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Question

Match List I with the List II and select the correct answer using the code given below the lists : List IList II(A)Let f:[−1,∞)→(0,∞) be defined by f(x)=ex2+|x|. Then f is (P)one-one(B)Let f:(1,∞)→[3,∞) be defined by f(x)=√10−2x+x2. Then f is(Q)into(C)Let f:R→Z be defined by f(x)=tan5π[x2+2x+3] where [. ] denotes(R)many-onethe greatest integer function. Then f is(D)Let f:[3,4]→[4,6] be defined by f(x)=|x−1|+|x−2|+|x−3|+|x−4|.(S)onto Then f is (T)periodic Which of the following is a CORRECT combination?

A
(A)(Q), (S); (B)(S), (T)
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B
(A)(Q), (R); (B)(S), (T)
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C
(A)(Q), (S); (B)(P), (Q)
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D
(A)(Q), (R); (B)(P), (Q)
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Solution

The correct option is D (A)→(Q), (R); (B)→(P), (Q)(A) We have f(x)=ex2+|x|,x∈[−1,∞) Clearly, f(−1)=e2=f(1) Also, 0≤x2+|x|<∞ ∀ x∈[−1,∞) ⇒1≤ex2+|x|<∞ Hence, range of f is [1,∞) ∴f(x) is many-one, into. (A)→(Q), (R) (B) We have, f(x)=√10−2x+x2 f(x)=√(x−1)2+9 Clearly, range of f is (3,∞) and also f(x) is one-one. ∴f(x) is one-one, into. (B)→(P), (Q)

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