If p, q, r are in A.P. and x, y, z in G.P., then Xq - r × Yr - p × Z

1.	If p, q, r are in A.P. and x, y, z in G.P., then Xq - r × Yr - p × Zp - q is equal to ………………….. A) p + q + r B) x × y × z C) 1 D) px + qy + rz
Answer» Correct option is (C) 1 Given that p, q, r are in A.P. \(\therefore\) p+r = 2q _____________(1) Also given that x, y, z are in G.P. \(\therefore xz=y^2\) _____________(2) Now, \(x^{q-r}\times y^{r-p}\times z^{p-q}\) \(=x^{q-(2q-p)}\times y^{((2q-p)-p)}\times z^{p-q}\) (By putting r = 2q - p (from (1)) \(=x^{p-q}\times y^{2(q-p)}\times z^{p-q}\) \(=(xz)^{p-q}\times y^{2(q-p)}\) \((\because a^m.b^m=(ab)^m)\) \(=(y^2)^{p-q}\times y^{2(q-p)}\) (From (2)) \(=y^{2(p-q)}\times y^{-2(p-q)}\) \(=y^{2(p-q)-2(p-q)}\) \(=y^0=1\) Hence, \(x^{q-r}\times y^{r-p}\times z^{p-q}=1\) Correct option is C) 1

If p, q, r are in A.P. and x, y, z in G.P., then Xq - r × Yr - p × Zp - q is equal to ………………….. A) p + q + r B) x × y × z C) 1 D) px + qy + rz

Answer»

Correct option is (C) 1

Given that p, q, r are in A.P.

\(\therefore\) p+r = 2q _____________(1)

Also given that x, y, z are in G.P.

\(\therefore xz=y^2\) _____________(2)

Now, \(x^{q-r}\times y^{r-p}\times z^{p-q}\) \(=x^{q-(2q-p)}\times y^{((2q-p)-p)}\times z^{p-q}\)

(By putting r = 2q - p (from (1))

\(=x^{p-q}\times y^{2(q-p)}\times z^{p-q}\)

\(=(xz)^{p-q}\times y^{2(q-p)}\) \((\because a^m.b^m=(ab)^m)\)

\(=(y^2)^{p-q}\times y^{2(q-p)}\) (From (2))

\(=y^{2(p-q)}\times y^{-2(p-q)}\)

\(=y^{2(p-q)-2(p-q)}\)

\(=y^0=1\)

Hence, \(x^{q-r}\times y^{r-p}\times z^{p-q}=1\)

Correct option is C) 1