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in Sets, relations and functions by (53.3k points)

Match the following:

Column Ι Column ΙI
(A) If the smallest positive integral value of x for which x2 − x − sin–1(sin 2) < 0 is λ, then 3 + λ is equal to  (p) 1
(B) Number of solution(s) of 2[x] = x + 2 {x} is (where [⋅], {⋅} are the greatest integer and least integer functions, respectively. (q) 2
(C) least integer functions, respectively) (q) 1 (C) If x2 + y2 = 1 and maximum value of x + y is √2 λ/3, then λ is equal to (s) 0
(D) f(x +1/2) + f(x - 1/2) = f(x)  for all x ∈ R, then period of f(x) is (t) 3

1 Answer

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Correct option  (A) → (p), (B) → (t), (C) → (t), (D) → (t)

(B) 2[x] = x + 2 {x}

(i)  If x is an integer, then the equation becomes 2x = x + 0 That is, x = 0 is a solution

(ii)  If x Ι, the equation becomes

 Therefore, there are 3 solutions

(C)  Let x = cosθ , y = sin θ

Therefore, x + y = cos θ + sin θ.

 Therefore, maximum value of x + y is √2.

Therefore, f(x) is periodic with period 3.

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