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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)=sin1x+cos1x+cot1x is p. [0,π2)(π2,π]b. Range of f(x)=cot1x+tan1x+cosec1x is q. [π2,3π2] c. Range of f(x)=cot1x+tan1x+cos1x is r. {0,π} d. Range of f(x)=sec1x+cosec1x+sin1x is s. [3π4,5π4]

A
ap; bq; cr; ds
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B
as; br; cq; dp
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C
as; bp; cq; dr
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D
aq; bs; cp; dr
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Solution

The correct option is C as; bp; cq; dr
a. f(x)=sin1x+cos1x+cot1x=π2+cot1x,x[1,1]
For x[1,1],cot1x[π4,3π4]
π2+cot1x[3π4,5π4]

b. f(x)=cot1x+tan1x+cosec1x
=π2+cosec1x,where x(,1][1,)
Now cosec1x[π2,0)(0,π2]
π2+cosec1x[0,π2)(π2,π]

c. f(x)=cot1x+tan1x+cos1x
=π2+cos1x, where x[1,1]
π2+cos1x[π2,3π2]

d. sec1x+cosec1x+sin1x, where x{1,1}
=π2+sin1x, where x{1,1}
Hence, f(x){0,π}

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