Let \(\mathrm{a}=3 \sqrt{2}\) and \(\mathrm{b}=\frac{1}{5^{1 / 6} \sqrt{6}}\). If \(\mathrm{x}, \mathrm{y} \in \mathbb{R}\) are such that \[ \begin{aligned} & 3 x+2 y=\log _{a}(18)^{\frac{5}{4}} \text { and } \\ & 2 x-y=\log _{b}(\sqrt{1080}) \end{aligned} \] then \(4x + 5y\) equals .........

JEE Advanced · 2024 · 26 May 2024 · Paper 1 · Basic Mathematics

Let

a = 32

and

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

. If

x, y \in R

are such that

3 x + 2 y = lo g_{a} (18)^{\frac{5}{4}} and 2 x - y = lo g_{b} (1080)

then

4 x + 5 y

equals .........

Answer: 8

First equation:

3 x + 2 y = lo g_{32} (18)^{5/4} = lo g_{32} (2 \cdot 3^{2})^{5/4} = lo g_{32} (32)^{5/2} = \frac{5}{2}

6 x + 4 y = 5

... (1)

Second equation:

2 x - y = lo g_{\frac{1}{5 ^{1/6} 6}} 1080 = lo g_{\frac{1}{5 ^{1/6} 6}} (5^{1/6} 6)^{3} = - 3

2 x - y = - 3

... (2)

Multiplying (2) by 2:

4 x - 2 y = - 6

... (3)

Subtracting (3) from (1):

(6 x + 4 y) - (4 x - 2 y) = 5 - (- 6)

2 x + 6 y = 11

From (2):

y = 2 x + 3

Substituting in

2 x + 6 y = 11

2 x + 6 (2 x + 3) = 11

14 x = - 7 \Rightarrow x = - \frac{1}{2}

y = 2 (- \frac{1}{2}) + 3 = 2

Therefore:

4 x + 5 y = 4 (- \frac{1}{2}) + 5 (2) = - 2 + 10 = 8

JEE Advanced · 2024 · 26 May 2024 · Paper 1 · Integer answer