Inequalitiesअसमानताएँ questions and answersप्रश्न और उत्तर

How inequalities worksअसमानताएँ को समझिए

Link the statements into one chain through the common letter. A relation follows only if every sign between the two letters points the same way.

The result is strict (> or <) if at least one sign on the path is strict; it is ≥ or ≤ only when every sign on the path is ≥, ≤ or =. A single sign pointing the other way means no relation follows.

In coded questions, first decode the symbols into >, <, =, ≥ and ≤.

Each trick says when to use it, why it works and when it fails. Formula tricks are checked by code over their whole range.हर ट्रिक बताती है कि उसे कब इस्तेमाल करें, वह क्यों काम करती है और कब नहीं। सूत्र वाली ट्रिक्स को कोड से उनकी पूरी रेंज पर जाँचा गया है।

Answers and full solutions are shown. Hide them to test yourself first.उत्तर और पूरे हल दिखाए गए हैं। पहले खुद को परखने के लिए इन्हें छिपाइए।

1. Example 1उदाहरण 1 · Inequalities (direct and coded) · Easyआसान

   Statement: J ≥ W ≥ S. Which of the following is definitely true?

   1. W > S
   2. J ≤ S
   3. J ≥ S
   4. J > W
   Answer:उत्तर: J ≥ S

   1. Chain: J ≥ W ≥ S.
   2. Between J and S: ≥, ≥: all in the same direction.
   3. So J ≥ S.
   4. Every other option either has opposite signs on the way or claims more than the chain gives.

2. Example 2उदाहरण 2 · Inequalities (direct and coded) · Mediumमध्यम

   ‘A @ B’ means A is greater than B; ‘A # B’ means A is greater than or equal to B; ‘A $ B’ means A is equal to B; ‘A & B’ means A is smaller than B; ‘A ★ B’ means A is smaller than or equal to B. Statement: N $ Q, Q $ M, M $ P. Which of the following is definitely true?

   1. N & Q
   2. N & M
   3. Q $ P
   4. M @ P
   Answer:उत्तर: Q $ P

   1. Chain: N = Q = M = P.
   2. Between Q and P: =, =: all in the same direction.
   3. So Q = P, written as Q $ P.
   4. Every other option either has opposite signs on the way or claims more than the chain gives.

3. Example 3उदाहरण 3 · Inequalities (direct and coded) · Hardकठिन

   ‘A ★ B’ means A is greater than B; ‘A & B’ means A is greater than or equal to B; ‘A $ B’ means A is equal to B; ‘A @ B’ means A is smaller than B; ‘A # B’ means A is smaller than or equal to B. Statement: J & V, V & N, N ★ P, P ★ K. Which of the following is definitely true?

   1. V ★ P
   2. N $ K
   3. J $ N
   4. J @ P
   Answer:उत्तर: V ★ P

   1. Chain: J ≥ V ≥ N > P > K.
   2. Between V and P: ≥, >: all in the same direction, with a strict sign.
   3. So V > P, written as V ★ P.
   4. Every other option either has opposite signs on the way or claims more than the chain gives.

4. Example 4उदाहरण 4 · Inequalities (direct and coded) · Mediumमध्यम

   ‘A # B’ means A is greater than B; ‘A ★ B’ means A is greater than or equal to B; ‘A & B’ means A is equal to B; ‘A @ B’ means A is smaller than B; ‘A $ B’ means A is smaller than or equal to B. Statement: J ★ K, K $ S, S $ V. Which of the following is definitely true?

   1. J $ S
   2. K ★ S
   3. K & V
   4. K $ V
   Answer:उत्तर: K $ V

   1. Chain: J ≥ K ≤ S ≤ V.
   2. Between K and V: ≤, ≤: all in the same direction.
   3. So K ≤ V, written as K $ V.
   4. Every other option either has opposite signs on the way or claims more than the chain gives.

5. Example 5उदाहरण 5 · Inequalities (direct and coded) · Mediumमध्यम

   ‘A @ B’ means A is greater than B; ‘A & B’ means A is greater than or equal to B; ‘A ★ B’ means A is equal to B; ‘A $ B’ means A is smaller than B; ‘A # B’ means A is smaller than or equal to B. Statement: K ★ U, U & J, J $ T. Which of the following is definitely true?

   1. K # T
   2. J ★ T
   3. K # J
   4. K & J
   Answer:उत्तर: K & J

   1. Chain: K = U ≥ J < T.
   2. Between K and J: =, ≥: all in the same direction.
   3. So K ≥ J, written as K & J.
   4. Every other option either has opposite signs on the way or claims more than the chain gives.

6. Example 6उदाहरण 6 · Inequalities (direct and coded) · Mediumमध्यम

   ‘A # B’ means A is greater than B; ‘A $ B’ means A is greater than or equal to B; ‘A @ B’ means A is equal to B; ‘A & B’ means A is smaller than B; ‘A ★ B’ means A is smaller than or equal to B. Statement: N ★ P, P $ Q, Q $ T. Which of the following is definitely true?

   1. N $ P
   2. P $ T
   3. N $ T
   4. P @ Q
   Answer:उत्तर: P $ T

   1. Chain: N ≤ P ≥ Q ≥ T.
   2. Between P and T: ≥, ≥: all in the same direction.
   3. So P ≥ T, written as P $ T.
   4. Every other option either has opposite signs on the way or claims more than the chain gives.

Common mistakesआम गलतियाँ

• Concluding A > C from A > B < C.
• Missing the “either–or” case.

Frequently asked questionsअक्सर पूछे जाने वाले प्रश्न