B. (a) एक पूर्ण वर्ग नहीं है/(a) is not a perfect square
Step 1
Concept
If (a) is not a perfect square then \(\sqrt{a}\) can be irrational. In exams first identify perfect squares.
Step 2
Why this answer is correct
The correct answer is B. (a) एक पूर्ण वर्ग नहीं है / (a) is not a perfect square. If (a) is not a perfect square then \(\sqrt{a}\) can be irrational. In exams first identify perfect squares.
Step 3
Exam Tip
यदि (a) पूर्ण वर्ग नहीं है तो \(\sqrt{a}\) अपरिमेय हो सकता है। परीक्षा में पहले पूर्ण वर्ग पहचानें।
(13) is not a perfect square, so \( \sqrt{13} \) is irrational. An irrational number cannot be written as \( \frac{p}{q} \).
Step 2
Why this answer is correct
The correct answer is A. \( \sqrt{13} \). (13) is not a perfect square, so \( \sqrt{13} \) is irrational. An irrational number cannot be written as \( \frac{p}{q} \).
Step 3
Exam Tip
(13) पूर्ण वर्ग नहीं है, इसलिए \( \sqrt{13} \) अपरिमेय है। अपरिमेय संख्या \( \frac{p}{q} \) के रूप में नहीं लिखी जाती।
(8) is not a perfect square, so \( \sqrt{8} \) is irrational. Check whether the number under the radical is a perfect square.
Step 2
Why this answer is correct
The correct answer is A. अपरिमेय / Irrational. (8) is not a perfect square, so \( \sqrt{8} \) is irrational. Check whether the number under the radical is a perfect square.
Step 3
Exam Tip
(8) पूर्ण वर्ग नहीं है, इसलिए \( \sqrt{8} \) अपरिमेय है। वर्गमूल में अंदर की संख्या पूर्ण वर्ग है या नहीं, यह देखें।
(15) is not a perfect square, so \( \sqrt{15} \) is irrational. Do not treat recurring decimals as irrational.
Step 2
Why this answer is correct
The correct answer is B. \( \sqrt{15} \). (15) is not a perfect square, so \( \sqrt{15} \) is irrational. Do not treat recurring decimals as irrational.
Step 3
Exam Tip
(15) पूर्ण वर्ग नहीं है, इसलिए \( \sqrt{15} \) अपरिमेय है। आवर्ती दशमलव को अपरिमेय न मानें।
\( \sqrt{6} \) is irrational, so its decimal is non-terminating non-recurring. An irrational decimal neither terminates nor repeats.
Step 2
Why this answer is correct
The correct answer is C. \( \sqrt{6} \). \( \sqrt{6} \) is irrational, so its decimal is non-terminating non-recurring. An irrational decimal neither terminates nor repeats.
Step 3
Exam Tip
\( \sqrt{6} \) अपरिमेय है, इसलिए इसका दशमलव असांत अनावर्ती होगा। अपरिमेय संख्या का दशमलव न समाप्त होता है न दोहरता है।
(50) is not a perfect square, so \( \sqrt{50} \) is irrational. Only the square root of a perfect square is an integer.
Step 2
Why this answer is correct
The correct answer is A. यह अपरिमेय है / It is irrational. (50) is not a perfect square, so \( \sqrt{50} \) is irrational. Only the square root of a perfect square is an integer.
Step 3
Exam Tip
(50) पूर्ण वर्ग नहीं है, इसलिए \( \sqrt{50} \) अपरिमेय है। केवल पूर्ण वर्ग का वर्गमूल पूर्णांक होता है।
B. \( \sqrt{2} \) और \( \sqrt{3} \)/\( \sqrt{2} \) and \( \sqrt{3} \)
Step 1
Concept
(2) and (3) are not perfect squares, so \( \sqrt{2} \) and \( \sqrt{3} \) are irrational. Square roots of non-perfect squares are irrational.
Step 2
Why this answer is correct
The correct answer is B. \( \sqrt{2} \) और \( \sqrt{3} \) / \( \sqrt{2} \) and \( \sqrt{3} \). (2) and (3) are not perfect squares, so \( \sqrt{2} \) and \( \sqrt{3} \) are irrational. Square roots of non-perfect squares are irrational.
Step 3
Exam Tip
(2) और (3) पूर्ण वर्ग नहीं हैं, इसलिए \( \sqrt{2} \) और \( \sqrt{3} \) अपरिमेय हैं। अपूर्ण वर्गों के वर्गमूल अपरिमेय होते हैं।
\( \sqrt{18} \) is irrational, so its decimal expansion is non-terminating non-recurring. Recurring decimals are rational.
Step 2
Why this answer is correct
The correct answer is C. \( \sqrt{18} \). \( \sqrt{18} \) is irrational, so its decimal expansion is non-terminating non-recurring. Recurring decimals are rational.
