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GATE 2023 EE – Question 42

Engineering Mathematics · Calculus: Maxima and minima · 2 marks · Multiple select

Consider the following equation in a 2-D real-space.

$$|x_1|^p+|x_2|^p=1\quad\text{for }p>0$$

Which of the following statement(s) is/are true.

  1. When $p=2$, the area enclosed by the curve is $\pi$.
  2. When $p$ tends to $\infty$, the area enclosed by the curve tends to 4.
  3. When $p$ tends to 0, the area enclosed by the curve is 1.
  4. When $p=1$, the area enclosed by the curve is 2.

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Show answer and explanation

Correct answer: (A) When $p=2$, the area enclosed by the curve is $\pi$.; (B) When $p$ tends to $\infty$, the area enclosed by the curve tends to 4.; (D) When $p=1$, the area enclosed by the curve is 2.

Explanation

$p=2$ is the unit circle, of area $\pi$ (A true). As $p\to\infty$ the curve approaches the square $\max(|x_1|,|x_2|)=1$, of area 4 (B true). As $p\to0$ the region shrinks towards the axes and its area tends to 0, not 1 (C false). For $p=1$ it is the diamond $|x_1|+|x_2|=1$ with diagonals 2 and 2, of area 2 (D true).