यदि (p) एक अशून्य परिमेय संख्या है और (q) एक अपरिमेय संख्या है, तो (pq) के बारे में सही कथन कौन-सा है?
If (p) is a non-zero rational number and (q) is an irrational number, which statement about (pq) is correct?
#irrational numbers
#real numbers
#class 10
#hard
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A यह हमेशा परिमेय है / It is always rational
B यह हमेशा अपरिमेय है / It is always irrational
C यह कभी-कभी शून्य होता है / It is sometimes zero
D यह पूर्णांक होना ही चाहिए / It must be an integer
Explanation opens after your attempt
Correct Answer
B. यह हमेशा अपरिमेय है / It is always irrational
Step 1
Concept
अशून्य परिमेय संख्या से गुणा करने पर अपरिमेयता बनी रहती है। / Multiplying an irrational number by a non-zero rational number keeps it irrational.
Step 2
Why this answer is correct
यदि (pq) परिमेय मान लें, तो \(q=\frac{pq}{p}\) परिमेय हो जाएगा, जो गलत है। / If (pq) were rational, then \(q=\frac{pq}{p}\) would be rational, which contradicts the given condition.
Step 3
Exam Tip
परीक्षा में ध्यान रखें कि (p) शून्य नहीं होना चाहिए। / Always check that the rational multiplier is not zero.
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यदि \(a=\sqrt{2}+\sqrt{8}\), तो (a) किस प्रकार की संख्या है?
If \(a=\sqrt{2}+\sqrt{8}\), what type of number is (a)?
#surds
#simplification
#irrational numbers
#class 10
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A परिमेय / Rational
B अपरिमेय / Irrational
C पूर्णांक / Integer
D प्राकृतिक संख्या / Natural number
Explanation opens after your attempt
Correct Answer
B. अपरिमेय / Irrational
Step 1
Concept
\(\sqrt{8}=2\sqrt{2}\) होता है। / \(\sqrt{8}=2\sqrt{2}\).
Step 2
Why this answer is correct
इसलिए \(a=\sqrt{2}+2\sqrt{2}=3\sqrt{2}\), और \(\sqrt{2}\) अपरिमेय है। / So \(a=\sqrt{2}+2\sqrt{2}=3\sqrt{2}\), and \(\sqrt{2}\) is irrational.
Step 3
Exam Tip
समान मूल वाले पदों को पहले सरल करें। / Simplify like radical terms before deciding the type of number.
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यदि \(\sqrt{n}\) अपरिमेय है और (n) एक धनात्मक पूर्णांक है, तो (n) के बारे में सही बात क्या है?
If \(\sqrt{n}\) is irrational and (n) is a positive integer, what is true about (n)?
#square root
#perfect square
#irrational numbers
#class 10
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A (n) पूर्ण वर्ग है / (n) is a perfect square
B (n) पूर्ण वर्ग नहीं है / (n) is not a perfect square
C (n) सम होना चाहिए / (n) must be even
D (n) अभाज्य होना चाहिए / (n) must be prime
Explanation opens after your attempt
Correct Answer
B. (n) पूर्ण वर्ग नहीं है / (n) is not a perfect square
Step 1
Concept
किसी पूर्ण वर्ग का वर्गमूल पूर्णांक होता है। / The square root of a perfect square is an integer.
Step 2
Why this answer is correct
यदि (n) पूर्ण वर्ग नहीं है, तो \(\sqrt{n}\) अपरिमेय होता है। / If (n) is not a perfect square, then \(\sqrt{n}\) is irrational.
Step 3
Exam Tip
केवल सम या अभाज्य देखकर निर्णय न लें; पूर्ण वर्ग की जाँच करें। / Do not judge only by evenness or primality; check whether it is a perfect square.
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निम्न में से कौन-सा व्यंजक परिमेय है?
Which of the following expressions is rational?
#surds
#error detection
#irrational numbers
#class 10
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A \(\sqrt{3}+\sqrt{12}\)
B \(\sqrt{50}-\sqrt{8}\)
C \(\sqrt{18}-\sqrt{2}\)
D \(\sqrt{27}-\sqrt{3}\)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{50}-\sqrt{8}\)
Step 1
Concept
\(\sqrt{50}=5\sqrt{2}\) और \(\sqrt{8}=2\sqrt{2}\) हैं। / Simplify each surd carefully.
Step 2
Why this answer is correct
\(\sqrt{50}-\sqrt{8}=3\sqrt{2}\), यह अपरिमेय है; पर सही परिमेय विकल्प खोजने के लिए सभी सरल करें: \(\sqrt{18}-\sqrt{2}=3\sqrt{2}-\sqrt{2}=2\sqrt{2}\), \(\sqrt{27}-\sqrt{3}=3\sqrt{3}-\sqrt{3}=2\sqrt{3}\), और \(\sqrt{3}+\sqrt{12}=3\sqrt{3}\); कोई भी परिमेय नहीं है। / \(\sqrt{50}-\sqrt{8}=3\sqrt{2}\), \(\sqrt{18}-\sqrt{2}=2\sqrt{2}\), \(\sqrt{27}-\sqrt{3}=2\sqrt{3}\), and \(\sqrt{3}+\sqrt{12}=3\sqrt{3}\), all are irrational.
Step 3
Exam Tip
यहाँ दिए गए विकल्पों में परिमेय संख्या नहीं है, इसलिए प्रश्न में त्रुटि से बचने के लिए सही चयन नहीं बनता। / This item has no rational option, so it should be treated as an invalid question.
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यदि \(x=\sqrt{5}-2\), तो (x) के बारे में सही कथन कौन-सा है?
If \(x=\sqrt{5}-2\), which statement about (x) is correct?
#rational plus irrational
#surd
#class 10
#hard
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A (x) परिमेय है / (x) is rational
B (x) अपरिमेय है / (x) is irrational
C (x) पूर्णांक है / (x) is an integer
D (x=0) है / (x=0)
Explanation opens after your attempt
Correct Answer
B. (x) अपरिमेय है / (x) is irrational
Step 1
Concept
\(\sqrt{5}\) अपरिमेय है और (2) परिमेय है। / \(\sqrt{5}\) is irrational and (2) is rational.
Step 2
Why this answer is correct
अपरिमेय संख्या में से परिमेय संख्या घटाने पर परिणाम अपरिमेय रहता है। / Subtracting a rational number from an irrational number gives an irrational number.
Step 3
Exam Tip
किसी मूल से पूर्णांक घटाने पर भी मूल की प्रकृति पर ध्यान दें। / When an integer is subtracted from a surd, focus on the nature of the surd.
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किस विकल्प में दी गई दोनों संख्याएँ अपरिमेय हैं, पर उनका गुणनफल परिमेय है?
In which option are both numbers irrational but their product is rational?
#product of irrationals
#surds
#real numbers
#class 10
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A \(\sqrt{2},\sqrt{3}\)
B \(\sqrt{5},\sqrt{20}\)
C \(\sqrt{6},\sqrt{10}\)
D \(\sqrt{7},\sqrt{11}\)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{5},\sqrt{20}\)
Step 1
Concept
दोनों संख्याएँ अलग-अलग अपरिमेय हैं। / Both numbers are individually irrational.
Step 2
Why this answer is correct
\(\sqrt{5}\times\sqrt{20}=\sqrt{100}=10\), जो परिमेय है। / \(\sqrt{5}\times\sqrt{20}=\sqrt{100}=10\), which is rational.
Step 3
Exam Tip
ऐसे प्रश्नों में गुणन के बाद मूल के अंदर की संख्या पूर्ण वर्ग बन रही है या नहीं, यह देखें। / Check whether the product inside the square root becomes a perfect square.
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यदि (x) अपरिमेय है, तो (x+3) के परिमेय होने की संभावना के बारे में सही कथन क्या है?
If (x) is irrational, what is correct about the possibility of (x+3) being rational?
#proof idea
#irrational plus rational
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A यह हमेशा परिमेय होगा / It will always be rational
B यह कभी परिमेय नहीं हो सकता / It can never be rational
C यह तभी परिमेय होगा जब (x=3) हो / It will be rational only when (x=3)
D यह तभी परिमेय होगा जब (x) ऋणात्मक हो / It will be rational only when (x) is negative
Explanation opens after your attempt
Correct Answer
B. यह कभी परिमेय नहीं हो सकता / It can never be rational
Step 1
Concept
(3) एक परिमेय संख्या है। / (3) is rational.
Step 2
Why this answer is correct
यदि (x+3) परिमेय हो, तो (x=(x+3)-3) भी परिमेय होगा, जो दी गई बात के विरुद्ध है। / If (x+3) were rational, then (x=(x+3)-3) would also be rational, contradicting the given fact.
Step 3
Exam Tip
ऐसे प्रश्नों में विरोध विधि तेज काम करती है। / The contradiction method is useful in such questions.
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कौन-सी संख्या \(\sqrt{45}\) के बराबर है और अपरिमेय भी है?
Which number is equal to \(\sqrt{45}\) and is also irrational?
#surd simplification
#square root
#irrational numbers
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A \(3\sqrt{5}\)
B \(5\sqrt{3}\)
C \(9\sqrt{5}\)
D (15)
Explanation opens after your attempt
Correct Answer
A. \(3\sqrt{5}\)
Step 1
Concept
\(45=9\times5\) है। / \(45=9\times5\).
Step 2
Why this answer is correct
\(\sqrt{45}=\sqrt{9}\sqrt{5}=3\sqrt{5}\), और \(\sqrt{5}\) अपरिमेय है। / \(\sqrt{45}=\sqrt{9}\sqrt{5}=3\sqrt{5}\), and \(\sqrt{5}\) is irrational.
Step 3
Exam Tip
मूल को सरल करते समय सबसे बड़े पूर्ण वर्ग गुणनखंड को अलग करें। / Separate the largest perfect square factor while simplifying surds.
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निम्न में से कौन-सा युग्म दिखाता है कि दो अपरिमेय संख्याओं का योग परिमेय हो सकता है?
Which pair shows that the sum of two irrational numbers can be rational?
#counterexample
#sum of irrationals
#class 10
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A \(\sqrt{2},\sqrt{3}\)
B \(\sqrt{5},-\sqrt{5}\)
C \(\sqrt{7},\sqrt{7}\)
D \(\sqrt{11},1\)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{5},-\sqrt{5}\)
Step 1
Concept
\(\sqrt{5}\) और \(-\sqrt{5}\) दोनों अपरिमेय हैं। / \(\sqrt{5}\) and \(-\sqrt{5}\) are both irrational.
Step 2
Why this answer is correct
उनका योग (0) है, जो परिमेय संख्या है। / Their sum is (0), which is rational.
Step 3
Exam Tip
दो अपरिमेय संख्याओं के योग पर सामान्य नियम लगाने से पहले उदाहरण जाँचें। / Before applying a general rule for two irrationals, test possible counterexamples.
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यदि (m) और (n) धनात्मक पूर्णांक हैं तथा \(\sqrt{mn}\) परिमेय है, तो कौन-सी स्थिति पर्याप्त है?
If (m) and (n) are positive integers and \(\sqrt{mn}\) is rational, which condition is sufficient?
#perfect square
#irrational test
#class 10
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A (mn) पूर्ण वर्ग हो / (mn) is a perfect square
B (m+n) सम हो / (m+n) is even
C (m-n) विषम हो / (m-n) is odd
D (m) और (n) अभाज्य हों / (m) and (n) are prime
Explanation opens after your attempt
Correct Answer
A. (mn) पूर्ण वर्ग हो / (mn) is a perfect square
Step 1
Concept
किसी धनात्मक पूर्णांक का वर्गमूल परिमेय तभी होता है जब वह पूर्ण वर्ग हो। / The square root of a positive integer is rational when that integer is a perfect square.
Step 2
Why this answer is correct
इसलिए \(\sqrt{mn}\) परिमेय होने के लिए (mn) का पूर्ण वर्ग होना पर्याप्त है। / Hence (mn) being a perfect square is sufficient for \(\sqrt{mn}\) to be rational.
Step 3
Exam Tip
योग या अंतर की समता से वर्गमूल की प्रकृति तय नहीं होती। / Parity of the sum or difference does not decide the nature of the square root.
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कौन-सा व्यंजक अपरिमेय है?
Which expression is irrational?
#surds
#irrational expression
#class 10
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A (\(\sqrt{2}\)2 )
B \(\sqrt{2}\times\sqrt{8}\)
C \(\sqrt{3}+\sqrt{12}\)
D \(\sqrt{5}\times\sqrt{20}\)
Explanation opens after your attempt
Correct Answer
C. \(\sqrt{3}+\sqrt{12}\)
Step 1
Concept
(\(\sqrt{2}\)2 =2), \(\sqrt{2}\times\sqrt{8}=4\), और \(\sqrt{5}\times\sqrt{20}=10\) परिमेय हैं। / (\(\sqrt{2}\)2 =2), \(\sqrt{2}\times\sqrt{8}=4\), and \(\sqrt{5}\times\sqrt{20}=10\) are rational.
Step 2
Why this answer is correct
\(\sqrt{3}+\sqrt{12}=\sqrt{3}+2\sqrt{3}=3\sqrt{3}\), जो अपरिमेय है। / \(\sqrt{3}+\sqrt{12}=3\sqrt{3}\), which is irrational.
Step 3
Exam Tip
जोड़ और गुणन को अलग-अलग नियमों से समझें। / Treat addition and multiplication of surds differently.
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यदि \(2+\sqrt{7}\) परिमेय मान लिया जाए, तो कौन-सा निष्कर्ष विरोध पैदा करता है?
If \(2+\sqrt{7}\) is assumed rational, which conclusion creates a contradiction?
#contradiction proof
#irrational numbers
#class 10
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A \(\sqrt{7}\) परिमेय होगा / \(\sqrt{7}\) would be rational
B \(\sqrt{7}\) शून्य होगा / \(\sqrt{7}\) would be zero
C (2) अपरिमेय होगा / (2) would be irrational
D (7) ऋणात्मक होगा / (7) would be negative
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{7}\) परिमेय होगा / \(\sqrt{7}\) would be rational
Step 1
Concept
मान लें \(2+\sqrt{7}\) परिमेय है। / Assume \(2+\sqrt{7}\) is rational.
Step 2
Why this answer is correct
तब (\sqrt{7}=\(2+\sqrt{7}\)-2) परिमेय होगा, जबकि \(\sqrt{7}\) अपरिमेय है। / Then (\sqrt{7}=\(2+\sqrt{7}\)-2) would be rational, but \(\sqrt{7}\) is irrational.
