Question : For what value of k will the expression x6 – 18x3 + k be a perfect square?
Option 1: –9
Option 2: –81
Option 3: 9
Option 4: 81
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Correct Answer: 81
Solution :
Given:
x
6
– 18x
3
+ k
Let x
3
= y
The given expression becomes y
2
– 18y + k.
Now break the expression into the form of (a – b)
2
= a
2
– 2ab + b
2
So, y
2
– 18y + k = y
2
– 2(y)(9) + k
For the above expression to be a perfect square, k should be equal to (9)
2
= 81
= y
2
– 2(y)(9) + 9
2
= (y – 9)
2
Substituting the value of y in the above expression, we get,
(x
3
– 9)
2
, which is a perfect square.
Hence, the correct answer is 81.
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