Question

# If $$\mathrm{z}(z\neq 0)$$ p,q,r are distinct cube roots of a complex number and $$\mathrm{a}$$, b,c are complex numbers such that $$ap+bq+cr\neq 0$$ then $$\displaystyle \frac{(aq+br+cp)(ar+bp+cq)}{(ap+bq+cr)^{2}}=?$$

A
1
B
1
C
abc
D
pqr

Solution

## The correct option is A $$1$$We know that $$1, w$$ and $$w^2$$ are cube roots of unity.Thus, cube roots of any number $$k^3$$ will be $$k, kw$$ and $$kw^2.$$Now substituting these values for $$p,q$$ and $$r$$ respectively and using $$w^3 = 1,$$numerator becomes $$k^2 \times (aw + bw^2 + c)(aw^2 + b + cw) = k^2(a^2 + abw + acw^2 + abw + b^2w^2 + bc + acw^2 + bc + c^2w)$$Denominator becomes $$k^2 (c^2w + a^2 + abw + acw^2 + abw + b^2w^2 + bc + acw^2 + bc)$$Thus observing, the numerator and denominator are the same.Hence answer becomes 1.  Maths

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