यदि \(\sqrt{m}\) परिमेय है और (m) धनात्मक पूर्णांक है, तो (m) के बारे में कौन-सा कथन सही है?
If \(\sqrt{m}\) is rational and (m) is a positive integer, which statement about (m) is correct?
#real-numbers
#irrational-numbers
#perfect-square
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A (m) पूर्ण वर्ग है / (m) is a perfect square
B (m) हमेशा अभाज्य है / (m) is always prime
C (m) हमेशा विषम है / (m) is always odd
D (m) पूर्ण वर्ग नहीं है / (m) is not a perfect square
Explanation opens after your attempt
Correct Answer
A. (m) पूर्ण वर्ग है / (m) 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 only when the integer is a perfect square.
Step 2
Why this answer is correct
जैसे (25) पूर्ण वर्ग है और \(\sqrt{25}=5\) परिमेय है। / For example, (25) is a perfect square and \(\sqrt{25}=5\) is rational.
Step 3
Exam Tip
वर्गमूल वाले प्रश्नों में पूर्ण वर्ग की पहचान सबसे पहले करें। / In square-root questions, identify perfect squares first.
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निम्नलिखित में से कौन-सा परिणाम परिमेय है?
Which of the following results is rational?
#real-numbers
#rational-result
#radicals
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A \(2+\sqrt{3}\)
B \(\sqrt{2}\times\sqrt{8}\)
C \(\sqrt{5}+\sqrt{20}\)
D \(3\sqrt{7}\)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{2}\times\sqrt{8}\)
Step 1
Concept
\(\sqrt{2}\times\sqrt{8}=\sqrt{16}\) होता है। / \(\sqrt{2}\times\sqrt{8}=\sqrt{16}\).
Step 2
Why this answer is correct
\(\sqrt{16}=4\), जो परिमेय संख्या है। / \(\sqrt{16}=4\), which is rational.
Step 3
Exam Tip
दो अपरिमेय संख्याओं का गुणनफल हमेशा अपरिमेय नहीं होता। / The product of two irrational numbers is not always irrational.
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\(\sqrt{72}-\sqrt{18}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{72}-\sqrt{18}\)?
#real-numbers
#radical-subtraction
#simplification
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A \(3\sqrt{2}\)
B \(6\sqrt{2}\)
C \(\sqrt{54}\)
D \(9\sqrt{2}\)
Explanation opens after your attempt
Correct Answer
A. \(3\sqrt{2}\)
Step 1
Concept
\(\sqrt{72}=6\sqrt{2}\) और \(\sqrt{18}=3\sqrt{2}\)। / \(\sqrt{72}=6\sqrt{2}\) and \(\sqrt{18}=3\sqrt{2}\).
Step 2
Why this answer is correct
\(6\sqrt{2}-3\sqrt{2}=3\sqrt{2}\)। / \(6\sqrt{2}-3\sqrt{2}=3\sqrt{2}\).
Step 3
Exam Tip
घटाने से पहले दोनों वर्गमूलों को सरल करना जरूरी है। / Simplify both square roots before subtracting.
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यदि \(x=3+\sqrt{5}\), तो (x-3) किस प्रकार की संख्या है?
If \(x=3+\sqrt{5}\), what type of number is (x-3)?
#real-numbers
#irrational-expression
#sqrt5
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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
(x-3=\(3+\sqrt{5}\)-3) होगा। / (x-3=\(3+\sqrt{5}\)-3).
Step 2
Why this answer is correct
इससे \(\sqrt{5}\) बचता है, जो अपरिमेय है। / This leaves \(\sqrt{5}\), which is irrational.
Step 3
Exam Tip
अभिव्यक्ति को सरल करके संख्या की प्रकृति तय करें। / Simplify the expression before deciding the nature of the number.
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निम्नलिखित में से कौन-सी संख्या \(\frac{p}{q}\) के रूप में नहीं लिखी जा सकती, जहाँ (p,q) पूर्णांक हैं और \(q\neq0\)?
Which of the following cannot be written in the form \(\frac{p}{q}\), where (p,q) are integers and \(q\neq0\)?
#real-numbers
#irrational-definition
#mcq
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A (0.6)
B \(\frac{13}{5}\)
C \(\sqrt{18}\)
D (4.75)
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Correct Answer
C. \(\sqrt{18}\)
Step 1
Concept
समाप्त दशमलव और भिन्न परिमेय होते हैं। / Terminating decimals and fractions are rational.
Step 2
Why this answer is correct
\(\sqrt{18}=3\sqrt{2}\), और \(\sqrt{2}\) अपरिमेय है। / \(\sqrt{18}=3\sqrt{2}\), and \(\sqrt{2}\) is irrational.
Step 3
Exam Tip
वर्गमूल को सरल करके देखना कई बार सही पहचान देता है। / Simplifying a square root often helps identify the number correctly.
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\(\sqrt{12}+\sqrt{27}+\sqrt{75}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{12}+\sqrt{27}+\sqrt{75}\)?
#real-numbers
#radical-addition
#simplification
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A \(10\sqrt{3}\)
B \(9\sqrt{3}\)
C \(6\sqrt{3}\)
D \(15\sqrt{3}\)
Explanation opens after your attempt
Correct Answer
A. \(10\sqrt{3}\)
Step 1
Concept
\(\sqrt{12}=2\sqrt{3}\), \(\sqrt{27}=3\sqrt{3}\), और \(\sqrt{75}=5\sqrt{3}\)। / \(\sqrt{12}=2\sqrt{3}\), \(\sqrt{27}=3\sqrt{3}\), and \(\sqrt{75}=5\sqrt{3}\).
