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Question
Match the following by appropriately matching the lists based on the information given in Column I and Column II.
Column IColumn IIa. Range of f(x)=sin−1x+cos−1x+cot−1x is p. [0,π2)∪(π2,π]b. Range of f(x)=cot−1x+tan−1x+cosec−1x is q. [π2,3π2] c. Range of f(x)=cot−1x+tan−1x+cos−1x is r. {0,π} d. Range of f(x)=sec−1x+cosec−1x+sin−1x is s. [3π4,5π4]
Match the following by appropriately matching the lists based on the information given in Column I and Column II.
Column IColumn IIa. Range of f(x)=sin−1x+cos−1x+cot−1x is p. [0,π2)∪(π2,π]b. Range of f(x)=cot−1x+tan−1x+cosec−1x is q. [π2,3π2] c. Range of f(x)=cot−1x+tan−1x+cos−1x is r. {0,π} d. Range of f(x)=sec−1x+cosec−1x+sin−1x is s. [3π4,5π4]
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Solution
The correct option is C a−s; b−p; c−q; d−r
a. f(x)=sin−1x+cos−1x+cot−1x=π2+cot−1x,x∈[−1,1]
For x∈[−1,1],cot−1x∈[π4,3π4]
⇒π2+cot−1x∈[3π4,5π4]
b. f(x)=cot−1x+tan−1x+cosec−1x
=π2+cosec−1x,where x∈(−∞,−1]∪[1,∞)
Now cosec−1x∈[−π2,0)∪(0,π2]
⇒π2+cosec−1x∈[0,π2)∪(π2,π]
c. f(x)=cot−1x+tan−1x+cos−1x
=π2+cos−1x, where x∈[−1,1]
⇒π2+cos−1x∈[π2,3π2]
d. sec−1x+cosec−1x+sin−1x, where x∈{−1,1}
=π2+sin−1x, where x∈{−1,1}
Hence, f(x)∈{0,π}
a. f(x)=sin−1x+cos−1x+cot−1x=π2+cot−1x,x∈[−1,1]
For x∈[−1,1],cot−1x∈[π4,3π4]
⇒π2+cot−1x∈[3π4,5π4]
b. f(x)=cot−1x+tan−1x+cosec−1x
=π2+cosec−1x,where x∈(−∞,−1]∪[1,∞)
Now cosec−1x∈[−π2,0)∪(0,π2]
⇒π2+cosec−1x∈[0,π2)∪(π2,π]
c. f(x)=cot−1x+tan−1x+cos−1x
=π2+cos−1x, where x∈[−1,1]
⇒π2+cos−1x∈[π2,3π2]
d. sec−1x+cosec−1x+sin−1x, where x∈{−1,1}
=π2+sin−1x, where x∈{−1,1}
Hence, f(x)∈{0,π}
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