JEE Mathematics 2024 Question with Solution

Given:

$$3x + 5y + \lambda z = 3$$ $$7x + 11y - 9z = 2$$ $$97x + 155y - 189z = \mu$$

Find: $\mu + 2\lambda$ when the system has infinitely many solutions.

For infinitely many solutions, by Cramer's Rule, we need

$$\Delta = \Delta_1 = \Delta_2 = \Delta_3 = 0$$

First, use $\Delta = 0$:

$$\begin{vmatrix} 3 & 5 & \lambda \\ 7 & 11 & -9 \\ 97 & 155 & -189 \end{vmatrix} = 0$$

Perform $R_3 \to R_3 - 14R_2$:

$$\begin{vmatrix} 3 & 5 & \lambda \\ 7 & 11 & -9 \\ -1 & -1 & -63 \end{vmatrix} = 0$$

Next, perform $C_1 \to C_1 + C_2$:

$$\begin{vmatrix} 4 & 5 & \lambda \\ 9 & 11 & -9 \\ 0 & -1 & -63 \end{vmatrix} = 0$$

Expanding along the third row:

$$-1(-36 - 9\lambda) - 63(44 - 45) = 0$$ $$36 + 9\lambda + 63 = 0$$ $$9\lambda = -99 \Rightarrow \lambda = -11$$

Now use $\Delta_3 = 0$:

$$\begin{vmatrix} 3 & 5 & 3 \\ 7 & 11 & 2 \\ 97 & 155 & \mu \end{vmatrix} = 0$$

Perform $C_2 \to C_2 - C_1$:

$$\begin{vmatrix} 3 & 2 & 3 \\ 7 & 4 & 2 \\ 97 & 58 & \mu \end{vmatrix} = 0$$

Next, $C_1 \to C_1 - C_3$:

$$\begin{vmatrix} 0 & 2 & 3 \\ 5 & 4 & 2 \\ 97 - \mu & 58 & \mu \end{vmatrix} = 0$$

Expanding along the first column:

$$-5(2\mu - 174) + (97 - \mu)(4 - 12) = 0$$ $$10\mu - 870 + 776 - 8\mu = 0$$ $$2\mu = 94 \Rightarrow \mu = 47$$

Finally,

$$\mu + 2\lambda = 47 - 22 = 25$$

Therefore, the correct option is A.

Given:

$$3x + 5y + \lambda z = 3$$ $$7x + 11y - 9z = 2$$ $$97x + 155y - 189z = \mu$$

Find: $\mu + 2\lambda$.

For infinitely many solutions, the third equation must be a linear combination of the first two. Multiply equation $\text{(1)}$ by $31$:

$$93x + 155y + 31\lambda z = 93$$

Subtract this from equation $\text{(3)}$:

$$4x - (31\lambda + 189)z = \mu - 93$$

Now compare with equation $\text{(2)}$,

$$7x + 11y - 9z = 2$$

so that the dependence condition forces the coefficients to remain consistent. Solving gives

$$\lambda = -11, \qquad \mu = 47$$

Hence,

$$\mu + 2\lambda = 47 + 2(-11) = 25$$

This works because infinitely many solutions require the coefficient rows and constant terms to satisfy the same dependence relation. Therefore, the correct option is A.