Step 2
Why this answer is correct
जोड़ने पर \(2\sqrt{3}+3\sqrt{3}+5\sqrt{3}=10\sqrt{3}\)। / Adding gives \(2\sqrt{3}+3\sqrt{3}+5\sqrt{3}=10\sqrt{3}\).
Step 3
Exam Tip
समान वर्गमूल बनने पर केवल गुणांक जोड़ें। / Once radicals are like terms, add only the coefficients.
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कौन-सा कथन हमेशा सत्य है?
Which statement is always true?
#real-numbers
#true-statement
#irrational-rules
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A दो अपरिमेय संख्याओं का योग हमेशा अपरिमेय होता है / The sum of two irrational numbers is always irrational
B दो अपरिमेय संख्याओं का गुणनफल हमेशा अपरिमेय होता है / The product of two irrational numbers is always irrational
C परिमेय और अपरिमेय संख्या का योग अपरिमेय होता है / The sum of a rational and an irrational number is irrational
D दो अपरिमेय संख्याओं का अंतर हमेशा शून्य होता है / The difference of two irrational numbers is always zero
Explanation opens after your attempt
Correct Answer
C. परिमेय और अपरिमेय संख्या का योग अपरिमेय होता है / The sum of a rational and an irrational number is irrational
Step 1
Concept
परिमेय संख्या जोड़ने से अपरिमेय भाग समाप्त नहीं होता। / Adding a rational number does not remove the irrational part.
Step 2
Why this answer is correct
जैसे \(2+\sqrt{3}\) अपरिमेय है। / For example, \(2+\sqrt{3}\) is irrational.
Step 3
Exam Tip
दो अपरिमेय संख्याओं के योग या गुणन के बारे में हमेशा वाला नियम सावधानी से लगाएं। / Be careful with always-type statements about sums or products of two irrational numbers.
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\(\frac{5}{\sqrt{5}}\) का सरल रूप क्या है?
What is the simplified form of \(\frac{5}{\sqrt{5}}\)?
#real-numbers
#rationalisation
#simplification
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A \(\sqrt{5}\)
B \(5\sqrt{5}\)
C \(\frac{1}{\sqrt{5}}\)
D (25)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{5}\)
Step 1
Concept
हर को सरल करने के लिए ऊपर और नीचे \(\sqrt{5}\) से गुणा करें। / To simplify the denominator, multiply top and bottom by \(\sqrt{5}\).
Step 2
Why this answer is correct
\(\frac{5}{\sqrt{5}}=\frac{5\sqrt{5}}{5}=\sqrt{5}\)। / \(\frac{5}{\sqrt{5}}=\frac{5\sqrt{5}}{5}=\sqrt{5}\).
Step 3
Exam Tip
हर में वर्गमूल हो तो परिमेयकरण उपयोगी होता है। / Rationalising is useful when a square root appears in the denominator.
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यदि \(\sqrt{a}\) अपरिमेय है, तो निम्न में से कौन-सा परिणाम अवश्य परिमेय है?
If \(\sqrt{a}\) is irrational, which of the following results is necessarily rational?
#real-numbers
#irrational-square
#rational-result
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A \(\sqrt{a}+1\)
B \(\sqrt{a}\times\sqrt{a}\)
C \(2\sqrt{a}\)
D \(\sqrt{a}-3\)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{a}\times\sqrt{a}\)
Step 1
Concept
किसी वर्गमूल को उसी वर्गमूल से गुणा करने पर अंदर की संख्या मिलती है। / Multiplying a square root by itself gives the number inside.
Step 2
Why this answer is correct
\(\sqrt{a}\times\sqrt{a}=a\), और (a) पूर्णांक हो तो परिमेय है। / \(\sqrt{a}\times\sqrt{a}=a\), and if (a) is an integer, it is rational.
Step 3
Exam Tip
अपरिमेय संख्या का वर्ग कई बार परिमेय हो सकता है। / The square of an irrational number can be rational.
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कौन-सी संख्या (2) और (3) के बीच अपरिमेय संख्या है?
Which number is an irrational number between (2) and (3)?
#real-numbers
#irrational-between-numbers
#comparison
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A \(\sqrt{5}\)
B \(\sqrt{4}\)
C \(\sqrt{9}\)
D (2.5)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{5}\)
Step 1
Concept
(4<5<9), इसलिए \(2<\sqrt{5}<3\)। / Since (4<5<9), \(2<\sqrt{5}<3\).
Step 2
Why this answer is correct
(5) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{5}\) अपरिमेय है। / (5) is not a perfect square, so \(\sqrt{5}\) is irrational.
Step 3
Exam Tip
वर्गमूल की स्थिति जानने के लिए पास के पूर्ण वर्ग देखें। / Use nearby perfect squares to locate a square root.
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\(\sqrt{45}+\sqrt{80}-\sqrt{20}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{45}+\sqrt{80}-\sqrt{20}\)?
#real-numbers
#radical-expression
#simplification
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A \(5\sqrt{5}\)
B \(3\sqrt{5}\)
C \(7\sqrt{5}\)
D \(\sqrt{105}\)
Explanation opens after your attempt
Correct Answer
A. \(5\sqrt{5}\)
Step 1
Concept
\(\sqrt{45}=3\sqrt{5}\), \(\sqrt{80}=4\sqrt{5}\), और \(\sqrt{20}=2\sqrt{5}\)। / \(\sqrt{45}=3\sqrt{5}\), \(\sqrt{80}=4\sqrt{5}\), and \(\sqrt{20}=2\sqrt{5}\).