Step 3
Exam Tip
\( \sqrt{18} \) अपरिमेय है, इसलिए इसका दशमलव असांत अनावर्ती होगा। आवर्ती दशमलव परिमेय होते हैं।
\( \sqrt{14} \) is irrational, and every irrational number is also real. Real numbers include both rational and irrational numbers.
Step 2
Why this answer is correct
The correct answer is A. \( \sqrt{14} \). \( \sqrt{14} \) is irrational, and every irrational number is also real. Real numbers include both rational and irrational numbers.
Step 3
Exam Tip
\( \sqrt{14} \) अपरिमेय है और हर अपरिमेय संख्या वास्तविक संख्या भी होती है। वास्तविक संख्याओं में परिमेय और अपरिमेय दोनों आते हैं।
(27) is not a perfect square, so \( \sqrt{27} \) is irrational. Distinguish perfect squares from non-perfect squares.
Step 2
Why this answer is correct
The correct answer is C. \( \sqrt{27} \). (27) is not a perfect square, so \( \sqrt{27} \) is irrational. Distinguish perfect squares from non-perfect squares.
Step 3
Exam Tip
(27) पूर्ण वर्ग नहीं है, इसलिए \( \sqrt{27} \) अपरिमेय है। पूर्ण वर्गों और अपूर्ण वर्गों में फर्क करें।
\(0.\overline{45}\) is a non-terminating recurring decimal, so it is rational. Do not treat recurring decimals as irrational.
Step 2
Why this answer is correct
The correct answer is C. \(0.\overline{45}\). \(0.\overline{45}\) is a non-terminating recurring decimal, so it is rational. Do not treat recurring decimals as irrational.
Step 3
Exam Tip
\(0.\overline{45}\) असांत आवर्ती दशमलव है, इसलिए परिमेय है। आवर्ती दशमलव को अपरिमेय न मानें।
(32) is not a perfect square, so \( \sqrt{32} \) is irrational. Check perfect squares when seeing a square root.
Step 2
Why this answer is correct
The correct answer is B. यह अपरिमेय है / It is irrational. (32) is not a perfect square, so \( \sqrt{32} \) is irrational. Check perfect squares when seeing a square root.
Step 3
Exam Tip
(32) पूर्ण वर्ग नहीं है, इसलिए \( \sqrt{32} \) अपरिमेय है। वर्गमूल देखते समय पूर्ण वर्ग जांचें।
\( \sqrt{26} \) is the square root of a non-perfect square, so it is irrational. \( \frac{22}{7} \) is rational, but \( \pi \) is not.
Step 2
Why this answer is correct
The correct answer is C. \( \sqrt{26} \). \( \sqrt{26} \) is the square root of a non-perfect square, so it is irrational. \( \frac{22}{7} \) is rational, but \( \pi \) is not.
Step 3
Exam Tip
\( \sqrt{26} \) अपूर्ण वर्ग का वर्गमूल है, इसलिए अपरिमेय है। \( \frac{22}{7} \) परिमेय है, \( \pi \) नहीं।
The decimal expansion of an irrational number is non-terminating non-recurring. This is a key exam identification.
Step 2
Why this answer is correct
The correct answer is C. असांत अनावर्ती / Non-terminating non-recurring. The decimal expansion of an irrational number is non-terminating non-recurring. This is a key exam identification.
Step 3
Exam Tip
अपरिमेय संख्या का दशमलव विस्तार असांत अनावर्ती होता है। यह परीक्षा में सबसे महत्वपूर्ण पहचान है।
A. \( \sqrt{4} \) और \( \sqrt{7} \)/\( \sqrt{4} \) and \( \sqrt{7} \)
Step 1
Concept
\( \sqrt{4}=2 \) is rational and \( \sqrt{7} \) is irrational. Check the type of both numbers in a pair.
Step 2
Why this answer is correct
The correct answer is A. \( \sqrt{4} \) और \( \sqrt{7} \) / \( \sqrt{4} \) and \( \sqrt{7} \). \( \sqrt{4}=2 \) is rational and \( \sqrt{7} \) is irrational. Check the type of both numbers in a pair.
Step 3
Exam Tip
\( \sqrt{4}=2 \) परिमेय है और \( \sqrt{7} \) अपरिमेय है। जोड़ी में दोनों संख्याओं का प्रकार अलग-अलग जांचें।
\( \sqrt{5} \) lies approximately between (2) and (3) and is irrational. A square root of a non-perfect square is a good example.
Step 2
Why this answer is correct
The correct answer is A. \( \sqrt{5} \). \( \sqrt{5} \) lies approximately between (2) and (3) and is irrational. A square root of a non-perfect square is a good example.
