JEE Mathematics 2023 Question with Solution

Given:

$$x + 2y + 3z = 3$$ $$4x + 3y - 4z = 4$$ $$8x + 4y - \lambda z = 9 + \mu$$

Find: The ordered pair $(\lambda, \mu)$ for which the system has infinitely many solutions.

Eliminate $x$ from the first two equations:

$$4(i) - (ii) \Rightarrow 5y + 16z = 8 \quad \cdots (iv)$$

Now eliminate $x$ from the second and third equations:

$$2(ii) - (iii) \Rightarrow 2y + (\lambda - 8)z = -1 - \mu \quad \cdots (v)$$

Next, eliminate $y$ using equations $(iv)$ and $(v)$:

$$2(iv) - 5(v)$$ $$\Rightarrow \left(10y + 32z\right) - \left(10y + 5(\lambda - 8)z\right) = 16 - 5(-1 - \mu)$$ $$\Rightarrow \left(32 - 5(\lambda - 8)\right)z = 21 + 5\mu$$

For infinitely many solutions, this reduced equation must become an identity. Therefore both coefficient and constant term must be zero:

$$32 - 5(\lambda - 8) = 0$$ $$72 - 5\lambda = 0$$ $$\lambda = \frac{72}{5}$$

Also,

$$21 + 5\mu = 0$$ $$\mu = -\frac{21}{5}$$

Hence,

$$(\lambda, \mu) = \left(\frac{72}{5}, -\frac{21}{5}\right)$$

Therefore, the correct option is C. The solution incorrectly labels the option as A, but the derived ordered pair matches option C.

Given:

$$x + 2y + 3z = 3$$ $$4x + 3y - 4z = 4$$ $$8x + 4y - \lambda z = 9 + \mu$$

Find: Values of $\lambda$ and $\mu$ such that the system has infinitely many solutions.

From

$$4(x + 2y + 3z) - (4x + 3y - 4z) = 12 - 4$$

we get

$$5y + 16z = 8$$

From

$$2(4x + 3y - 4z) - (8x + 4y - \lambda z) = 8 - (9 + \mu)$$

we get

$$2y + (\lambda - 8)z = -1 - \mu$$

Multiply the second reduced equation by $5$ and the first by $2$:

$$10y + 5(\lambda - 8)z = -5 - 5\mu$$ $$10y + 32z = 16$$

Subtracting,

$$\left(10y + 32z\right) - \left(10y + 5(\lambda - 8)z\right) = 16 - (-5 - 5\mu)$$ $$\left(32 - 5\lambda + 40\right)z = 21 + 5\mu$$ $$(72 - 5\lambda)z = 21 + 5\mu$$

For infinitely many solutions, this must hold for all permissible values in the dependent system, so

$$72 - 5\lambda = 0, \qquad 21 + 5\mu = 0$$

Thus,

$$\lambda = \frac{72}{5}, \qquad \mu = -\frac{21}{5}$$

Therefore,

$$(\lambda, \mu) = \left(\frac{72}{5}, -\frac{21}{5}\right)$$

So the correct option is C.