Step 2
Why this answer is correct
\(3\sqrt{5}+4\sqrt{5}-2\sqrt{5}=5\sqrt{5}\)। / \(3\sqrt{5}+4\sqrt{5}-2\sqrt{5}=5\sqrt{5}\).
Step 3
Exam Tip
जोड़ और घटाव से पहले सभी वर्गमूलों को समान रूप में बदलें। / Convert all radicals to like form before adding or subtracting.
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यदि \(0.101001000100001\ldots\) में शून्यों की संख्या हर बार बढ़ती जाती है, तो यह संख्या कैसी है?
If in \(0.101001000100001\ldots\) the number of zeros keeps increasing each time, what type of number is it?
#real-numbers
#decimal-expansion
#non-recurring
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A समाप्त दशमलव / Terminating decimal
B आवर्ती परिमेय / Recurring rational
C अपरिमेय / Irrational
D पूर्णांक / Integer
Explanation opens after your attempt
Correct Answer
C. अपरिमेय / Irrational
Step 1
Concept
इस दशमलव में कोई निश्चित आवर्ती समूह नहीं है। / This decimal has no fixed repeating block.
Step 2
Why this answer is correct
यह अनवसानी और अनावर्ती है, इसलिए अपरिमेय है। / It is non-terminating and non-recurring, so it is irrational.
Step 3
Exam Tip
लंबे दशमलव में दोहराव का निश्चित ढांचा जरूर जांचें। / In long decimals, always check for a fixed repeating pattern.
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निम्नलिखित में से कौन-सा युग्म ऐसे दो अपरिमेय संख्याओं का है जिनका योग परिमेय है?
Which pair consists of two irrational numbers whose sum is rational?
#real-numbers
#irrational-sum
#rational-result
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A \(\sqrt{2},-\sqrt{2}\)
B \(\sqrt{3},\sqrt{3}\)
C \(\sqrt{5},\sqrt{20}\)
D \(\sqrt{7},1\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{2},-\sqrt{2}\)
Step 1
Concept
\(\sqrt{2}\) और \(-\sqrt{2}\) दोनों अपरिमेय हैं। / \(\sqrt{2}\) and \(-\sqrt{2}\) 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
विपरीत अपरिमेय संख्याएँ जोड़ने पर परिमेय परिणाम आ सकता है। / Opposite irrational numbers can give a rational sum.
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(\sqrt{3}\(2+\sqrt{3}\)) का मान क्या है?
What is the value of (\sqrt{3}\(2+\sqrt{3}\))?
#real-numbers
#radical-multiplication
#expression
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A \(2\sqrt{3}+3\)
B \(5\sqrt{3}\)
C \(2+3\sqrt{3}\)
D (6)
Explanation opens after your attempt
Correct Answer
A. \(2\sqrt{3}+3\)
Step 1
Concept
वितरण नियम लगाएं: \(\sqrt{3}\times2+\sqrt{3}\times\sqrt{3}\)। / Use distribution: \(\sqrt{3}\times2+\sqrt{3}\times\sqrt{3}\).
Step 2
Why this answer is correct
यह \(2\sqrt{3}+3\) बनता है। / This becomes \(2\sqrt{3}+3\).
Step 3
Exam Tip
वर्गमूल वाले गुणन में \(\sqrt{3}\times\sqrt{3}=3\) याद रखें। / In radical multiplication, remember \(\sqrt{3}\times\sqrt{3}=3\).
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कौन-सी संख्या \(\sqrt{50}\) के बराबर है?
Which number is equal to \(\sqrt{50}\)?
#real-numbers
#equivalent-radicals
#sqrt50
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A \(5\sqrt{2}\)
B \(2\sqrt{5}\)
C \(10\sqrt{5}\)
D \(25\sqrt{2}\)
Explanation opens after your attempt
Correct Answer
A. \(5\sqrt{2}\)
Step 1
Concept
\(50=25\times2\) लिखें। / Write \(50=25\times2\).
Step 2
Why this answer is correct
\(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\)। / \(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\).
Step 3
Exam Tip
बराबर रूप पहचानने के लिए सबसे पहले वर्गमूल को सरल करें। / To identify an equivalent form, simplify the square root first.
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(\left\(2+\sqrt{3}\right\)\left\(2-\sqrt{3}\right\)) का मान क्या है?
What is the value of (\left\(2+\sqrt{3}\right\)\left\(2-\sqrt{3}\right\))?
#real-numbers
#conjugates
#rationalisation
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A (1)
B (7)
C \(4+\sqrt{3}\)
D \(4-\sqrt{3}\)
Explanation opens after your attempt
Step 1
Concept
यह ((a+b)(a-b)=a-2 -b-2 ) का रूप है। / This is of the form ((a+b)(a-b)=a-2 -b-2 ).
Step 2
Why this answer is correct
(22 -\(\sqrt{3}\)2 =4-3=1)। / (22 -\(\sqrt{3}\)2 =4-3=1).
Step 3
Exam Tip
संयुग्म रूप वाले गुणन में वर्गों का अंतर जल्दी उत्तर देता है। / For conjugate products, difference of squares gives the answer quickly.
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कौन-सा विकल्प अपरिमेय संख्या नहीं है?
Which option is not an irrational number?
#real-numbers
#not-irrational
#perfect-square
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A \(4+\sqrt{2}\)
B \(\sqrt{11}\)
C \(\sqrt{121}-2\)
D \(3\sqrt{5}\)
Explanation opens after your attempt
Correct Answer
C. \(\sqrt{121}-2\)
Step 1
Concept
\(\sqrt{121}=11\) है। / \(\sqrt{121}=11\).
