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{:(,"List-I",,"List-II",),((P),"The number of integral values of k for which the ",(1),1,),(,"equation" sin^(01)x^(-2)+tan^(-1)x^(2)k+1 "has a solution",,,),((Q),"Let" sum_(K=1)^(oo)cot^(-1)(k^(2)/(8))=(p)/(q)pi where" (p)/(q) "is rational in its lowest",(2),2,),(,"form then find" |q-p|,,,),((R),"If" tan^(-1)(sin^(2)theta-2 sin theta +3)+cot^(-1) (5^(sec^(2)_(y)+1)=(pi)/(2), then the",(3),3,),(,"value of" cos^(2)y-sin theta,,,),((S),"Minimum positive integral value of x such that f(x)",(4),0,),(,"is defined" f(x)=sqrt(sec^(-1)((1-|x|)/(2))),(4),0,):} |
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Answer» `{:(P,Q,R,S),(3,1,2,2):}` function and domain is `[-1.1]` `f(x)|._(min) =f(0)=0` `f(x)|_(min) =f(1) =(pi)/(2)+(pi)/(4)=(3pi)/(4)` `f(x)in[0.(3pi)/(4)]` `if K=0,2k+1=1`, possible if `k=1, 2k +1=3gt(3pi)/(4)` Not possible i.e only one integral value of k is possible (Q) `T_(K)tan^(-1)((8)/(k^(2)))=tan^(-1),(2)/(k^(2)/(4))=tan^(-1),(2)/(1+((k)/(2)-1)((k)/(2)+1))` `T_(K)=tan^(-1)((k)/(2)+1)-tan^(-1)((k)/(2)-1)` `T_(1)=tan^(7).(3)/(2)-tan^(-1)((-1)/(2))` `T_(2)=tan^(-1).(4)/(2)-tan^(-1)0` `T_(3)=tan^(-1).(5)/(2)-tan^(-1).(1)/(2)` `T_(4)=tan^(-1).(6)/(2)-tan_(-1),(2)/(2)` `T_(5)=tan^(-1).(7)/(2)-tan^(-1).(3)/(2)` : : `Sum =` `2pi-(-tan^(-1).(1)/(2)+tan_(-1)/(2)+tan^(-1)1)=(7pi)/(4)impliesp=7,q=4` `(2pi us sum of last four term)` we know `tan^(-1)x + cot^(-1) y=(pi)/(2)impliesx=y` according to question `underset([2.6])ubrace((sintheta-1)^(2)+2)=underset(ge6)ubrace(5sec^(2)y+1)` Only possible `implies sintheta -1 =-1` `sin theta=-1` ` sec^(2)y=1implies cos^(2)y=1` `cos^(2)y-sin theta=2` `(S) sec^(-1)( (1-|x|)/(2))ge0implies (1-|x|)/(2)ge1` or `(1-|x|)/(2)le-1` `1|x|ge2 or 3le|x|` `-1ge|x|0r x epsilon (-oo,-3]uu[3,oo)` Not possible least positve integer is 3 |
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