Column-I | Column-II |
(A)If for some real x, the equation x + = 2 cos θ holds, then cos θ is equal to | (P) 2 |
(B)If sin θ + cosec θ = 2 then sin2008 θ + cosec2008θ is equal to | (Q) 1 |
(C)Maximum value of sin4θ + cos4θ is | (R) 0 |
(D)Least value of 2 sin2θ + 3 cos2θ is | (S) – 1 |
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
→ (Q, S)
→ (P)
→ (Q)
→ (P)
Sol. By using A.M. ≥ G.M. ⇒ x + 
= 2 cos θ ≥ 2 or ≤ – 2
⇒ cos θ = 1 or –1
sin θ + cosec θ = 2 ⇒ By using A.M. ≥ G.M.
sin θ + 
≥ 2 or ≤ – 2
but given that sin θ + cosec θ = 2 ⇒ sin θ + 
= 2
which is possible only when sin θ = 1
∴ sin 2008 θ + cosec 2008 θ = sin 2008 θ + 
= 1 + 1 = 2
sin 4 θ + cos 4 θ = (sin 2 θ + cos 2 θ ) 2 – 2 sin 2 θ cos 2 θ = 1– 
sin 2 2 θ
0 ≤ sin 2 2 θ ≤ 1 ∴ 
≤ 1– 
sin 2 2 θ ≤ 1
∴ maximum value = 1
2 sin 2 θ + 3 cos 2 θ = 2 sin 2 θ + 3 – 3 sin 2 θ = 3 – sin 2 θ
0 ≤ sin 2 θ ≤ 1 ∴ 2 ≤ 3 – sin 2 θ ≤ 3 ∴ least value = 2
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