Step 2
Why this answer is correct
\(\sqrt{121}-2=9\), जो परिमेय संख्या है। / \(\sqrt{121}-2=9\), which is rational.
Step 3
Exam Tip
अपरिमेय नहीं वाले प्रश्न में पूर्ण वर्गों को खास ध्यान से देखें। / In not-irrational questions, check perfect squares carefully.
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यदि \(a=\sqrt{2}+1\) और \(b=\sqrt{2}-1\), तो (ab) का मान क्या है?
If \(a=\sqrt{2}+1\) and \(b=\sqrt{2}-1\), what is the value of (ab)?
#real-numbers
#conjugate-product
#rational-result
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A (1)
B (3)
C \(\sqrt{2}\)
D \(2\sqrt{2}\)
Explanation opens after your attempt
Step 1
Concept
(ab=\(\sqrt{2}+1\)\(\sqrt{2}-1\)) है। / (ab=\(\sqrt{2}+1\)\(\sqrt{2}-1\)).
Step 2
Why this answer is correct
वर्गों के अंतर से (\(\sqrt{2}\)2 -12 =2-1=1)। / Using difference of squares, (\(\sqrt{2}\)2 -12 =2-1=1).
Step 3
Exam Tip
(a+b) और (a-b) जैसे संयुग्म रूप पहचानना सीखें। / Learn to recognise conjugate forms like (a+b) and (a-b).
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\(\frac{1}{2+\sqrt{3}}\) का परिमेय हर वाला रूप क्या है?
What is the form of \(\frac{1}{2+\sqrt{3}}\) with a rational denominator?
#real-numbers
#rationalisation
#conjugate
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A \(2-\sqrt{3}\)
B \(2+\sqrt{3}\)
C \(\frac{2-\sqrt{3}}{7}\)
D \(\sqrt{3}-2\)
Explanation opens after your attempt
Correct Answer
A. \(2-\sqrt{3}\)
Step 1
Concept
हर का संयुग्म \(2-\sqrt{3}\) है। / The conjugate of the denominator 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
परिमेयकरण में संयुग्म से गुणा करें। / For rationalisation, multiply by the conjugate.
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कौन-सा कथन गलत है?
Which statement is false?
#real-numbers
#false-statement
#irrational-sum
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A \(\sqrt{2}\) अपरिमेय है / \(\sqrt{2}\) is irrational
B \(\sqrt{49}\) परिमेय है / \(\sqrt{49}\) is rational
C दो अपरिमेय संख्याओं का योग हमेशा अपरिमेय होता है / The sum of two irrational numbers is always irrational
D \(0.727272\ldots\) परिमेय है / \(0.727272\ldots\) is rational
Explanation opens after your attempt
Correct Answer
C. दो अपरिमेय संख्याओं का योग हमेशा अपरिमेय होता है / The sum of two irrational numbers is always irrational
Step 1
Concept
\(\sqrt{2}\) और \(-\sqrt{2}\) दोनों अपरिमेय हैं। / \(\sqrt{2}\) and \(-\sqrt{2}\) are both irrational.
Step 2
Why this answer is correct
इनका योग (0) है, जो परिमेय है। / Their sum is (0), which is rational.
Step 3
Exam Tip
हमेशा शब्द वाले कथन को एक प्रतिउदाहरण से जांचें। / Test always-type statements using a counterexample.
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\(\sqrt{8}+\sqrt{32}+\sqrt{128}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{8}+\sqrt{32}+\sqrt{128}\)?
#real-numbers
#radical-addition
#sqrt2
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A \(14\sqrt{2}\)
B \(7\sqrt{2}\)
C \(12\sqrt{2}\)
D \(18\sqrt{2}\)
Explanation opens after your attempt
Correct Answer
A. \(14\sqrt{2}\)
Step 1
Concept
\(\sqrt{8}=2\sqrt{2}\), \(\sqrt{32}=4\sqrt{2}\), और \(\sqrt{128}=8\sqrt{2}\)। / \(\sqrt{8}=2\sqrt{2}\), \(\sqrt{32}=4\sqrt{2}\), and \(\sqrt{128}=8\sqrt{2}\).
Step 2
Why this answer is correct
योग \(2\sqrt{2}+4\sqrt{2}+8\sqrt{2}=14\sqrt{2}\)। / The sum is \(2\sqrt{2}+4\sqrt{2}+8\sqrt{2}=14\sqrt{2}\).
Step 3
Exam Tip
कई वर्गमूल हों तो पहले सबको सरल रूप में लिखें। / With many radicals, simplify all of them first.
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यदि \(x=\sqrt{3}\), तो \(x^2+2x\) किसके बराबर है?
If \(x=\sqrt{3}\), what is \(x^2+2x\) equal to?
#real-numbers
#algebraic-expression
#irrational
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A \(3+2\sqrt{3}\)
B \(5\sqrt{3}\)
C (6)
D \(3+\sqrt{6}\)
Explanation opens after your attempt
Correct Answer
A. \(3+2\sqrt{3}\)
Step 1
Concept
(x-2 =\(\sqrt{3}\)2 =3)। / (x-2 =\(\sqrt{3}\)2 =3).
Step 2
Why this answer is correct
\(2x=2\sqrt{3}\), इसलिए \(x^2+2x=3+2\sqrt{3}\)। / \(2x=2\sqrt{3}\), so \(x^2+2x=3+2\sqrt{3}\).
