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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+^jis √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?

A
P  B
Q  C
R  D
S  E
T  Solution

The correct options are A P B QOption (A): The projection vector of the vector α^i+β^j on √3^i+^j is (α^i+β^j)⋅√3^i+^j2⇒∣∣ ∣∣α^i+β^j)⋅√3^i+^j2∣∣ ∣∣=√3⇒√3α+β=±2√3     ⋯(1)α=2+√3β     ⋯(2) Solving (1) and (2), we get ⇒α=2,−1|α|=2,1Mathematics

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