Question

The number of ordered 4-tuple (x, y, z, w) (x, y, z, w Ī

[0,10]) which satisfies the inequality

$2^{\sin^{2}x}3^{\cos^{2}y}4^{\sin^{2}z}5^{\cos^{2}w} \geq 120$ is –

• A. 0
• B. 144
• C. 81
• D. Infinite

Answer 1

Answer: (B)

144

Answer 2

Q x, y, z, w Ī [0, 10]

Q $2^{\sin^{2}x}3^{\cos^{2}y}4^{\sin^{2}z}5^{\cos^{2}w} \geq 120$

Ž $2^{\sin^{2}x}3^{\cos^{2}y}4^{\sin^{2}z}5^{\cos^{2}w} \geq 2.3.4.5$

Taking logarithm both sides we have

Ž sin²x log 2 + cos²y log3 + sin²z log 4 + cos² w log 5 ³ log 2 + log 3 + log 4 + log 5

Ž cos²x log 2 + sin²y log 3 + cos² z log 4 + sin²w log 5 £ 0

which is possible only when

cos² x = 0 Ž x = mp + p/2, m Ī I

sin² y = 0 Ž y = np, n Ī I

cos² z = 0 Ž z = rp + p/2, r Ī I

sin² w = 0 Ž w = pp, p Ī I

Q x, y, z, w Ī [0, 10]

Ž x = p/2, 3p/2, 5p/2 (three solutions)

Ž y = 0, p, 2p, 3p (four solutions)

Ž z = p/2, 3p/2, 5p/2 (three solutions)

Ž w = 0, p, 2p, 3p (four solutions)

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