The right side has factor (2) so \(a^2\) is even. First identify evenness of the square in the proof.
Step 2
Why this answer is correct
The correct answer is A. \(a^2\) सम है / \(a^2\) is even. The right side has factor (2) so \(a^2\) is even. First identify evenness of the square in the proof.
Step 3
Exam Tip
दाएँ पक्ष में (2) का गुणनखंड है इसलिए \(a^2\) सम है। प्रमाण में पहले वर्ग की समता पहचानें।
If \(p^2\) is divisible by (3), then (p) is also divisible by (3). Remember this prime factor rule.
Step 2
Why this answer is correct
The correct answer is B. (p) (3) से विभाज्य है / (p) is divisible by (3). If \(p^2\) is divisible by (3), then (p) is also divisible by (3). Remember this prime factor rule.
Step 3
Exam Tip
यदि \(p^2\) (3) से विभाज्य है तो (p) भी (3) से विभाज्य होगा। अभाज्य गुणनखंड का यह नियम याद रखें।
A. क्योंकि वे सहभाज्य नहीं रह सकते/Because they cannot remain coprime
Step 1
Concept
If both are even, (2) is a common factor. This contradicts the coprime condition of lowest form.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि वे सहभाज्य नहीं रह सकते / Because they cannot remain coprime. If both are even, (2) is a common factor. This contradicts the coprime condition of lowest form.
Step 3
Exam Tip
दोनों सम होने पर (2) सामान्य गुणनखंड होगा। यह सरलतम रूप की सहभाज्य शर्त के विरुद्ध है।
C. परिमेय मानना फिर वर्ग करना फिर विरोधाभास/Assume rational then square then contradiction
Step 1
Concept
In contradiction method we first assume rationality. Then squaring gives the contradiction that both are even.
Step 2
Why this answer is correct
The correct answer is C. परिमेय मानना फिर वर्ग करना फिर विरोधाभास / Assume rational then square then contradiction. In contradiction method we first assume rationality. Then squaring gives the contradiction that both are even.
Step 3
Exam Tip
विरोधाभास विधि में पहले परिमेय मानते हैं। फिर वर्ग करके दोनों सम होने का विरोधाभास मिलता है।
A. \(\sqrt{2}\) की अपरिमेयता/Irrationality of \(\sqrt{2}\)
Step 1
Concept
In the proof of \(\sqrt{2}\), (a) is concluded even from \(a^2\) being even. This is the main argument.
Step 2
Why this answer is correct
The correct answer is A. \(\sqrt{2}\) की अपरिमेयता / Irrationality of \(\sqrt{2}\). In the proof of \(\sqrt{2}\), (a) is concluded even from \(a^2\) being even. This is the main argument.
Step 3
Exam Tip
\(\sqrt{2}\) के प्रमाण में \(a^2\) सम होने से (a) सम निकाला जाता है। यही मुख्य तर्क है।
B. \(\sqrt{3}\) की अपरिमेयता/Irrationality of \(\sqrt{3}\)
Step 1
Concept
In the proof of \(\sqrt{3}\), divisibility of (p) by (3) is derived from \(p^2\). This leads to contradiction.
Step 2
Why this answer is correct
The correct answer is B. \(\sqrt{3}\) की अपरिमेयता / Irrationality of \(\sqrt{3}\). In the proof of \(\sqrt{3}\), divisibility of (p) by (3) is derived from \(p^2\). This leads to contradiction.
Step 3
Exam Tip
\(\sqrt{3}\) के प्रमाण में \(p^2\) से (p) का (3) से विभाज्य होना निकाला जाता है। यही आगे विरोधाभास देता है।
C. ताकि सहभाज्य शर्त से विरोधाभास दिख सके/So that contradiction with coprime condition can be shown
Step 1
Concept
In lowest form (a) and (b) are coprime. Later both becoming even creates the contradiction.
Step 2
Why this answer is correct
The correct answer is C. ताकि सहभाज्य शर्त से विरोधाभास दिख सके / So that contradiction with coprime condition can be shown. In lowest form (a) and (b) are coprime. Later both becoming even creates the contradiction.
