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CAT 2019 Slot 1 — QA Question 6

BasicsEasy

Let S be the set of all points (x, y) in the x-y plane such that |x| + |y| ≤ 2 and |x| ≥ 1. Then, the area, in square units, of the region represented by S equals

Answer & solution

Correct answer: 2

Answer: 2

Solution

Easy

∣x∣+∣y∣≤2|x|+|y|\le 2 is a square (a "diamond") with vertices on the axes at distance 22. The extra condition ∣x∣≥1|x|\ge 1 keeps only the parts with x≥1x\ge 1 or x≤−1x\le -1, slicing off the central band. By symmetry the leftover is two congruent triangles; add their areas.

(2,0) x=1
1

Describe the right piece (x≥1x\ge 1). On the line x=1x=1, ∣y∣≤2−1=1|y|\le 2-1=1, so the boundary runs from (1,1)(1,1) down to (1,−1)(1,-1). The diamond's right vertex is (2,0)(2,0). These three points form a triangle.

vertices: (1,1), (1,−1), (2,0)\begin{aligned} &\text{vertices: }(1,1),\ (1,-1),\ (2,0) \end{aligned}
2

Area of one triangle. Base is the vertical segment from (1,1)(1,1) to (1,−1)(1,-1), length 22; height is the horizontal distance from x=1x=1 to x=2x=2, which is 11.

Aright=12×2×1=1\begin{aligned} &A_{\text{right}} = \frac12\times 2\times 1 = 1 \end{aligned}
3

Add the symmetric left piece. The x≤−1x\le -1 region is the mirror image, with the same area.

A=Aright+Aleft=1+1=2\begin{aligned} &A = A_{\text{right}} + A_{\text{left}} = 1 + 1 = 2 \end{aligned}
Area=2 square units\text{Area} = 2 \text{ square units}

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CAT 2019 Slot 1 QA Q6: Let S be the set of all points (x, y) in the x-y plane such that |x| + |y| ≤ 2 and |x| ≥ 1. Then, the ar — Solution | TheCATExam