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Question

Assertion :If a, b, c ϵR and abc and x, y, z are non zero. Then the system of equations ax+by+cz=0
bx+cy+az=0
cx+ay+bz=0 has infinite solutions. Reason: If the homogeneous system of equations has non trivial solution, then it has infinitely many solutions.

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

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
Assertion:
Given system of equations can be written as
AX=O
Where O is zero matrix .
Here, A=abcbcacab
Now, D=∣ ∣abcbcacab∣ ∣
D=3abca3b3c3
D=(a3+b3+c33abc)=0
Clearly, D1=∣ ∣0bc0ca0ab∣ ∣=0
Similarly, D2=D3=0
Hence, D=D1=D2=D3=0
Hence, the system has infinite many solution.
Reason is also correct . A homogeneous system of eqn having non-trivial solution has infinite many solutions

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