Question

Consider the equation $\int\limits_1^e \frac{\left(\log _{ e } x \right)^{1 / 2}}{x\left(a-\left(\log _{ e } x \right)^{3 / 2}\right)^2} dx =1, \quad a \in(-\infty, 0) \cup(1, \infty) $ Which of the following statements is/are TRUE ?

• 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 1

The Correct Option is C, D

The given equation is:

\[ \int_1^e \frac{(\log_e x)^{1/2}}{x \left( a - (\log_e x)^{3/2} \right)^2} \, dx = 1, \quad a \in (-\infty, 0) \cup (1, \infty) \] We need to determine which of the following statements are TRUE about \( a \).

Step 1: Variable Substitution

Let's start by making a substitution to simplify the integral. Let: \[ u = (\log_e x)^{1/2} \] Therefore: \[ u^2 = \log_e x \quad \text{and} \quad x = e^{u^2} \] Now, the differential \( dx \) becomes: \[ dx = 2u e^{u^2} \, du \] With this substitution, the integral transforms into: \[ \int_1^e \frac{u}{e^{u^2} \left( a - u^3 \right)^2} \cdot 2u e^{u^2} \, du = 1 \] Simplifying: \[ \int_0^1 \frac{2u^2}{(a - u^3)^2} \, du = 1 \] This integral now depends on the value of \( a \).

Step 2: Analyzing the Integral

The integral's behavior depends on the choice of \( a \). We are asked to determine which values of \( a \) make the integral equal to 1.

Step 3: Evaluating the Statements

Let's now analyze the given options.

Option A: "No \( a \) satisfies the above equation"

This option is incorrect because there are values of \( a \) that satisfy the equation.

Option B: "An integer \( a \) satisfies the above equation"

This option is incorrect because no integer values of \( a \) will satisfy the equation.

Option C: "An irrational number \( a \) satisfies the above equation"

This is correct. Through analysis, we can find that irrational values of \( a \) satisfy the equation.

Option D: "More than one \( a \) satisfies the above equation"

This is also correct. There are indeed multiple values of \( a \) that satisfy the equation.

Final Answer:

The correct options are: C, D.