GATE Mechanical 2017 Set 1 | GA | Question: 8

Question

Let $S_{1}$ be the plane figure consisting of the points $(x, y)$ given by the inequalities $\mid x - 1 \mid  \leq 2$ and $ \mid y+2 \mid \leq 3$. Let $S_{2}$ be the plane figure given by the inequalities $x-y \geq -2, y \geq 1$, and $x \leq 3$. Let $S$ be the union of $S_{1}$ and $S_{2}$. The area of $S$ is.

• A. $26$
• B. $28$
• C. $32$
• D. $34$

Comments

This is from Linear programming.

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Is this topic in gate cse syllabus?

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is there any faster way to do this question ?

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Attempting such Questions in the last would be better option

Answer 1

For $'S_1'$ plane

1. $|x - 1| \le 2$

      A.  when $(x-1) \ge0 \Rightarrow x \ge 1$ 

           Then $+(x - 1) \le 2 \Rightarrow x \le 3$     $\{$ here $(x \ge 1) \cap (x \le 3) \Rightarrow 1 \le x \le 3 \}$

      B. When $(x -1) < 0 \Rightarrow x < 1$

          Then $-(x - 1) \le 2 \Rightarrow x \ge -1$      $\{$ here $(x <1) \cap (x \ge -1) \Rightarrow -1 \le x < 1 \}$

Union $(A \cup B) \Rightarrow -1 \le x \le 3$

2. $|y + 2| \le 3$

      A.  when $(y+2) \ge0 \Rightarrow y \ge -2$ 

           Then $(y + 2) \le 3 \Rightarrow y \le 1$     $\{$ here $(y \ge -2) \cap (y \le 1) \Rightarrow -2 \le y \le 1 \}$

      B. When $(y+2) < 0 \Rightarrow y < -2$

          Then $-(y+2) \le 3 \Rightarrow y \ge -5$    $\{$ here $(y <-2) \cap (y\ge -5) \Rightarrow -5 \le y < -2 \}$

Union $(A \cup B) \Rightarrow -5 \le y \le 1$

So,  $ 'S_1'$  plane will be for intervals : $ -1 \le x \leq 3  \text{ and } -5 \le y \leq 1$

For $'S_2' $plane ,

1. $x - y \ge -2 \Rightarrow y \le x +2$

2. $y \ge 1$

3. $x \le 3$

Here, $S = S_{1} \cup S_2$

So, Area of $S=$  Area of $S_1$ $+$ Area of $S_2$

$\Rightarrow$ Area of $S = 6*4 + \frac{1}{2}*4*4$

$\Rightarrow$ Area of $S= 24 + 8$

$\Rightarrow$ Area of $S= 32$

So, Answer is (C)