Question : If $A + \frac{1}{A} = - 1$ , then find the value of $\frac{A^{6} + A^{3} - 1}{A^{9} + A^{3} - 1}$ .
Option 1: 0
Option 2: –1
Option 3: 2
Option 4: 1

Question

Question : If $N + \frac{1}{N} = \sqrt{3}$ , then find the value of $N^{6} + \frac{1}{N^{6}} + 11$ .
Option 1: –11
Option 2: –2
Option 3: 11
Option 4: 9

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Answer 1

Correct Answer: 1

Solution : $A + \frac{1}{A} = - 1$ ⇒ $A^{2} + A + 1 = 0$
Now,
$A^{3} - 1^{3} = (A - 1) (A^{2} + A + 1)$
⇒ $A^{3} - 1 = (A - 1) \times 0$
⇒ $A^{3} - 1 = 0$
⇒ $A^{3} = 1$
So, $\frac{A^{6} + A^{3} - 1}{A^{9} + A^{3} - 1}$
$= \frac{{(A^{3})}^{2} + A^{3} - 1}{{(A^{3})}^{3} + A^{3} - 1}$
$= \frac{1^{2} + 1 - 1}{1^{3} + 1 - 1}$
$= 1$
Hence, the correct answer is 1.

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Question : If A+1A=−1, then find the value of A6+A3−1 A9+A3−1.Option 1: 0Option 2: –1Option 3: 2Option 4: 1

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Question : If $A + \frac{1}{A} = - 1$ , then find the value of $\frac{A^{6} + A^{3} - 1}{A^{9} + A^{3} - 1}$ .
Option 1: 0
Option 2: –1
Option 3: 2
Option 4: 1