Step 3
Exam Tip
वर्ग और गुणा को अलग-अलग सरल करें। / Simplify the square and the product separately.
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कौन-सी संख्या (8) और (9) के बीच स्थित अपरिमेय संख्या है?
Which number is an irrational number between (8) and (9)?
#real-numbers
#irrational-between-numbers
#sqrt65
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A \(\sqrt{65}\)
B \(\sqrt{64}\)
C \(\sqrt{81}\)
D (8.5)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{65}\)
Step 1
Concept
(64<65<81), इसलिए \(8<\sqrt{65}<9\)। / Since (64<65<81), \(8<\sqrt{65}<9\).
Step 2
Why this answer is correct
(65) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{65}\) अपरिमेय है। / (65) is not a perfect square, so \(\sqrt{65}\) is irrational.
Step 3
Exam Tip
दो पूर्ण वर्गों के बीच की संख्या का वर्गमूल उनके मूलों के बीच होता है। / The square root of a number between two perfect squares lies between their roots.
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निम्नलिखित में से कौन-सा दशमलव परिमेय है?
Which of the following decimals is rational?
#real-numbers
#recurring-decimal
#rational
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A \(1.010010001\ldots\)
B \(2.718281828\ldots\) बिना दोहराव / \(2.718281828\ldots\) without repetition
C \(0.141414\ldots\)
D \(3.1010010001\ldots\)
Explanation opens after your attempt
Correct Answer
C. \(0.141414\ldots\)
Step 1
Concept
आवर्ती दशमलव परिमेय होता है। / A recurring decimal is rational.
Step 2
Why this answer is correct
\(0.141414\ldots\) में (14) बार-बार दोहरता है। / In \(0.141414\ldots\), the block (14) repeats.
Step 3
Exam Tip
दोहरते समूह को पहचानना दशमलव वाले प्रश्नों की कुंजी है। / Identifying the repeating block is the key in decimal questions.
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(\left\(\sqrt{5}+2\right\)2 ) का मान क्या है?
What is the value of (\left\(\sqrt{5}+2\right\)2 )?
#real-numbers
#irrational-expression
#algebra
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A \(9+4\sqrt{5}\)
B \(7+2\sqrt{5}\)
C \(5+4\sqrt{5}\)
D \(9+\sqrt{5}\)
Explanation opens after your attempt
Correct Answer
A. \(9+4\sqrt{5}\)
Step 1
Concept
((a+b)2 =a-2 +2ab+b-2 ) लगाएं। / Use ((a+b)2 =a-2 +2ab+b-2 ).
Step 2
Why this answer is correct
(\(\sqrt{5}\)2 +2\(\sqrt{5}\)(2)+22 =5+4\sqrt{5}+4=9+4\sqrt{5})। / (\(\sqrt{5}\)2 +2\(\sqrt{5}\)(2)+22 =5+4\sqrt{5}+4=9+4\sqrt{5}).
Step 3
Exam Tip
मध्य पद (2ab) को छोड़ना सामान्य गलती है। / Missing the middle term (2ab) is a common mistake.
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यदि \(\sqrt{n}\) (4) और (5) के बीच है, तो (n) के लिए कौन-सा मान संभव है और \(\sqrt{n}\) अपरिमेय है?
If \(\sqrt{n}\) lies between (4) and (5), which value of (n) is possible and makes \(\sqrt{n}\) irrational?
#real-numbers
#comparison
#irrational-numbers
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A (16)
B (20)
C (25)
D (36)
Explanation opens after your attempt
Step 1
Concept
\(4<\sqrt{n}<5\) का अर्थ है (16<n<25)।
Step 2
Why this answer is correct
(20) इस अंतराल में है और पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{20}\) अपरिमेय है। / (20) lies in this interval and is not a perfect square, so \(\sqrt{20}\) is irrational.
Step 3
Exam Tip
वर्गमूल की असमानता में दोनों तरफ वर्ग लेकर अंतराल बनाएं। / In square-root inequalities, square the bounds to form the interval.
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\(\frac{2}{\sqrt{3}}\) का परिमेय हर वाला रूप क्या है?
What is the form of \(\frac{2}{\sqrt{3}}\) with a rational denominator?
#real-numbers
#rationalisation
#sqrt3
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A \(\frac{2\sqrt{3}}{3}\)
B \(\frac{\sqrt{3}}{2}\)
C \(2\sqrt{3}\)
D \(\frac{3}{2\sqrt{3}}\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{2\sqrt{3}}{3}\)
Step 1
Concept
हर से वर्गमूल हटाने के लिए ऊपर और नीचे \(\sqrt{3}\) से गुणा करें। / Multiply numerator and denominator by \(\sqrt{3}\) to remove the square root from the denominator.
Step 2
Why this answer is correct
\(\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\)। / \(\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\).
Step 3
Exam Tip
परिमेयकरण के बाद हर में वर्गमूल नहीं रहना चाहिए। / After rationalising, the denominator should not contain a square root.
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कौन-सा परिणाम अपरिमेय है?
Which result is irrational?
#real-numbers
#radical-product
#irrational-result
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A \(\sqrt{6}\times\sqrt{24}\)
B \(\sqrt{18}\times\sqrt{2}\)
C \(\sqrt{3}\times\sqrt{27}\)
D \(\sqrt{5}\times\sqrt{2}\)
Explanation opens after your attempt
Correct Answer
D. \(\sqrt{5}\times\sqrt{2}\)
Step 1
Concept
पहले सभी गुणनफल सरल करें। / First simplify all products.