Step 3
Exam Tip
विरोध सिद्ध करने में ज्ञात परिमेय संख्या को घटाकर मूल को अलग करें। / In contradiction proofs, isolate the surd using rational operations.
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यदि \(x=\sqrt{3}\) और \(y=2\sqrt{3}\), तो (x+y) के बारे में सही कथन क्या है?
If \(x=\sqrt{3}\) and \(y=2\sqrt{3}\), which statement about (x+y) is correct?
#like surds
#irrational sum
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A यह \(3\sqrt{3}\) है और अपरिमेय है / It is \(3\sqrt{3}\) and irrational
B यह \(3\sqrt{3}\) है और परिमेय है / It is \(3\sqrt{3}\) and rational
C यह \(5\sqrt{3}\) है और अपरिमेय है / It is \(5\sqrt{3}\) and irrational
D यह (6) है और परिमेय है / It is (6) and rational
Explanation opens after your attempt
Correct Answer
A. यह \(3\sqrt{3}\) है और अपरिमेय है / It is \(3\sqrt{3}\) and irrational
Step 1
Concept
समान मूल वाले पदों के गुणांक जोड़े जाते हैं। / Add the coefficients of like surds.
Step 2
Why this answer is correct
\(\sqrt{3}+2\sqrt{3}=3\sqrt{3}\), और \(\sqrt{3}\) अपरिमेय है। / \(\sqrt{3}+2\sqrt{3}=3\sqrt{3}\), and \(\sqrt{3}\) is irrational.
Step 3
Exam Tip
गुणांक जुड़ते हैं, मूल के अंदर की संख्या नहीं बदलती। / Coefficients add; the number inside the radical remains unchanged.
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कौन-सा विकल्प गलत कथन है?
Which option is a false statement?
#decimal classification
#false statement
#class 10
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A हर सांत दशमलव परिमेय होता है / Every terminating decimal is rational
B हर आवर्ती दशमलव परिमेय होता है / Every recurring decimal is rational
C हर असांत दशमलव अपरिमेय होता है / Every non-terminating decimal is irrational
D हर असांत अनावर्ती दशमलव अपरिमेय होता है / Every non-terminating non-recurring decimal is irrational
Explanation opens after your attempt
Correct Answer
C. हर असांत दशमलव अपरिमेय होता है / Every non-terminating decimal is irrational
Step 1
Concept
असांत दशमलव दो प्रकार के हो सकते हैं, आवर्ती और अनावर्ती। / Non-terminating decimals can be recurring or non-recurring.
Step 2
Why this answer is correct
असांत आवर्ती दशमलव जैसे \(0.\overline{3}\) परिमेय होता है। / A non-terminating recurring decimal like \(0.\overline{3}\) is rational.
Step 3
Exam Tip
अपरिमेय के लिए असांत के साथ अनावर्ती होना भी जरूरी है। / For an irrational decimal, it must be both non-terminating and non-recurring.
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यदि (r) परिमेय और (s) अपरिमेय है, तो (r-s) कब अपरिमेय होगा?
If (r) is rational and (s) is irrational, when will (r-s) be irrational?
#rational minus irrational
#closure
#class 10
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A हमेशा / Always
B केवल जब (r=0) हो / Only when (r=0)
C केवल जब (s>0) हो / Only when (s>0)
D कभी नहीं / Never
Explanation opens after your attempt
Correct Answer
A. हमेशा / Always
Step 1
Concept
परिमेय संख्या में से अपरिमेय संख्या घटाने पर परिणाम अपरिमेय रहता है। / A rational number minus an irrational number is irrational.
Step 2
Why this answer is correct
यदि (r-s) परिमेय हो, तो (s=r-(r-s)) परिमेय हो जाएगा, जो असंभव है। / If (r-s) were rational, then (s=r-(r-s)) would be rational, which is impossible.
Step 3
Exam Tip
घटाव में भी वही सोच रखें जो योग में रखते हैं। / Use the same reasoning for subtraction as for addition.
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निम्न में से कौन-सा उदाहरण सिद्ध करता है कि दो अपरिमेय संख्याओं का गुणनफल परिमेय हो सकता है?
Which example proves that the product of two irrational numbers can be rational?
#counterexample
#product of irrationals
#class 10
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A \(\sqrt{2}\times\sqrt{2}=2\)
B \(\sqrt{2}+\sqrt{2}=2\sqrt{2}\)
C \(\sqrt{2}\times2=2\sqrt{2}\)
D \(\sqrt{2}+2=2+\sqrt{2}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{2}\times\sqrt{2}=2\)
Step 1
Concept
\(\sqrt{2}\) एक अपरिमेय संख्या है। / \(\sqrt{2}\) is irrational.
Step 2
Why this answer is correct
दो समान अपरिमेय संख्याओं का गुणन \(\sqrt{2}\times\sqrt{2}=2\) देता है, जो परिमेय है। / Multiplying the two irrational numbers gives \(\sqrt{2}\times\sqrt{2}=2\), which is rational.
Step 3
Exam Tip
किसी सामान्य कथन को गलत दिखाने के लिए एक सही उदाहरण काफी होता है। / One valid counterexample is enough to disprove a universal statement.
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\(0.202002000200002\ldots\) किस प्रकार की संख्या है?
What type of number is \(0.202002000200002\ldots\)?
#non terminating decimal
#irrational
#class 10
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A सांत परिमेय / Terminating rational
B असांत आवर्ती परिमेय / Non-terminating recurring rational
C असांत अनावर्ती अपरिमेय / Non-terminating non-recurring irrational
D पूर्णांक / Integer
Explanation opens after your attempt
Correct Answer
C. असांत अनावर्ती अपरिमेय / Non-terminating non-recurring irrational
Step 1
Concept
यह दशमलव समाप्त नहीं हो रहा है। / The decimal does not terminate.
Step 2
Why this answer is correct
इसमें अंकों का कोई स्थिर आवर्तन नहीं है, क्योंकि शून्यों की संख्या बदलती जाती है। / It has no fixed recurring block because the number of zeros keeps changing.
Step 3
Exam Tip
असांत अनावर्ती दशमलव को अपरिमेय माना जाता है। / A non-terminating non-recurring decimal is irrational.
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कौन-सा व्यंजक (5) के बराबर है?
Which expression is equal to (5)?
#square root
#surd misconception
#class 10
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A (\(\sqrt{5}\)2 )
B \(\sqrt{5}+\sqrt{5}\)
C \(\sqrt{25}+\sqrt{5}\)
D \(\sqrt{10}-\sqrt{5}\)
Explanation opens after your attempt
Correct Answer
A. (\(\sqrt{5}\)2 )
Step 1
Concept
किसी धनात्मक संख्या के वर्गमूल का वर्ग वही संख्या देता है। / The square of the square root of a positive number gives the number itself.
Step 2
Why this answer is correct
इसलिए (\(\sqrt{5}\)2 =5), जो परिमेय है। / Hence (\(\sqrt{5}\)2 =5), which is rational.
Step 3
Exam Tip
\(\sqrt{5}+\sqrt{5}\) को (10) समझने की गलती न करें। / Do not mistake \(\sqrt{5}+\sqrt{5}\) for (10).
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यदि \(\sqrt{a}=4\sqrt{3}\), तो (a) का मान क्या होगा?
If \(\sqrt{a}=4\sqrt{3}\), what is the value of (a)?
#surd equation
#square root
#class 10
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A (12)
B (24)
C (48)
D (64)
Explanation opens after your attempt
Step 1
Concept
दोनों ओर वर्ग करें। / Square both sides.
Step 2
Why this answer is correct
(a=\(4\sqrt{3}\)2 =16\times3=48)। / (a=\(4\sqrt{3}\)2 =16\times3=48).
Step 3
Exam Tip
(\(k\sqrt{n}\)2 =k-2 n) का ध्यान रखें। / Remember that (\(k\sqrt{n}\)2 =k-2 n).
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कौन-सा विकल्प \(2\sqrt{6}\) के बराबर है?
Which option is equal to \(2\sqrt{6}\)?
#surd conversion
#irrational numbers
#class 10
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A \(\sqrt{24}\)
B \(\sqrt{12}\)
C \(\sqrt{36}\)
D \(\sqrt{48}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{24}\)
Step 1
Concept
\(2\sqrt{6}=\sqrt{4}\sqrt{6}\) लिखा जा सकता है। / \(2\sqrt{6}\) can be written as \(\sqrt{4}\sqrt{6}\).
Step 2
Why this answer is correct
इसलिए \(2\sqrt{6}=\sqrt{24}\) है। / Therefore \(2\sqrt{6}=\sqrt{24}\).
Step 3
Exam Tip
गुणांक को मूल के अंदर ले जाते समय उसका वर्ग अंदर जाता है। / When moving a coefficient inside a square root, square the coefficient.
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कौन-सा कथन \(\sqrt{12}\) के लिए सही है?
Which statement is correct for \(\sqrt{12}\)?
#simplifying radicals
#irrational numbers
#class 10
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A यह (6) के बराबर है / It is equal to (6)
B यह \(2\sqrt{3}\) के बराबर है और अपरिमेय है / It is equal to \(2\sqrt{3}\) and irrational
C यह \(3\sqrt{2}\) के बराबर है / It is equal to \(3\sqrt{2}\)
D यह परिमेय संख्या है / It is rational
Explanation opens after your attempt
Correct Answer
B. यह \(2\sqrt{3}\) के बराबर है और अपरिमेय है / It is equal to \(2\sqrt{3}\) and irrational
Step 1
Concept
\(12=4\times3\) है। / \(12=4\times3\).
Step 2
Why this answer is correct
\(\sqrt{12}=2\sqrt{3}\), और \(\sqrt{3}\) अपरिमेय है। / \(\sqrt{12}=2\sqrt{3}\), and \(\sqrt{3}\) is irrational.
Step 3
Exam Tip
मूल को सरल करने के बाद भी अंदर पूर्ण वर्ग न बचे तो संख्या अपरिमेय रहती है। / After simplification, if a non-square remains inside the root, the number stays irrational.
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यदि (x) एक अपरिमेय संख्या है, तो (-x) के बारे में सही कथन क्या है?
If (x) is an irrational number, which statement about (-x) is correct?
#negative irrational
#number type
#class 10
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A यह हमेशा परिमेय है / It is always rational
B यह हमेशा अपरिमेय है / It is always irrational
C यह हमेशा धनात्मक है / It is always positive
D यह हमेशा पूर्णांक है / It is always an integer
Explanation opens after your attempt
Correct Answer
B. यह हमेशा अपरिमेय है / It is always irrational
Step 1
Concept
ऋण चिह्न केवल संख्या की दिशा बदलता है, उसकी परिमेयता नहीं। / A negative sign changes direction, not rationality.
Step 2
Why this answer is correct
यदि (-x) परिमेय हो, तो (x) भी परिमेय होगा, जो गलत है। / If (-x) were rational, then (x) would also be rational, which is false.
Step 3
Exam Tip
संख्या के चिह्न और संख्या की प्रकृति को अलग-अलग समझें। / Treat the sign of a number and its type separately.
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निम्न में से कौन-सा परिमेय और अपरिमेय संख्या के बीच अंतर को सही बताता है?
Which option correctly states the difference between rational and irrational numbers?
#definition
#rational irrational
#class 10
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A परिमेय संख्या केवल पूर्णांक होती है / Rational numbers are only integers
B अपरिमेय संख्या को \(\frac{p}{q}\) के रूप में लिखा जा सकता है / Irrational numbers can be written as \(\frac{p}{q}\)
C परिमेय संख्या \(\frac{p}{q}\) के रूप में लिखी जा सकती है, जहाँ \(q\neq0\) / Rational numbers can be written as \(\frac{p}{q}\), where \(q\neq0\)
D हर वास्तविक संख्या अपरिमेय होती है / Every real number is irrational
Explanation opens after your attempt
Correct Answer
C. परिमेय संख्या \(\frac{p}{q}\) के रूप में लिखी जा सकती है, जहाँ \(q\neq0\) / Rational numbers can be written as \(\frac{p}{q}\), where \(q\neq0\)
Step 1
Concept
परिमेय संख्या वह है जिसे \(\frac{p}{q}\) के रूप में लिखा जा सके, जहाँ (p,q) पूर्णांक और \(q\neq0\) हों। / A rational number can be written as \(\frac{p}{q}\), where (p,q) are integers and \(q\neq0\).
Step 2
Why this answer is correct
अपरिमेय संख्या ऐसे रूप में नहीं लिखी जा सकती। / An irrational number cannot be written in that form.
Step 3
Exam Tip
परिभाषा के प्रश्न में \(q\neq0\) अवश्य देखें। / In definition questions, always check the condition \(q\neq0\).
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किस संख्या का दशमलव प्रसार असांत और अनावर्ती होगा?
Which number will have a non-terminating and non-recurring decimal expansion?
#decimal expansion
#square root
#class 10
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A \(\frac{7}{8}\)
B \(\frac{2}{3}\)
C \(\sqrt{17}\)
D (4.25)
Explanation opens after your attempt
Correct Answer
C. \(\sqrt{17}\)
Step 1
Concept
\(\frac{7}{8}\) और (4.25) सांत दशमलव देते हैं। / \(\frac{7}{8}\) and (4.25) are terminating decimals.
Step 2
Why this answer is correct
\(\frac{2}{3}\) असांत आवर्ती दशमलव देता है। \(\sqrt{17}\) अपरिमेय है, इसलिए उसका दशमलव असांत अनावर्ती होगा। / \(\frac{2}{3}\) is non-terminating recurring. \(\sqrt{17}\) is irrational, so its decimal expansion is non-terminating and non-recurring.
Step 3
Exam Tip
अपूर्ण वर्ग के वर्गमूल को तुरंत पहचानें। / Quickly identify square roots of non-perfect squares.
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यदि \(x=\sqrt{2}+\sqrt{3}\), तो \(x^2\) के बारे में सही कथन क्या है?
If \(x=\sqrt{2}+\sqrt{3}\), which statement about \(x^2\) is correct?
