GATE AE 2022 • Q3 MCQ • 1 Mark General Aptitude

General Aptitude

Quantitative Aptitude

Subtopic: Arithmetic & Numerical Estimation

Question Statement:

If $f(x) = 2 \ln\left(\sqrt{e^x}\right)$, what is the area bounded by $f(x)$ for the interval $[0, 2]$ on the $x$-axis?
Question figure for GATE AE 2022 Q3

Scientific schematic diagram reproduced from official IIT paper

Options:

  • (A) $\frac{1}{2}$
  • (B) 1
  • (C) 2
  • (D) 4

Official Answer Key:

C

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2022 Q.3)** **1. Simplification of $f(x)$**: Using properties of logarithms and powers: $$\sqrt{e^x} = e^{x/2}$$ $$f(x) = 2 \ln\left(e^{x/2}\right) = 2 \times \frac{x}{2} = x$$ **2. Area Integration**: The area bounded by $f(x) = x$ over $[0, 2]$ is: $$\text{Area} = \int_0^2 x \, dx = \left[ \frac{x^2}{2} \right]_0^2 = \frac{4}{2} - 0 = 2$$ - **Official Answer Key**: **C**
← Previous (Q2) All 2022 Questions Next (Q4) →

Related Quantitative Aptitude Solved Questions

View All Quantitative Aptitude PYQs →
GATE AE 2026 Q2 MCQ • 1M

In the sequence of tiles shown below, the missing tile indicated by the question mark should be

GATE AE 2026 Q3 MCQ • 1M

A school has 100 students distributed among 1st to 10th standards. Based on this, which one of the following statements is always correct?

GATE AE 2026 Q4 MCQ • 1M

How many 3-digit numbers can be formed using three distinct single digit prime numbers?

GATE AE 2026 Q5 MCQ • 1M

In a group of students, 10 students like Mathematics, 12 students like English, 4 students like both Mathematics and English, and 6 students like neither Mathematics nor English. The number of students in the group is ____

Practice This in Real CBT Exam Interface

Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.

Launch CBT Mock Test