Let z be a complex number such that the minimum value of -z-z-1-2z-7- is - - If y - 2 -x -3 - 3 -x- - - where - denotes the greatest integer function- then the value of -x-y - is

|z|+|z 1|+|7 2z|≥|z+z 1+7 2z|=6 ⇒ |z|+|z 1|+|2z 7|≥6 ∴λ =6 Now, y = 2 [x ]+3 = 3 [x 6 ] ⇒ 2 [x ] +3 = 3[x ] 18 (∵ [x+m]=[x]+m for m∈ℤ) ⇒[x ] = 21 ∴ ...

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Standard XI

Mathematics

Properties of Modulus

Let z be a co...

Question

Let $z$ be a complex number such that the minimum value of $| z | + | z - 1 | + | 2 z - 7 |$ is $λ .$ If $y = 2 [x] + 3 = 3 [x - λ],$ where $[.]$ denotes the greatest integer function, then the value of $[x + y]$ is

48

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66

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76

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72

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Solution

The correct option is B $66$
$| z | + | z - 1 | + | 7 - 2 z | \geq | z + z - 1 + 7 - 2 z | = 6$
$\Rightarrow | z | + | z - 1 | + | 2 z - 7 | \geq 6$
$∴ λ = 6$

Now, $y = 2 [x] + 3 = 3 [x - 6]$
$\Rightarrow 2 [x] + 3 = 3 [x] - 18 (∵ [x + m] = [x] + m for m \in Z)$
$\Rightarrow [x] = 21$

$∴ y = 2 [x] + 3 = 42 + 3 = 45$
$[x + y] = [x] + y = 21 + 45 = 66$

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Properties of Modulus

Standard XI Mathematics

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Let z be a complex number such that the minimum value of |z|+|z−1|+|2z−7| is λ. If y=2[x]+3=3[x−λ], where [.] denotes the greatest integer function, then the value of [x+y] is

The correct option is B 66|z|+|z−1|+|7−2z|≥|z+z−1+7−2z|=6 ⇒|z|+|z−1|+|2z−7|≥6 ∴λ=6 Now, y=2[x]+3=3[x−6] ⇒2[x]+3=3[x]−18 (∵[x+m]=[x]+m for m∈Z) ⇒[x]=21 ∴y=2[x]+3=42+3=45 [x+y]=[x]+y=21+45=66

Let $z$ be a complex number such that the minimum value of $| z | + | z - 1 | + | 2 z - 7 |$ is $λ .$ If $y = 2 [x] + 3 = 3 [x - λ],$ where $[.]$ denotes the greatest integer function, then the value of $[x + y]$ is