 x, y, z, w ∈ [0, 10]



⇒

Taking logarithm both sides we have

⇒ sin 2 x log 2 + cos 2 y log3 + sin 2 z log 4 + cos 2 w log 5 ≥ log 2 + log 3 + log 4 + log 5

⇒ cos 2 x log 2 + sin 2 y log 3 + cos 2 z log 4 + sin 2 w log 5 ≤ 0

which is possible only when

cos 2 x = 0 ⇒ x = m π + π /2, m ∈ I

sin 2 y = 0 ⇒ y = n π , n ∈ I

cos 2 z = 0 ⇒ z = r π + π /2, r ∈ I

sin 2 w = 0 ⇒ w = p π , p ∈ I

 x, y, z, w ∈ [0, 10]

⇒ x = π /2, 3 π /2, 5 π /2 (three solutions)

⇒ y = 0, π , 2 π , 3 π (four solutions)

⇒ z = π /2, 3 π /2, 5 π /2 (three solutions)

⇒ w = 0, π , 2 π , 3 π (four solutions)

Hence the number of ordered 4-tuple (x, y, z, w) is 3. 4. 3. 4. = 144.