edited by
2,264 views
8 votes
8 votes

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.

  1. $26$
  2. $28$
  3. $32$
  4. $34$
edited by
Migrated from GO Mechanical 4 years ago by Arjun

1 Answer

Best answer
11 votes
11 votes

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)

edited by
Answer:

Related questions

5 votes
5 votes
2 answers
3
Arjun asked Feb 26, 2017
2,316 views
What is the sum of the missing digits in the subtraction problem below?$\begin{array}{cccccc}&5&\_&\_&\_&\_&\\-&4&8&\_&8&9\\\rlap{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\\&&1&1...
5 votes
5 votes
1 answer
4
Arjun asked Feb 26, 2017
1,291 views
In a company with $100$ employees, $45$ earn $Rs. 20,000$ per month, $25$ earn $Rs. 30000$, $20$ earn $Rs. 40000$, $8$ earn $Rs. 60000$, and $2$ earn $Rs. 150,000$. The m...