If u , vand w (all positive) are the p^ th , q^ th and r^ th terms of a GP, the determinant of the matrix | array lll In u & p & 1 In v & q & 1 In w & r & 1 array | is

Step-by-step solution

Let u, v and w are p ^ th , q ^ th and r ^ th term of the G.P. with first term ' a ' and common ratio ' d '. then, aligned & u=a .(d)^ p-1 In u= In (a)+(p-1) In (d) & v=a .(d)^ q-1 In v= In (a)+(q-1) In (d) & w=a .(d)^ r-1 In w= In (a)+(r-1) In (d) aligned Now, In u- In v=(p-q) In (d) In u- In w=(p-r) In (d) | array lll In u & p & 1 In v & q & 1 In w & r & 1 array | | array ccc In u & p & 1 In u- In v & (p-q) & 0 In u- In w & (p-r) & 0 array | . Applying R ₂ R ₁- R ₂ and R ₃ R ₁- R ₃ or, aligned | array ccc In u &