Correct Answer - Option 2 : 1
Given:
m + 1/m = √3
Concept Used:
If x + 1/x = a,
then, x3 + 1/x3 = a3 - 3a
Calculations:
m + 1/m = √3
⇒ m3 + 1/m3 = (√3)3 - 3 × √3
⇒ m3 + 1/m3 = 3√3 - 3√3 = 0
⇒ m6 + 1 = 0 ----(1)
⇒ m6 = -1 ----(2)
Now, m28 + m22 + m20 + m14 + m12
⇒ m22 × (m6 + 1) + m14 × (m6 + 1) + m12
Putting (1) and (2) in this expression, we get
⇒ 22 × 0 + m14 × 0 + (-1)2
⇒ 0 + 0 + 1 = 1
∴ The value of (m28 + m22 + m20 + m14 + m12 ) is 1.
Short Trick/Topper's Approach:
If x + 1/x = √3,
then x6 + 1 = 0
Now, m28 + m22 + m20 + m14 + m12
⇒ m22 × (m6 + 1) + m14 × (m6 + 1) + m12
Putting (1) and (2) in this expression, we get
⇒ 22 × 0 + m14 × 0 + (-1)2
⇒ 0 + 0 + 1 = 1
∴ The value of (m28 + m22 + m20 + m14 + m12 ) is 1.