Let $\mathrm{f}(\mathrm{x})=2 \mathrm{x}^{\mathrm{n}}+\lambda, \lambda \in \mathbb{R}, \mathrm{n} \in \mathbb{N}$, and $\mathrm{f}(4)=133$, $f(5)=255$. Then the sum of all the positive integer divisors of ( $\mathrm{f}(3)-\mathrm{f}(2)$ ) is

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Permutations and Combinations

Let

f (x) = 2 x^{n} + λ, λ \in R, n \in N

, and

f (4) = 133

f (5) = 255

. Then the sum of all the positive integer divisors of (

f (3) - f (2)

) is

Answer: (B)

From

f (4) = 133

and

f (5) = 255

, subtraction gives

2 (5^{n} - 4^{n}) = 122

, so

5^{n} - 4^{n} = 61

, satisfied by

n = 3

since

125 - 64 = 61

; then

λ = 133 - 2 \cdot 64 = 5

. Now

f (3) - f (2) = 2 (3^{3} - 2^{3}) = 2 \cdot 19 = 38 = 2 \times 19

, whose positive divisors

1, 2, 19, 38

sum to

60

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Single correct option