If `A = int_(1)^(sintheta) (t)/(1 + r^(2)) dt and B = int_(1)^("cosec"theta) (1)/(t(1 +t^(2))) dt`, then the value of the determinant `|(A,A^(2),B),(e^(A +B),B^(2),-1),(1,A^(2) + B^(2) ,-1)|` isA. `sin theta`B. `cosec theta`C. 0D. 1

If `A = int_(1)^(sintheta) (t)/(1 + r^(2)) dt and B = int_(1)^("cosec"

1.	If `A = int_(1)^(sintheta) (t)/(1 + r^(2)) dt and B = int_(1)^("cosec"theta) (1)/(t(1 +t^(2))) dt`, then the value of the determinant `\|(A,A^(2),B),(e^(A +B),B^(2),-1),(1,A^(2) + B^(2) ,-1)\|` isA. `sin theta`B. `cosec theta`C. 0D. 1
Answer» Correct Answer - C We have, `A + B = underset(1)overset(sin theta) int (t)/(1 +t^(2)) dt + underset(1)overset("cosec" theta)int (1)/(t(1 + t^(2))) dt` `rArr A + B = underset(1)overset(sin theta)int (t)/(1 +t^(2)) dt + underset(1)overset(sin theta)int - (u)/(1 + u^(2)) du`, where `u = (1)/(t)` `rArr A + B = 0` `rArr B = - A` `:. \|(A,A^(2),B),(e^(A +B),B^(2),-1),(1,A^(2) + B^(2),-1)\|` `= \|(A,A^(2),-A),(1,A^(2),-1),(1,2A^(2),-1)\| = -\|(A,A^(2),A),(1,A^(2),1),(1,2A^(2),1)\| = 0`