JEE Main 2022 · 27 Jun · Shift 1 · Mathematics Q22

← JEE Main 2022 · 27 Jun · Shift 1

MathematicsInteger+4 / -1Question 22 of 90

If the sum of all the roots of the equation $e^{2x}-11e^x-45e^{-x}+\dfrac{81}2=0$ is $\log_ep$ , then $p$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 45

Cubic in $t=e^x$ . Multiplying by $e^x$ : $2t^3-22t^2+81t-90=0$ .

Keyed reasoning. Treating all three roots, $t_1t_2t_3=45$ , so $\sum x=\log_e45$ and $p=45$ .

Note. $t=2$ is a root and the other factor $2t^2-18t+45$ has no real roots, so the only real root is $x=\log_e2$ . The key's 45 counts the complex roots too.

Answer: 45

Topics:Theory of equations (higher degree)Logarithms

← Previous question Next question →

JEE Main 2022 · 27 Jun · Shift 1, Mathematics question 22