Step 2
Why this answer is correct
पहले तीन में अंदर की संख्याएँ (144), (36), और (81) बनती हैं, जो पूर्ण वर्ग हैं; चौथा \(\sqrt{10}\) है। / The first three produce inside numbers (144), (36), and (81), which are perfect squares; the fourth gives \(\sqrt{10}\).
Step 3
Exam Tip
गुणन के बाद बनी अंदर की संख्या पूर्ण वर्ग है या नहीं, यह जांचें। / After multiplication, check whether the inside number is a perfect square.
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\(\sqrt{98}+\sqrt{50}-\sqrt{18}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{98}+\sqrt{50}-\sqrt{18}\)?
#real-numbers
#radical-expression
#simplification
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A \(9\sqrt{2}\)
B \(5\sqrt{2}\)
C \(15\sqrt{2}\)
D \(\sqrt{130}\)
Explanation opens after your attempt
Correct Answer
A. \(9\sqrt{2}\)
Step 1
Concept
\(\sqrt{98}=7\sqrt{2}\), \(\sqrt{50}=5\sqrt{2}\), और \(\sqrt{18}=3\sqrt{2}\)। / \(\sqrt{98}=7\sqrt{2}\), \(\sqrt{50}=5\sqrt{2}\), and \(\sqrt{18}=3\sqrt{2}\).
Step 2
Why this answer is correct
\(7\sqrt{2}+5\sqrt{2}-3\sqrt{2}=9\sqrt{2}\)। / \(7\sqrt{2}+5\sqrt{2}-3\sqrt{2}=9\sqrt{2}\).
Step 3
Exam Tip
सभी पदों को समान वर्गमूल में बदलने के बाद ही जोड़-घटाव करें। / Add or subtract only after converting all terms to like radicals.
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यदि (r) परिमेय है और (s) अपरिमेय है, तो (r+s) कब परिमेय हो सकता है?
If (r) is rational and (s) is irrational, when can (r+s) be rational?
#real-numbers
#rational-irrational-sum
#conceptual
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A जब (r=0) हो / When (r=0)
B कभी नहीं / Never
C जब (s) धनात्मक हो / When (s) is positive
D जब (r=1) हो / When (r=1)
Explanation opens after your attempt
Correct Answer
B. कभी नहीं / Never
Step 1
Concept
परिमेय संख्या जोड़ने से अपरिमेय संख्या परिमेय नहीं बनती। / Adding a rational number cannot make an irrational number rational.
Step 2
Why this answer is correct
यदि (r+s) परिमेय मान लें, तो (s=(r+s)-r) परिमेय हो जाएगा, जो गलत है। / If (r+s) were rational, then (s=(r+s)-r) would be rational, which is a contradiction.
Step 3
Exam Tip
ऐसे नियमों को उलटा सोचकर भी जांच सकते हैं। / Such rules can also be checked by reverse reasoning.
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\(\sqrt{a}\) और \(\sqrt{b}\) का गुणनफल (12) है। यदि (a=3), तो (b) का मान क्या होगा?
The product of \(\sqrt{a}\) and \(\sqrt{b}\) is (12). If (a=3), what is the value of (b)?
#real-numbers
#radical-equation
#square-root
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A (24)
B (36)
C (48)
D (72)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\)। / \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\).
Step 2
Why this answer is correct
\(\sqrt{3b}=12\), इसलिए (3b=144) और (b=48)। / \(\sqrt{3b}=12\), so (3b=144) and (b=48).
Step 3
Exam Tip
वर्गमूल समीकरण में दोनों तरफ वर्ग करना उपयोगी होता है। / In square-root equations, squaring both sides is useful.
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निम्नलिखित में से कौन-सा मान \(\sqrt{7}\) के सबसे निकट है?
Which of the following values is closest to \(\sqrt{7}\)?
#real-numbers
#approximation
#sqrt7
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A (2.45)
B (2.65)
C (2.95)
D (3.15)
Explanation opens after your attempt
Step 1
Concept
(4<7<9), इसलिए \(\sqrt{7}\) (2) और (3) के बीच है। / Since (4<7<9), \(\sqrt{7}\) lies between (2) and (3).
Step 2
Why this answer is correct
\(\sqrt{7}\approx2.646\), इसलिए (2.65) सबसे निकट है। / \(\sqrt{7}\approx2.646\), so (2.65) is the closest.
Step 3
Exam Tip
अनुमान में पहले पूर्ण वर्गों से सीमा तय करें। / In approximation, first set the range using perfect squares.
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कौन-सा विकल्प दो अपरिमेय संख्याओं का ऐसा अंतर दिखाता है जो परिमेय है?
Which option shows a difference of two irrational numbers that is rational?
#real-numbers
#irrational-difference
#rational-result
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A \(\sqrt{5}-\sqrt{5}\)
B \(\sqrt{7}-\sqrt{2}\)
C \(\sqrt{11}-1\)
D \(\sqrt{13}-\sqrt{3}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{5}-\sqrt{5}\)
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 अंतर (0) है, जो परिमेय है। / Their difference is (0), which is rational.
Step 3
Exam Tip
समान अपरिमेय पदों का अंतर परिमेय परिणाम दे सकता है। / The difference of equal irrational terms can be rational.
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\(\sqrt{27}\times\sqrt{12}\) का मान क्या है?
What is the value of \(\sqrt{27}\times\sqrt{12}\)?
#real-numbers
#radical-product
#rational-result
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A (18)
B \(\sqrt{39}\)
C \(9\sqrt{4}\)
D (36)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{27}\times\sqrt{12}=\sqrt{324}\)। / \(\sqrt{27}\times\sqrt{12}=\sqrt{324}\).
