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If [z] denotes the least integer greater
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24 May 2017, 11:50
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Q. If [z] denotes the least integer greater than or equal to z and [\(z^2\)] = 2, which of the following could be the value of [z]? Answer Choices A. I only B. II only C. III only D. I and II only E. II and III only
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Re: If [z] denotes the least integer greater
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31 May 2017, 06:09
Official Solution Given: • The function [z] • \([z^2] = 2\) To find: Can [z] be {2, 1, 2}? Approach: • To find which values of [z] are possible and which are not, we need to first know the range of possible values of z. Once we know what z can be, we’ll be able to find what [z] can be. We’ll get an idea of the possible values of z from the fact that \([z^2] = 2\) Working Out: • \([z^2] = 2\)
• This means the least integer that is greater than or equal to \(z^2\) is 2.
• We can write: \(1 < z^2 ≤ 2\)
• The above inequality contains 2 inequalities: \(z^2 > 1\) AND \(z^2 ≤ 2\)
So now, we’ll solve these inequalities one by one, and then find the values of z that satisfy both these inequalities • Solving \(z^2 > 1\)
• \(z^2 – 1 > 0\)
• \((z+1)(z1) > 0\) This means, \(z <  1\) or \(z > 1\) . . . (1) • Solving \(z^2 ≤ 2\)
• \(z^2 – 2 ≤ 0\)
• \((z + √2)(z√2) ≤ 0\) That is, \(√2 ≤ z ≤ √2\) . . . (2) Combining (1) and (2): Combining them to find their overlap zones and finding those values of z that satisfy both the inequalities • So, either \(√2 ≤ z < 1\) or \(1 < z ≤ √2\)
• Either \(√2 ≤ z < 1\)
• \(1.4 ≤ z < 1\)
• In this case, \([z] = 1\)
• Or \(1 < z ≤ √2\)
• \(1 < z ≤ 1.4 \) • \([z] = 2\)
Thus, we see that [z] is either equal to 1 or equal to 2. Out of the 3 given values, only value I (which is 2) is therefore possible. Looking at the answer choices, we see that the correct answer is Option A Thanks, Saquib Quant Expert eGMATRegister for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the realtime guidance of our Experts
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If [z] denotes the least integer greater
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24 May 2017, 14:13
\([z^2] = 2\) means that
\(1<z^2\leq{2}\)
Now, Taking Square Root of both the sides.
If Z is positive, then
\(1<z\leq{\sqrt{2}}\)
then [z] = 2
If Z is negative, then
\(1>z\geq{\sqrt{2}}\)
then [z] = 1
Answer is A. I only



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Re: If [z] denotes the least integer greater than or equal to z and [z2] =
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25 May 2017, 14:05
niteshwaghray wrote: If [z] denotes the least integer greater than or equal to z and [\(z^2\)] = 2, which of the following could be the value of [z]?
I) 2 II) 1 III) 2
A. I only B. II only C. III only D. I and II only E. II and III only Given function rounds up a number to the nearest integer. For example, [1.3] = 2, because 2 is the least integer greater than or equal to 1.3. [\(z^2\)] = 2; \(1 < z^2 \leq 2\); \(1 <z \leq \sqrt{2}\) (\(\sqrt{2} \approx 1.4\)). \(1 < z \leq \sqrt{2}\) or \(1 > z \geq \sqrt{2}\) [z] can be 2 (if \(1 < z \leq \sqrt{2}\)) or 1 (if \(1 > z \geq \sqrt{2}\)). Answer: A.
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If [z] denotes the least integer greater than or equal to z and [z^2]
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05 Mar 2018, 07:01
If [z] denotes the least integer greater than or equal to z and [z^2] = 2, which of the following could be the value of [z]? I. 2 II. 1 III. 2 (A) I only (B) II only (C) III only (D) I and II only (E) II and III only
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Re: If [z] denotes the least integer greater than or equal to z and [z^2]
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05 Mar 2018, 07:25
itisSheldon wrote: If [z] denotes the least integer greater than or equal to z and [z^2] = 2, which of the following could be the value of [z]?
I. 2 II. 1 III. 2
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only Good question... if [z^2] =2, \(z^2\) lies between 2 and 3 therefore z lies between \(\sqrt{2}=1.4\) and \(\sqrt{3}=1.7\) or between \(\sqrt{2}=1.4\) and \(\sqrt{3}=1.7\) if z is negative.. so when z is positive it lies between 1.4 and 1.7 and the NEXT greater integer is 2 so [z]=2 but when z is negative it lies between 1.4 and 1.7 and the NEXT greater integer is 1 so [z]=1.. Be careful here and do not take 2Only 2 is given ans A
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Re: If [z] denotes the least integer greater
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05 Mar 2018, 07:30
itisSheldon wrote: If [z] denotes the least integer greater than or equal to z and [z^2] = 2, which of the following could be the value of [z]?
I. 2 II. 1 III. 2
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only Merging topics. Please search before posting. Thank you.
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Re: If [z] denotes the least integer greater
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02 Jul 2019, 09:03
I don't understand, where does the z > 1 and z < 1 come from?



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Re: If [z] denotes the least integer greater
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03 Aug 2019, 13:20
Is my reasoning correct:
We know that [z] will take any number, (ex 1.4), and will round it up to the next highest integer, (2 in the case of 1.4) unless Z is already an integer.
Therefore, as others mentioned,
1<z^2 ≤ 2
Which means that √(Z^2), or Z is either larger than √1, which si 1 or 1, or smaller than √2 or √2 (1.4, or 1.4)
1 (or 1) <Z ≤ 1.4 ( or 1.4).
The different possibilities for [Z] are thus
1) 0, because if Z> 1 , the closest higher integer is 0, 2) 2, because if Z>1, Z < 1.4 , the closest integers are 2, 3) 1, because if Z> 1.4, the closest highest integer is 1.
The only option that appears in the answer choices is 2, thus the answer is A.




Re: If [z] denotes the least integer greater
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03 Aug 2019, 13:20






