If [z] denotes the least integer greater than or equal to z and

Question

If   denotes the least integer greater than or equal to z and  z^2 ] = 2, which of the following could be the value of  ?

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

EgmatQuantExpert · Accepted Answer

Official Solution Given : • The function • = 2 To find : Can be {2, 1, -2}? Approach : • To find which values of 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 can be. We’ll get an idea of the possible values of z from the fact that = 2 Working Out : • = 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)(z-1) > 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, = -1 • Or 1 < z ≤ √2 • 1 < z ≤ 1.4 • = 2 Thus, we see that 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 e-GMAT Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts

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Bunuel · Answer

Given function rounds up a number to the nearest integer. For example, = 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} can be 2 (if 1 < z \leq \sqrt{2} ) or -1 (if -1 > z \geq \sqrt{2} ). Answer: A.

quantumliner · Answer

= 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   = 2

If Z is negative, then

-1>z\geq{-\sqrt{2}}

then   = -1

Answer is A. I only

chetan2u · Answer

Good question...
if   =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  =2
but when z is negative it lies between -1.4 and -1.7 and the NEXT greater integer is -1 so  =-1..   Be careful here and do not take -2

Only 2 is given 
ans A

georgethomps · Answer

I don't understand, where does the z > 1 and z < -1 come from?

TheMBmonster · Answer

Is my reasoning correct: We know that 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 -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.

Bambi2021 · Answer

I: If z = 1,2 -->   = 2 -->   = 2

II: If z = 0,5 -->   = 1 -->   = 1

III: If z = -2,2 -->   = -2 -->   = 4

Only I works.

GmatKnightTutor · Answer

Nice question. Might be good to not follow a heavy paper/pad approach with this one.

Crytiocanalyst · Answer

let us assume z^2 = 1.96 then z= 1.4 and greatest integer function will lead to z= 2

now alternatively in the neagtive axis z^2 =1.96 and z=-1.4 then the greatest integer will be -1 ...........a

and we can safely eleminate the other 2 option with the reasoning at a

therefore IMO a

Kinshook · Answer

If   denotes the least integer greater than or equal to z and  z^2 ] = 2, which of the following could be the value of  ?

z^2 ] = 2
1.00000...1<z^2 < 1.99.....9
1.00000....1 < z < 1.34
-1.34 < z < -1

I. 2: z = 1.1;   = 2
II. 1: No value of z can be found
III. -2: No value of z can be found

IMO A

AbhishekP220108 · Answer

The first line gives us the definition that [Z]>=Z, where [Z] has an integer value but Z can be an integer or non integer value which means that lets suppose Z= 2.1 then [z] is next integer value greater than or equal to Z which will be 3, and if z=2.9 then also [z] will be 3, [z]can't be 2 as per the definition. And the range of values will be 2<[z]<=3

The next statement says that [z^2]=2. Now as per above we know that z will be less than equal to 2 and greater than 1, z can be negative or positive because of square of variable. Now as per example given we can write [z^2] range as 1<[z^2]<=2

So it follows two inequality z^2>1 and z^2<=2

When we solve this we will get intervals, we can't take negative intervals for the values of [z^2] because of definition given because if take z as -1.41 then next integer value will be -1 which doesn't satisfy the condition given. Hence for positive z can take only values for 1<z<=sqrt2.

Now check for any value of this intervals of z=1.2, z^2= 1.44 hence as per question [z^2]=2 satisfied

EgmatQuantExpert

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

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If [z] denotes the least integer greater than or equal to z and

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