Let M and m respectively be the maximum and the minimum values of
f(x)=|1+sin ² x, cos ² x, 4 sin 4 x; sin ² x, 1+cos ² x, 4 sin 4 x; sin ² x, cos ² x, 1+4 sin 4 x|, x ∈ R
Then M⁴ - m⁴ is equal to :

Solution

Maths sir · 4 steps

Add all three columns into the first one: every row gives the same sum, so the determinant collapses fast.

1. Same row sums. Apply C₁ → C₁ + C₂ + C₃. Each row then has first entry sin² x + cos² x + 1 + 4 sin 4x = 2 + 4 sin 4x. Take this common factor out of C₁, which leaves 1 in every entry of C₁.
2. Clear the rows. Now apply R₂ → R₂ - R₁ and R₃ → R₃ - R₁. Row 2 becomes (0, 1, 0) and row 3 becomes (0, 0, 1). The remaining determinant is upper triangular with diagonal entries 1, 1 and 1, so its value is 1.

   f(x) = (2 + 4 sin 4x) |1, cos² x, 4 sin 4x; 0, 1, 0; 0, 0, 1| = 2 + 4 sin 4x

3. Range of f. Since -1 ≤ sin 4x ≤ 1, the value of f runs from 2 - 4 = -2 to 2 + 4 = 6. So M = 6 and m = -2.
4. So. M⁴ - m⁴ = 6⁴ - (-2)⁴ = 1296 - 16 = 1280, which is option (a).

The idea to keep. When the rows of a determinant all add up to the same expression, add every column into C₁ and take that sum out. The determinant that is left is usually a plain number.

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