N}, and f(4)=133f(4) = 133, f(5)=255f(5) = 255. Then the sum of all the positive integer divisors of (f(3)−f(2))(f(3) - f(2)) is
  1. A.
    61
  2. B.
    58
  3. C.
    59
  4. D.
    60
Show answer and solution

Correct option: D

Solution

From f(4) = 133: 2·4n + λ = 133, and f(5) = 255: 2·5n + λ = 255.

Subtracting: 2(5n - 4n) = 122, so 5n - 4n = 61.

Testing n = 3: 125 - 64 = 61. ✓

Then λ = 133 - 2·64 = 133 - 128 = 5.

So f(3) - f(2) = 2·33 - 2·23 = 2·27 - 2·8 = 54 - 16 = 38.

Wait — λ cancels: f(3) - f(2) = 2(27 - 8) = 2·19 = 38.

Divisors of 38: 1, 2, 19, 38. Sum = 60.

[JEE Main 2023 · 25 Jan · Shift 2]

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