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Question

# Answer the following by appropriately matching the lists based on the information given in Column I and Column II ​​​​​​Column 1Column 2a. f:R→[3π4,π) and f(x)=cot−1(2x−x2−2),then f is p. one-oneb. f:R→R and f(x)=epxsinqx where p,q∈R+,then f is q. into c. f:R+→[4,∞) and f(x)=4+3x2, then f is r. many-one d. f:R→R and f(f(x))=x, ∀ x∈R then f is s. onto

A
ap,r; bq,r; cp,s; dq,s
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B
as,r; br,s; cp,q; dp,s
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C
aq,r; bp,s; cp,r; dq,s
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D
ap,q; bq,s; cr,s; dp,r
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Solution

## The correct option is B a−s,r; b−r,s; c−p,q; d−p,sa. f(x)=cot−1(2x−x2−2) =cot−1(−1−(x−1)2)≤−1 ∴f(0)=f(2) Hence, f(x) is many-one. Thus, cot−1(2x−x2−2)∈[3π4,π) Hence, f(x) is onto. Also, f(3)=f(−1). Hence, function is many-one. b. Clearly, from the graph, f(x) is many-one and onto. c. Clearly, from the graph, f(x) is one-one and into. d. Let X={x1,x2,x3,…,xn} Let f(x1)=x2 or, f(f(x1))=f(x2)=x1 Thus, f(x) is one-one and onto.

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