QJEEAdv24Mad23b: JEE Advanced 2024 Mathematics - Class 11 JEEAdv24Mad23b Solution: Let and . If are such that 3 x + 2 y = log a ⁡ ( 18 ) 5 4…

Given:

$a = 3\sqrt{2} \Rightarrow a^2 = 18$

Simplification of $\log_a(18)^{\frac{5}{4}}$

Since $a^2 = 18$ , we have:

$\log_a(18) = \log_a(a^2) = 2$

Therefore:

$\log_a(18)^{\frac{5}{4}} = \frac{5}{4} \log_a(18) = \frac{5}{4} \cdot 2 = \frac{5}{2}$

So the equation becomes:

$3x + 2y = \frac{5}{2}$

Simplification of $\log_b(\sqrt{1080})$

Given:

$1080 = 2^3 \cdot 3^3 \cdot 5 = 6^3 \cdot 5$

$b = \frac{1}{5^{1/6} \sqrt{6}}$

$\Rightarrow \frac{1}{b} = 5^{1/6} \sqrt{6}$

$\Rightarrow 1080^{1/6} = 5^{1/6} \cdot 6^{1/2} = \frac{1}{b}$

Taking the square root of both sides:

$\sqrt{1080} = \frac{1}{b^3}$

Thus:

$\log_b(\sqrt{1080}) = \log_b\left(\frac{1}{b^3}\right) = \log_b(b^{-3}) = -3$

So the second equation becomes:

$2x - y = -3$

Solving the System of Equations

Now we have:

$3x + 2y = \frac{5}{2}$

$2x - y = -3$

Multiply the second equation by 2:

$4x - 2y = -6$

Add this to the first equation:

$3x + 2y + 4x - 2y = \frac{5}{2} - 6$

$7x = \frac{5}{2} - 6$

$7x = \frac{5}{2} - \frac{12}{2}$

$7x = \frac{5 - 12}{2}$

$7x = -\frac{7}{2}$

$x = -\frac{1}{2}$

Substitute $x$ back into $2x - y = -3$ :

$2\left(-\frac{1}{2}\right) - y = -3$

$-1 - y = -3$

$-y = -2$

$y = 2$

Finding $4x + 5y$

$4x + 5y = 4 \left(-\frac{1}{2}\right) + 5 \cdot 2$

$= -2 + 10$

$= 8$

Thus, the value of $4x + 5y$ is $\boxed{8}$ .