In a triangle XYZ let a,b,and c be the lengthsof the sides opposite to the angles X,Y and Zrespectively. If $$2a2$$−$$b2=c2$$ and λ$$=sinX$$−YsinZ ,then possible $$value(s)$$ of n for which cosnπλ$$=0is$$ (are)In$$ a triangle XYZ let a,b,and c be the lengthsof the sides opposite to the angles X,Y and Zrespectively. If $$1+cos2X$$−$$2cos2Y=2sinXsinY$$, then possible $$value(s) of ab is (are)In$$ $$R2$$ let $$3i^+j^$$,$$i^+3j^$$ and β$$i^+1$$−β$$j^$$ bethe position vectors of X,Yand Z with respectto the origin O, respectively.If the distance ofZ from the bisector of acute angle of OX→withOY→ is $$32$$ then possible $$value(s) of β is (are)Suppose$$ that Fα denotes the area of theregion bounded by $$x=0$$,$$x=2$$,$$y2=4x$$ and $$y=$$αx−$$1+$$αx−$$2+$$αx,where α∈$$0$$,$$1$$ ,Then the $$value(s) of Fα$$+832$$,when α$$=0$$ andα$$=1$$ is (are)

Question

In a triangle XYZ let a,b,and c be the lengthsof the sides opposite to the  angles X,Y and Zrespectively. If $$2a2$$−$$b2=c2$$  and λ$$=sinX$$−YsinZ ,then  possible $$value(s)$$ of n for which cosnπλ$$=0is$$ (are)In$$ a triangle XYZ  let a,b,and c be the lengthsof the sides opposite to the  angles X,Y and Zrespectively. If $$1+cos2X$$−$$2cos2Y=2sinXsinY$$, then possible $$value(s) of  ab is (are)In$$  $$R2$$ let $$3i^+j^$$,$$i^+3j^$$  and β$$i^+1$$−β$$j^$$ bethe position  vectors of X,Yand Z with respectto the origin O, respectively.If  the distance ofZ from the bisector of acute angle of  OX→withOY→ is $$32$$  then possible $$value(s) of  β is (are)Suppose$$ that Fα  denotes  the area of theregion bounded by $$x=0$$,$$x=2$$,$$y2=4x$$  and $$y=$$αx−$$1+$$αx−$$2+$$αx,where α∈$$0$$,$$1$$ ,Then the $$value(s) of Fα$$+832$$,when α$$=0$$ andα$$=1$$  is (are)

Infinity Learn · Accepted Answer

A $$2a2$$−$$b2=c2$$⇒$$2sin2X$$−$$sin2Y=sin2Z$$⇒$$2sinX+YsinX$$−$$Y=sin2Z$$⇒sinX−$$YsinZ=12$$⇒λ$$=12cosn$$πλ$$=0$$  ⇒cosn$$π$$$$2=0n=1$$,$$3$$,$$5B$$ $$1+cos2X$$−$$2cos2Y=2sinXsinY$$⇒$$1+1$$−$$2sin2X$$−$$21$$−$$2sin2Y=2sinXsinY$$⇒$$sin2X+sinXsinY$$−$$2sin2Y=0sinXsinY=$$−$$2$$  $$or1but$$ −$$2$$  is not possible⇒$$sinXsinY=1$$⇒$$ab=1C$$ Bisectors of OX→&OY→  are $$x=y$$  or $$x+y=0$$∴β$$-1-$$β$$2=32$$⇒$$2$$β$$-1=3$$⇒β$$=1$$,$$2$$ (D) For α$$=1$$                   $$y=x$$−$$1+x$$−$$2+x=3$$−x  ;  x<$$11+x$$  ;   $$1$$≤x<$$23x$$−$$3$$;   x≥$$2$$                   ∴Area $$=F1=122+3$$×$$1+122+3$$×$$1$$− $$022xdx=5$$−$$823$$⇒$$F1+823=5$$                       For α$$=0$$, $$y=3$$.                         ∴$$Area=F0=2$$×$$3$$−∫$$0222dx=6$$−$$823$$⇒$$F0+823=6$$

Start JEE / NEET / Foundation preparation at rupees 99/day !!

Detailed Solution

Get Expert Academic Guidance – Connect with a Counselor Today!