JEE Advanced 2022 · Paper 2 · Mathematics Q4

MathematicsInteger+3 / -1Question 4 of 54

The product of all positive real values of $x$ satisfying the equation
$x^{\left(16(\log_5 x)^3 - 68\log_5 x\right)} = 5^{-16}$
is _____.

Your answer

Answer and worked solution

Official answer: 1

Take $\log_5$ . Let $t=\log_5x$ . Then
$(16t^3-68t)\,t=-16\implies4t^4-17t^2+4=0\implies(4t^2-1)(t^2-4)=0.$

Roots. $t=\pm\frac12,\ \pm2$ , all real, so there are four positive $x=5^t$ .

Product.
$\prod x=5^{\sum t}=5^0=1.$

JEE Advanced 2022 · Paper 2, Mathematics question 4