Around the World in 80 Questions (Day 2): If 3^(-k/2) 1/81

Question

If  3^{(-\frac{k}{2})} > \frac{1}{81} , what is the value of k ?

(1)  k  is the square of a prime number.

(2)  \frac{k}{2}  is a prime number.

nivivacious · Answer

Lets look at the question statement: 
3^(−k/2) > 1/81 Simplifying we get:
3^(−k/2) > 3^(-4)
-k/2 > -4. 
Therefore: k < 8.

Let's evaluate the statements:
(1) k is the square of a prime number.
The square of a prime number could be 4, 9, 25, etc. Only 4 is less than 8, so this statement alone is not sufficient to determine the value of k.

(2) k/2 is a prime number.
If k/2 is a prime number, then k could be 2, 4, 6, 10, etc. Only 2, 4, and 6 are less than 8, so this statement alone is also not sufficient to determine the value of k.

Therefore, neither statement alone is sufficient to determine the value of k.

However, if we combine the two statements, we find that the only value that satisfies both is k = 4 (since 4 is the square of a prime number and 4/2 is a prime number). 
So, the value of k is 4 when both statements are combined.
Hence answer is C (Both)

Kinshook · Answer

Asked: If 3^{(-\frac{k}{2})} > \frac{1}{81} , what is the value of k ? 3^{(-\frac{k}{2})} > \frac{1}{81}= 3^{-4} k/2 <4 k < 8 (1) k is the square of a prime number. k = 4 = 2^2 is only possible value of k since k<8 SUFFICIENT (2) \frac{k}{2} is a prime number. k = 4 or 6 NOT SUFFICIENT IMO A

divyagolyan · Answer

We know that 1/81 = 1/3^4 = 3^-4
Expressing both sides in terms of power of 3:

-k/2 > -4
k/2 < 4
k < 8

From (1), 
If k is the square of a prime number, k = 2^2, 3^2, 5^2, .... = 4, 9, 25, .....

But from the given statement, k < 8, hence k = 4 is the only possible solution. (1) is sufficient

From (2),
k/2 is a prime number. So k = 4, 6, 10, .... so that k/2 = 2, 3, 5, .... (primes)

Not sufficient

Ans. (1)

missionmba2025 · Answer

3^{(-\frac{k}{2})} > \frac{1}{81} 3^{(-\frac{k}{2})} > 3^{-4} -\frac{k}{2} < -4 - k > -8 k < 8 1) Only possible value of k = 4 The statement is sufficient. 2) k can be 4 or 6 as 4/2 = 6 and 6/2 = 3 The statement is not sufficient. IMO A

Niddd · Answer

3^{(-\frac{k}{2})} > \frac{1}{81} \frac{1}{3^\frac{k}{2}} > \frac{1}{3^4} As we are dealing with fractions, for the left hand side to be a bigger we need the denominator to be smaller. Therefore with dropping the base of 3: \frac{k}{2}<4 k<8 (1) k is the square of a prime number. Given the parameters of k, the only possible answer for k is 2. 2^2 = 4 works. 3^2 = 9 exceeds the inequality k<8 SUFFICIENT (2) k/2 is a prime number. If k is 6: \frac{6}{2}= 3 If k is 4: \frac{4}{2}= 2 One cannot come to a single value for k. INSUFFICIENT Answer A

Rahulbasu007 · Answer

Given:
If 3^(−k/2)>1/81, what is the value of k ?
(1) k is the square of a prime number.
(2) k/2 is a prime number.

Let's evaluate:
=> 3^(-k/2) > 1/(3^4)
=> 3^(-k/2) > 3 (-4)

Therefore, we can write:
=> - k/2 = -4
=> -k > -8
=> k < 8

Statement 1: k is the square of a prime number.
2^2 = 4
3^2 = 9
So, here, we can definitely say that k is 4. (k<8)
Also, we should remember that square of any number is a positive number.

(Sufficient)

Statement 2:
k/2 is a prime number.

Plug in a few values:
k = 4 => 4/2 = 2 (prime)
k = 6 => 6/2 = 3 (prime)
So we don't have a single answer.
(Not sufficient.)

So that means, Statement 1 alone is sufficient.
I will go with option A.

animsaha27 · Answer

3^ (−k/2)> 1/81
3^ (−k/2)> 3^(-4)
(−k/2)> (-4)
k/2 < 4
k < 8

(1) k is the square of a prime number.
2^2 = 4

(2) k / 2 is a prime number.
6/2 = 3
4/2 = 2

So, 1 alone is sufficient, but 2 alone is not
Ans : A

iamchinu97 · Answer

So here is my ans:

When we solve the equation we get k< 8

a) k is square of prime number

sqre of 2 = 4
sqre of 3 = 9

so we know that there can be only one option which is k=4

b) k/2 is prime number 
so that means k is a even number
so k can be 4,6
when k = 4 =>4/2=2 prime number
when k = 6 =>6/2=3 prime number
so both the answers are possible.

Hence only a is sufficient

NamanPro · Answer

If 3^(-k/2) > 1/81. Then K=?

We can rewrite Question as:-

3^(-k/2) > 1/3^4
3^(-k/2 + 4) > 1
so for 3^x to be greater than 1. x should be greater than 0.

-k/2 + 4 > 0
k < 8
Question :- if K<8 what is k?

Statement 1: k is the square of a prime number.

i.e. K can take the values -> 2^2 , 3^2 , 5^2....... -> 4,9,25...
As K<8 only possible value is 4 so S1: SUFFICIENT

Statement 2: k/2 is a prime number.

i.e. K can take the values -> 2*2 , 3*2 , 5*2....... -> 4,6,10...
As K<8 k can take 4 and 6 so S2: INSUFFICIENT

IMO A

vjbhagat · Answer

We can write 3^(-k/2) = 1/(3^(k/2))

Hence, 1/(3^(k/2)) > 1/81.

We can cross multiple the denominators for simplification and this wont change the sign between them, since "81" is positive and "3^(k/2)" is also positive (independent whether "k" is positive or negative)

Therefore:- 3^(k/2) < 81
So, (k/2) < 4, hence  k < 8

(1) k is square of a prime number:-
Square of prime numbers are 4, 9, 25,.... and only "4" satisfies our condition of k < 8.

Hence statement (1) alone is sufficient.

(2) k/2 is a prime number:-
Using this condition, the possible values of "k" are 4, 6, 10, 14,....
Here "4" and "6" both satisfies the condition of k < 8. So using statement (2) we still cannot uniquely find the value of "k".

Hence statement (2) alone is NOT sufficient.

vishal_gupta · Answer

Q:- 3^{(-\frac{k}{2})} > 3^{(-4)} -k/2 > -4 k/2 < 4 k < 8 s1 :- k = square of a prime number prime number less than \sqrt{8} = 2 So, k = 2 Therefore, statement 1 is sufficient. s2:- k/2 = prime number prime numbers valid for above cases = 2 , 3 Therefore, s2 is not sufficient. Hence, A is correct answer.

BottomJee · Answer

3^{(-\frac{k}{2})} > \frac{1}{81} => 3^{(-\frac{k}{2})} > 3^(-4) => -(\frac{k}{2}) > -4 => \frac{k}{2 } < 4 => k < 8 From Statement 1, we get Squares of prime numbers are 2^2 , 3^2 , 5^2 , and so on But k < 8 , this implies that k = 4. Statement 1 is sufficient From Statement 2, we get \frac{k}{2} = Prime number = n => k = 2 * n Since k < 8 => 2*n < 8 => n < 4 Therefore, n could be 2 or 3. Statement 2 is not sufficient. IMO OA should be A.

Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81

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