If a-b-c are nonzero real numbers such that bc ca ab ca ab bc ab bc ca -0- then

The correct options are A 1 a + 1 bω + 1 cω^ 2 =0 C 1 a + 1 bω^ 2 + 1 cω =0 D 1 a ω + 1 bω^ 2 + 1 c =0 bc amp; ca amp ...

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Byju's Answer

Standard XII

Mathematics

Cramer's Rule

If a,b,c ar...

Question

If $a, b, c$ are nonzero real numbers such that $∣ ∣ ∣ ∣ \begin{matrix} b c & c a & a b c a & a b & b c a b & b c & c a \end{matrix} ∣ ∣ ∣ ∣ = 0$ , then

\frac{1}{a} + \frac{1}{b ω} + \frac{1}{c ω^{2}} = 0

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\frac{1}{a} + \frac{1}{b ω^{2}} + \frac{1}{c ω} = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\frac{1}{a ω} + \frac{1}{b ω^{2}} + \frac{1}{c} = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

none of these

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Solution

The correct options are
A $\frac{1}{a} + \frac{1}{b ω} + \frac{1}{c ω^{2}} = 0$
C $\frac{1}{a} + \frac{1}{b ω^{2}} + \frac{1}{c ω} = 0$
D $\frac{1}{a ω} + \frac{1}{b ω^{2}} + \frac{1}{c} = 0$
$∣ ∣ ∣ ∣ \begin{matrix} b c & c a & a b c a & a b & b c a b & b c & c a \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow {(a b)}^{3} + {(b c)}^{3} + {(c a)}^{3} - 3 (a b) (b c) (c a) = 0 \Rightarrow (a b + b c ω^{2} + c a ω) (a b ω + b c ω^{2} + c a) (a b ω^{2} + b c ω + c a) = 0$
$\Rightarrow a b + b c ω^{2} + c a ω = 0$ or $a b ω + b c ω^{2} + c a = 0$ or $a b ω^{2} + b c ω + c a = 0$
$\Rightarrow \frac{1}{a} + \frac{1}{b ω} + \frac{1}{c ω^{2}} = 0$ or $\frac{1}{a} + \frac{1}{b ω^{2}} + \frac{1}{c ω} = 0$ or $\frac{1}{a ω} + \frac{1}{b ω^{2}} + \frac{1}{c} = 0$

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If a,b,c are nonzero real numbers such that ∣∣ ∣∣bccaabcaabbcabbcca∣∣ ∣∣=0, then

If $a, b, c$ are nonzero real numbers such that $∣ ∣ ∣ ∣ \begin{matrix} b c & c a & a b c a & a b & b c a b & b c & c a \end{matrix} ∣ ∣ ∣ ∣ = 0$ , then