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GATE 2022 BT – Question 2

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 1 mark · Multiple choice

Two straight lines pass through $(0,0)$ and respectively $(1,3)$ and $(1,2)$. What is the area enclosed between them for $x\in[0,1]$?

  1. 0.5
  2. 1.0
  3. 1.5
  4. 2.0

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Show answer and explanation

Correct answer: (A) 0.5

Explanation

**Step 1: equations of the lines.** A line through the origin and $(1,3)$ has slope 3, so $y=3x$. A line through the origin and $(1,2)$ has slope 2, so $y=2x$.

**Step 2: area between them** for $x\in[0,1]$. The line $y=3x$ is above $y=2x$ for $x>0$:
$$A=\int_0^1(3x-2x)\,dx=\int_0^1x\,dx=\left[\frac{x^2}{2}\right]_0^1=\mathbf{0.5}.$$

(Check with a triangle: the vertical side at $x=1$ has length $3-2=1$ and the base is 1, so the area is $\tfrac12\cdot1\cdot1=0.5$.)