Mathematics

Mathematics

Problem

Match the statement of Column I with values of Column II.
Column IColumn II
(A) If θ+ϕ=π2\theta + \phi = \frac{\pi}{2} , where θ\theta and ϕ\phi are positive, then (sin⁡θ+sin⁡ϕ)sin⁡(π4)(\sin \theta + \sin \phi) \sin \left(\frac{\pi}{4}\right) is always less than(p) 1
(B) If sin⁡θ−sin⁡ϕ=a\sin \theta - \sin \phi = a and cos⁡θ+cos⁡ϕ=b\cos \theta + \cos \phi = b , then a2+b2a^2 + b^2 cannot exceed(q) 2
(C) If 3sin⁡θ+5cos⁡θ=53 \sin \theta + 5 \cos \theta = 5 , ( θ≠0\theta \neq 0 ) then the value of 5sin⁡θ−3cos⁡θ5 \sin \theta - 3 \cos \theta is(r) 3
(s) 4
(t) 5
Intuitive explanation
INTUITIVE EXPLANATION

Each statement uses a constraint to bound or determine a trigonometric expression. For (A), use the complementary angle relationship ϕ=π2−θ\phi = \frac{\pi}{2} - \theta to simplify the sum and find its upper bound. For (B), recognize that a2+b2a^2 + b^2 represents the squared magnitude of a vector built from sum-to-product identities—it's bounded by the range of sine and cosine. For (C), treat the two equations as a system: square both and add them to find a relationship, then solve for the unknown expression using the constraint.

Intuitive explanation
INTUITIVE EXPLANATION

Each statement uses a constraint to bound or determine a trigonometric expression. For (A), use the complementary angle relationship ϕ=π2−θ\phi = \frac{\pi}{2} - \theta to simplify the sum and find its upper bound. For (B), recognize that a2+b2a^2 + b^2 represents the squared magnitude of a vector built from sum-to-product identities—it's bounded by the range of sine and cosine. For (C), treat the two equations as a system: square both and add them to find a relationship, then solve for the unknown expression using the constraint.

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