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CAT 2017 Slot 1 — QA Question 15

BasicsEasy

The area of the closed region bounded by the equation | x | + | y | = 2 in the two-dimensional plane is

Answer & solution

Correct answer: 8

  • A

    4π

  • B

    4

  • 8

  • D

    2π

Solution

Easy

The equation ∣x∣+∣y∣=2|x|+|y|=2 describes a square (a tilted "diamond") whose vertices sit on the axes. Find the intercepts to get the diagonals, then use the rhombus/square area formula 12d1d2\tfrac12 d_1 d_2.

(0,2) (2,0) (0,-2) (-2,0)
1

Find the intercepts. Set y=0y=0, then x=0x=0:

∣x∣=2⇒x=±2(x-intercepts)∣y∣=2⇒y=±2(y-intercepts)\begin{aligned} &|x|=2\Rightarrow x=\pm 2 \quad\text{(x-intercepts)}\\ &|y|=2\Rightarrow y=\pm 2 \quad\text{(y-intercepts)} \end{aligned}
2

Identify the figure and its diagonals. The four vertices (±2,0),(0,±2)(\pm2,0),(0,\pm2) form a square with both diagonals along the axes.

d1=d2=2−(−2)=4\begin{aligned} &d_1=d_2=2-(-2)=4 \end{aligned}
3

Apply the diagonal area formula. For a square (rhombus) area =12d1d2=\tfrac12 d_1 d_2:

Area=12×4×4=8\begin{aligned} &\text{Area}=\tfrac12\times 4\times 4=8 \end{aligned}
Area=8 sq units\text{Area}=8\ \text{sq units}

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CAT 2017 Slot 1 QA Q15: The area of the closed region bounded by the equation | x | + | y | = 2 in the two-dimensional plane is — Solution | TheCATExam