  Question

# Column Matching: Column (I)Column (II)(A)  In R2, if the magnitude of the projectionvector of the vector α^i+β^j on √3^i+^j√3 and if α=2+√3β, then possiblevalue(s) of |α| is (are)(P)     1(B)  Let a and b be real numbers such thatthe functionf(x)={−3ax2−2,  x<1bx+a2,       x≥1 is differentiable for all x∈R. Thenpossible value(s) of a is (are) (Q)     2(C)  Let ω≠1 be a complex cube root ofunity. If (3−3ω+2ω2)4n+3+(2+3ω−3ω2)4n+3+(−3+2ω+3ω2)4n+3=0,then possible value(s) of n is (are)(R)     3(D)  Let the harmonic mean of two positive realnumbers a and b be 4. If q is a positive realnumber such that a,5,q,b is an arithmeticprogression, then the value(s) of |q−a| is (are)(S)     4(T)     5 Option (D) matches with which of the elements of right hand column?PQRST

Solution

## The correct options are A P B QFunction is continuous. So limx→1f(x)=f(1) ⇒−3a−2=b+a2     ⋯(1) For function to be differentiable,  −6a=b     ⋯(2) From (1) and (2), we get ⇒−3a−2=−6a+a2 ⇒a2−3a+2=0⇒a=1,2  Suggest corrections   