#surd square
#irrational expression
#class 10
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A \(x^2=5+2\sqrt{6}\), अपरिमेय / \(x^2=5+2\sqrt{6}\), irrational
B \(x^2=5\), परिमेय / \(x^2=5\), rational
C \(x^2=6\), परिमेय / \(x^2=6\), rational
D \(x^2=2\sqrt{5}\), अपरिमेय / \(x^2=2\sqrt{5}\), irrational
Explanation opens after your attempt
Correct Answer
A. \(x^2=5+2\sqrt{6}\), अपरिमेय / \(x^2=5+2\sqrt{6}\), irrational
Step 1
Concept
((a+b)2 =a-2 +2ab+b-2 ) का प्रयोग करें। / Use ((a+b)2 =a-2 +2ab+b-2 ).
Step 2
Why this answer is correct
\(x^2=2+2\sqrt{6}+3=5+2\sqrt{6}\), जिसमें अपरिमेय भाग है। / \(x^2=2+2\sqrt{6}+3=5+2\sqrt{6}\), which has an irrational part.
Step 3
Exam Tip
दो मूलों के योग का वर्ग करते समय बीच वाला पद न भूलें। / Do not forget the middle term when squaring a sum of surds.
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कौन-सा विकल्प \(\frac{1}{\sqrt{3}}\) का परिमेय हर वाला रूप है?
Which option is the rationalized form of \(\frac{1}{\sqrt{3}}\)?
#rationalization
#surd denominator
#class 10
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A \(\frac{\sqrt{3}}{3}\)
B \(\frac{3}{\sqrt{3}}\)
C \(\sqrt{3}\)
D \(\frac{1}{3\sqrt{3}}\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{\sqrt{3}}{3}\)
Step 1
Concept
हर को परिमेय बनाने के लिए ऊपर और नीचे \(\sqrt{3}\) से गुणा करें। / Multiply numerator and denominator by \(\sqrt{3}\).
Step 2
Why this answer is correct
\(\frac{1}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)। / \(\frac{1}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{3}\).
Step 3
Exam Tip
हर को मूल से मुक्त करना परीक्षा में साफ उत्तर देता है। / Rationalizing the denominator gives a cleaner exam answer.
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यदि \(a=3+\sqrt{2}\), तो (a) की प्रकृति क्या है?
If \(a=3+\sqrt{2}\), what is the nature of (a)?
#rational plus irrational
#real numbers
#class 10
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A परिमेय / Rational
B अपरिमेय / Irrational
C पूर्ण वर्ग / Perfect square
D ऋणात्मक पूर्णांक / Negative integer
Explanation opens after your attempt
Correct Answer
B. अपरिमेय / Irrational
Step 1
Concept
(3) परिमेय है और \(\sqrt{2}\) अपरिमेय है। / (3) is rational and \(\sqrt{2}\) is irrational.
Step 2
Why this answer is correct
परिमेय और अपरिमेय का योग अपरिमेय होता है। / The sum of a rational and an irrational number is irrational.
Step 3
Exam Tip
पूर्णांक जोड़ने से मूल की अपरिमेयता समाप्त नहीं होती। / Adding an integer does not remove the irrational nature of the surd.
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किस विकल्प में संख्या निश्चित रूप से अपरिमेय है?
In which option is the number definitely irrational?
#zero multiplier
#irrational numbers
#class 10
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A \(0\times\sqrt{7}\)
B \(4\sqrt{7}\)
C \(\sqrt{7}\times\sqrt{7}\)
D \(\sqrt{28}\div\sqrt{7}\)
Explanation opens after your attempt
Correct Answer
B. \(4\sqrt{7}\)
Step 1
Concept
\(0\times\sqrt{7}=0\), \(\sqrt{7}\times\sqrt{7}=7\), और \(\sqrt{28}\div\sqrt{7}=2\) परिमेय हैं। / \(0\times\sqrt{7}=0\), \(\sqrt{7}\times\sqrt{7}=7\), and \(\sqrt{28}\div\sqrt{7}=2\) are rational.
Step 2
Why this answer is correct
\(4\sqrt{7}\) में अशून्य परिमेय गुणक और अपरिमेय मूल है, इसलिए यह अपरिमेय है। / \(4\sqrt{7}\) is a non-zero rational multiple of an irrational number, so it is irrational.
Step 3
Exam Tip
शून्य से गुणा वाले विकल्प को जल्दी परिमेय पहचानें। / Quickly identify multiplication by zero as rational.
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यदि \(\sqrt{p}\) परिमेय है और (p) धनात्मक पूर्णांक है, तो (p) कैसा होगा?
If \(\sqrt{p}\) is rational and (p) is a positive integer, what must (p) be?
#perfect square
#rational square root
#class 10
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A पूर्ण वर्ग / Perfect square
B अभाज्य संख्या / Prime number
C विषम संख्या / Odd number
D अपरिमेय संख्या / Irrational number
Explanation opens after your attempt
Correct Answer
A. पूर्ण वर्ग / Perfect square
Step 1
Concept
धनात्मक पूर्णांक का परिमेय वर्गमूल तभी मिलता है जब वह पूर्ण वर्ग हो। / A positive integer has a rational square root only when it is a perfect square.
Step 2
Why this answer is correct
जैसे (16) का वर्गमूल (4) है, पर (18) का वर्गमूल अपरिमेय है। / For example, \(\sqrt{16}=4\), but \(\sqrt{18}\) is irrational.
Step 3
Exam Tip
वर्गमूल की प्रकृति जानने के लिए पूर्ण वर्ग जाँचें। / Check perfect squares to decide the nature of a square root.
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निम्न में से कौन-सा कथन \(\sqrt{2}\) की अपरिमेयता के प्रमाण में प्रयोग होता है?
Which statement is used in proving the irrationality of \(\sqrt{2}\)?
#proof of sqrt 2
#parity
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A यदि \(a^2\) सम है, तो (a) सम है / If \(a^2\) is even, then (a) is even
B यदि (a) सम है, तो (a) अभाज्य है / If (a) is even, then (a) is prime
C हर विषम संख्या पूर्ण वर्ग है / Every odd number is a perfect square
D हर सम संख्या अपरिमेय है / Every even number is irrational
Explanation opens after your attempt
Correct Answer
A. यदि \(a^2\) सम है, तो (a) सम है / If \(a^2\) is even, then (a) is even
Step 1
Concept
\(\sqrt{2}\) के प्रमाण में मानते हैं कि \(\sqrt{2}=\frac{a}{b}\) है। / In the proof for \(\sqrt{2}\), we assume \(\sqrt{2}=\frac{a}{b}\).
Step 2
Why this answer is correct
इससे \(a^2=2b^2\) मिलता है, इसलिए \(a^2\) सम है और (a) सम होगा। / This gives \(a^2=2b^2\), so \(a^2\) is even and hence (a) is even.
Step 3
Exam Tip
समता वाला यह तर्क विरोध तक पहुँचाता है। / This parity argument leads to a contradiction.
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किस विकल्प में अपरिमेय संख्या का वर्ग परिमेय है?
In which option is the square of an irrational number rational?
#square of irrational
#counterexample
#class 10
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A (\(\sqrt{13}\)2 )
B (\(1+\sqrt{2}\)2 )
C (\(\sqrt{2}+\sqrt{3}\)2 )
D (\(2+\sqrt{5}\)2 )
Explanation opens after your attempt
Correct Answer
A. (\(\sqrt{13}\)2 )
Step 1
Concept
\(\sqrt{13}\) अपरिमेय है। / \(\sqrt{13}\) is irrational.
Step 2
Why this answer is correct
इसका वर्ग (\(\sqrt{13}\)2 =13) परिमेय है। / Its square is (\(\sqrt{13}\)2 =13), which is rational.
Step 3
Exam Tip
हर अपरिमेय संख्या का वर्ग अपरिमेय नहीं होता, इसलिए उदाहरण ध्यान से देखें। / The square of an irrational number is not always irrational, so examine examples carefully.
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कौन-सी संख्या \(\sqrt{72}\) का सरल रूप है?
Which number is the simplified form of \(\sqrt{72}\)?
#surd simplification
#irrational
#class 10
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A \(6\sqrt{2}\)
B \(8\sqrt{2}\)
C \(12\sqrt{2}\)
D \(3\sqrt{8}\)
Explanation opens after your attempt
Correct Answer
A. \(6\sqrt{2}\)
Step 1
Concept
\(72=36\times2\) है। / \(72=36\times2\).
Step 2
Why this answer is correct
\(\sqrt{72}=\sqrt{36}\sqrt{2}=6\sqrt{2}\), जो अपरिमेय है। / \(\sqrt{72}=\sqrt{36}\sqrt{2}=6\sqrt{2}\), which is irrational.
Step 3
Exam Tip
सबसे बड़ा पूर्ण वर्ग गुणनखंड लेने से सरल रूप जल्दी मिलता है। / Use the largest perfect square factor for quick simplification.
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यदि \(x=\frac{2}{\sqrt{5}}\), तो (x) की प्रकृति क्या है?
If \(x=\frac{2}{\sqrt{5}}\), what is the nature of (x)?
#rationalization
#irrational fraction
#class 10
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A परिमेय / Rational
B अपरिमेय / Irrational
C पूर्णांक / Integer
D शून्य / Zero
Explanation opens after your attempt
Correct Answer
B. अपरिमेय / Irrational
Step 1
Concept
हर में \(\sqrt{5}\) है, जो अपरिमेय है। / The denominator has \(\sqrt{5}\), which is irrational.
Step 2
Why this answer is correct
हर को परिमेय बनाने पर \(x=\frac{2\sqrt{5}}{5}\) मिलेगा, जो अशून्य परिमेय गुणक के साथ अपरिमेय है। / Rationalizing gives \(x=\frac{2\sqrt{5}}{5}\), a non-zero rational multiple of an irrational number.
Step 3
Exam Tip
हर परिमेय बनाने से संख्या की प्रकृति साफ दिखती है। / Rationalizing the denominator often reveals the number type clearly.
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निम्न में से कौन-सा योग परिमेय है?
Which of the following sums is rational?
#cancellation of surds
#rational sum
#class 10
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A (\(2+\sqrt{3}\)+\(5-\sqrt{3}\))
B (\(1+\sqrt{2}\)+\(1+\sqrt{2}\))
C \(\sqrt{5}+\sqrt{20}\)
D \(\sqrt{7}+2\)
Explanation opens after your attempt
Correct Answer
A. (\(2+\sqrt{3}\)+\(5-\sqrt{3}\))
Step 1
Concept
पहले समान अपरिमेय पदों को देखें। / First look for like irrational terms.
Step 2
Why this answer is correct
(\(2+\sqrt{3}\)+\(5-\sqrt{3}\)=7), क्योंकि \(\sqrt{3}\) और \(-\sqrt{3}\) कट जाते हैं। / (\(2+\sqrt{3}\)+\(5-\sqrt{3}\)=7) because \(\sqrt{3}\) and \(-\sqrt{3}\) cancel.
Step 3
Exam Tip
विपरीत अपरिमेय पदों से परिमेय उत्तर बन सकता है। / Opposite irrational terms can produce a rational result.
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यदि (x) अपरिमेय है और \(x^2=7\), तो (x) का संभावित मान कौन-सा है?
If (x) is irrational and \(x^2=7\), which can be the value of (x)?
#square equation
#irrational root
#class 10
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A (7)
B \(\sqrt{7}\)
C \(\frac{7}{2}\)
D (49)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{7}\)
Step 1
Concept
\(x^2=7\) से \(x=\sqrt{7}\) या \(x=-\sqrt{7}\) हो सकता है। / From \(x^2=7\), \(x=\sqrt{7}\) or \(x=-\sqrt{7}\).
Step 2
Why this answer is correct
दिए गए विकल्पों में \(\sqrt{7}\) है, जो अपरिमेय है। / Among the options, \(\sqrt{7}\) is present and it is irrational.
Step 3
Exam Tip
वर्ग समीकरण में धन और ऋण दोनों मूल याद रखें, पर विकल्प के अनुसार चुनें। / Remember both positive and negative roots, then match the given options.
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किस विकल्प में \(\sqrt{a}+\sqrt{b}\) निश्चित रूप से परिमेय है?
In which option is \(\sqrt{a}+\sqrt{b}\) definitely rational?
#sum of roots
#perfect square
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A (a=2,b=8)
B (a=9,b=16)
C (a=3,b=12)
D (a=5,b=20)
Explanation opens after your attempt
Correct Answer
B. (a=9,b=16)
Step 1
Concept
\(\sqrt{9}=3\) और \(\sqrt{16}=4\) हैं। / \(\sqrt{9}=3\) and \(\sqrt{16}=4\).
Step 2
Why this answer is correct
इनका योग (7) परिमेय है। / Their sum is (7), which is rational.
Step 3
Exam Tip
दोनों मूल यदि पूर्ण वर्गों के हों, तो योग आसानी से परिमेय बनता है। / If both radicands are perfect squares, the sum is easily rational.
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कौन-सा विकल्प \(3-\sqrt{2}\) की प्रकृति सही बताता है?
Which option correctly describes the nature of \(3-\sqrt{2}\)?
#rational minus irrational
#number type
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A परिमेय, क्योंकि (3) परिमेय है / Rational because (3) is rational
B अपरिमेय, क्योंकि परिमेय में से अपरिमेय घटा है / Irrational because an irrational is subtracted from a rational
C पूर्णांक, क्योंकि घटाव किया गया है / Integer because subtraction is done
D शून्य, क्योंकि दोनों पद कट जाते हैं / Zero because both terms cancel
Explanation opens after your attempt
Correct Answer
B. अपरिमेय, क्योंकि परिमेय में से अपरिमेय घटा है / Irrational because an irrational is subtracted from a rational
Step 1
Concept
(3) परिमेय है और \(\sqrt{2}\) अपरिमेय है। / (3) is rational and \(\sqrt{2}\) is irrational.
Step 2
Why this answer is correct
परिमेय संख्या में से अपरिमेय घटाने पर परिणाम अपरिमेय रहता है। / A rational number minus an irrational number remains irrational.
Step 3
Exam Tip
केवल परिमेय पद देखकर पूरे व्यंजक को परिमेय न मानें। / Do not classify the whole expression by looking only at the rational part.
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निम्न में से कौन-सा मान \(1+\sqrt{2}\) और \(1-\sqrt{2}\) के गुणनफल के बराबर है?
Which value equals the product of \(1+\sqrt{2}\) and \(1-\sqrt{2}\)?
#conjugates
#difference of squares
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A (1)
B (3)
C (-1)
D \(2\sqrt{2}\)
Explanation opens after your attempt
Step 1
Concept
यह संयुग्मी संख्याओं का गुणन है। / This is a product of conjugates.
