EXPLANATION.

⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².

As we know that,

Range of sinθ = [-1,1].

There are three digit exists between [-1,1].

⇒ - 1, 0, 1.

⇒ [x² + 1/3].

in this particular equation, we get.

Only two POSSIBILITY exists = 0,1.

⇒ [x² - 2/3].

In this particular case, we get.

Three possibility exists = -1, 0, 1.

Total we can SAY that,

There are six [6] possibility exists.

⇒ [x² + 1/3]     [x² - 2/3].

⇒ 0                  -1.

⇒ 0                   0.

⇒ 0                    1.

⇒ 1                     -1.

⇒ 1                     0.

⇒ 1                     1.

Now, we can equation as,

⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².

PUT the value (0, -1) in the equation, we get.

⇒ sin⁻¹[(0)² + 1/3] + cos⁻¹[(-1)² - 2/3] = x².

⇒ sin⁻¹(0) + cos⁻¹(1) = x².

⇒ 0 + π = x². [L.H.S.]

Now again put the value of x² = π in the equation, we get.

⇒ sin⁻¹[π + 1/3] + cos⁻¹[π - 2/3] = π.

⇒ sin⁻¹[3.14 + 0.33] + cos⁻¹[3.14 - 0.6667] = π. [R.H.S.]

As we can see that,

⇒ L.H.S ≠ R.H.S.

So 1st possibility is rejected.

Put the value (0,0) in the equation, we get.

⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².

⇒ sin⁻¹[(0)² + 1/3] + cos⁻¹[(0)² - 2/3] = x².

⇒ sin⁻¹(0) + cos⁻¹[0] = x².

⇒ 0 + π/2 = x². [L.H.S.]

Put the value of π/2 in the equation, we get.

⇒ sin⁻¹[π/2 + 1/3] + cos⁻¹[π/2 - 2/3] = π/2.

⇒ sin⁻¹[1.57 + 0.33] + cos⁻¹[1.57 - 0.667] = 1.57. [R.H.S.]

As we can see that,

⇒ L.H.S. ≠ R.H.S.

So 2nd possibility is ALSO rejected.

Put the value (0,-1) in the equation, we get.

⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².

⇒ sin⁻¹[(0)² + 1/3] + cos⁻¹[(-1)² - 2/3] = x².

⇒ sin⁻¹(0) + cos⁻¹(1) = x².

⇒ 0 + 0 = x². [L.H.S.].

Put the value of x² = 0 in the equation, we get.

⇒ sin⁻¹[0 + 1/3] + cos⁻¹[0 - 2/3] = 0.

⇒ sin⁻¹(0) + cos⁻¹(-1) = 0. [R.H.S].

As we can see that,

⇒ L.H.S. ≠ R.H.S.

So 3rd possibility is also rejected.

Put the value (1, -1) in the equation, we get.

⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².

⇒ sin⁻¹[(1)² + 1/3] + cos⁻¹[(-1)² - 2/3] = x².

⇒ sin⁻¹(1) + cos⁻¹(-1) = x².

⇒ π/2 + π = x².

⇒ 3π/2 = x². [L.H.S.].

Put the value of x² = 3π/2 in the equation, we get.

⇒ sin⁻¹[3π/2 + 1/3] + cos⁻¹[3π/2 - 2/3] = 3π/2.

⇒ sin⁻¹[9π + 2/6] + cos⁻¹[9π - 4/6] = 3π/2. [R.H.S.].

As we can see that,

⇒ L.H.S. ≠ R.H.S.

So, 4th possibility is also rejected.

Put the value (1,0) in the equation, we get.

⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².

⇒ sin⁻¹[(1)² + 1/3] + cos⁻¹[(0)² - 2/3] = x².

⇒ sin⁻¹(1) + cos⁻¹(0) = x².

⇒ π/2 + π/2 = x².

⇒ x² = π. [L.H.S.].

Put the value of x² = π in the equation, we get.

⇒ sin⁻¹[π + 1/3] + cos⁻¹[π - 2/3] = π.

⇒ sin⁻¹[3.14 + 0.33] + cos⁻¹[3.14 - 0.667] = π. [R.H.S.].

As we can see that,

⇒ L.H.S. ≠ R.H.S.

So 5th possibility is also rejected.

Put the value (1,1) in the equation, we get.

⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².

⇒ sin⁻¹[(1)² + 1/3] + cos⁻¹[(1)² - 2/3] = x².

⇒ sin⁻¹(1) + cos⁻¹(1) = x².

⇒ π/2 = x². [L.H.S.].

Put the value of x² = π/2 in the equation, we get.

⇒ sin⁻¹[π/2 + 1/3] + cos⁻¹[π/2 - 2/3] = π/2.

⇒ sin⁻¹[1.57 + 1.33] + cos⁻¹[1.57 - 0.667] = 1.57. [R.H.S.].

As we can see that,

⇒ L.H.S. ≠ R.H.S.

So 6th possibility is also rejected.

As we can observe all the possibility and see that no one possibility is matched.

Hence, Number of solutions = 0.

Option [A] is correct answer.