Step 3
Exam Tip
\( \sqrt{5} \) लगभग (2) और (3) के बीच है और अपरिमेय है। अपूर्ण वर्ग का वर्गमूल अच्छा उदाहरण होता है।
Adding a rational number to an irrational number gives an irrational number. (3) is rational and \( \sqrt{2} \) is irrational.
Step 2
Why this answer is correct
The correct answer is B. अपरिमेय / Irrational. Adding a rational number to an irrational number gives an irrational number. (3) is rational and \( \sqrt{2} \) is irrational.
Step 3
Exam Tip
परिमेय संख्या में अपरिमेय संख्या जोड़ने पर परिणाम अपरिमेय रहता है। (3) परिमेय है और \( \sqrt{2} \) अपरिमेय है।
Multiplying non-zero rational (5) by irrational \( \sqrt{3} \) gives an irrational number. The zero case is different.
Step 2
Why this answer is correct
The correct answer is C. अपरिमेय / Irrational. Multiplying non-zero rational (5) by irrational \( \sqrt{3} \) gives an irrational number. The zero case is different.
Step 3
Exam Tip
गैर-शून्य परिमेय संख्या (5) को अपरिमेय \( \sqrt{3} \) से गुणा करने पर अपरिमेय संख्या मिलती है। गुणन में शून्य का मामला अलग होता है।
A. पूर्ण वर्ग का वर्गमूल/Square root of a perfect square
Step 1
Concept
The square root of a perfect square is an integer, so it is rational. For example, \( \sqrt{64}=8 \).
Step 2
Why this answer is correct
The correct answer is A. पूर्ण वर्ग का वर्गमूल / Square root of a perfect square. The square root of a perfect square is an integer, so it is rational. For example, \( \sqrt{64}=8 \).
Step 3
Exam Tip
पूर्ण वर्ग का वर्गमूल पूर्णांक होता है, इसलिए परिमेय होता है। जैसे \( \sqrt{64}=8 \)।
\( \sqrt{3} \) is irrational, so its decimal is non-terminating non-recurring. The decimal of an irrational number is never recurring.
Step 2
Why this answer is correct
The correct answer is C. असांत अनावर्ती / Non-terminating non-recurring. \( \sqrt{3} \) is irrational, so its decimal is non-terminating non-recurring. The decimal of an irrational number is never recurring.
Step 3
Exam Tip
\( \sqrt{3} \) अपरिमेय है, इसलिए इसका दशमलव असांत अनावर्ती होता है। अपरिमेय संख्या का दशमलव कभी आवर्ती नहीं होता।
A. \( \sqrt{6} \), \( \sqrt{10} \), \( \sqrt{11} \)
Step 1
Concept
(6), (10), and (11) are not perfect squares, so all three square roots are irrational. Check each number separately.
Step 2
Why this answer is correct
The correct answer is A. \( \sqrt{6} \), \( \sqrt{10} \), \( \sqrt{11} \). (6), (10), and (11) are not perfect squares, so all three square roots are irrational. Check each number separately.
Step 3
Exam Tip
(6), (10) और (11) पूर्ण वर्ग नहीं हैं, इसलिए तीनों वर्गमूल अपरिमेय हैं। सभी संख्याओं को अलग-अलग जांचें।
\( \sqrt{2}\times\sqrt{2}=2 \), so it is rational. The product of two irrational numbers is not always irrational.
Step 2
Why this answer is correct
The correct answer is B. परिमेय / Rational. \( \sqrt{2}\times\sqrt{2}=2 \), so it is rational. The product of two irrational numbers is not always irrational.
Step 3
Exam Tip
\( \sqrt{2}\times\sqrt{2}=2 \) है, इसलिए परिमेय है। दो अपरिमेय संख्याओं का गुणन हमेशा अपरिमेय नहीं होता।
(2) is rational and \( \sqrt{5} \) is irrational, so the sum is irrational. Adding rational and irrational gives irrational.
Step 2
Why this answer is correct
The correct answer is B. यह अपरिमेय है / It is irrational. (2) is rational and \( \sqrt{5} \) is irrational, so the sum is irrational. Adding rational and irrational gives irrational.
Step 3
Exam Tip
(2) परिमेय है और \( \sqrt{5} \) अपरिमेय है, इसलिए योग अपरिमेय है। परिमेय में अपरिमेय जोड़ने से अपरिमेय मिलता है।
\(48=16\times3\), so \( \sqrt{48}=4\sqrt{3} \). Taking out the largest perfect square gives the simplest answer.
Step 2
Why this answer is correct
The correct answer is B. \(4\sqrt{3}\). \(48=16\times3\), so \( \sqrt{48}=4\sqrt{3} \). Taking out the largest perfect square gives the simplest answer.