Step 2
Why this answer is correct
(\(1+\sqrt{2}\)\(1-\sqrt{2}\)=1-\(\sqrt{2}\)2 =1-2=-1)। / (\(1+\sqrt{2}\)\(1-\sqrt{2}\)=1-\(\sqrt{2}\)2 =1-2=-1).
Step 3
Exam Tip
संयुग्मी गुणन में बीच के अपरिमेय पद कट जाते हैं। / In conjugate multiplication, the middle irrational terms cancel.
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कौन-सा विकल्प असांत आवर्ती दशमलव है और इसलिए परिमेय है?
Which option is a non-terminating recurring decimal and hence rational?
#recurring decimal
#rational number
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A \(0.123123123\ldots\)
B \(0.1020030004\ldots\)
C \(0.1010010001\ldots\)
D \(0.1234567891011\ldots\)
Explanation opens after your attempt
Correct Answer
A. \(0.123123123\ldots\)
Step 1
Concept
\(0.123123123\ldots\) में (123) बार-बार आ रहा है। / In \(0.123123123\ldots\), the block (123) repeats.
Step 2
Why this answer is correct
आवर्ती दशमलव परिमेय होता है। / A recurring decimal is rational.
Step 3
Exam Tip
केवल असांत देखकर अपरिमेय न मानें; आवर्तन जरूर जाँचें। / Do not call a decimal irrational just because it is non-terminating; check repetition.
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यदि (a) अपरिमेय है और (b) अपरिमेय है, तो कौन-सा निष्कर्ष हमेशा सही नहीं है?
If (a) is irrational and (b) is irrational, which conclusion is not always correct?
#closure misconception
#irrational numbers
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A (a+b) अपरिमेय है / (a+b) is irrational
B (a-b) परिमेय हो सकता है / (a-b) can be rational
C (ab) परिमेय हो सकता है / (ab) can be rational
D \(\frac{a}{b}\) परिमेय हो सकता है, यदि \(b\neq0\) / \(\frac{a}{b}\) can be rational if \(b\neq0\)
Explanation opens after your attempt
Correct Answer
A. (a+b) अपरिमेय है / (a+b) is irrational
Step 1
Concept
दो अपरिमेय संख्याओं का योग कभी परिमेय भी हो सकता है। / The sum of two irrational numbers can be rational.
Step 2
Why this answer is correct
उदाहरण (\sqrt{2}+\(-\sqrt{2}\)=0) है। इसलिए (a+b) हमेशा अपरिमेय कहना गलत है। / For example, (\sqrt{2}+\(-\sqrt{2}\)=0). Therefore, saying (a+b) is always irrational is false.
Step 3
Exam Tip
दो अपरिमेय संख्याओं पर हमेशा वाले नियम बहुत सावधानी से लगाएँ। / Be careful with universal statements about two irrational numbers.
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कौन-सी संख्या \(\sqrt{27}+\sqrt{12}\) का सरल रूप है?
Which number is the simplified form of \(\sqrt{27}+\sqrt{12}\)?
#addition of surds
#common mistake
#class 10
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A \(5\sqrt{3}\)
B \(3\sqrt{5}\)
C (15)
D \(\sqrt{39}\)
Explanation opens after your attempt
Correct Answer
A. \(5\sqrt{3}\)
Step 1
Concept
\(\sqrt{27}=3\sqrt{3}\) और \(\sqrt{12}=2\sqrt{3}\)। / \(\sqrt{27}=3\sqrt{3}\) and \(\sqrt{12}=2\sqrt{3}\).
Step 2
Why this answer is correct
योग \(3\sqrt{3}+2\sqrt{3}=5\sqrt{3}\), जो अपरिमेय है। / The sum is \(3\sqrt{3}+2\sqrt{3}=5\sqrt{3}\), which is irrational.
Step 3
Exam Tip
अलग-अलग मूलों को सीधे जोड़कर \(\sqrt{39}\) न लिखें। / Do not combine separate square roots as \(\sqrt{39}\).
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यदि \(\sqrt{2}\) को \(\frac{p}{q}\) के रूप में लिखा जाए, जहाँ (p) और (q) सहअभाज्य हैं, तो प्रमाण में अंततः क्या विरोध मिलता है?
If \(\sqrt{2}\) is written as \(\frac{p}{q}\), where (p) and (q) are coprime, what contradiction appears in the proof?
#sqrt 2 proof
#coprime
#irrationality
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A (p) और (q) दोनों सम निकलते हैं / Both (p) and (q) turn out even
B (p) और (q) दोनों विषम निकलते हैं / Both (p) and (q) turn out odd
C (p) और (q) दोनों शून्य निकलते हैं / Both (p) and (q) turn out zero
D (p) और (q) दोनों अभाज्य निकलते हैं / Both (p) and (q) turn out prime
Explanation opens after your attempt
Correct Answer
A. (p) और (q) दोनों सम निकलते हैं / Both (p) and (q) turn out even
Step 1
Concept
सहअभाज्य मानने का अर्थ है कि (p) और (q) में (1) के अलावा कोई समान गुणनखंड नहीं है। / Coprime means (p) and (q) have no common factor except (1).
Step 2
Why this answer is correct
\(\sqrt{2}\) के प्रमाण में (p) और (q) दोनों सम निकलते हैं, यानी उनमें (2) समान गुणनखंड है। / In the proof of \(\sqrt{2}\), both (p) and (q) turn out even, so they have common factor (2).
Step 3
Exam Tip
यही विरोध सिद्ध करता है कि \(\sqrt{2}\) परिमेय नहीं है। / This contradiction proves that \(\sqrt{2}\) is not rational.
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कौन-सा विकल्प \(\sqrt{32}-\sqrt{2}\) का सही सरल रूप और प्रकृति बताता है?
Which option gives the correct simplified form and nature of \(\sqrt{32}-\sqrt{2}\)?
#subtraction of surds
#irrational numbers
#class 10
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A \(3\sqrt{2}\), अपरिमेय / \(3\sqrt{2}\), irrational
B \(4\sqrt{2}\), अपरिमेय / \(4\sqrt{2}\), irrational
C \(5\sqrt{2}\), अपरिमेय / \(5\sqrt{2}\), irrational
D (30), परिमेय / (30), rational
Explanation opens after your attempt
Correct Answer
A. \(3\sqrt{2}\), अपरिमेय / \(3\sqrt{2}\), irrational
Step 1
Concept
\(\sqrt{32}=4\sqrt{2}\) है। / \(\sqrt{32}=4\sqrt{2}\).
Step 2
Why this answer is correct
\(\sqrt{32}-\sqrt{2}=4\sqrt{2}-\sqrt{2}=3\sqrt{2}\), जो अपरिमेय है। / \(\sqrt{32}-\sqrt{2}=4\sqrt{2}-\sqrt{2}=3\sqrt{2}\), which is irrational.
Step 3
Exam Tip
समान मूल वाले पदों में केवल गुणांक घटाएँ। / For like surds, subtract only the coefficients.
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यदि \(x=\sqrt{11}+\sqrt{44}\), तो (x) का सरल रूप और प्रकृति क्या है?
If \(x=\sqrt{11}+\sqrt{44}\), what is the simplified form and nature of (x)?
#surds
#irrational numbers
#like radicals
#class 10
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A \(3\sqrt{11}\), अपरिमेय / \(3\sqrt{11}\), irrational
B \(5\sqrt{11}\), अपरिमेय / \(5\sqrt{11}\), irrational
C (55), परिमेय / (55), rational
D \(\sqrt{55}\), अपरिमेय / \(\sqrt{55}\), irrational
Explanation opens after your attempt
Correct Answer
A. \(3\sqrt{11}\), अपरिमेय / \(3\sqrt{11}\), irrational
Step 1
Concept
\(\sqrt{44}=\sqrt{4\times11}=2\sqrt{11}\) होता है। / \(\sqrt{44}=\sqrt{4\times11}=2\sqrt{11}\).
Step 2
Why this answer is correct
इसलिए \(x=\sqrt{11}+2\sqrt{11}=3\sqrt{11}\), और \(\sqrt{11}\) अपरिमेय है। / Hence \(x=\sqrt{11}+2\sqrt{11}=3\sqrt{11}\), and \(\sqrt{11}\) is irrational.
Step 3
Exam Tip
समान मूल वाले पदों में केवल गुणांक जोड़ें, मूल के अंदर की संख्या नहीं। / For like surds, add only the coefficients, not the numbers inside the roots.
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किस विकल्प में दिया गया दशमलव परिमेय है, यद्यपि वह समाप्त नहीं होता?
Which option gives a rational decimal even though it does not terminate?
#recurring decimal
#rational decimal
#irrational numbers
#class 10
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A \(0.1101001000100001\ldots\)
B \(0.37373737\ldots\)
C \(0.1234567891011\ldots\)
D \(0.101001000100001\ldots\)
Explanation opens after your attempt
Correct Answer
B. \(0.37373737\ldots\)
Step 1
Concept
असांत दशमलव परिमेय भी हो सकता है, यदि उसमें आवर्तन हो। / A non-terminating decimal can still be rational if it is recurring.
Step 2
Why this answer is correct
\(0.37373737\ldots\) में (37) बार-बार आ रहा है, इसलिए यह परिमेय है। / In \(0.37373737\ldots\), the block (37) repeats, so it is rational.
Step 3
Exam Tip
असांत देखकर तुरंत अपरिमेय न मानें; आवर्ती भाग को ध्यान से पहचानें। / Do not call a decimal irrational just because it is non-terminating; check for a repeating block.
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यदि \(a=\sqrt{3}+2\) और \(b=\sqrt{3}-2\), तो (ab) की प्रकृति क्या है?
If \(a=\sqrt{3}+2\) and \(b=\sqrt{3}-2\), what is the nature of (ab)?
#conjugate surds
#rational product
#class 10
#hard
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A परिमेय और ऋणात्मक / Rational and negative
B अपरिमेय और धनात्मक / Irrational and positive
C परिमेय और धनात्मक / Rational and positive
D अपरिमेय और ऋणात्मक / Irrational and negative
Explanation opens after your attempt
Correct Answer
A. परिमेय और ऋणात्मक / Rational and negative
Step 1
Concept
(a) और (b) संयुग्मी रूप में हैं। / (a) and (b) are conjugates.
Step 2
Why this answer is correct
(ab=\(\sqrt{3}\)2 -22 =3-4=-1), जो परिमेय और ऋणात्मक है। / (ab=\(\sqrt{3}\)2 -22 =3-4=-1), which is rational and negative.
Step 3
Exam Tip
संयुग्मी गुणन में बीच के अपरिमेय पद कट जाते हैं। / In conjugate multiplication, the middle irrational terms cancel.
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निम्न में से कौन-सा व्यंजक निश्चित रूप से अपरिमेय है?
Which of the following expressions is definitely irrational?
#surd subtraction
#irrational expression
#class 10
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A \(\sqrt{18}-3\sqrt{2}\)
B \(\sqrt{50}-5\sqrt{2}\)
C \(\sqrt{75}-4\sqrt{3}\)
D \(\sqrt{98}-7\sqrt{2}\)
Explanation opens after your attempt
Correct Answer
C. \(\sqrt{75}-4\sqrt{3}\)
Step 1
Concept
पहले हर मूल को सरल करें। / Simplify each radical first.
Step 2
Why this answer is correct
\(\sqrt{75}=5\sqrt{3}\), इसलिए \(\sqrt{75}-4\sqrt{3}=\sqrt{3}\), जो अपरिमेय है। / \(\sqrt{75}=5\sqrt{3}\), so \(\sqrt{75}-4\sqrt{3}=\sqrt{3}\), which is irrational.
Step 3
Exam Tip
जिन विकल्पों में समान पद पूरी तरह कट रहे हों, वे शून्य परिमेय दे सकते हैं। / Options where like terms cancel completely may give rational zero.
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यदि \(\frac{5}{\sqrt{k}}\) अपरिमेय है और (k) धनात्मक पूर्णांक है, तो कौन-सा (k) संभव है?
If \(\frac{5}{\sqrt{k}}\) is irrational and (k) is a positive integer, which (k) is possible?
#perfect square
#irrational fraction
#square root
#class 10
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A (4)
B (9)
C (16)
D (18)
Explanation opens after your attempt
Step 1
Concept
यदि (k) पूर्ण वर्ग हो, तो \(\sqrt{k}\) परिमेय होगा और भिन्न परिमेय बन जाएगी। / If (k) is a perfect square, then \(\sqrt{k}\) is rational and the fraction becomes rational.
Step 2
Why this answer is correct
(18) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{18}\) अपरिमेय है और \(\frac{5}{\sqrt{18}}\) भी अपरिमेय रहेगा। / (18) is not a perfect square, so \(\sqrt{18}\) is irrational and \(\frac{5}{\sqrt{18}}\) remains irrational.
Step 3
Exam Tip
ऐसे प्रश्नों में पहले (k) के पूर्ण वर्ग होने की जाँच करें। / In such questions, first check whether (k) is a perfect square.
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कौन-सा विकल्प दो अलग-अलग अपरिमेय संख्याओं का उदाहरण है जिनका भागफल परिमेय है?
Which option is an example of two different irrational numbers whose quotient is rational?
#quotient of irrationals
#counterexample
#class 10
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A \(\frac{\sqrt{2}}{\sqrt{3}}\)
B \(\frac{\sqrt{12}}{\sqrt{3}}\)
C \(\frac{\sqrt{5}}{\sqrt{20}}\)
D \(\frac{\sqrt{7}}{2}\)
Explanation opens after your attempt
Correct Answer
C. \(\frac{\sqrt{5}}{\sqrt{20}}\)
Step 1
Concept
\(\sqrt{5}\) और \(\sqrt{20}=2\sqrt{5}\) दोनों अपरिमेय हैं और अलग-अलग हैं। / \(\sqrt{5}\) and \(\sqrt{20}=2\sqrt{5}\) are both irrational and different.
Step 2
Why this answer is correct
\(\frac{\sqrt{5}}{\sqrt{20}}=\frac{\sqrt{5}}{2\sqrt{5}}=\frac{1}{2}\), जो परिमेय है। / \(\frac{\sqrt{5}}{\sqrt{20}}=\frac{\sqrt{5}}{2\sqrt{5}}=\frac{1}{2}\), which is rational.
