Let and let the equation E be Then the largest element in the set is an integer solution of E is _________ . [2023]

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Solution

Q.
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 $}$ is _________ . [2023]

Ans.
(5)

$| x |^{2}$ $- 2$ $| x |$ $+$ $| λ - 3 |$ $= 0$

${(| x | - 1)}^{2} +$ $| λ - 3 |$ $- 1 = 0$ $...(i)$

Given that $x$ is an integer $\Rightarrow$ $| x |$ $- 1$ is an integer

${(| x | - 1)}^{2} - 1$ is also an integer. Thus $| λ - 3 |$ is also an integer. Therefore $λ$ is also an integer, ${(| x | - 1)}^{2} \leq 1$ .

$Case I:$ $| x |$ $- 1 = 0 \Rightarrow x = \pm 1$

$∴$ $| λ - 3 |$ $= 1 \Rightarrow λ = 4, 2$

$Possible values of λ + x = 5, 3, 1 .$

$Case II:$ $| x |$ $- 1 = 1 \Rightarrow x = \pm 2$

$∴$ $| λ - 3 |$ $= 0 \Rightarrow λ = 3$

$Possible values of x + λ = 5, 1 .$

$Considering all possible cases, the maximum value of λ + x = 5 .$

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