Let \(\lambda \in \mathbb{R}\) and let the equation E be \(|\mathrm{x}|^{2}-2|\mathrm{x}|+|\lambda-3|=0\). Then the largest element in the set \(\mathrm{S}=\{\mathrm{x}+\lambda: \mathrm{x}\) is an integer solution of E\(\}\) is

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Basic Mathematics

Let

λ \in R

and let the equation E be

∣ x ∣^{2} - 2∣ x ∣ + ∣ λ - 3∣ = 0

. Then the largest element in the set

S = {x + λ : x

is an integer solution of E

}

Answer: 5

Write the equation as

(∣ x ∣ - 1)^{2} = 1 - ∣ λ - 3∣

, which requires

∣ λ - 3∣ \leq 1

, i.e.

2 \leq λ \leq 4

. For

λ = 3

the integer solutions are

x \in {- 2, 0, 2}

; for

λ = 2

λ = 4

they are

x \in {- 1, 1}

. Forming

x + λ

across all cases gives the set

{1, 3, 5}

, whose largest element is

5.

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Integer answer