JEE Advanced 2024 · Paper 1 · Mathematics Q8

← JEE Advanced 2024 · Paper 1

MathematicsInteger+4Question 8 of 51

Let $a = 3\sqrt{2}$ and $b = \dfrac{1}{5^{1/6}\sqrt{6}}$ . If $x, y \in \mathbb{R}$ are such that

$3x + 2y = \log_a (18)^{\frac{5}{4}} \quad\text{and}$
$2x - y = \log_b\left(\sqrt{1080}\right),$

then $4x + 5y$ is equal to _____.

Your answer

Answer and worked solution

Official answer: 8

First equation. $a = 3\sqrt2 = 18^{1/2}$ , so $\log_a 18^{5/4} = \dfrac{5/4}{1/2} = \dfrac52$ . Hence $3x + 2y = \dfrac52$ .

Second equation. $1080 = 2^3\cdot3^3\cdot5$ , so $\sqrt{1080} = 6^{3/2}\,5^{1/2}$ . Also $b = 5^{-1/6}\,6^{-1/2}$ , so $b^{-3} = 5^{1/2}\,6^{3/2} = \sqrt{1080}$ . Hence $2x - y = -3$ .

Solve. $y = 2x + 3$ , so $3x + 4x + 6 = \frac52$ , giving $x = -\frac12$ and $y = 2$ .

$4x + 5y = -2 + 10 = 8$ .

Topics:Logarithms

← Previous question Next question →

JEE Advanced 2024 · Paper 1, Mathematics question 8