Question
Consider the equation ∫ 1 e log e x 1 2 x a - log e x 3 2 2 d x = 1 , a ∈ - ∞ , 0 ∪ 1 , ∞ . Which of the following statements is/are TRUE?
JEE Advanced
Mathematics
Definite Integration
2022
JEE Advanced 2022 (Paper 1)
JEE Advanced Mathematics Question (2022) — Solution
Options
- A. No a satisfies the above equation
- B. An integer a satisfies the above equation
- C. An irrational number a satisfies the above equation
- D. More than one a satisfy the above equation
Answer
D. More than one a satisfy the above equation
Step-by-step solution
Let I = ∫ 1 e ln x 1 2 d x x a - ln x 3 2 2 Now put a - ln x 3 2 = t ⇒ - 3 2 ln x 1 2 · 1 x d x = d t So, the integral becomes I = ∫ a a - 1 - 2 3 d t t 2 ⇒ I = - 2 3 t - 2 + 1 - 2 + 1 a a - 1 ⇒ I = 2 3 t a a - 1 ⇒ I = 2 3 1 a - 1 - 1 a Now given I = 2 3 1 a a - 1 = 1 ⇒ 2 = 3 a 2 - 3 a ⇒ 3 a 2 - 3 a - 2 = 0 ⇒   a = 3 ± 9 - 4 3 - 2 6 So, a = 3 + 33 6 , 3 - 33 6
Related: Mathematics — Definite Integration · All PYQ Banks