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UPSC NDA II 2018 Mathematics Solved Paper

Question : 21 of 120

Marks: +1, -0

2x + y - 3z = 5

z = 5

5x - 3y - z = 16

Solution:

Concept:

For a system of linear equations

AX = B

, if

\det(A) \neq 0

, the system is consistent and has a unique solution.

Explanation:

Write the given equations in matrix form

AX = B

Coefficient matrix

A = \begin{bmatrix} 2 & 1 & -3 \\ 3 & -2 & 2 \\ 5 & -3 & -1 \end{bmatrix}

Calculate determinant

|A|

using expansion along first row.

|A| = 2[(-2)(-1) - (2)(-3)] - 1[(3)(-1) - (2)(5)] + (-3)[(3)(-3) - (5)(-2)]

= 2[(2) - (-6)] - 1[(-3) - (10)] - 3[(-9) - (-10)]

= 2(2+6) - 1(-3-10) - 3(-9+10)

= 2(8) - 1(-13) - 3(1)

= 16 + 13 - 3 = 26

Since

|A| = 26 \neq 0

, the system is consistent with a unique solution.

Answer:

Option B: is consistent, with a unique solution.

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