Question

Statement-1: if `thetane2npi+(pi)/(2),n inZ,` then
`(sec^(2)theta+tantheta)/(sec^(2)theta-tantheta)"lies between"1/3 and 3.`
Statement-2: If `x inR, then 1/3le(x^(2)-x+1)/(x^(2)+x+1)le3.`
A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement -1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer 1

Correct Answer - A
Let `y=(x^(2)-x+1)/(x^(2)+x+1).Then,`
`x^(2)(y-1)+x(y+1)+(y-1)=0`
`implies(y+1)^(2)-4(y-1)^(2)ge0" "[becausex inRthereforeDiscge0]`
`implies(3y-1)(y-3)le0`
`1/3leyle3`
`implies1/2le(x^(2)-x+1)/(x^(2)+x+1)le3"for all"x inR.`
We have,
`(sec^(2)theta+tantheta)/(sec^(2)theta-tantheta)=(tan^(2)theta+tantheta-1)/(tan^(2)theta-tantheta+1)`
Using statement-2, we have
`1/3le (tan^(2)theta+tantheta+1)/(tan^(2)theta-tantheta+1)le3"for all"theta(ne(2n+1)(pi)/(2)),n inZ`
`implies1/3le(sec^(2)theta+tantheta)/(sec^(2)theta-tantheta)le3"for all"theta ne2npi+-(pi)/(2)`
Hence, both the statements are true and statement-2 is a correct explanation for statement-1.