Step 3
Exam Tip
सरलतम रूप में (a) और (b) सहभाज्य होते हैं। बाद में दोनों सम निकलना इसी से विरोधाभास बनाता है।
A. (a) और (b) दोनों सम हैं जबकि वे सहभाज्य माने गए थे/Both (a) and (b) are even although they were assumed coprime
Step 1
Concept
Both even gives common factor (2). This directly contradicts the coprime assumption.
Step 2
Why this answer is correct
The correct answer is A. (a) और (b) दोनों सम हैं जबकि वे सहभाज्य माने गए थे / Both (a) and (b) are even although they were assumed coprime. Both even gives common factor (2). This directly contradicts the coprime assumption.
Step 3
Exam Tip
दोनों सम होने से सामान्य गुणनखंड (2) मिलता है। यह सहभाज्य मान्यता से सीधा विरोधाभास है।
B. (p) और (q) दोनों (3) से विभाज्य हैं जबकि वे सहभाज्य माने गए थे/Both (p) and (q) are divisible by (3) although they were assumed coprime
Step 1
Concept
Both being divisible by (3) gives common factor (3). This breaks the coprime condition.
Step 2
Why this answer is correct
The correct answer is B. (p) और (q) दोनों (3) से विभाज्य हैं जबकि वे सहभाज्य माने गए थे / Both (p) and (q) are divisible by (3) although they were assumed coprime. Both being divisible by (3) gives common factor (3). This breaks the coprime condition.
Step 3
Exam Tip
दोनों का (3) से विभाज्य होना सामान्य गुणनखंड (3) देता है। यह सहभाज्य शर्त को तोड़ता है।
C. क्योंकि \(b^2=2r^2\) से \(b^2\) सम है/Because \(b^2=2r^2\) makes \(b^2\) even
Step 1
Concept
From \(b^2=2r^2\), \(b^2\) is even. Therefore (b) is also even.
Step 2
Why this answer is correct
The correct answer is C. क्योंकि \(b^2=2r^2\) से \(b^2\) सम है / Because \(b^2=2r^2\) makes \(b^2\) even. From \(b^2=2r^2\), \(b^2\) is even. Therefore (b) is also even.
Step 3
Exam Tip
\(b^2=2r^2\) से \(b^2\) सम मिलता है। इसलिए (b) भी सम होगा।
A. क्योंकि \(q^2=3k^2\) से \(q^2\) (3) से विभाज्य है/Because \(q^2=3k^2\) makes \(q^2\) divisible by (3)
Step 1
Concept
After getting \(q^2=3k^2\), \(q^2\) is divisible by (3). Therefore (q) is also divisible by (3).
Step 2
Why this answer is correct
The correct answer is A. क्योंकि \(q^2=3k^2\) से \(q^2\) (3) से विभाज्य है / Because \(q^2=3k^2\) makes \(q^2\) divisible by (3). After getting \(q^2=3k^2\), \(q^2\) is divisible by (3). Therefore (q) is also divisible by (3).
Step 3
Exam Tip
\(q^2=3k^2\) मिलने पर \(q^2\) (3) से विभाज्य है। इसलिए (q) भी (3) से विभाज्य है।
A. \(\sqrt{2}\) में (2) से समता और \(\sqrt{3}\) में (3) से विभाज्यता उपयोग होती है/\(\sqrt{2}\) uses evenness by (2) and \(\sqrt{3}\) uses divisibility by (3)
Step 1
Concept
In \(\sqrt{2}\), (2) is the key factor, and in \(\sqrt{3}\), (3) is the key factor. Identify the difference by the factor.
Step 2
Why this answer is correct
The correct answer is A. \(\sqrt{2}\) में (2) से समता और \(\sqrt{3}\) में (3) से विभाज्यता उपयोग होती है / \(\sqrt{2}\) uses evenness by (2) and \(\sqrt{3}\) uses divisibility by (3). In \(\sqrt{2}\), (2) is the key factor, and in \(\sqrt{3}\), (3) is the key factor. Identify the difference by the factor.
Step 3
Exam Tip
\(\sqrt{2}\) में (2) मुख्य गुणनखंड है और \(\sqrt{3}\) में (3) मुख्य गुणनखंड है। अंतर को गुणनखंड से पहचानें।
A. सरलतम रूप में उन्हें सहभाज्य मानना चाहिए/They should be assumed coprime in lowest form
Step 1
Concept
At the start the fraction is in lowest form so (a) and (b) are assumed coprime. Both even is the final contradiction.