Step 3
Exam Tip
\(48=16\times3\), इसलिए \( \sqrt{48}=4\sqrt{3} \) है। सबसे बड़ा पूर्ण वर्ग निकालने से उत्तर सरल मिलता है।
\( \sqrt{3}\times\sqrt{12}=\sqrt{36}=6 \), so it is rational. The product of two irrational numbers is not always irrational.
Step 2
Why this answer is correct
The correct answer is B. परिमेय / Rational. \( \sqrt{3}\times\sqrt{12}=\sqrt{36}=6 \), so it is rational. The product of two irrational numbers is not always irrational.
Step 3
Exam Tip
\( \sqrt{3}\times\sqrt{12}=\sqrt{36}=6 \) है, इसलिए परिमेय है। दो अपरिमेय संख्याओं का गुणन हमेशा अपरिमेय नहीं होता।
\( \sqrt{11} \) is irrational, and its negative remains irrational. Changing the sign does not change rational or irrational type.
Step 2
Why this answer is correct
The correct answer is C. अपरिमेय / Irrational. \( \sqrt{11} \) is irrational, and its negative remains irrational. Changing the sign does not change rational or irrational type.
Step 3
Exam Tip
\( \sqrt{11} \) अपरिमेय है और उसका ऋणात्मक भी अपरिमेय रहता है। चिन्ह बदलने से संख्या का परिमेय-अपरिमेय प्रकार नहीं बदलता।
\(1.01001000100001\ldots\) has no fixed repetition and does not terminate. Such a decimal is irrational.
Step 2
Why this answer is correct
The correct answer is D. \(1.01001000100001\ldots\). \(1.01001000100001\ldots\) has no fixed repetition and does not terminate. Such a decimal is irrational.
Step 3
Exam Tip
\(1.01001000100001\ldots\) में निश्चित दोहराव नहीं है और यह समाप्त भी नहीं होता। ऐसा दशमलव अपरिमेय होता है।
\( \frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2} \), so it is irrational. If the denominator is irrational, first think of the simplified form.
Step 2
Why this answer is correct
The correct answer is B. अपरिमेय / Irrational. \( \frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2} \), so it is irrational. If the denominator is irrational, first think of the simplified form.
Step 3
Exam Tip
\( \frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2} \) है, इसलिए यह अपरिमेय है। हर में अपरिमेय हो तो पहले सरल रूप सोचें।
Subtracting irrational \( \sqrt{6} \) from rational (4) gives an irrational number. The sum or difference of rational and irrational is generally irrational.
Step 2
Why this answer is correct
The correct answer is B. अपरिमेय / Irrational. Subtracting irrational \( \sqrt{6} \) from rational (4) gives an irrational number. The sum or difference of rational and irrational is generally irrational.
Step 3
Exam Tip
परिमेय संख्या (4) में से अपरिमेय \( \sqrt{6} \) घटाने पर परिणाम अपरिमेय रहता है। परिमेय और अपरिमेय का योग या अंतर सामान्यतः अपरिमेय होता है।
\(108=36\times3\), so \( \sqrt{108}=6\sqrt{3} \). While simplifying radicals, take out the largest perfect square.
Step 2
Why this answer is correct
The correct answer is A. \(6\sqrt{3}\). \(108=36\times3\), so \( \sqrt{108}=6\sqrt{3} \). While simplifying radicals, take out the largest perfect square.
Step 3
Exam Tip
\(108=36\times3\), इसलिए \( \sqrt{108}=6\sqrt{3} \) है। वर्गमूल सरल करते समय सबसे बड़ा पूर्ण वर्ग निकालें।
Since (9<13<16), \(3<\sqrt{13}<4\), and (13) is not a perfect square. Therefore, \( \sqrt{13} \) is irrational.
Step 2
Why this answer is correct
The correct answer is D. \( \sqrt{13} \). Since (9<13<16), \(3<\sqrt{13}<4\), and (13) is not a perfect square. Therefore, \( \sqrt{13} \) is irrational.
Step 3
Exam Tip
(9<13<16), इसलिए \(3<\sqrt{13}<4\) और (13) पूर्ण वर्ग नहीं है। इसलिए \( \sqrt{13} \) अपरिमेय है।
\( \sqrt{18}=3\sqrt{2} \), so \( \frac{\sqrt{18}}{3}=\sqrt{2} \). First simplify the radical and then cancel common factors.
Step 2
Why this answer is correct
The correct answer is C. \( \sqrt{2} \). \( \sqrt{18}=3\sqrt{2} \), so \( \frac{\sqrt{18}}{3}=\sqrt{2} \). First simplify the radical and then cancel common factors.
Step 3
Exam Tip
\( \sqrt{18}=3\sqrt{2} \), इसलिए \( \frac{\sqrt{18}}{3}=\sqrt{2} \) है। पहले वर्गमूल सरल करें और फिर समान गुणक काटें।
