NIMCET 2011 — Mathematics PYQ

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Question

NIMCET 2011 — Mathematics PYQ

NIMCET | Mathematics | 2011

The area enclosed within the lines $|x| + |y| = 1$ is:

Choose the correct answer:

Explanation

1. Identify the four lines:

Quadrant I ( $x \ge 0, y \ge 0$ ): $x + y = 1$
Quadrant II ( $x < 0, y \ge 0$ ): $-x + y = 1$
Quadrant III ( $x < 0, y < 0$ ): $-x - y = 1$
Quadrant IV ( $x \ge 0, y < 0$ ): $x - y = 1$

2. Visualize the shape:

These four lines form a square (often called a diamond shape) with vertices at:

(1, 0), (0, 1), (-1, 0), \text{ and } (0, -1)

3. Calculate the Area:

There are two easy ways to find the area:

Method A (Area of 4 Triangles):

The square is composed of 4 identical right-angled triangles (one in each quadrant). Each triangle has a base of 1 unit and a height of 1 unit.

$\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}$

$\text{Total Area} = 4 \times \left(\frac{1}{2}\right) = 2$
Method B (Formula for $|x| + |y| = k$ ):

For any equation of the form $|x| + |y| = k$ , the enclosed area is always:

$\text{Area} = 2k^2$

Here $k = 1$ , so:

$\text{Area} = 2(1)^2 = 2$

The correct option is (b) 2.