Step 2
Why this answer is correct
The correct answer is A. सरलतम रूप में उन्हें सहभाज्य मानना चाहिए / They should be assumed coprime in lowest form. At the start the fraction is in lowest form so (a) and (b) are assumed coprime. Both even is the final contradiction.
Step 3
Exam Tip
शुरुआत में भिन्न सरलतम रूप में होती है इसलिए (a) और (b) सहभाज्य माने जाते हैं। दोनों सम होना अंत का विरोधाभास है।
B. शुरुआत में (p) और (q) सहभाज्य माने जाते हैं/At the start (p) and (q) are assumed coprime
Step 1
Concept
At the start \(\frac{p}{q}\) is in lowest form. Both divisible by (3) is derived later as a contradiction.
Step 2
Why this answer is correct
The correct answer is B. शुरुआत में (p) और (q) सहभाज्य माने जाते हैं / At the start (p) and (q) are assumed coprime. At the start \(\frac{p}{q}\) is in lowest form. Both divisible by (3) is derived later as a contradiction.
Step 3
Exam Tip
शुरुआत में \(\frac{p}{q}\) सरलतम रूप में होता है। दोनों का (3) से विभाज्य होना बाद में विरोधाभास के रूप में निकलता है।
A. यदि \(a^2\) सम है तो (a) सम है/If \(a^2\) is even then (a) is even
Step 1
Concept
This fact is used after \(a^2=2b^2\). Then (a) is written in the form (2r).
Step 2
Why this answer is correct
The correct answer is A. यदि \(a^2\) सम है तो (a) सम है / If \(a^2\) is even then (a) is even. This fact is used after \(a^2=2b^2\). Then (a) is written in the form (2r).
Step 3
Exam Tip
यह तथ्य \(a^2=2b^2\) के बाद उपयोग होता है। इससे (a) को (2r) रूप में लिखा जाता है।
B. \(\sqrt{2}\) अपरिमेय है/\(\sqrt{2}\) is irrational
Step 1
Concept
Both being even contradicts the rational assumption. Therefore \(\sqrt{2}\) is irrational.
Step 2
Why this answer is correct
The correct answer is B. \(\sqrt{2}\) अपरिमेय है / \(\sqrt{2}\) is irrational. Both being even contradicts the rational assumption. Therefore \(\sqrt{2}\) is irrational.
Step 3
Exam Tip
दोनों सम होने से परिमेय मान्यता में विरोधाभास आता है। इसलिए \(\sqrt{2}\) अपरिमेय है।
C. \(\sqrt{3}\) अपरिमेय है/\(\sqrt{3}\) is irrational
Step 1
Concept
A common factor (3) contradicts the coprime condition. Therefore \(\sqrt{3}\) is irrational.
Step 2
Why this answer is correct
The correct answer is C. \(\sqrt{3}\) अपरिमेय है / \(\sqrt{3}\) is irrational. A common factor (3) contradicts the coprime condition. Therefore \(\sqrt{3}\) is irrational.
Step 3
Exam Tip
दोनों में सामान्य गुणनखंड (3) मिलना सहभाज्य शर्त से विरोधाभास है। इसलिए \(\sqrt{3}\) अपरिमेय है।
In the proof of \(\sqrt{2}\), the chain uses evenness by (2). This chain shows (b) is also even.
Step 2
Why this answer is correct
The correct answer is A. \(a^2=2b^2\), (a=2r), \(b^2=2r^2\). In the proof of \(\sqrt{2}\), the chain uses evenness by (2). This chain shows (b) is also even.
Step 3
Exam Tip
\(\sqrt{2}\) के प्रमाण में (2) से समता की कड़ी बनती है। यह कड़ी (b) को भी सम दिखाती है।
In the proof of \(\sqrt{3}\), the chain uses divisibility by (3). It shows (q) is also divisible by (3).
Step 2
Why this answer is correct
The correct answer is C. \(p^2=3q^2\), (p=3k), \(q^2=3k^2\). In the proof of \(\sqrt{3}\), the chain uses divisibility by (3). It shows (q) is also divisible by (3).
Step 3
Exam Tip
\(\sqrt{3}\) के प्रमाण में (3) से विभाज्यता की कड़ी चलती है। इससे (q) भी (3) से विभाज्य निकलता है।
A. (a) और (b) सहभाज्य और \(b\neq0\)/(a) and (b) coprime and \(b\neq0\)
Step 1
Concept
A rational number is written in lowest form as a ratio of coprime integers. The denominator cannot be zero.
