Question

Assertion :If $a,b,c$ are in Arithmetic Progression, then the system of equations:
$3x+4y+5z=a$ ------- (1)
$4x+5y+6z=b$ -------(2)
$5x+6y+7z=c$ ------(3) are consistent.
Reason: If $|A|\ne 0$, the system of equations $AX=B$ is consistent.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

If $|A|\ne 0$, A is invertible and we can write $AX=B$ as $X=A^{-1}B$.
$\therefore AX=B$ has a unique solution and hence is consistent.
Subtracting (2) from (3) and (1) from (2), we get the system of equation as
$3x+4y+5z=a$ -------(4)
$x+y+z=b-a$ -------(5)
$x+y+z=c-b$ -------(6)
As $a,b,c$ are in A.P. $b-a=c-b$
$\therefore$ the last two equations are identical.
From (4) and (5) we obtain
$x=4b-5a+k$
$y=4a-3b-2k$
$z=k$
where $k$ is an arbitrary complex number. Thus, the system of equations in statement-1 is consistent.
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.