In a plane, if the magnitude of the projection vector Of the vector α $$i^+$$β$$j^$$ on $$3i^+j^$$ is $$3and$$ if α$$=2+3$$β, then possible $$value(s)$$ of αis (are)Let$$ a and b be real numbers such that the function is $$fx=$$−$$3ax2$$−$$2$$,x

Question

In a plane, if the magnitude of the projection vector Of the vector α $$i^+$$β$$j^$$  on $$3i^+j^$$ is $$3and$$ if α$$=2+3$$β, then possible $$value(s)$$ of  αis (are)Let$$ a and b be real numbers such that the function  is $$fx=$$−$$3ax2$$−$$2$$,x<$$1bx+a2$$,x≥$$1differentiable$$ for all real values of x. Then $$possiblevalue(s) of  a $$is(are)Let$$  ω≠$$1be$$ a complex cube root of unity. If $$3$$−$$3$$ω$$+2$$ω$$24n+3+2+3$$ω−$$3$$ω$$24n+3+$$−$$3+2$$ω$$+3$$ω$$24n+3=0$$, then possible $$value(s)of$$ n $$is(are)Let$$ the harmonic mean of two positive real numbers a and b be $$4$$. If q is a positive real number such that  a,$$5$$,q,bis an arithmetic progression, then the $$value(s)$$ of q−a $$is(are)$$

Infinity Learn · Accepted Answer

A The projection of α $$i^+$$β$$j^$$ on the vector $$3i^+j^$$  is $$3Hence$$, $$3$$α $$+$$β$$2=33$$α $$+$$β$$=23substitute$$ α $$=2+3$$β$$32+3$$β$$+$$β$$=2323+3$$β$$+$$β$$=23$$β$$=0$$  and α $$=2B$$ $$fx=$$−$$3ax2$$−$$2x$$<$$1bx+a2x$$≥$$1Ltx$$→$$1$$−$$fx=Ltx$$→$$1+fx$$−$$3a$$−$$2=b+a2$$−$$a2$$−$$3a$$−$$2=bf1x=$$−$$6axx$$<$$1bx$$>$$1f11$$−$$=f11+$$−$$6a=b$$−$$a2$$−$$3a$$−$$2=$$−$$6aa2$$−$$3a+2=0a=1$$,$$a=2C$$ $$3$$−$$3$$ω$$+2$$ω$$24n+3+2+3$$ω−$$3$$ω$$24n+3+$$−$$3+2$$ω$$+3$$ω$$24n+3=0$$ $$3$$−$$3$$ω$$+2$$ω$$2=21+$$ω$$2+1$$−$$3$$ω$$=$$−$$5$$ω$$+12+3$$ω−$$3$$ω$$2=21+$$ω$$+$$ω−$$3$$ω$$2=$$ω−$$5$$ω$$2=$$ω−$$5$$ω$$+1$$−$$3+2$$ω$$+3$$ω$$2=$$−$$3+2$$ω$$+$$ω$$2+$$ω$$2=$$−$$5+$$ω$$2=$$ω$$2$$−$$5$$ω$$+1$$−$$5$$ω$$+14n+31+$$ω$$4n+3+$$ω$$24n+3=0n=1$$,$$n=2$$,$$n=4$$,$$n=5D$$ $$2aba+b=4$$⇒$$ab=2a+b$$⇒$$b=2aa$$−$$2a$$,$$5$$,q,b  are in $$APa+d=5$$⇒$$d=5$$−$$ab=a+3d=2aa$$−$$2$$⇒$$a+35$$−$$a=2aa$$−$$2$$⇒$$2a2$$−$$17a+30=0$$⇒$$a=6$$  or $$52q$$−$$a=2$$,$$5$$

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