The number of solutions of the equation
sin ⁻¹[ x² + 1/3 ] + cos ⁻¹[ x² - 2/3 ] = x², for x∈[-1, 1], and [x] denotes the greatest integer less than or equal to x, is :

Solution

Maths sir · 3 steps

The brackets turn both inverse functions into constants, so first find what the left side really equals.

1. Range of x squared. For x ∈ [-1, 1] we have x² ∈ [0, 1]. So x² + 1/3 ∈ [1/3, 4/3] and x² - 2/3 ∈ [-2/3, 1/3]. The greatest integer function can only give a few values on these intervals.

   x² lies in	[x² + 1/3]	[x² - 2/3]	Left side
   [0, 2/3)	0	-1	sin⁻¹ 0 + cos⁻¹(-1) = 0 + π = π
   [2/3, 1]	1	0	sin⁻¹ 1 + cos⁻¹ 0 = π/2 + π/2 = π

2. Compare both sides. In every case the left side equals π ≈ 3.14. The right side is x², which is at most 1. A number that is never more than 1 can never equal π.
3. So. The equation has no solution in [-1, 1]. The number of solutions is 0, option (a).

The idea to keep. When greatest integer brackets sit inside inverse trig functions, split the domain where each bracket changes value. The left side usually collapses to a constant, then compare it with the range of the right side.

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