Animated Solution for Mathematics - Matrices and Determinants: For the system of linear equations $2x+4y+2az=b$, $x+2y+3z=4$, $2x+5y+2z=8$ which of the following is NOT correct?

Imagine you are standing in a vast, three-dimensional room. You have three sheets of glass, each representing one of our linear equations: $P_1:2x+4y+2az=b$ $P_2:x+2y+3z=4$ $P_3:2x+5y+2z=8$

These are planes. The question of whether this system has a solution is simply a question of whether these three planes share a common point of intersection.

If they meet at a single point, we have a unique solution. If they meet along a line, we have infinitely many. If they never meet, we have no solution. This is the physical reality behind the algebra.

The Determinant $\Delta$

To unlock the secrets of these planes, we need the master key: the coefficient determinant, $\Delta$. This value tells us if the system is 'well-behaved' (unique solution) or 'singular' (potentially problematic).

We define it as:

Let's expand this along the first row. It is a dance of arithmetic:

Simplifying this, we get:

This simplifies beautifully to:

Factoring out the two, we find $\Delta =2(a-3)$. This is our critical threshold. If $a eq 3$, then $\Delta eq 0$, and the system is guaranteed to have a unique solution. This immediately validates Option 1 ($a=6,b=6$) and Option 4 ($a=8,b=8$).

When $a=3$

Now, what happens when $a=3$? The determinant $\Delta$ becomes zero. The planes are no longer intersecting at a single point.

They are either parallel, or they intersect along a line, or they form a triangular prism shape with no common intersection. To distinguish between 'Infinite Solutions' and 'No Solution', we must look at the auxiliary determinants.

Let's calculate ${\Delta }_z$ by replacing the $z$-coefficients with the constants $b,4,8$:

Expanding this along the first row:

We are now armed with the truth. For the system to have infinitely many solutions, we need $\Delta =0$ and ${\Delta }_z=0$.

If $a=3$ and $b=8$, then $\Delta =0$ and ${\Delta }_z=0$. This confirms Option 3 is correct.

However, look at Option 2: $a=3$ and $b=6$. Here, $\Delta =0$, but ${\Delta }_z=6-8=-2$.

Since $\Delta_z eq 0$, the system is inconsistent. It has NO solution. Therefore, the claim in Option 2 that it has 'infinitely many solutions' is mathematically false.

We have successfully navigated the geometry of the planes and identified the trap. Remember, in JEE Advanced, the determinant is not just a number; it is the heartbeat of the system.