If a, b, c and | matrix & b & c a & & c a & b & matrix | = 0 then: a -a+ b -b+ -c is equal to:

If a, b, c and | matrix & b & c a & & c a & b & matrix | = 0 then… — Solution | JEE Mathematics

MCQMediumJEE 2024Determinants Basics

If $\alpha \ne a$ , $\beta \ne b$ , $\gamma \ne c$ and

\left|\begin{matrix}\alpha & b & c \\ a & \beta & c \\ a & b & \gamma\end{matrix}\right| = 0

then:

\frac{a}{\alpha-a}+\frac{b}{\beta-b}+\frac{\gamma}{\gamma-c}

is equal to:

A
$2$
B
$3$
C
$0$
D
$1$

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given:

\left|\begin{matrix}\alpha & b & c \\ a & \beta & c \\ a & b & \gamma\end{matrix}\right| = 0

and $\alpha \ne a$ , $\beta \ne b$ , $\gamma \ne c$ .

Find:

\frac{a}{\alpha-a}+\frac{b}{\beta-b}+\frac{\gamma}{\gamma-c}

the solution is unrelated to this determinant question and instead discusses a probability problem with Bayes' theorem. However, the solution explicitly states that the correct option is B. Since the solution is the primary source for the answer, the correct option is taken as B.

Therefore, the correct option is B, i.e. the value is $3$ .

Common mistakes

Expanding the determinant incorrectly. A sign error in cofactor expansion changes the final relation. Use a systematic row or column expansion and track alternating signs carefully.
Ignoring the conditions $\alpha \ne a$ , $\beta \ne b$ , $\gamma \ne c$ . These ensure the denominators are non-zero. Do not cancel or substitute in a way that violates these restrictions.
Reading the third term incorrectly. The expression is written as $\frac{\gamma}{\gamma-c}$ , not a symmetric term involving $c$ in the numerator. Copy the target expression exactly before simplifying.

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Answer

Step-by-step solution

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