Take a=1,b=2,c=3

Det will come as 2

Only b will be factor(1-2=-1)

Det will come as 2

Only b will be factor(1-2=-1)

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+18 votes

Consider the following determinant $\Delta = \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix}$

Which of the following is a factor of $\Delta$?

- $a+b$
- $a-b$
- $a+b+c$
- $abc$

+1

take some random example with a b and c values.

Solve determinant

use options to see what can be factor

Solve determinant

use options to see what can be factor

+6

just small addition to your approach , whenever you take numbers to solve question vy hit and trial , first check if all options yield different results so that you don't need to do backtrack with some other set of numbers. here for 1,2,3, both c and d gives 6 , better to select 2,3,4 for this question

+22 votes

Best answer

$\begin{vmatrix} 1 &a &bc \\ 1& b& ca\\ 1& c& ab \end{vmatrix}$

$\begin{vmatrix} 1 &a &bc \\ 0& b-a& ca-bc\\ 0& c-a& ab-bc \end{vmatrix} \ \ \ \ \ \ \ \ R_{2}\rightarrow R_{2}- R_{1} \ \ , \ R_{3}\rightarrow R_{3}- R_{1}$

$(b-a)( ab-bc) - (c-a)(ca-bc)$

$-(a-b) \ \ b \ \ ( a-c) + (a-c)\ \ c \ \ (a-b)$

$(a-b)( a-c) (c-b)$

Option$(B)a-b$ is the correct choice.

$\begin{vmatrix} 1 &a &bc \\ 0& b-a& ca-bc\\ 0& c-a& ab-bc \end{vmatrix} \ \ \ \ \ \ \ \ R_{2}\rightarrow R_{2}- R_{1} \ \ , \ R_{3}\rightarrow R_{3}- R_{1}$

$(b-a)( ab-bc) - (c-a)(ca-bc)$

$-(a-b) \ \ b \ \ ( a-c) + (a-c)\ \ c \ \ (a-b)$

$(a-b)( a-c) (c-b)$

Option$(B)a-b$ is the correct choice.

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