Let x * y = x² + y³ and (x * 1) * 1 = x * (1 * 1).
Then a value of 2sin ⁻¹( (x⁴ + x² - 2)/(x⁴ + x² + 2) ) is :

Solution

Maths sir · 4 steps

Expand the star operation one step at a time, find x², and the inverse sine gives a standard angle.

1. Left side. First x ∗ 1 = x² + 1. Then (x ∗ 1) ∗ 1 = (x² + 1)² + 1³ = x⁴ + 2x² + 2.
2. Right side. Here 1 ∗ 1 = 1 + 1 = 2. So x ∗ 2 = x² + 2³ = x² + 8. The second entry is cubed, so it gives 8, not 4.

   x⁴ + 2x² + 2 = x² + 8 ⇒ x⁴ + x² - 6 = 0 ⇒ (x² + 3)(x² - 2) = 0

3. Pick x squared. x² = -3 is impossible for real x, so x² = 2 and x⁴ = 4. The fraction becomes (4 + 2 - 2)/(4 + 2 + 2) = 4/8 = 1/2.
4. So. 2 sin⁻¹(1/2) = 2 × π/6 = π/3, which is option (b).

The idea to keep. With a custom operation, substitute in strict order and note which slot is squared and which is cubed. Then solve for x², not x, because only x² and x⁴ appear.

Answer from the NTA final answer key. Solution by PrepWiser.