Step 3
Exam Tip
भागफल में समान अपरिमेय गुणनखंड कट सकता है। / A common irrational factor can cancel in a quotient.
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यदि \(x=4+\sqrt{6}\), तो (x-4) की प्रकृति क्या होगी?
If \(x=4+\sqrt{6}\), what will be the nature of (x-4)?
#irrational expression
#simplification
#class 10
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A परिमेय / Rational
B अपरिमेय / Irrational
C पूर्णांक / Integer
D सांत दशमलव / Terminating decimal
Explanation opens after your attempt
Correct Answer
B. अपरिमेय / Irrational
Step 1
Concept
(x-4=\(4+\sqrt{6}\)-4) है। / (x-4=\(4+\sqrt{6}\)-4).
Step 2
Why this answer is correct
सरल करने पर \(x-4=\sqrt{6}\), और (6) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{6}\) अपरिमेय है। / On simplifying, \(x-4=\sqrt{6}\), and since (6) is not a perfect square, \(\sqrt{6}\) is irrational.
Step 3
Exam Tip
व्यंजक में परिमेय पद कट जाए तो बचे हुए मूल की प्रकृति देखें। / When rational terms cancel, check the nature of the remaining radical.
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यदि (r) एक अशून्य परिमेय संख्या है और (x) अपरिमेय संख्या है, तो \(\frac{x}{r}\) के बारे में सही कथन कौन-सा है?
If (r) is a non-zero rational number and (x) is an irrational number, which statement about \(\frac{x}{r}\) is correct?
#irrational numbers
#rational divisor
#class 10
#hard
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A यह हमेशा परिमेय है / It is always rational
B यह हमेशा अपरिमेय है / It is always irrational
C यह हमेशा पूर्णांक है / It is always an integer
D यह शून्य हो सकता है / It can be zero
Explanation opens after your attempt
Correct Answer
B. यह हमेशा अपरिमेय है / It is always irrational
Step 1
Concept
अशून्य परिमेय संख्या से भाग देना उसी के व्युत्क्रम से गुणा करने जैसा है। / Dividing by a non-zero rational number is the same as multiplying by its reciprocal.
Step 2
Why this answer is correct
\(\frac{1}{r}\) भी अशून्य परिमेय है, इसलिए \(\frac{x}{r}\) अपरिमेय रहेगा। / \(\frac{1}{r}\) is also a non-zero rational number, so \(\frac{x}{r}\) remains irrational.
Step 3
Exam Tip
भाग वाले प्रश्न में हर के शून्य न होने की शर्त जरूर देखें। / In division questions, always check that the denominator is not zero.
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कौन-सा विकल्प \(\sqrt{2}+\sqrt{18}\) का सही सरल रूप और प्रकृति बताता है?
Which option gives the correct simplified form and nature of \(\sqrt{2}+\sqrt{18}\)?
#surds
#like radicals
#irrational numbers
#class 10
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A \(4\sqrt{2}\), अपरिमेय / \(4\sqrt{2}\), irrational
B \(5\sqrt{2}\), अपरिमेय / \(5\sqrt{2}\), irrational
C (20), परिमेय / (20), rational
D \(\sqrt{20}\), अपरिमेय / \(\sqrt{20}\), irrational
Explanation opens after your attempt
Correct Answer
A. \(4\sqrt{2}\), अपरिमेय / \(4\sqrt{2}\), irrational
Step 1
Concept
\(\sqrt{18}=3\sqrt{2}\) होता है। / \(\sqrt{18}=3\sqrt{2}\).
Step 2
Why this answer is correct
\(\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}\), जो अपरिमेय है। / \(\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}\), which is irrational.
Step 3
Exam Tip
समान मूल वाले पदों में केवल बाहर के गुणांक जोड़ें। / For like surds, add only the outside coefficients.
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निम्न में से कौन-सा दशमलव अपरिमेय संख्या दर्शाता है?
Which of the following decimals represents an irrational number?
#decimal expansion
#non recurring
#irrational numbers
#class 10
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A \(2.454545\ldots\)
B (3.125)
C \(1.01001000100001\ldots\)
D (4.6000)
Explanation opens after your attempt
Correct Answer
C. \(1.01001000100001\ldots\)
Step 1
Concept
सांत और आवर्ती दशमलव परिमेय होते हैं। / Terminating and recurring decimals are rational.
Step 2
Why this answer is correct
\(1.01001000100001\ldots\) असांत है और इसमें कोई स्थिर आवर्तन नहीं है। / \(1.01001000100001\ldots\) is non-terminating and has no fixed repeating block.
Step 3
Exam Tip
अपरिमेय दशमलव पहचानने के लिए असांत और अनावर्ती दोनों बातें देखें। / To identify an irrational decimal, check both non-termination and non-repetition.
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यदि \(x=\sqrt{3}+2\) है, तो \(x-\sqrt{3}\) का मान और प्रकृति क्या होगी?
If \(x=\sqrt{3}+2\), what will be the value and nature of \(x-\sqrt{3}\)?
#cancellation of surds
#rational result
#class 10
#hard
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A (2), परिमेय / (2), rational
B \(\sqrt{3}\), अपरिमेय / \(\sqrt{3}\), irrational
C \(2\sqrt{3}\), अपरिमेय / \(2\sqrt{3}\), irrational
D (5), परिमेय / (5), rational
Explanation opens after your attempt
Correct Answer
A. (2), परिमेय / (2), rational
Step 1
Concept
दिए गए (x) का मान रखें। / Substitute the given value of (x).
Step 2
Why this answer is correct
(x-\sqrt{3}=\(\sqrt{3}+2\)-\sqrt{3}=2), जो परिमेय है। / (x-\sqrt{3}=\(\sqrt{3}+2\)-\sqrt{3}=2), which is rational.
Step 3
Exam Tip
समान अपरिमेय पद कट सकते हैं, इसलिए सरल करने के बाद ही प्रकृति तय करें। / Like irrational terms may cancel, so decide the nature only after simplifying.
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किस विकल्प में दी गई संख्या निश्चित रूप से अपरिमेय है?
In which option is the given number definitely irrational?
#surd simplification
#irrational test
#class 10
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A \(\sqrt{64}\)
B \(\frac{\sqrt{45}}{3}\)
C \(\sqrt{12}\times\sqrt{3}\)
D \(\sqrt{50}\div\sqrt{2}\)
Explanation opens after your attempt
Correct Answer
B. \(\frac{\sqrt{45}}{3}\)
Step 1
Concept
\(\frac{\sqrt{45}}{3}=\frac{3\sqrt{5}}{3}=\sqrt{5}\) है। / \(\frac{\sqrt{45}}{3}=\frac{3\sqrt{5}}{3}=\sqrt{5}\).
Step 2
Why this answer is correct
(5) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{5}\) अपरिमेय है। / Since (5) is not a perfect square, \(\sqrt{5}\) is irrational.
Step 3
Exam Tip
भाग और गुणन वाले विकल्पों को सरल किए बिना उत्तर न चुनें। / Do not choose an answer in multiplication or division of surds without simplifying.
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यदि \(\sqrt{n}\) परिमेय है और (n) एक धनात्मक पूर्णांक है, तो (n=72) के बारे में सही निर्णय क्या है?
If \(\sqrt{n}\) is rational when (n) is a positive integer, what is the correct decision for (n=72)?
#perfect square
#square root
#irrational numbers
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A \(\sqrt{72}\) परिमेय है क्योंकि (72) सम है / \(\sqrt{72}\) is rational because (72) is even
B \(\sqrt{72}\) अपरिमेय है क्योंकि (72) पूर्ण वर्ग नहीं है / \(\sqrt{72}\) is irrational because (72) is not a perfect square
C \(\sqrt{72}\) पूर्णांक है / \(\sqrt{72}\) is an integer
D \(\sqrt{72}\) शून्य है / \(\sqrt{72}\) is zero
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{72}\) अपरिमेय है क्योंकि (72) पूर्ण वर्ग नहीं है / \(\sqrt{72}\) is irrational because (72) is not a perfect square
Step 1
Concept
धनात्मक पूर्णांक का वर्गमूल परिमेय तभी होता है जब वह पूर्ण वर्ग हो। / The square root of a positive integer is rational only when the integer is a perfect square.
Step 2
Why this answer is correct
(72) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{72}\) अपरिमेय है। / (72) is not a perfect square, so \(\sqrt{72}\) is irrational.
Step 3
Exam Tip
सम संख्या होना वर्गमूल को परिमेय नहीं बनाता। / Being even does not make a square root rational.
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यदि \(a=5+\sqrt{7}\) और \(b=5-\sqrt{7}\), तो (ab) का मान क्या है?
If \(a=5+\sqrt{7}\) and \(b=5-\sqrt{7}\), what is the value of (ab)?
#conjugate surds
#rational product
#class 10
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A (18)
B \(25+\sqrt{7}\)
C (32)
D \(10\sqrt{7}\)
Explanation opens after your attempt
Step 1
Concept
यह संयुग्मी संख्याओं का गुणन है। / This is multiplication of conjugates.
Step 2
Why this answer is correct
(ab=52 -\(\sqrt{7}\)2 =25-7=18)। / (ab=52 -\(\sqrt{7}\)2 =25-7=18).
Step 3
Exam Tip
संयुग्मी गुणन में बीच के अपरिमेय पद कट जाते हैं। / In conjugate multiplication, the middle irrational terms cancel.
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कौन-सी संख्या (3) और (4) के बीच एक अपरिमेय संख्या है?
Which number is an irrational number between (3) and (4)?
#number line
#irrational between integers
#class 10
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A \(\sqrt{9}\)
B \(\sqrt{10}\)
C \(\frac{7}{2}\)
D (3.75)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{10}\)
Step 1
Concept
\(3=\sqrt{9}\) और \(4=\sqrt{16}\) हैं। / \(3=\sqrt{9}\) and \(4=\sqrt{16}\).
Step 2
Why this answer is correct
(10) पूर्ण वर्ग नहीं है और (9<10<16), इसलिए \(\sqrt{10}\) (3) और (4) के बीच अपरिमेय संख्या है। / (10) is not a perfect square and (9<10<16), so \(\sqrt{10}\) is an irrational number between (3) and (4).
Step 3
Exam Tip
दो पूर्णांकों के बीच अपरिमेय खोजने में बीच का अपूर्ण वर्ग उपयोगी होता है। / A non-perfect square between two square numbers helps find an irrational number between two integers.
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यदि \(\frac{1}{x}\) परिमेय है और \(x\neq0\), तो (x) के अपरिमेय होने के बारे में सही निष्कर्ष क्या है?
If \(\frac{1}{x}\) is rational and \(x\neq0\), what is the correct conclusion about (x) being irrational?
#reciprocal
#rational irrational
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A (x) अवश्य अपरिमेय है / (x) must be irrational
B (x) अवश्य परिमेय है / (x) must be rational
C (x) अवश्य शून्य है / (x) must be zero
D (x) अवश्य ऋणात्मक है / (x) must be negative
Explanation opens after your attempt
Correct Answer
B. (x) अवश्य परिमेय है / (x) must be rational
Step 1
Concept
यदि \(\frac{1}{x}\) परिमेय है और शून्य नहीं है, तो उसका व्युत्क्रम भी परिमेय होगा। / If \(\frac{1}{x}\) is rational and non-zero, then its reciprocal is also rational.
Step 2
Why this answer is correct
इसलिए (x) परिमेय होगा, अपरिमेय नहीं। / Therefore (x) is rational, not irrational.
Step 3
Exam Tip
व्युत्क्रम वाले प्रश्नों में शून्य की शर्त जरूर देखें। / In reciprocal questions, always check the non-zero condition.
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कौन-सा विकल्प \(\frac{3}{2+\sqrt{5}}\) का परिमेय हर वाला रूप है?
Which option is the rationalized form of \(\frac{3}{2+\sqrt{5}}\)?
#rationalization
#conjugate surds
#class 10
#hard
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A (3\(2-\sqrt{5}\))
B (3\(\sqrt{5}-2\))
C \(\frac{3}{2-\sqrt{5}}\)
D \(\frac{2+\sqrt{5}}{3}\)
Explanation opens after your attempt
Correct Answer
B. (3\(\sqrt{5}-2\))
Step 1
Concept
हर का संयुग्मी \(2-\sqrt{5}\) है। / The conjugate of the denominator is \(2-\sqrt{5}\).
Step 2
Why this answer is correct
(\frac{3}{2+\sqrt{5}}\times\frac{2-\sqrt{5}}{2-\sqrt{5}}=\frac{3\(2-\sqrt{5}\)}{4-5}=3\(\sqrt{5}-2\))। / (\frac{3}{2+\sqrt{5}}\times\frac{2-\sqrt{5}}{2-\sqrt{5}}=\frac{3\(2-\sqrt{5}\)}{4-5}=3\(\sqrt{5}-2\)).
Step 3
Exam Tip
संयुग्मी से गुणा करते समय हर में अंतर के वर्ग का प्रयोग करें। / Use the difference of squares in the denominator when multiplying by a conjugate.
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यदि \(x=\sqrt{5}+\sqrt{20}\), तो \(x^2\) का मान क्या है?
If \(x=\sqrt{5}+\sqrt{20}\), what is the value of \(x^2\)?
#surd square
#simplification
#class 10
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A (125)
B (45)
C (25)
D \(50\sqrt{5}\)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{20}=2\sqrt{5}\), इसलिए \(x=3\sqrt{5}\)। / \(\sqrt{20}=2\sqrt{5}\), so \(x=3\sqrt{5}\).
Step 2
Why this answer is correct
(x-2 =\(3\sqrt{5}\)2 =9\times5=45)। / (x-2 =\(3\sqrt{5}\)2 =9\times5=45).
Step 3
Exam Tip
वर्ग करने से पहले मूल वाले पदों को सरल करें। / Simplify surd terms before squaring.
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निम्न में से कौन-सा विकल्प \(\sqrt{98}-\sqrt{50}\) का सही सरल रूप है?
Which option is the correct simplified form of \(\sqrt{98}-\sqrt{50}\)?
#surd subtraction
#irrational numbers
#class 10
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A \(2\sqrt{2}\)
B \(\sqrt{48}\)
C \(12\sqrt{2}\)
D \(7\sqrt{2}\)
Explanation opens after your attempt
Correct Answer
A. \(2\sqrt{2}\)
Step 1
Concept
\(\sqrt{98}=7\sqrt{2}\) और \(\sqrt{50}=5\sqrt{2}\)। / \(\sqrt{98}=7\sqrt{2}\) and \(\sqrt{50}=5\sqrt{2}\).