Step 2
Why this answer is correct
The correct answer is A. (a) और (b) सहभाज्य और \(b\neq0\) / (a) and (b) coprime and \(b\neq0\). A rational number is written in lowest form as a ratio of coprime integers. The denominator cannot be zero.
Step 3
Exam Tip
परिमेय संख्या सरलतम रूप में सहभाज्य पूर्णांकों के अनुपात में लिखी जाती है। हर शून्य नहीं हो सकता।
B. (p) और (q) सहभाज्य और \(q\neq0\)/(p) and (q) coprime and \(q\neq0\)
Step 1
Concept
In lowest rational form, numerator and denominator are coprime. This condition later gives the contradiction.
Step 2
Why this answer is correct
The correct answer is B. (p) और (q) सहभाज्य और \(q\neq0\) / (p) and (q) coprime and \(q\neq0\). In lowest rational form, numerator and denominator are coprime. This condition later gives the contradiction.
Step 3
Exam Tip
सरलतम परिमेय रूप में अंश और हर सहभाज्य होते हैं। इसी शर्त से बाद में विरोधाभास मिलता है।
A. क्योंकि विषम संख्या का वर्ग विषम होता है/Because the square of an odd number is odd
Step 1
Concept
If (a) were odd then \(a^2\) would also be odd. Therefore if \(a^2\) is even, (a) is even.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि विषम संख्या का वर्ग विषम होता है / Because the square of an odd number is odd. If (a) were odd then \(a^2\) would also be odd. Therefore if \(a^2\) is even, (a) is even.
Step 3
Exam Tip
यदि (a) विषम होता तो \(a^2\) भी विषम होता। इसलिए \(a^2\) सम होने पर (a) सम होगा।
A. क्योंकि (3) अभाज्य गुणनखंड है/Because (3) is a prime factor
Step 1
Concept
A prime factor appears in a square only if it appears in the original number. Therefore (p) is divisible by (3).
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (3) अभाज्य गुणनखंड है / Because (3) is a prime factor. A prime factor appears in a square only if it appears in the original number. Therefore (p) is divisible by (3).
Step 3
Exam Tip
अभाज्य गुणनखंड किसी वर्ग में तभी आता है जब मूल संख्या में आता है। इसलिए (p) (3) से विभाज्य है।
A. (a=2r) रखने से \(b^2=2r^2\) मिलता है/Taking (a=2r) gives \(b^2=2r^2\)
Step 1
Concept
From \(b^2=2r^2\), \(b^2\) is even. Then (b) is also even.
Step 2
Why this answer is correct
The correct answer is A. (a=2r) रखने से \(b^2=2r^2\) मिलता है / Taking (a=2r) gives \(b^2=2r^2\). From \(b^2=2r^2\), \(b^2\) is even. Then (b) is also even.
Step 3
Exam Tip
\(b^2=2r^2\) से \(b^2\) सम मिलता है। फिर (b) भी सम होगा।
B. (p=3k) रखने से \(q^2=3k^2\) मिलता है/Taking (p=3k) gives \(q^2=3k^2\)
Step 1
Concept
From \(q^2=3k^2\), \(q^2\) is divisible by (3). Therefore (q) is also divisible by (3).
Step 2
Why this answer is correct
The correct answer is B. (p=3k) रखने से \(q^2=3k^2\) मिलता है / Taking (p=3k) gives \(q^2=3k^2\). From \(q^2=3k^2\), \(q^2\) is divisible by (3). Therefore (q) is also divisible by (3).
Step 3
Exam Tip
\(q^2=3k^2\) से \(q^2\) (3) से विभाज्य है। इसलिए (q) भी (3) से विभाज्य है।
A. क्योंकि प्रमाण समता और सहभाज्य विरोधाभास पर आधारित है/Because the proof is based on evenness and coprime contradiction
Step 1
Concept
This proof does not need decimals. Evenness and the lowest fraction condition are enough.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि प्रमाण समता और सहभाज्य विरोधाभास पर आधारित है / Because the proof is based on evenness and coprime contradiction. This proof does not need decimals. Evenness and the lowest fraction condition are enough.
