GATE 2022 PH – Question 2

Correct answer: (A) 0.5

Explanation

**Equations of the lines.** The line through the origin and $(1,3)$ has slope 3, so $y=3x$. The line through the origin and $(1,2)$ has slope 2, so $y=2x$.

**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 $\tfrac12\cdot1\cdot1=0.5$.)