Step 2
Why this answer is correct
अंतर \(7\sqrt{2}-5\sqrt{2}=2\sqrt{2}\), जो अपरिमेय है। / The difference is \(7\sqrt{2}-5\sqrt{2}=2\sqrt{2}\), which is irrational.
Step 3
Exam Tip
समान मूल वाले पदों में केवल गुणांक घटाएँ। / For like surds, subtract only the coefficients.
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यदि (p) और (q) सहअभाज्य धनात्मक पूर्णांक हैं और \(\sqrt{3}=\frac{p}{q}\) मान लिया जाए, तो प्रमाण में कौन-सा विरोध मिलेगा?
If (p) and (q) are coprime positive integers and \(\sqrt{3}=\frac{p}{q}\) is assumed, what contradiction appears in the proof?
#irrational proof
#square root of 3
#coprime
#class 10
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A (p) और (q) दोनों (3) से विभाज्य निकलते हैं / Both (p) and (q) turn out divisible by (3)
B (p) और (q) दोनों विषम निकलते हैं / Both (p) and (q) turn out odd
C (p) और (q) दोनों शून्य निकलते हैं / Both (p) and (q) turn out zero
D (p) और (q) दोनों (2) से विभाज्य निकलते हैं / Both (p) and (q) turn out divisible by (2)
Explanation opens after your attempt
Correct Answer
A. (p) और (q) दोनों (3) से विभाज्य निकलते हैं / Both (p) and (q) turn out divisible by (3)
Step 1
Concept
\(\sqrt{3}=\frac{p}{q}\) मानने पर \(p^2=3q^2\) मिलता है। / Assuming \(\sqrt{3}=\frac{p}{q}\) gives \(p^2=3q^2\).
Step 2
Why this answer is correct
इससे (p) और फिर (q) दोनों (3) से विभाज्य निकलते हैं, जबकि वे सहअभाज्य माने गए थे। / This makes both (p) and (q) divisible by (3), contradicting that they are coprime.
Step 3
Exam Tip
ऐसे प्रमाण में समान गुणनखंड मिलना ही विरोध बनाता है। / In such proofs, finding a common factor creates the contradiction.
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कौन-सा युग्म दिखाता है कि दो अपरिमेय संख्याओं का भागफल परिमेय हो सकता है?
Which pair shows that the quotient of two irrational numbers can be rational?
#quotient of irrationals
#rational result
#class 10
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A \(\sqrt{6}\) और \(\sqrt{3}\) / \(\sqrt{6}\) and \(\sqrt{3}\)
B \(\sqrt{12}\) और \(\sqrt{3}\) / \(\sqrt{12}\) and \(\sqrt{3}\)
C \(\sqrt{10}\) और \(\sqrt{2}\) / \(\sqrt{10}\) and \(\sqrt{2}\)
D \(\sqrt{7}\) और \(\sqrt{5}\) / \(\sqrt{7}\) and \(\sqrt{5}\)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{12}\) और \(\sqrt{3}\) / \(\sqrt{12}\) and \(\sqrt{3}\)
Step 1
Concept
\(\sqrt{12}=2\sqrt{3}\) और \(\sqrt{3}\) दोनों अपरिमेय हैं। / \(\sqrt{12}=2\sqrt{3}\) and \(\sqrt{3}\) are both irrational.
Step 2
Why this answer is correct
\(\frac{\sqrt{12}}{\sqrt{3}}=\sqrt{4}=2\), जो परिमेय है। / \(\frac{\sqrt{12}}{\sqrt{3}}=\sqrt{4}=2\), which is rational.
Step 3
Exam Tip
भागफल में मूल के अंदर का भाग पूर्ण वर्ग बन रहा है या नहीं, यह देखें। / In quotients, check whether the value inside the root becomes a perfect square.
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किस विकल्प में दिया गया कथन गलत है?
Which given statement is false?
#decimal classification
#false statement
#class 10
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A हर सांत दशमलव परिमेय होता है / Every terminating decimal is rational
B हर असांत आवर्ती दशमलव परिमेय होता है / Every non-terminating recurring decimal is rational
C हर असांत दशमलव अपरिमेय होता है / Every non-terminating decimal is irrational
D हर असांत अनावर्ती दशमलव अपरिमेय होता है / Every non-terminating non-recurring decimal is irrational
Explanation opens after your attempt
Correct Answer
C. हर असांत दशमलव अपरिमेय होता है / Every non-terminating decimal is irrational
Step 1
Concept
असांत दशमलव आवर्ती भी हो सकता है। / A non-terminating decimal can also be recurring.
Step 2
Why this answer is correct
जैसे \(0.\overline{6}\) असांत है, फिर भी परिमेय है। / For example, \(0.\overline{6}\) is non-terminating but rational.
Step 3
Exam Tip
अपरिमेय के लिए असांत के साथ अनावर्ती होना जरूरी है। / For irrational decimals, non-repetition is also necessary.
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यदि \(y=\sqrt{8}+\sqrt{32}\), तो \(\frac{y}{\sqrt{2}}\) का मान क्या है?
If \(y=\sqrt{8}+\sqrt{32}\), what is the value of \(\frac{y}{\sqrt{2}}\)?
#surd simplification
#rational quotient
#class 10
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A (6)
B (8)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{8}=2\sqrt{2}\) और \(\sqrt{32}=4\sqrt{2}\)। / \(\sqrt{8}=2\sqrt{2}\) and \(\sqrt{32}=4\sqrt{2}\).
Step 2
Why this answer is correct
\(y=6\sqrt{2}\), इसलिए \(\frac{y}{\sqrt{2}}=6\)। / \(y=6\sqrt{2}\), so \(\frac{y}{\sqrt{2}}=6\).
Step 3
Exam Tip
पहले समान मूल वाले पदों को जोड़ें, फिर भाग दें। / First add like surds, then divide.
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कौन-सा विकल्प \(2\sqrt{15}\) के बराबर है?
Which option is equal to \(2\sqrt{15}\)?
#surd conversion
#radicals
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A \(\sqrt{30}\)
B \(\sqrt{60}\)
C \(\sqrt{45}\)
D \(\sqrt{75}\)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{60}\)
Step 1
Concept
\(2\sqrt{15}=\sqrt{4}\sqrt{15}\) लिखा जा सकता है। / \(2\sqrt{15}=\sqrt{4}\sqrt{15}\).
Step 2
Why this answer is correct
इसलिए \(2\sqrt{15}=\sqrt{60}\)। / Therefore \(2\sqrt{15}=\sqrt{60}\).
Step 3
Exam Tip
गुणांक को मूल के अंदर ले जाते समय उसका वर्ग अंदर जाता है। / When moving a coefficient inside a square root, its square goes inside.
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यदि \(a=\sqrt{13}-\sqrt{52}\), तो (a) का सरल रूप क्या है?
If \(a=\sqrt{13}-\sqrt{52}\), what is the simplified form of (a)?
#negative surd
#subtraction of surds
#class 10
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A \(-\sqrt{13}\)
B \(-2\sqrt{13}\)
C \(-3\sqrt{13}\)
D (0)
Explanation opens after your attempt
Correct Answer
A. \(-\sqrt{13}\)
Step 1
Concept
\(\sqrt{52}=2\sqrt{13}\) है। / \(\sqrt{52}=2\sqrt{13}\).
Step 2
Why this answer is correct
\(a=\sqrt{13}-2\sqrt{13}=-\sqrt{13}\), जो अपरिमेय है। / \(a=\sqrt{13}-2\sqrt{13}=-\sqrt{13}\), which is irrational.
Step 3
Exam Tip
ऋण चिह्न आने पर भी अपरिमेयता नहीं बदलती। / A negative sign does not change irrationality.
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किस विकल्प में संख्या परिमेय है?
In which option is the number rational?
#conjugate product
#rational number
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A \(\sqrt{7}+\sqrt{28}\)
B (\(\sqrt{11}+1\)\(\sqrt{11}-1\))
C \(\sqrt{2}+5\)
D \(\frac{3}{\sqrt{7}}\)
Explanation opens after your attempt
Correct Answer
B. (\(\sqrt{11}+1\)\(\sqrt{11}-1\))
Step 1
Concept
(\(\sqrt{11}+1\)\(\sqrt{11}-1\)) संयुग्मी गुणन है। / (\(\sqrt{11}+1\)\(\sqrt{11}-1\)) is a conjugate product.
Step 2
Why this answer is correct
इसका मान (11-1=10), जो परिमेय है। / Its value is (11-1=10), which is rational.
Step 3
Exam Tip
जहाँ संयुग्मी रूप हो, वहाँ अपरिमेय पद कट सकते हैं। / In conjugate forms, irrational terms can cancel.
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यदि (x) अपरिमेय है, तो (x+0) के बारे में सही कथन क्या है?
If (x) is irrational, which statement about (x+0) is correct?
#zero property
#irrational number
#class 10
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A यह परिमेय है / It is rational
B यह अपरिमेय है / It is irrational
C यह पूर्णांक है / It is an integer
D यह सांत दशमलव है / It is a terminating decimal
Explanation opens after your attempt
Correct Answer
B. यह अपरिमेय है / It is irrational
Step 1
Concept
(0) परिमेय संख्या है। / (0) is rational.
Step 2
Why this answer is correct
(x+0=x), इसलिए संख्या की प्रकृति वही रहेगी और वह अपरिमेय होगी। / (x+0=x), so the nature remains the same and it is irrational.
Step 3
Exam Tip
शून्य जोड़ने से मान और प्रकृति दोनों नहीं बदलते। / Adding zero does not change either the value or the type of a number.
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कौन-सा विकल्प \(\sqrt{5}+\sqrt{45}-\sqrt{20}\) का सही सरल रूप है?
Which option is the correct simplified form of \(\sqrt{5}+\sqrt{45}-\sqrt{20}\)?
#surds
#addition subtraction
#irrational numbers
#class 10
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A \(2\sqrt{5}\)
B \(3\sqrt{5}\)
C \(4\sqrt{5}\)
D \(5\sqrt{5}\)
Explanation opens after your attempt
Correct Answer
A. \(2\sqrt{5}\)
Step 1
Concept
\(\sqrt{45}=3\sqrt{5}\) और \(\sqrt{20}=2\sqrt{5}\)। / \(\sqrt{45}=3\sqrt{5}\) and \(\sqrt{20}=2\sqrt{5}\).
Step 2
Why this answer is correct
\(\sqrt{5}+3\sqrt{5}-2\sqrt{5}=2\sqrt{5}\)। / \(\sqrt{5}+3\sqrt{5}-2\sqrt{5}=2\sqrt{5}\).
Step 3
Exam Tip
कई मूलों वाले प्रश्न में पहले सभी पदों को समान मूल में बदलें। / In questions with many radicals, first convert all terms to like surds when possible.
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किस विकल्प में \(x^2\) परिमेय है, पर (x) अपरिमेय है?
In which option is \(x^2\) rational but (x) irrational?
#square of irrational
#rational square
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A \(x=\sqrt{19}\)
B (x=4)
C \(x=\frac{3}{5}\)
D (x=0.25)
Explanation opens after your attempt
Correct Answer
A. \(x=\sqrt{19}\)
Step 1
Concept
\(\sqrt{19}\) अपरिमेय है क्योंकि (19) पूर्ण वर्ग नहीं है। / \(\sqrt{19}\) is irrational because (19) is not a perfect square.
Step 2
Why this answer is correct
(\(\sqrt{19}\)2 =19), जो परिमेय है। / (\(\sqrt{19}\)2 =19), which is rational.
Step 3
Exam Tip
किसी अपरिमेय संख्या का वर्ग कभी-कभी परिमेय हो सकता है। / The square of an irrational number can sometimes be rational.
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निम्न में से कौन-सा असांत अनावर्ती दशमलव बनाने का सही तरीका है?
Which is a correct way to form a non-terminating non-recurring decimal?
#decimal pattern
#non recurring
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A एक ही अंक को लगातार दोहराना / Repeating the same digit again and again
B एक निश्चित समूह को लगातार दोहराना / Repeating a fixed block again and again
C अंकों की लंबाई को बदलते हुए कोई स्थिर आवर्तन न रखना / Changing the length of digit groups without a fixed repetition
D दशमलव को कुछ अंकों के बाद रोक देना / Stopping the decimal after a few digits
Explanation opens after your attempt
Correct Answer
C. अंकों की लंबाई को बदलते हुए कोई स्थिर आवर्तन न रखना / Changing the length of digit groups without a fixed repetition
Step 1
Concept
अपरिमेय दशमलव में न तो समाप्ति होती है और न निश्चित आवर्तन। / An irrational decimal neither terminates nor has a fixed repeating block.
Step 2
Why this answer is correct
बदलती हुई लंबाई वाले अंकों से स्थिर आवर्तन नहीं बनता। / Digit groups with changing lengths do not form a fixed repetition.
Step 3
Exam Tip
आवर्तन दिखते ही दशमलव परिमेय की ओर जाता है। / Once a fixed repetition appears, the decimal becomes rational.
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यदि \(x=\sqrt{6}+\sqrt{24}\), तो (x) किसके बराबर है?
If \(x=\sqrt{6}+\sqrt{24}\), what is (x) equal to?
#surd addition
#simplification
#class 10
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A \(2\sqrt{6}\)
B \(3\sqrt{6}\)
C \(4\sqrt{6}\)
D (30)
Explanation opens after your attempt
Correct Answer
B. \(3\sqrt{6}\)
Step 1
Concept
\(\sqrt{24}=2\sqrt{6}\) है। / \(\sqrt{24}=2\sqrt{6}\).
Step 2
Why this answer is correct
इसलिए \(x=\sqrt{6}+2\sqrt{6}=3\sqrt{6}\), जो अपरिमेय है। / So \(x=\sqrt{6}+2\sqrt{6}=3\sqrt{6}\), which is irrational.
Step 3
Exam Tip
मूल को सरल करके समान पद बनाएँ, फिर जोड़ें। / Simplify radicals to like terms before adding.
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कौन-सा विकल्प \(0.15155155515555\ldots\) की प्रकृति सही बताता है?
Which option correctly describes the nature of \(0.15155155515555\ldots\)?