Step 3
Exam Tip
इस प्रमाण में दशमलव की जरूरत नहीं होती। समता और सरलतम भिन्न की शर्त पर्याप्त है।
A. क्योंकि दशमलव अनुमान प्रमाण नहीं देता/Because decimal approximation does not give a proof
Step 1
Concept
A decimal can only approximate. For proof, divisibility by (3) and contradiction are needed.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि दशमलव अनुमान प्रमाण नहीं देता / Because decimal approximation does not give a proof. A decimal can only approximate. For proof, divisibility by (3) and contradiction are needed.
Step 3
Exam Tip
दशमलव केवल अनुमान दे सकता है। प्रमाण के लिए (3) से विभाज्यता और विरोधाभास चाहिए।
A. यह असंभव है क्योंकि (2) सामान्य गुणनखंड है/This is impossible because (2) is a common factor
Step 1
Concept
In lowest form, numerator and denominator should be coprime. Both even is a contradiction.
Step 2
Why this answer is correct
The correct answer is A. यह असंभव है क्योंकि (2) सामान्य गुणनखंड है / This is impossible because (2) is a common factor. In lowest form, numerator and denominator should be coprime. Both even is a contradiction.
Step 3
Exam Tip
सरलतम रूप में अंश और हर सहभाज्य होने चाहिए। दोनों सम मिलना विरोधाभास है।
B. यह सरलतम रूप से विरोधाभास है/This contradicts lowest form
Step 1
Concept
Both will have common factor (3). So the fraction cannot be in lowest form.
Step 2
Why this answer is correct
The correct answer is B. यह सरलतम रूप से विरोधाभास है / This contradicts lowest form. Both will have common factor (3). So the fraction cannot be in lowest form.
Step 3
Exam Tip
दोनों में सामान्य गुणनखंड (3) होगा। इसलिए भिन्न सरलतम रूप में नहीं हो सकती।
A. \(\sqrt{2}\) परिमेय है/\(\sqrt{2}\) is rational
Step 1
Concept
The contradiction makes the rationality assumption false. Therefore the conclusion is that \(\sqrt{2}\) is irrational.
Step 2
Why this answer is correct
The correct answer is A. \(\sqrt{2}\) परिमेय है / \(\sqrt{2}\) is rational. The contradiction makes the rationality assumption false. Therefore the conclusion is that \(\sqrt{2}\) is irrational.
Step 3
Exam Tip
विरोधाभास परिमेय होने की मान्यता को गलत करता है। इसलिए \(\sqrt{2}\) अपरिमेय निष्कर्ष है।
B. \(\sqrt{3}\) परिमेय है/\(\sqrt{3}\) is rational
Step 1
Concept
The rational assumption makes both (p) and (q) divisible by (3). Thus the assumption is rejected.
Step 2
Why this answer is correct
The correct answer is B. \(\sqrt{3}\) परिमेय है / \(\sqrt{3}\) is rational. The rational assumption makes both (p) and (q) divisible by (3). Thus the assumption is rejected.
Step 3
Exam Tip
परिमेय मान्यता से (p) और (q) दोनों (3) से विभाज्य निकलते हैं। इससे मान्यता अस्वीकार होती है।
A. अपरिमेय क्योंकि परिमेय मान्यता से दोनों सम मिलते हैं/Irrational because rational assumption makes both even
Step 1
Concept
Both becoming even contradicts the coprime condition. Therefore \(\sqrt{2}\) is irrational.
Step 2
Why this answer is correct
The correct answer is A. अपरिमेय क्योंकि परिमेय मान्यता से दोनों सम मिलते हैं / Irrational because rational assumption makes both even. Both becoming even contradicts the coprime condition. Therefore \(\sqrt{2}\) is irrational.
Step 3
Exam Tip
दोनों सम मिलना सहभाज्य शर्त से विरोधाभास देता है। इसलिए \(\sqrt{2}\) अपरिमेय है।
B. अपरिमेय क्योंकि परिमेय मान्यता से दोनों (3) से विभाज्य मिलते हैं/Irrational because rational assumption makes both divisible by (3)
Step 1
Concept
Both being divisible by (3) contradicts the coprime condition. Therefore \(\sqrt{3}\) is irrational.
