JEE Mathematics 2023 Question with Solution

Given: The system is

$$x - y + z = 5$$ $$2x + 2y + \alpha z = 8$$ $$3x - y + 4z = \beta$$

Find: The condition on $\alpha$ and $\beta$ corresponding to the correct option shown in the solution.

For infinitely many solutions, the determinant of the coefficient matrix must be zero:

$$\begin{vmatrix} 1 & -1 & 1 \\ 2 & 2 & \alpha \\ 3 & -1 & 4 \end{vmatrix} = 0$$

Expanding as given in the solution,

$$8 + \alpha - 3\alpha - 6 = 0$$ $$\Rightarrow \alpha = 1$$

Now substitute the obtained value into the relation stated in the solution for $\beta$. The working states the final value as $\beta = 56$.

Thus, the resulting quadratic indicated by the solution corresponds to option A. The solution and the listed options are inconsistent, because the displayed option text for A is $x^2 - 10x + 16 = 0$ while the answer key says option $3$.

Given: the solution explicitly says The Correct Option is A.

It also states:

• determinant condition is used,
• a value of $\alpha$ is obtained from

$$8 + \alpha - 3\alpha - 6 = 0$$

• and then $\beta = 56$ is written.

However, solving the displayed equation gives

$$2 - 2\alpha = 0 \Rightarrow \alpha = 1$$

not $\alpha = 4$ as also written in the working. Therefore, the source solution itself contains inconsistencies.

Following the instruction that the solution is the primary source, the answer is taken as A because that is the explicit conclusion printed on the solution.