#irrational decimal
#pattern
#class 10
#hard
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A सांत परिमेय / Terminating rational
B असांत आवर्ती परिमेय / Non-terminating recurring rational
C असांत अनावर्ती अपरिमेय / Non-terminating non-recurring irrational
D पूर्णांक / Integer
Explanation opens after your attempt
Correct Answer
C. असांत अनावर्ती अपरिमेय / Non-terminating non-recurring irrational
Step 1
Concept
यह दशमलव समाप्त नहीं हो रहा है। / This decimal does not terminate.
Step 2
Why this answer is correct
इसमें (5) की संख्या बढ़ती जाती है, इसलिए कोई स्थिर आवर्तन नहीं बनता। / The number of (5)'s keeps increasing, so no fixed repeating block is formed.
Step 3
Exam Tip
असांत अनावर्ती दशमलव अपरिमेय होता है। / A non-terminating non-recurring decimal is irrational.
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यदि \(\sqrt{a}=5\sqrt{2}\), तो (a) का मान क्या है?
If \(\sqrt{a}=5\sqrt{2}\), what is the value of (a)?
#surd equation
#square root
#class 10
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A (10)
B (25)
C (50)
D (100)
Explanation opens after your attempt
Step 1
Concept
दोनों ओर वर्ग करें। / Square both sides.
Step 2
Why this answer is correct
(a=\(5\sqrt{2}\)2 =25\times2=50)। / (a=\(5\sqrt{2}\)2 =25\times2=50).
Step 3
Exam Tip
(\(k\sqrt{m}\)2 =k-2 m) को सही ढंग से लगाएँ। / Apply (\(k\sqrt{m}\)2 =k-2 m) correctly.
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किस विकल्प में दोनों संख्याएँ अपरिमेय हैं और उनका योग भी अपरिमेय है?
In which option are both numbers irrational and their sum is also irrational?
#sum of irrationals
#like surds
#class 10
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A \(\sqrt{2}\) और \(-\sqrt{2}\) / \(\sqrt{2}\) and \(-\sqrt{2}\)
B \(\sqrt{3}\) और \(2\sqrt{3}\) / \(\sqrt{3}\) and \(2\sqrt{3}\)
C \(\sqrt{5}\) और \(-\sqrt{5}\) / \(\sqrt{5}\) and \(-\sqrt{5}\)
D \(\sqrt{7}\) और \(1-\sqrt{7}\) / \(\sqrt{7}\) and \(1-\sqrt{7}\)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{3}\) और \(2\sqrt{3}\) / \(\sqrt{3}\) and \(2\sqrt{3}\)
Step 1
Concept
\(\sqrt{3}\) और \(2\sqrt{3}\) दोनों अपरिमेय हैं। / \(\sqrt{3}\) and \(2\sqrt{3}\) are both irrational.
Step 2
Why this answer is correct
उनका योग \(3\sqrt{3}\) है, जो अपरिमेय है। / Their sum is \(3\sqrt{3}\), which is irrational.
Step 3
Exam Tip
योग वाले प्रश्नों में कटने वाले और जुड़ने वाले समान मूल अलग-अलग पहचानें। / In sum questions, identify whether like surds cancel or combine.
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यदि \(x=2-\sqrt{3}\), तो \(\frac{1}{x}\) का परिमेय हर वाला रूप क्या है?
If \(x=2-\sqrt{3}\), what is the rationalized form of \(\frac{1}{x}\)?
#rationalization
#conjugate
#irrational numbers
#class 10
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A \(2+\sqrt{3}\)
B \(2-\sqrt{3}\)
C \(\sqrt{3}-2\)
D \(\frac{2+\sqrt{3}}{7}\)
Explanation opens after your attempt
Correct Answer
A. \(2+\sqrt{3}\)
Step 1
Concept
\(\frac{1}{2-\sqrt{3}}\) में हर का संयुग्मी \(2+\sqrt{3}\) है। / The conjugate of the denominator in \(\frac{1}{2-\sqrt{3}}\) is \(2+\sqrt{3}\).
Step 2
Why this answer is correct
\(\frac{1}{2-\sqrt{3}}\times\frac{2+\sqrt{3}}{2+\sqrt{3}}=\frac{2+\sqrt{3}}{4-3}=2+\sqrt{3}\)। / \(\frac{1}{2-\sqrt{3}}\times\frac{2+\sqrt{3}}{2+\sqrt{3}}=\frac{2+\sqrt{3}}{4-3}=2+\sqrt{3}\).
Step 3
Exam Tip
हर में संयुग्मी से गुणा करने पर मूल हट जाता है। / Multiplying by the conjugate removes the radical from the denominator.
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किस विकल्प में \(\sqrt{a}\) अपरिमेय है?
In which option is \(\sqrt{a}\) irrational?
#perfect square
#irrational root
#class 10
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A (a=49)
B (a=81)
C (a=90)
D (a=121)
Explanation opens after your attempt
Step 1
Concept
(49), (81) और (121) पूर्ण वर्ग हैं। / (49), (81), and (121) are perfect squares.
Step 2
Why this answer is correct
(90) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{90}\) अपरिमेय है। / (90) is not a perfect square, so \(\sqrt{90}\) is irrational.
Step 3
Exam Tip
वर्गमूल की प्रकृति जानने के लिए सबसे पहले पूर्ण वर्ग जाँचें। / To decide the nature of a square root, first check perfect squares.
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कौन-सा विकल्प (\sqrt{3}\(2+\sqrt{3}\)) का सही मान देता है?
Which option gives the correct value of (\sqrt{3}\(2+\sqrt{3}\))?
#distributive law
#surd multiplication
#class 10
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A \(2\sqrt{3}+3\)
B \(5\sqrt{3}\)
C (6)
D \(2+3\sqrt{3}\)
Explanation opens after your attempt
Correct Answer
A. \(2\sqrt{3}+3\)
Step 1
Concept
वितरण नियम लगाएँ। / Use the distributive law.
Step 2
Why this answer is correct
(\sqrt{3}\(2+\sqrt{3}\)=2\sqrt{3}+\(\sqrt{3}\)2 =2\sqrt{3}+3)। / (\sqrt{3}\(2+\sqrt{3}\)=2\sqrt{3}+\(\sqrt{3}\)2 =2\sqrt{3}+3).
Step 3
Exam Tip
\(\sqrt{3}\times\sqrt{3}\) को (3) लिखना न भूलें। / Remember that \(\sqrt{3}\times\sqrt{3}=3\).
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यदि \(2+\sqrt{n}\) अपरिमेय है और (n) धनात्मक पूर्णांक है, तो कौन-सा (n) उपयुक्त है?
If \(2+\sqrt{n}\) is irrational and (n) is a positive integer, which (n) is suitable?
#irrational expression
#perfect square
#class 10
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A (16)
B (25)
C (36)
D (40)
Explanation opens after your attempt
Step 1
Concept
(2) परिमेय है। इसलिए \(2+\sqrt{n}\) अपरिमेय तभी होगा जब \(\sqrt{n}\) अपरिमेय हो। / (2) is rational. So \(2+\sqrt{n}\) is irrational when \(\sqrt{n}\) is irrational.
Step 2
Why this answer is correct
(40) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{40}\) अपरिमेय है। / (40) is not a perfect square, so \(\sqrt{40}\) is irrational.
Step 3
Exam Tip
दिए गए पूर्णांकों में पूर्ण वर्गों को पहले हटाएँ। / First eliminate perfect squares from the given integers.
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कौन-सा विकल्प \(\sqrt{75}\) और \(\sqrt{27}\) के योग को सही बताता है?
Which option correctly gives the sum of \(\sqrt{75}\) and \(\sqrt{27}\)?
#addition of surds
#common mistake
#class 10
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A \(8\sqrt{3}\)
B \(6\sqrt{3}\)
C \(4\sqrt{3}\)
D \(\sqrt{102}\)
Explanation opens after your attempt
Correct Answer
A. \(8\sqrt{3}\)
Step 1
Concept
\(\sqrt{75}=5\sqrt{3}\) और \(\sqrt{27}=3\sqrt{3}\)। / \(\sqrt{75}=5\sqrt{3}\) and \(\sqrt{27}=3\sqrt{3}\).
Step 2
Why this answer is correct
योग \(5\sqrt{3}+3\sqrt{3}=8\sqrt{3}\) है। / The sum is \(5\sqrt{3}+3\sqrt{3}=8\sqrt{3}\).
Step 3
Exam Tip
अलग-अलग मूलों को सीधे जोड़कर एक मूल न बनाएं। / Do not combine separate square roots directly into one root.
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यदि \(a=\sqrt{2}+\sqrt{5}\), तो \(a^2\) की प्रकृति क्या होगी?
If \(a=\sqrt{2}+\sqrt{5}\), what will be the nature of \(a^2\)?
#square of surd sum
#irrational expression
#class 10
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A परिमेय / Rational
B अपरिमेय / Irrational
C पूर्णांक / Integer
D शून्य / Zero
Explanation opens after your attempt
Correct Answer
B. अपरिमेय / Irrational
Step 1
Concept
(\(\sqrt{2}+\sqrt{5}\)2 =2+5+2\sqrt{10}) है। / (\(\sqrt{2}+\sqrt{5}\)2 =2+5+2\sqrt{10}).
Step 2
Why this answer is correct
यह \(7+2\sqrt{10}\) है, जिसमें अपरिमेय भाग मौजूद है। / This is \(7+2\sqrt{10}\), which has an irrational part.
Step 3
Exam Tip
दो अलग मूलों के योग का वर्ग करते समय बीच वाले पद पर ध्यान दें। / When squaring a sum of two different surds, pay attention to the middle term.
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निम्न में से कौन-सी संख्या (4) और (5) के बीच नहीं है?
Which of the following numbers is not between (4) and (5)?
#comparison of surds
#number line
#class 10
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A \(\sqrt{17}\)
B \(\sqrt{20}\)
C \(\sqrt{24}\)
D \(\sqrt{26}\)
Explanation opens after your attempt
Correct Answer
D. \(\sqrt{26}\)
Step 1
Concept
\(4=\sqrt{16}\) और \(5=\sqrt{25}\) हैं। / \(4=\sqrt{16}\) and \(5=\sqrt{25}\).
Step 2
Why this answer is correct
(17), (20) और (24) (16) और (25) के बीच हैं, पर (26) (25) से बड़ा है। / (17), (20), and (24) lie between (16) and (25), but (26) is greater than (25).
Step 3
Exam Tip
धनात्मक वर्गमूलों की तुलना में वर्गों की तुलना आसान होती है। / For positive square roots, comparing squares is easier.
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यदि \(x=\sqrt{7}\), तो (3x-2) की प्रकृति क्या है?
If \(x=\sqrt{7}\), what is the nature of (3x-2)?
#linear expression
#irrational numbers
#class 10
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A परिमेय / Rational
B अपरिमेय / Irrational
C पूर्णांक / Integer
D प्राकृतिक संख्या / Natural number
Explanation opens after your attempt
Correct Answer
B. अपरिमेय / Irrational
Step 1
Concept
\(3x-2=3\sqrt{7}-2\) है। / \(3x-2=3\sqrt{7}-2\).
Step 2
Why this answer is correct
\(3\sqrt{7}\) अपरिमेय है और उसमें से परिमेय संख्या घटाने पर परिणाम अपरिमेय रहता है। / \(3\sqrt{7}\) is irrational, and subtracting a rational number keeps it irrational.
Step 3
Exam Tip
अशून्य परिमेय गुणक से गुणा करने पर मूल की अपरिमेयता बनी रहती है। / A non-zero rational multiple of a surd remains irrational.
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कौन-सा विकल्प \(\frac{\sqrt{27}}{\sqrt{3}}\) का सही मान देता है?
Which option gives the correct value of \(\frac{\sqrt{27}}{\sqrt{3}}\)?
#division of radicals
#rational result
#class 10
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A (3)
B \(\sqrt{9}\) और इसलिए (3) / \(\sqrt{9}\) and hence (3)
C \(9\sqrt{3}\)
D \(\sqrt{24}\)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{9}\) और इसलिए (3) / \(\sqrt{9}\) and hence (3)
Step 1
Concept
\(\frac{\sqrt{27}}{\sqrt{3}}=\sqrt{\frac{27}{3}}\) लिखा जा सकता है। / \(\frac{\sqrt{27}}{\sqrt{3}}=\sqrt{\frac{27}{3}}\).
Step 2
Why this answer is correct
यह \(\sqrt{9}=3\) है, जो परिमेय है। / This is \(\sqrt{9}=3\), which is rational.
Step 3
Exam Tip
भाग में मूलों को एक साथ सरल करना जल्दी तरीका है। / In division, simplifying the radicals together is a quick method.
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किस विकल्प में दी गई संख्या (0) के बराबर है?
Which option is equal to (0)?
#cancellation
#surd simplification
#class 10
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A \(\sqrt{8}-2\sqrt{2}\)
B \(\sqrt{18}-2\sqrt{2}\)
C \(\sqrt{20}-2\sqrt{5}\)
D \(\sqrt{27}-2\sqrt{3}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{8}-2\sqrt{2}\)
Step 1
Concept
\(\sqrt{8}=2\sqrt{2}\) है। / \(\sqrt{8}=2\sqrt{2}\).
Step 2
Why this answer is correct
इसलिए \(\sqrt{8}-2\sqrt{2}=0\), जो परिमेय है। / Therefore \(\sqrt{8}-2\sqrt{2}=0\), which is rational.
Step 3
Exam Tip
कभी-कभी अपरिमेय जैसे दिखने वाले पद पूरी तरह कट जाते हैं। / Sometimes terms that look irrational cancel completely.
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यदि \(x=1+\sqrt{2}\), तो \(x^2-2x\) का मान क्या है?
If \(x=1+\sqrt{2}\), what is the value of \(x^2-2x\)?
#algebraic expression
#surds
#class 10
#hard
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A (1)
B (-1)
C \(2\sqrt{2}\)
D \(\sqrt{2}\)
Explanation opens after your attempt
Step 1
Concept
(x-2 -2x=x(x-2)) है। / (x-2 -2x=x(x-2)).
Step 2
Why this answer is correct
\(x=1+\sqrt{2}\) रखने पर \(x-2=\sqrt{2}-1\), इसलिए गुणन (\(1+\sqrt{2}\)\(\sqrt{2}-1\)=1) मिलता है। / With \(x=1+\sqrt{2}\), \(x-2=\sqrt{2}-1\), so the product (\(1+\sqrt{2}\)\(\sqrt{2}-1\)=1).
