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26 Jun 2023, 03:00
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
47% (02:45) correct
53%
(02:37)
wrong
based on 100
sessions
History
Date
Time
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Not Attempted Yet
What is the sum of all possible solutions of the equation \((3^n-9)^3+(9^n-3)^3=(3^n+9^n-12)^3\), where \(n\) is a positive integer?
A. \(1\)
B. \(2\)
C. \(3\)
D. \(4 \)
E. \(5\)
A. \(1\)
B. \(2\)
C. \(3\)
D. \(4 \)
E. \(5\)
26 Jun 2023, 04:22
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Bunuel
\((a + b)^3 = a^3 + b^3 + 3ab(a+b)\)
\(a = 3^n-9\)
\(b = 9^n-3\)
\((3^n+9^n-12)^3\) ⇒
\((3^n-9+9^n-3)^3 = (3^n-9)^3 + (9^n-3)^3 + 3(3^n-9)(9^n-3)(3^n-9+9^n-3)\)
\((3^n-9)^3+(9^n-3)^3 = (3^n-9)^3 + (9^n-3)^3 + 3(3^n-9)(9^n-3)(3^n-9+9^n-3)\)
Cancelling \((3^n-9)^3+(9^n-3)^3\)
\(3(3^n-9)(9^n-3)(3^n+9^n-12) = 0\)
Possible Values
- \((3^n-9) = 0\)
⇒ \(n = 2\) - \((9^n-3) = 0\)
⇒ \(n = \frac{1}{2}\) - \((3^n+9^n-12) = 0\)
\((3^n+9^n) = 12\)
⇒ \(n = 1\)
As n is a positive integer, we can discard the value of \(\frac{1}{2}\).
Sum = 2 + 1 = 3
Option C
udvats
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27 Jun 2023, 01:17
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Is it better to just plug in values in such type of questions? I can see max sum is 5 according to options so just need to check for n = 1,2 & 3
Option C
Option C