Step 2
Why this answer is correct
The correct answer is B. अपरिमेय क्योंकि परिमेय मान्यता से दोनों (3) से विभाज्य मिलते हैं / Irrational because rational assumption makes both divisible by (3). Both being divisible by (3) contradicts the coprime condition. Therefore \(\sqrt{3}\) is irrational.
Step 3
Exam Tip
दोनों का (3) से विभाज्य होना सहभाज्य शर्त से विरोधाभास है। इसलिए \(\sqrt{3}\) अपरिमेय है।
A. दोनों का सामान्य गुणनखंड (2) होना/Both having common factor (2)
Step 1
Concept
Coprime numbers have only (1) as common factor. Therefore common factor (2) is impossible.
Step 2
Why this answer is correct
The correct answer is A. दोनों का सामान्य गुणनखंड (2) होना / Both having common factor (2). Coprime numbers have only (1) as common factor. Therefore common factor (2) is impossible.
Step 3
Exam Tip
सहभाज्य संख्याओं का सामान्य गुणनखंड केवल (1) होता है। इसलिए (2) सामान्य होना असंभव है।
B. दोनों का सामान्य गुणनखंड (3) होना/Both having common factor (3)
Step 1
Concept
Coprime numbers have no common factor except (1). Therefore (3) cannot be common.
Step 2
Why this answer is correct
The correct answer is B. दोनों का सामान्य गुणनखंड (3) होना / Both having common factor (3). Coprime numbers have no common factor except (1). Therefore (3) cannot be common.
Step 3
Exam Tip
सहभाज्य संख्याओं में (1) के अलावा कोई सामान्य गुणनखंड नहीं होता। इसलिए (3) सामान्य नहीं हो सकता।
A. क्योंकि (a) सम सिद्ध होता है/Because (a) is proved even
Step 1
Concept
Since \(a^2\) is even, (a) is even. An even number is written as (2r).
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (a) सम सिद्ध होता है / Because (a) is proved even. Since \(a^2\) is even, (a) is even. An even number is written as (2r).
Step 3
Exam Tip
\(a^2\) सम होने से (a) सम है। सम संख्या को (2r) के रूप में लिखा जाता है।
B. क्योंकि (p) (3) से विभाज्य सिद्ध होता है/Because (p) is proved divisible by (3)
Step 1
Concept
When \(p^2\) is divisible by (3), (p) is also divisible by (3). Therefore (p=3k) is written.
Step 2
Why this answer is correct
The correct answer is B. क्योंकि (p) (3) से विभाज्य सिद्ध होता है / Because (p) is proved divisible by (3). When \(p^2\) is divisible by (3), (p) is also divisible by (3). Therefore (p=3k) is written.
Step 3
Exam Tip
\(p^2\) (3) से विभाज्य होने पर (p) भी (3) से विभाज्य होता है। इसलिए (p=3k) लिखा जाता है।
A. \(\sqrt{2}=\frac{a}{b}\), जहाँ (a,b) सहभाज्य हैं और \(b\neq0\)/\(\sqrt{2}=\frac{a}{b}\), where (a,b) are coprime and \(b\neq0\)
Step 1
Concept
A rational number is written as a ratio of two integers in lowest form. The denominator is not zero.
Step 2
Why this answer is correct
The correct answer is A. \(\sqrt{2}=\frac{a}{b}\), जहाँ (a,b) सहभाज्य हैं और \(b\neq0\) / \(\sqrt{2}=\frac{a}{b}\), where (a,b) are coprime and \(b\neq0\). A rational number is written as a ratio of two integers in lowest form. The denominator is not zero.
Step 3
Exam Tip
परिमेय संख्या को दो पूर्णांकों के अनुपात में सरलतम रूप में लिखा जाता है। हर शून्य नहीं होता।
B. \(\sqrt{3}=\frac{p}{q}\), जहाँ (p,q) सहभाज्य हैं और \(q\neq0\)/\(\sqrt{3}=\frac{p}{q}\), where (p,q) are coprime and \(q\neq0\)
Step 1
Concept
When assumed rational, \(\sqrt{3}\) is written in lowest fractional form. Therefore (p) and (q) are coprime.
Step 2
Why this answer is correct
The correct answer is B. \(\sqrt{3}=\frac{p}{q}\), जहाँ (p,q) सहभाज्य हैं और \(q\neq0\) / \(\sqrt{3}=\frac{p}{q}\), where (p,q) are coprime and \(q\neq0\). When assumed rational, \(\sqrt{3}\) is written in lowest fractional form. Therefore (p) and (q) are coprime.
