JAMIA 2023 — Mathematics PYQ

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Question

JAMIA 2023 — Mathematics PYQ

JAMIA | Mathematics | 2023

The system of linear equations
a+2b+3c=7
2a+4b+c=12
3a+ 6b+4c=20

Choose the correct answer:

Explanation

1. Given System of Equations

a + 2b + 3c = 7 \quad \dots (1)

2a + 4b + c = 12 \quad \dots (2)

3a + 6b + 4c = 20 \quad \dots (3)

2. Matrix Representation

We can write the system as $AX = B$ :

\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 1 \\ 3 & 6 & 4 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} 7 \\ 12 \\ 20 \end{bmatrix}

3. Row Operations (Gaussian Elimination)

We will perform operations on the Augmented Matrix $[A|B]$ :

[A|B] = \begin{bmatrix} 1 & 2 & 3 & | & 7 \\ 2 & 4 & 1 & | & 12 \\ 3 & 6 & 4 & | & 20 \end{bmatrix}

Perform $R_2 \to R_2 - 2R_1$ and $R_3 \to R_3 - 3R_1$ :

\begin{bmatrix} 1 & 2 & 3 & | & 7 \\ 0 & 0 & -5 & | & -2 \\ 0 & 0 & -5 & | & -1 \end{bmatrix}

Perform $R_3 \to R_3 - R_2$ :

\begin{bmatrix} 1 & 2 & 3 & | & 7 \\ 0 & 0 & -5 & | & -2 \\ 0 & 0 & 0 & | & 1 \end{bmatrix}

4. Analyzing the Result

Look at the third row:

0a + 0b + 0c = 1

0 = 1

This is a contradiction. Mathematically, this means that the planes represented by these equations do not have a common point of intersection.

5. Final Conclusion

Since we arrived at an impossible statement ( $0 = 1$ ), the system of equations is Inconsistent.

Result:

The system has No Solution.