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MathsHardMCQ2021 · 16 Mar Shift 2

Q68. sin2 x 1 + cos2 x cos 2x The maximum value of f(x) = 1 + sin2 x cos2 x cos 2x , x ∈R is sin2 x cos2 x sin 2x (1) √7 (2) 34 (3) √5 (4) 5

What This Question Tests

This question involves evaluating a trigonometric determinant and then finding its maximum value, requiring simplification using trigonometric identities.

Concepts Tested

Determinant of a matrixTrigonometric identities

Formulas Used

Determinant properties

cos(2x) = cos²x - sin²x = 1 - 2sin²x

sin²x + cos²x = 1

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9.17 (a) sin i¢c = 1.44/1.68 which gives i¢c = 59°. Total internal reflection takes place when i > 59° or when r < rmax = 31°. Now, (sin i /sin r max max ) = 1.68 , which gives imax ~ 60°. Thus, all incident rays of angles in the range 0 < i < 60° will suffer total internal reflections in the pipe. (If the length of the pipe is finite, which it is in practice, there will be a lower limit on i determined by the ratio of the diameter to the length of the pipe.) (b) If there is no outer coating, i¢c = sin–1(1/1.68) = 36.5°. Now, i = 90° will have r = 36.5° and i¢ = 53.5° which is greater than i¢c. Thus, all incident rays (in the range 53.5° < i < 90°) will suffer total internal reflections.