Prove that `|(a_(1)alpha_(1)+b_(1)beta_(1),a_(1)alpha_(2)+b_(1)beta_(2),a_(1)alpha_(3)+b_(1)beta_(3)),(a_(2)alpha_(1)+b_(2)beta_(1), a_(2)alpha_(2)+b_(2)beta_(2), a_(2)alpha_(3)+b_(2)beta_(3)),(a_(3)alpha_(1)+b_(3)beta_(1), a_(3)alpha_(2)+b_(3)beta_(2),a_(3)alpha_(3)+b_(3)beta_(3))|=0`

If Delta_(1)|(a_(1)^(2)+b_(1)+c_(1),a_(1)a_(2)+b_(2)+c_(2),a_(1)a_(3)+b_(3)+c_(3)),(b_(1)b_(2)+c_(1),b_(2)^(2)+c_(2),b_(2)b_(3)+c_(3)),(c_(3)c_(1),c_(3)c_(2),c_(3)^(2))|andDelta_(2)=|(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| then Delta_(1) // Delta_(2) is equal to a) a_(1)b_(2)c_(3) b) a_(1)a_(2)a_(3) c) a_(3)b_(2)c_(1) d) a_(1)b_(1)c_(1) +a_(2)b_(2)c_(2)+a_(3)b_(3)c_(3)

Suppose Delta = |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))|andDelta'=|(a_(1)+pb_(1),b_(1)+qc_(1),c_(1)+ra_(1)),(a_(2)+pb_(2),b_(2)+qc_(2),c_(2)+ra_(2)),(a_(3)+pb_(3),b_(3)+qc_(3),c_(3)+ra_(3))| then (i) Δ'=Δ (ii) Δ'=Δ(1−pqr) (iii) Δ'=Δ(1+P+q+r) (iv) Δ'=Δ(1+pqr)

Let a_(1),a_(2),a_(3),a_(4) and a_(5) be such that a_(1), a_(2) and a_(3) are in A.P. a_(2) ,a_(3) and a_(4) are in G.P and a_(3) ,a_(4) and a_(5) are in H.P . Then a_(1) ,a_(3) and a_(5) are in

If omega is a complex cube root of unity, then value of Delta=|(a_(1)+b_(1)omega,a_(1)omega^(2)+b_(1),c_(1)+b_(1)omega),(a_(2)+b_(2)omega,a_(2)omega^(2)+b_(2),c_(2)+b_(2)omega),(a_(3)+b_(3)omega,a_(3)omega^(2)+b_(3),c_(3)+b_(3)omega)| is a)0 b)-1 c)2 d)None of these

If a_(1), a_(2), a_(3),…….,a_(20) are in AP and a_(1) + a_(20) = 45 , then a_(1) + a_(2) + a_(3)+……+a_(20) is equal to

If a_(1), a_(2), …, a_(50) are in GP, then " "(a_(1)-a_(3)+a_(5)-…+a_(49))/(a_(2)-a_(4)+a_(6)-…+a_(50)) is equal to :

If cot alpha cot beta=2 , show that (cos (alpha+beta))/(cos (alpha-beta))=(1)/(3)

If alpha ne beta, alpha^(2)=5 alpha-3, beta^(2)=5beta-3 , then the equation having (alpha)/(beta) and (beta)/(alpha) as its roots is

If a_(1), a_(2), a_(3), a_(4) are in AP, then (1)/(sqrt(a_(1)) + sqrt(a_(2))) + (1)/(sqrt(a_(2)) + sqrt(a_(3))) + (1)/(sqrt(a_(3)) + sqrt(a_(4))) is equal to