Step 3
Exam Tip
परिमेय मानने पर \(\sqrt{3}\) को सरलतम भिन्न में लिखा जाता है। इसलिए (p) और (q) सहभाज्य होते हैं।
D. (a) और (b) दोनों सम हों तब भी वे सहभाज्य हैं/Even if both (a) and (b) are even, they are coprime
Step 1
Concept
If both are even, (2) is a common factor. Therefore they cannot be coprime.
Step 2
Why this answer is correct
The correct answer is D. (a) और (b) दोनों सम हों तब भी वे सहभाज्य हैं / Even if both (a) and (b) are even, they are coprime. If both are even, (2) is a common factor. Therefore they cannot be coprime.
Step 3
Exam Tip
दोनों सम हों तो (2) सामान्य गुणनखंड होता है। इसलिए वे सहभाज्य नहीं हो सकते।
C. (p) और (q) दोनों (3) से विभाज्य हों तब भी वे सहभाज्य हैं/Even if both (p) and (q) are divisible by (3), they are coprime
Step 1
Concept
If both are divisible by (3), common factor (3) exists. Therefore they cannot be coprime.
Step 2
Why this answer is correct
The correct answer is C. (p) और (q) दोनों (3) से विभाज्य हों तब भी वे सहभाज्य हैं / Even if both (p) and (q) are divisible by (3), they are coprime. If both are divisible by (3), common factor (3) exists. Therefore they cannot be coprime.
Step 3
Exam Tip
दोनों (3) से विभाज्य हों तो सामान्य गुणनखंड (3) है। इसलिए वे सहभाज्य नहीं हो सकते।
Assuming either rational gives a contradiction with the coprime condition. Therefore both are irrational.
Step 2
Why this answer is correct
The correct answer is B. दोनों अपरिमेय हैं / Both are irrational. Assuming either rational gives a contradiction with the coprime condition. Therefore both are irrational.
Step 3
Exam Tip
दोनों को परिमेय मानने पर सहभाज्य शर्त से विरोधाभास मिलता है। इसलिए दोनों अपरिमेय हैं।
\(q^2\) is divisible by (3) so (q) is also divisible by (3). This contradicts the coprime assumption.
Step 2
Why this answer is correct
The correct answer is B. (q) (3) से विभाज्य है / (q) is divisible by (3). \(q^2\) is divisible by (3) so (q) is also divisible by (3). This contradicts the coprime assumption.
Step 3
Exam Tip
\(q^2\) (3) से विभाज्य है इसलिए (q) भी (3) से विभाज्य होगा। यही सहभाज्य मान्यता से विरोधाभास देता है।
C. (a) और (b) दोनों सम होकर भी सहभाज्य हैं/(a) and (b) are coprime even though both are even
Step 1
Concept
If both are even then (2) is a common factor. Therefore they cannot be coprime.
Step 2
Why this answer is correct
The correct answer is C. (a) और (b) दोनों सम होकर भी सहभाज्य हैं / (a) and (b) are coprime even though both are even. If both are even then (2) is a common factor. Therefore they cannot be coprime.
Step 3
Exam Tip
दोनों सम हों तो (2) सामान्य गुणनखंड है। इसलिए वे सहभाज्य नहीं हो सकते।
A. (p) और (q) दोनों (3) से विभाज्य हैं इसलिए वे सहभाज्य नहीं हो सकते/Both (p) and (q) are divisible by (3), so they cannot be coprime
Step 1
Concept
Having common factor (3) breaks the condition of lowest fraction. Therefore \(\sqrt{3}\) cannot be rational.
Step 2
Why this answer is correct
The correct answer is A. (p) और (q) दोनों (3) से विभाज्य हैं इसलिए वे सहभाज्य नहीं हो सकते / Both (p) and (q) are divisible by (3), so they cannot be coprime. Having common factor (3) breaks the condition of lowest fraction. Therefore \(\sqrt{3}\) cannot be rational.
Step 3
Exam Tip
दोनों में सामान्य गुणनखंड (3) होना सरलतम भिन्न की शर्त को तोड़ता है। इसलिए \(\sqrt{3}\) परिमेय नहीं हो सकता।
