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CAT 2021 Slot 3 — QA Question 16

ModulusEasy

If 3x + 2|y| + y = 7 and x + |x| + 3y = 1, then x + 2y is

Answer & solution

  • A

    8/3

  • B

    1

  • 0

  • D

    -4/3

Solution

Easy

The terms 2∣y∣2|y| and ∣x∣|x| change form by sign. Split into sign cases for xx and yy, solve the resulting linear system in each, and keep only the case whose solution matches its sign assumption.

1

Try x≥0, y≥0x\ge 0,\ y\ge 0. Then ∣x∣=x, ∣y∣=y|x|=x,\ |y|=y.

3x+3y=7,2x+3y=1⇒ x=6, y=−113(rejected: y<0)\begin{aligned} &3x+3y=7,\quad 2x+3y=1\\ &\Rightarrow\ x=6,\ y=-\tfrac{11}{3}\quad\text{(rejected: }y<0\text{)} \end{aligned}
2

Try x≥0, y<0x\ge 0,\ y<0. Then ∣x∣=x, ∣y∣=−y|x|=x,\ |y|=-y.

3x−y=7,2x+3y=1⇒ x=2, y=−1(accepted: signs match)\begin{aligned} &3x - y = 7,\quad 2x+3y=1\\ &\Rightarrow\ x=2,\ y=-1\quad\text{(accepted: signs match)} \end{aligned}
3

Remaining cases fail. For x<0,y≥0x<0,y\ge0: 3y=1, 3x+3y=7⇒x=2>03y=1,\,3x+3y=7\Rightarrow x=2>0 (rejected). For x<0,y<0x<0,y<0: 3y=1⇒y>03y=1\Rightarrow y>0 (rejected). So the unique solution is x=2, y=−1x=2,\ y=-1.

x+2y=2+2(−1)=0\begin{aligned} &x+2y = 2 + 2(-1) = 0 \end{aligned}
x+2y=0x+2y = 0
CAT 2021 Slot 3 QA Q16: If 3x + 2|y| + y = 7 and x + |x| + 3y = 1, then x + 2y is — Solution | TheCATExam