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 ∴ ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

66

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

76

No worries! We‘ve got your back. Try BYJU‘S free classes today!

72

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

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

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

Suggest Corrections

Similar questions

Q. Let

λ

be an integer. If the shortest distance between the lines

x - λ = 2 y - 1 = - 2 z

and

x = y + 2 λ = z - λ

\frac{\sqrt{7}}{2 \sqrt{2}},

then the value of

| λ |

Q. Let

0 \leq x < 4

- 2 \leq y < 3

and

- 1 \leq z < 5

. If [a] denotes the greatest integer

\leq a

, then maximum possible value of

Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} [x + 2] & [y] & [z] [x] & [y + 1] & [z] [x] & [y] & [z + 1] \end{matrix} ∣ ∣ ∣ ∣ ∣

2 x - y - 2 z = 2, x - 2 y + z = - 4, x + y + λ z = 4

then the value of

λ

such that system of equations has no solution, is

Q. Let

x = a_{2} z + a_{3} y, y = a_{1} z + a_{3} x

and

z = a_{2} x + a_{1} y

, where x, y, z are not all zero. If a1 = k - [k], k being a non-integral constant, where [.] denotes greatest integer function, then

a_{1} a_{2} a_{3}

Q. Given

\begin{matrix} 2 x - y + 2 z = 2 x - 2 y + z = - 4 x + y + λ z = 4 \end{matrix}

then the value of

λ

such that the given system of equations has no solution, is ?

Join BYJU'S Learning Program

Explore more

Properties of Modulus

Standard XI Mathematics

Join BYJU'S Learning Program

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