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JEE Main 20248 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
8 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Determinants

JEE Main 2024 · 8 April · Evening Shift · Mathematics · Determinants

If α ≠ a, β ≠ b, γ ≠ c and | array lll α & b & c a & β & c a & b &...

If \alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c} and \left|αbcaβcabγ\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0, then aα−a+bβ−b+γγ−c\frac{\mathrm{a}}{\alpha-\mathrm{a}}+\frac{\mathrm{b}}{\beta-\mathrm{b}}+\frac{\gamma}{\gamma-\mathrm{c}} is equal to:

Options

  1. A3
  2. B0
  3. C1
  4. D2

Correct answer

B. 0

Step-by-step solution

R1→R1−R2,R2→R2−R3∣α−ab−β00β−bc−γabγ∣=0(α−a)(γ(β−b)−b(c−γ))−(b−β)(−a(c−γ))=0γ(α−a)(β−b)−b(α−a)(c−γ)+a(b−β)(c−γ)γγ−c+bβ−b+aα−a=0\begin{aligned} & \mathrm{R}_1 \rightarrow \mathrm{R}_1-\mathrm{R}_2, \mathrm{R}_2 \rightarrow \mathrm{R}_2-\mathrm{R}_3 \\ & \left|\begin{array}{ccc}\alpha-\mathrm{a} & \mathrm{b}-\beta & 0 \\ 0 & \beta-\mathrm{b} & \mathrm{c}-\gamma \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0 \\ & (\alpha-\mathrm{a})(\gamma(\beta-\mathrm{b})-\mathrm{b}(\mathrm{c}-\gamma))-(\mathrm{b}-\beta)(-\mathrm{a}(\mathrm{c}-\gamma))=0 \\ & \gamma(\alpha-\mathrm{a})(\beta-\mathrm{b})-\mathrm{b}(\alpha-\mathrm{a})(\mathrm{c}-\gamma)+\mathrm{a}(\mathrm{b}-\beta)(\mathrm{c}-\gamma) \\ & \frac{\gamma}{\gamma-\mathrm{c}}+\frac{\mathrm{b}}{\beta-\mathrm{b}}+\frac{\mathrm{a}}{\alpha-\mathrm{a}}=0\end{aligned}
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