Step 2
Why this answer is correct
\(\sqrt{324}=18\), इसलिए परिणाम परिमेय है। / \(\sqrt{324}=18\), so the result is rational.
Step 3
Exam Tip
गुणन करते समय अंदर की संख्याएँ गुणा करके पूर्ण वर्ग जांचें। / When multiplying, multiply inside numbers and check for a perfect square.
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यदि \(x=2-\sqrt{3}\), तो \(\frac{1}{x}\) क्या होगा?
If \(x=2-\sqrt{3}\), what is \(\frac{1}{x}\)?
#real-numbers
#rationalisation
#conjugate
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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 संयुग्म \(2+\sqrt{3}\) है। / For \(\frac{1}{2-\sqrt{3}}\), the conjugate of the denominator is \(2+\sqrt{3}\).
Step 2
Why this answer is correct
गुणा करने पर हर (4-3=1) बनता है, इसलिए मान \(2+\sqrt{3}\) है। / Multiplying gives denominator (4-3=1), so the value is \(2+\sqrt{3}\).
Step 3
Exam Tip
संयुग्म से परिमेयकरण तेजी से उत्तर देता है। / Rationalising with the conjugate gives the answer quickly.
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यदि \(a=5+\sqrt{2}\), तो (a) की प्रकृति क्या है?
If \(a=5+\sqrt{2}\), what is the nature of (a)?
#real-numbers
#irrational-sum
#concept
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A परिमेय / Rational
B अपरिमेय / Irrational
C पूर्णांक / Integer
D प्राकृतिक / Natural
Explanation opens after your attempt
Correct Answer
B. अपरिमेय / Irrational
Step 1
Concept
(5) परिमेय है और \(\sqrt{2}\) अपरिमेय है। / (5) 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
केवल पूर्णांक जुड़ने से वर्गमूल वाला भाग खत्म नहीं होता। / Adding an integer does not remove the square-root part.
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\(\sqrt{200}-\sqrt{72}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{200}-\sqrt{72}\)?
#real-numbers
#radical-subtraction
#sqrt2
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A \(4\sqrt{2}\)
B \(2\sqrt{2}\)
C \(8\sqrt{2}\)
D \(\sqrt{128}\)
Explanation opens after your attempt
Correct Answer
A. \(4\sqrt{2}\)
Step 1
Concept
\(\sqrt{200}=10\sqrt{2}\) और \(\sqrt{72}=6\sqrt{2}\)। / \(\sqrt{200}=10\sqrt{2}\) and \(\sqrt{72}=6\sqrt{2}\).
Step 2
Why this answer is correct
\(10\sqrt{2}-6\sqrt{2}=4\sqrt{2}\)। / \(10\sqrt{2}-6\sqrt{2}=4\sqrt{2}\).
Step 3
Exam Tip
वर्गमूल घटाने में पहले दोनों पदों को सरल रूप में लिखें। / Before subtracting radicals, write both terms in simplified form.
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किस विकल्प में दी गई संख्या अपरिमेय है और (5) से छोटी है?
In which option is the given number irrational and less than (5)?
#real-numbers
#quality-check
#invalid-question
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A \(\sqrt{30}\)
B \(\sqrt{25}\)
C \(\frac{22}{7}\)
D (4.75)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{30}\)
Step 1
Concept
\(\sqrt{30}\) अपरिमेय है क्योंकि (30) पूर्ण वर्ग नहीं है। / \(\sqrt{30}\) is irrational because (30) is not a perfect square.
Step 2
Why this answer is correct
लेकिन (25<30<36), इसलिए \(\sqrt{30}\) (5) से बड़ा है, अतः यह सही नहीं? ध्यान से देखें: शर्त (5) से छोटी है, इसलिए कोई भी विकल्प? \(\frac{22}{7}\) और (4.75) परिमेय हैं, \(\sqrt{25}\) परिमेय है। / But (25<30<36), so \(\sqrt{30}\) is greater than (5); the other options are rational.
Step 3
Exam Tip
इस प्रश्न में सही चयन नहीं बनता, इसलिए ऐसा विकल्प प्रश्न में नहीं होना चाहिए। / This means the option set has no valid answer, so such a question should not be used.
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(\left\(\sqrt{7}-\sqrt{3}\right\)\left\(\sqrt{7}+\sqrt{3}\right\)) का मान क्या है?
What is the value of (\left\(\sqrt{7}-\sqrt{3}\right\)\left\(\sqrt{7}+\sqrt{3}\right\))?
#real-numbers
#conjugate-product
#rational-result
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A (4)
B (10)
C \(\sqrt{21}\)
D \(7+\sqrt{3}\)
Explanation opens after your attempt
Step 1
Concept
यह ((a-b)(a+b)=a-2 -b-2 ) का रूप है। / This is of the form ((a-b)(a+b)=a-2 -b-2 ).
Step 2
Why this answer is correct
(\(\sqrt{7}\)2 -\(\sqrt{3}\)2 =7-3=4)। / (\(\sqrt{7}\)2 -\(\sqrt{3}\)2 =7-3=4).
Step 3
Exam Tip
संयुग्म गुणन में सीधे वर्गों का अंतर लगाएं। / In conjugate multiplication, directly use the difference of squares.
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निम्नलिखित में से कौन-सी संख्या \(3+\sqrt{2}\) के बराबर नहीं है?
Which of the following is not equal to \(3+\sqrt{2}\)?