Step 3
Exam Tip
संयुग्मी जैसे रूपों को पहचानने से गणना छोटी होती है। / Recognizing conjugate-like forms makes calculation shorter.
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कौन-सी संख्या \(\sqrt{3}\) और \(2\sqrt{3}\) के बीच है?
Which number lies between \(\sqrt{3}\) and \(2\sqrt{3}\)?
#comparison
#number line
#irrational numbers
#class 10
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A (1)
B (2)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{3}\) लगभग (1.732) है और \(2\sqrt{3}\) लगभग (3.464) है। / \(\sqrt{3}\) is about (1.732), and \(2\sqrt{3}\) is about (3.464).
Step 2
Why this answer is correct
(2) इन दोनों के बीच आता है। / (2) lies between these two values.
Step 3
Exam Tip
तुलना में अनुमान या वर्ग दोनों तरीकों का उपयोग कर सकते हैं। / For comparison, you may use estimation or squaring.
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निम्न में से कौन-सा व्यंजक परिमेय नहीं है?
Which of the following expressions is not rational?
#rational irrational
#surd expressions
#class 10
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A (\(\sqrt{6}\)2 )
B \(\sqrt{12}\times\sqrt{3}\)
C \(\sqrt{6}+\sqrt{24}\)
D (\(\sqrt{5}+2\)\(\sqrt{5}-2\))
Explanation opens after your attempt
Correct Answer
C. \(\sqrt{6}+\sqrt{24}\)
Step 1
Concept
पहले प्रत्येक विकल्प को सरल करें। / Simplify each option first.
Step 2
Why this answer is correct
\(\sqrt{6}+\sqrt{24}=\sqrt{6}+2\sqrt{6}=3\sqrt{6}\), जो अपरिमेय है। / \(\sqrt{6}+\sqrt{24}=\sqrt{6}+2\sqrt{6}=3\sqrt{6}\), which is irrational.
Step 3
Exam Tip
गुणन में मूल कट सकते हैं, पर योग में हमेशा ऐसा नहीं होता। / Radicals may cancel in multiplication, but not always in addition.
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यदि (m) एक धनात्मक पूर्णांक है और (m) पूर्ण वर्ग नहीं है, तो \(\sqrt{m}+4\) के बारे में सही कथन क्या है?
If (m) is a positive integer and (m) is not a perfect square, which statement about \(\sqrt{m}+4\) is correct?
#general statement
#perfect square
#irrational numbers
#class 10
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A यह हमेशा परिमेय है / It is always rational
B यह हमेशा अपरिमेय है / It is always irrational
C यह हमेशा पूर्णांक है / It is always an integer
D यह हमेशा शून्य है / It is always zero
Explanation opens after your attempt
Correct Answer
B. यह हमेशा अपरिमेय है / It is always irrational
Step 1
Concept
(m) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{m}\) अपरिमेय है। / Since (m) is not a perfect square, \(\sqrt{m}\) is irrational.
Step 2
Why this answer is correct
(4) परिमेय है, और अपरिमेय में परिमेय जोड़ने पर परिणाम अपरिमेय होता है। / (4) is rational, and adding it to an irrational number gives an irrational number.
Step 3
Exam Tip
पूर्ण वर्ग न होने की जानकारी को सीधे वर्गमूल की प्रकृति से जोड़ें। / Connect the non-perfect-square condition directly with the nature of the square root.
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कौन-सा विकल्प \(\sqrt{a}\times\sqrt{b}\) को परिमेय बनाने के लिए सही उदाहरण है?
Which option is a correct example that makes \(\sqrt{a}\times\sqrt{b}\) rational?
#product of roots
#perfect square
#class 10
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A (a=2,b=7)
B (a=3,b=12)
C (a=5,b=6)
D (a=7,b=8)
Explanation opens after your attempt
Correct Answer
B. (a=3,b=12)
Step 1
Concept
\(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\) होता है। / \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\).
Step 2
Why this answer is correct
(a=3,b=12) पर (ab=36), इसलिए \(\sqrt{36}=6\), जो परिमेय है। / For (a=3,b=12), (ab=36), so \(\sqrt{36}=6\), which is rational.
Step 3
Exam Tip
गुणन के बाद अंदर की संख्या पूर्ण वर्ग बन रही है या नहीं, यह देखें। / Check whether the product inside the radical becomes a perfect square.
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कौन-सी संख्या \(\frac{2}{\sqrt{2}+1}\) के बराबर है?
Which number is equal to \(\frac{2}{\sqrt{2}+1}\)?
#rationalization
#denominator
#conjugate
#class 10
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A \(2\sqrt{2}-2\)
B \(2\sqrt{2}+2\)
C \(\sqrt{2}-1\)
D \(\sqrt{2}+1\)
Explanation opens after your attempt
Correct Answer
A. \(2\sqrt{2}-2\)
Step 1
Concept
हर का संयुग्मी \(\sqrt{2}-1\) है। / The conjugate of the denominator is \(\sqrt{2}-1\).
Step 2
Why this answer is correct
(\frac{2}{\sqrt{2}+1}\times\frac{\sqrt{2}-1}{\sqrt{2}-1}=\frac{2\(\sqrt{2}-1\)}{2-1}=2\sqrt{2}-2)। / (\frac{2}{\sqrt{2}+1}\times\frac{\sqrt{2}-1}{\sqrt{2}-1}=\frac{2\(\sqrt{2}-1\)}{2-1}=2\sqrt{2}-2).
Step 3
Exam Tip
संयुग्मी का सही चिह्न चुनना बहुत जरूरी है। / Choosing the correct conjugate sign is very important.
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यदि \(x=\sqrt{5}-\sqrt{2}\), तो \(x^2\) कौन-सा होगा?
If \(x=\sqrt{5}-\sqrt{2}\), what is \(x^2\)?
#square of difference
#surds
#class 10
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A \(7-2\sqrt{10}\)
B (3)
C \(7+2\sqrt{10}\)
D \(\sqrt{3}\)
Explanation opens after your attempt
Correct Answer
A. \(7-2\sqrt{10}\)
Step 1
Concept
((a-b)2 =a-2 -2ab+b-2 ) का प्रयोग करें। / Use ((a-b)2 =a-2 -2ab+b-2 ).
Step 2
Why this answer is correct
\(x^2=5-2\sqrt{10}+2=7-2\sqrt{10}\)। / \(x^2=5-2\sqrt{10}+2=7-2\sqrt{10}\).
Step 3
Exam Tip
अंतर के वर्ग में बीच वाले पद का ऋण चिह्न न भूलें। / Do not forget the negative sign in the middle term when squaring a difference.
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किस विकल्प में (x+y) परिमेय है, जबकि (x) और (y) दोनों अपरिमेय हैं?
In which option is (x+y) rational while both (x) and (y) are irrational?
#sum of irrationals
#rational result
#class 10
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A \(x=\sqrt{8},y=-2\sqrt{2}\)
B \(x=\sqrt{3},y=\sqrt{12}\)
C \(x=\sqrt{5},y=\sqrt{20}\)
D \(x=\sqrt{6},y=\sqrt{24}\)
Explanation opens after your attempt
Correct Answer
A. \(x=\sqrt{8},y=-2\sqrt{2}\)
Step 1
Concept
\(\sqrt{8}=2\sqrt{2}\), इसलिए (x) और \(y=-2\sqrt{2}\) दोनों अपरिमेय हैं। / \(\sqrt{8}=2\sqrt{2}\), so (x) and \(y=-2\sqrt{2}\) are both irrational.
Step 2
Why this answer is correct
उनका योग \(2\sqrt{2}-2\sqrt{2}=0\), जो परिमेय है। / Their sum is \(2\sqrt{2}-2\sqrt{2}=0\), which is rational.
Step 3
Exam Tip
विपरीत अपरिमेय पदों के योग से परिमेय उत्तर मिल सकता है। / Opposite irrational terms can give a rational sum.
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यदि (0<x<1) और (x) अपरिमेय है, तो कौन-सा उदाहरण उपयुक्त है?
If (0<x<1) and (x) is irrational, which example is suitable?
#irrational between zero and one
#class 10
#number line
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A \(\frac{\sqrt{2}}{2}\)
B \(\frac{1}{2}\)
C (0.75)
D \(\sqrt{4}\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{\sqrt{2}}{2}\)
Step 1
Concept
\(\sqrt{2}\) अपरिमेय है और अशून्य परिमेय (2) से भाग देने पर अपरिमेयता बनी रहती है। / \(\sqrt{2}\) is irrational, and dividing by non-zero rational (2) keeps it irrational.
Step 2
Why this answer is correct
\(\frac{\sqrt{2}}{2}\) लगभग (0.707) है, इसलिए यह (0) और (1) के बीच है। / \(\frac{\sqrt{2}}{2}\) is about (0.707), so it lies between (0) and (1).
Step 3
Exam Tip
अंतराल और संख्या की प्रकृति दोनों शर्तें साथ-साथ जाँचें। / Check both the interval condition and the nature of the number.
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कौन-सा विकल्प \(\sqrt{3}+\sqrt{12}+\sqrt{27}\) का सही सरल रूप है?
Which option is the correct simplified form of \(\sqrt{3}+\sqrt{12}+\sqrt{27}\)?
#multiple surds
#addition
#class 10
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A \(6\sqrt{3}\)
B \(5\sqrt{3}\)
C \(3\sqrt{6}\)
D \(\sqrt{42}\)
Explanation opens after your attempt
Correct Answer
A. \(6\sqrt{3}\)
Step 1
Concept
\(\sqrt{12}=2\sqrt{3}\) और \(\sqrt{27}=3\sqrt{3}\)। / \(\sqrt{12}=2\sqrt{3}\) and \(\sqrt{27}=3\sqrt{3}\).
Step 2
Why this answer is correct
कुल योग \(\sqrt{3}+2\sqrt{3}+3\sqrt{3}=6\sqrt{3}\) है। / The total sum is \(\sqrt{3}+2\sqrt{3}+3\sqrt{3}=6\sqrt{3}\).
Step 3
Exam Tip
सभी पदों को समान मूल में बदलने से जोड़ आसान हो जाता है। / Converting all terms into like surds makes addition easy.
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किस विकल्प में \(\sqrt{p}\) की अपरिमेयता को गलत तरीके से समझाया गया है?
Which option explains the irrationality of \(\sqrt{p}\) incorrectly?
#concept check
#perfect square
#irrationality
#class 10
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A यदि (p) पूर्ण वर्ग नहीं है, तो \(\sqrt{p}\) अपरिमेय होगा / If (p) is not a perfect square, \(\sqrt{p}\) is irrational
B यदि (p) अभाज्य है, तो \(\sqrt{p}\) अपरिमेय होगा / If (p) is prime, \(\sqrt{p}\) is irrational
C यदि (p=36), तो \(\sqrt{p}\) अपरिमेय होगा / If (p=36), \(\sqrt{p}\) is irrational
D यदि (p=50), तो \(\sqrt{p}\) अपरिमेय होगा / If (p=50), \(\sqrt{p}\) is irrational
Explanation opens after your attempt
Correct Answer
C. यदि (p=36), तो \(\sqrt{p}\) अपरिमेय होगा / If (p=36), \(\sqrt{p}\) is irrational
Step 1
Concept
(36) पूर्ण वर्ग है। / (36) is a perfect square.
Step 2
Why this answer is correct
\(\sqrt{36}=6\), जो परिमेय है, इसलिए (p=36) वाला कथन गलत है। / \(\sqrt{36}=6\), which is rational, so the statement for (p=36) is incorrect.
Step 3
Exam Tip
वर्गमूल पर निर्णय लेने में पूर्ण वर्ग की जाँच सबसे सुरक्षित तरीका है। / Checking perfect squares is the safest way to decide the nature of a square root.
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यदि \(a=\sqrt{2}+\sqrt{3}\) और \(b=\sqrt{3}-\sqrt{2}\), तो (ab) का मान क्या है?
If \(a=\sqrt{2}+\sqrt{3}\) and \(b=\sqrt{3}-\sqrt{2}\), what is the value of (ab)?
#conjugate surds
#difference of squares
#class 10
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A (1)
B (5)
C \(\sqrt{6}\)
D \(2\sqrt{6}\)
Explanation opens after your attempt
Step 1
Concept
(ab=\(\sqrt{3}+\sqrt{2}\)\(\sqrt{3}-\sqrt{2}\)) के रूप में देखा जा सकता है। / View (ab) as (\(\sqrt{3}+\sqrt{2}\)\(\sqrt{3}-\sqrt{2}\)).
Step 2
Why this answer is correct
यह (\(\sqrt{3}\)2 -\(\sqrt{2}\)2 =3-2=1) है। / This equals (\(\sqrt{3}\)2 -\(\sqrt{2}\)2 =3-2=1).
Step 3
Exam Tip
क्रम बदलने से योग नहीं बदलता, इसलिए संयुग्मी रूप पहचानें। / Since addition order does not change the sum, recognize the conjugate form.
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यदि (A=\(3+\sqrt{2}\)2 -\(3-\sqrt{2}\)2 ), तो (A) का सही मान और प्रकृति क्या है?
If (A=\(3+\sqrt{2}\)2 -\(3-\sqrt{2}\)2 ), what is the correct value and nature of (A)?
#algebraic identity
#surds
#irrational numbers
#class 10
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A \(12\sqrt{2}\), अपरिमेय / \(12\sqrt{2}\), irrational
B (18), परिमेय / (18), rational
C \(6\sqrt{2}\), अपरिमेय / \(6\sqrt{2}\), irrational
D (22), परिमेय / (22), rational
Explanation opens after your attempt
Correct Answer
A. \(12\sqrt{2}\), अपरिमेय / \(12\sqrt{2}\), irrational
Step 1
Concept
((a+b)2 -(a-b)2 =4ab) का प्रयोग करें। / Use ((a+b)2 -(a-b)2 =4ab).
Step 2
Why this answer is correct
यहाँ (a=3) और \(b=\sqrt{2}\) हैं, इसलिए \(A=4\times3\times\sqrt{2}=12\sqrt{2}\), जो अपरिमेय है। / Here (a=3) and \(b=\sqrt{2}\), so \(A=4\times3\times\sqrt{2}=12\sqrt{2}\), which is irrational.
Step 3
Exam Tip
ऐसे प्रश्न में दोनों वर्गों को पूरा फैलाने के बजाय पहचान वाला सूत्र लगाएँ। / In such questions, use the identity instead of expanding both squares fully.
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