For the system of equations x + y + z = 6 x + 2y + z = 10 x + 3y + 5z = which one of the following is NOT true:

Given: The system of equations is aligned x + y + z &= 6 x + 2y + z &= 10 x + 3y + 5z &= aligned Find: Which statement is not true. Using the coefficient matrix shown in the solution, pmatrix 1 & 1 & 1 1 & 2 & 1 1 & 3 & 5 pmatrix we analyze the determinant to determine the nature of the solution. The extracted working states: = | matrix 1 & 1 & 1 1 & 2 & 1 1 & 3 & 5 matrix | and then gives = 6(10 - 3) - (50 - 13) + (30 - 2) = 40 - 18 + - 2 From the solution, the stated conclusions are: - For infinite solutions, = 0 and x = y = z = 0. - This gives = 3 and = 14. - For a unique solution, 3. Now compare these statements with the options: 1. Unique solution for = 3, 14 2. Unique solution for = -3, = 14 3. No solution for = 3, = 24 4. Infinitely many solutions for = 3, = 14 Since the extracted solution says unique solution occurs for 3, any option claiming a unique solution at = 3 must be false. Option A says the system has a unique solution for = 3, 14, which contradicts the stated condition. Therefore, the statement that is not true is A. Therefore, the correct option is A.

For the system of equations x + y + z = 6 x + 2y + z = 10 x + 3y… — Solution | JEE Mathematics

MCQMediumJEE 2023Solving Linear Equations (Matrix Method)

For the system of equations

$x + y + z = 6$

$x + 2y + z = 10$

$x + 3y + 5z = \beta$

which one of the following is NOT true:

A
System has a unique solution for $\alpha = 3, \beta \neq 14$
B
System has a unique solution for $\alpha = -3, \beta = 14$
C
System has no solution for $\alpha = 3, \beta = 24$
D
System has infinitely many solutions for $\alpha = 3, \beta = 14$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: The system of equations is

\begin{aligned} x + y + z &= 6 \\ x + 2y + z &= 10 \\ x + 3y + 5z &= \beta \end{aligned}

Find: Which statement is not true.

Using the coefficient matrix shown in the solution,

\begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 3 & 5 \end{pmatrix}

we analyze the determinant to determine the nature of the solution.

The extracted working states:

\Delta = \left| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 3 & 5 \end{matrix} \right|

and then gives

\Delta = 6(10 - 3\alpha) - (50 - \alpha 13) + (30 - 2\beta)

= 40 - 18\alpha + \alpha\beta - 2\beta

Conclusion from the extracted determinant conditions

From the solution, the stated conclusions are:

For infinite solutions, $\Delta = 0$ and $\Delta x = \Delta y = \Delta z = 0$ .
This gives $\alpha = 3$ and $\beta = 14$ .
For a unique solution, $\alpha \neq 3$ .

Now compare these statements with the options:

Unique solution for $\alpha = 3, \beta \neq 14$
Unique solution for $\alpha = -3, \beta = 14$
No solution for $\alpha = 3, \beta = 24$
Infinitely many solutions for $\alpha = 3, \beta = 14$

Direct option check

Since the extracted solution says unique solution occurs for $\alpha \neq 3$ , any option claiming a unique solution at $\alpha = 3$ must be false.

Option A says the system has a unique solution for $\alpha = 3, \beta \neq 14$ , which contradicts the stated condition. Therefore, the statement that is not true is A.

Therefore, the correct option is A.

Common mistakes

Assuming that only $\beta$ determines the nature of the system. This is wrong because the extracted solution states the key condition for uniqueness in terms of $\alpha$ . Check the determinant condition first.
Confusing the cases of no solution and infinitely many solutions when the determinant becomes zero. A zero determinant alone is not enough; one must also examine the corresponding numerator determinants or consistency conditions.
Reading the question as asking which statement is true instead of which statement is NOT true. This reverses the final choice, so identify the required logical direction before evaluating options.

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Answer

Step-by-step solution

Common mistakes

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