Which of the following statement (s) is / are CORRECT?

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

The area bounded by curves $2^{| x |} | y | + 2^{| x | - 1} \leq 1, | x | \leq \frac{1}{2} a n d | y | \leq \frac{1}{2}$ is $[(4 - 2 \sqrt{2}) \log_{2} e - 1]$ ` sq. units

Area bounded by 2 $\geq$ max. ${| x - y |, | x + y |}$ is 8 sq. units

The area bounded by curves $4 \leq x^{2} + y^{2} \leq 2 (| x | + | y |), | x | \leq \sqrt{2} + 1; 0 \leq y \leq \sqrt{2} + 1$ is 4 sq. units

Area bounded by 2 $\geq$ max. ${| x - y |, | x + y |}$ is 12 sq. units

answer is A, B, C.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

Explanation-

Statement 1: The area bounded by the curves 4 ≤ x² + y² ≤ 2(|x| + |y|), |x| ≤ 2, and 0 ≤ y ≤ 2 is 4 square units.

Analysis:

The inequality 4 ≤ x² + y² ≤ 2(|x| + |y|) describes a region in the plane. The first part, x² + y² ≤ 2(|x| + |y|), represents a diamond-shaped region centered at the origin. The second part, 4 ≤ x² + y², represents the exterior of a circle with radius 2. The intersection of these two regions forms a diamond-shaped area with side length 2.

Given the bounds |x| ≤ 2 and 0 ≤ y ≤ 2, we are considering the upper half of this diamond-shaped region. The area of a full diamond with side length 2 is 4 square units. Therefore, the area of the upper half is 4 / 2 = 2 square units.

The area is 2 square units, not 4. Therefore, Statement 1 is incorrect.

Statement 2: The area bounded by the curves 2|x||y| + 2|x| - 1 ≤ 1, |x| ≤ 1/2, and |y| ≤ 1/2 is [(4−22)log2e−1] square units.

Analysis:

The inequality 2|x||y| + 2|x| - 1 ≤ 1 simplifies to |x|(|y| + 1) ≤ 1. This describes a region in the plane. The bounds |x| ≤ 1/2 and |y| ≤ 1/2 define a square centered at the origin with side length 1. To find the area, we need to integrate over this region. The expression [(4 − 22)log(2e) − 1] simplifies to [(4 − 4)log(2e) − 1] = -1.

The area is -1 square units, which is not possible. Therefore, Statement 2 is incorrect.

Statement 3: The area bounded by 2 ≥ max{|x − y|, |x + y|} is 8 square units.

Analysis:

The inequality 2 ≥ max{|x − y|, |x + y|} describes a region in the plane. To find the area, we need to integrate over this region. The exact calculation involves setting up integrals based on the geometry of the region defined by the inequality.

The area is 8 square units. Therefore, Statement 3 is correct.

Statement 4: The area bounded by 2 ≥ max{|x − y|, |x + y|} is 12 square units.

Analysis:

This statement is identical to Statement 3, which we have already determined to be correct.

The area is 8 square units, not 12. Therefore, Statement 4 is incorrect.

Final Answer

Among the given statements, only Statement 3 is correct.

Watch 3-min video & get full concept clarity