CAT 2025 Slot 3 QA Question 22

Multiple choice (+3 / −1) · Algebra · Identities · Try it, then check the answer and solution below.

If \left(x^2+

\frac{1}{x^2}

\right)=25

and

x>0 , then the value of \left(x^7+

\frac{1}{x^7}

\right)$ is

⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution

Official Convenor Answer Key: D

Answer and solution

Answer: D) $44853\sqrt{3}$

📌 Core Concept & Formula

We use the identity for powers of sums:

\left(x + \frac{1}{x}\right)^n = x^n + \frac{1}{x^n} + \text{terms involving lower powers}

🔢 Step-by-Step Solution

Given:

x^2 + \frac{1}{x^2} = 25

Find

x +

\frac{1}{x}$ :

Let

y = x + \frac{1}{x}

y^2 = x^2 + 2 + \frac{1}{x^2} = 25 + 2 = 27

\( y = \

\sqrt{27}

= 3

Keep going