# If a2 + 1 = a, then the value of a12 + a6 + 1 is

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1. -3
2. 1
3. 2
4. 3

Option 4 : 3
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## Detailed Solution

Given:

a2 + 1 = a

The value of a12 + a6 + 1 = ?

Calculation:

a2 + 1 = a

⇒ (a + 1/a) = 1

Squaring on both sides,

⇒ (a + 1/a)2 = 1

⇒ a2 + 1/a2 + 2 = 1

⇒ a2 + 1/a2 = -1 ....(1)

Now, Cubing on both sides,

⇒ (a2 + 1/a2)3 = (-1)3

⇒ a6 + 1/a6 + 3a2 × 1/a2(a2 + 1/a2) = -1

From equation (1),

⇒ a6 + 1/a6 + 3 × (-1) = -1

⇒ a6 + 1/a6 + 1 = 3  ...(2)

If we take a = 1, it satisfies the equation (2).

So, a12 + a6 + 1 = (1)12 + (1)6 + 1 = 1 + 1 + 1 = 3

Hence, the correct answer is "3".

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