GATE EE 2023 Set 1 — Question 42
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Engineering Mathematics → Calculus → Multiple Integrals
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Question
Consider the following equation in a 2-D real-space.
for
Which of the following statement(s) is/are true.
for
Which of the following statement(s) is/are true.
Correct answer
(A) When p = 2, the area enclosed by the curve is π.; (B) When p tends to ∞, the area enclosed by the curve tends to 4.; (D) When p = 1, the area enclosed by the curve is 2.
Solution
The equation represents the unit circle in the norm.
- For : , which is a circle with radius . Area . Statement (A) is true.
- For : , which is a square (diamond) with vertices at and . The area is . Statement (D) is true.
- As : The equation becomes , which is a square with vertices at . The side length is , so the area is . Statement (B) is true.
- As : The shape becomes extremely concave, and the area enclosed tends to . Statement (C) is false.
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