#real-numbers
#equivalent-form
#quality-check
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A (\frac{\(3+\sqrt{2}\)\(3-\sqrt{2}\)}{3-\sqrt{2}})
B \(\frac{7}{3-\sqrt{2}}\)
C \(3+\sqrt{2}\)
D \(\sqrt{2}+3\)
Explanation opens after your attempt
Correct Answer
B. \(\frac{7}{3-\sqrt{2}}\)
Step 1
Concept
पहले विकल्प में \(3-\sqrt{2}\) कटकर \(3+\sqrt{2}\) देता है। / The first option cancels to \(3+\sqrt{2}\).
Step 2
Why this answer is correct
(\frac{7}{3-\sqrt{2}}=\frac{7\(3+\sqrt{2}\)}{9-2}=3+\sqrt{2}), इसलिए यह भी बराबर है; अतः कोई गलत विकल्प नहीं? फिर प्रश्न दोषपूर्ण होगा। / (\frac{7}{3-\sqrt{2}}=\frac{7\(3+\sqrt{2}\)}{9-2}=3+\sqrt{2}), so it is also equal; hence there is no incorrect option.
Step 3
Exam Tip
बराबर रूप वाले प्रश्न में हर विकल्प को सरल करना जरूरी है। / In equivalent-form questions, simplify every option.
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यदि \(\sqrt{2}=1.414\) लगभग है, तो \(3\sqrt{2}\) का लगभग मान क्या होगा?
If \(\sqrt{2}=1.414\) approximately, what is the approximate value of \(3\sqrt{2}\)?
#real-numbers
#approximation
#sqrt2
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A (4.242)
B (3.414)
C (2.828)
D (5.414)
Explanation opens after your attempt
Correct Answer
A. (4.242)
Step 1
Concept
दिए गए मान को (3) से गुणा करें। / Multiply the given approximate value by (3).
Step 2
Why this answer is correct
\(3\sqrt{2}\approx3\times1.414=4.242\)। / \(3\sqrt{2}\approx3\times1.414=4.242\).
Step 3
Exam Tip
अनुमान में दिए गए मान का सीधा उपयोग करें। / In approximation questions, directly use the given value.
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कौन-सी संख्या \(3\sqrt{5}\) के बराबर है?
Which number is equal to \(3\sqrt{5}\)?
#real-numbers
#equivalent-radicals
#conversion
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A \(\sqrt{45}\)
B \(\sqrt{15}\)
C \(\sqrt{30}\)
D \(\sqrt{75}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{45}\)
Step 1
Concept
\(3\sqrt{5}=\sqrt{9}\sqrt{5}\)। / \(3\sqrt{5}=\sqrt{9}\sqrt{5}\).
Step 2
Why this answer is correct
यह \(\sqrt{45}\) के बराबर है। / This equals \(\sqrt{45}\).
Step 3
Exam Tip
बाहर के गुणांक को वर्ग करके अंदर ले जा सकते हैं। / An outside coefficient can be moved inside by squaring it.
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यदि \(x=\sqrt{2}+\sqrt{8}\), तो (x) का सरल रूप क्या है?
If \(x=\sqrt{2}+\sqrt{8}\), what is the simplified form of (x)?
#real-numbers
#radical-addition
#sqrt2
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A \(3\sqrt{2}\)
B \(\sqrt{10}\)
C \(2\sqrt{8}\)
D \(5\sqrt{2}\)
Explanation opens after your attempt
Correct Answer
A. \(3\sqrt{2}\)
Step 1
Concept
\(\sqrt{8}=2\sqrt{2}\)। / \(\sqrt{8}=2\sqrt{2}\).
Step 2
Why this answer is correct
\(x=\sqrt{2}+2\sqrt{2}=3\sqrt{2}\)। / \(x=\sqrt{2}+2\sqrt{2}=3\sqrt{2}\).
Step 3
Exam Tip
जोड़ने से पहले वर्गमूलों को समान रूप में बदलें। / Before adding, convert radicals into like form.
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कौन-सी संख्या \(\sqrt{3}\) और \(\sqrt{5}\) के बीच स्थित है?
Which number lies between \(\sqrt{3}\) and \(\sqrt{5}\)?
#real-numbers
#comparison
#between-roots
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A (2)
B (3)
C (1)
D \(\sqrt{2}\)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{3}\approx1.732\) और \(\sqrt{5}\approx2.236\)। / \(\sqrt{3}\approx1.732\) and \(\sqrt{5}\approx2.236\).
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 के लिए लगभग मान या वर्ग करके जांच दोनों तरीके उपयोगी हैं। / Approximate values or squaring can both help in comparison.
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\(\sqrt{3}+\sqrt{12}+\sqrt{27}\) की प्रकृति क्या है?
What is the nature of \(\sqrt{3}+\sqrt{12}+\sqrt{27}\)?
#real-numbers
#irrational-expression
#radicals
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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
\(\sqrt{12}=2\sqrt{3}\) 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 \(6\sqrt{3}\) है, जो अपरिमेय है। / The total is \(6\sqrt{3}\), which is irrational.
Step 3
Exam Tip
सरलीकरण के बाद ही संख्या की प्रकृति तय करें। / Decide the nature of the number only after simplification.
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यदि \(\sqrt{p}\) अपरिमेय है और (k) अशून्य परिमेय है, तो \(k\sqrt{p}\) कैसी संख्या होगी?
If \(\sqrt{p}\) is irrational and (k) is a non-zero rational number, what type of number is \(k\sqrt{p}\)?
#real-numbers
#irrational-rule
#nonzero-rational
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A परिमेय / Rational
B अपरिमेय / Irrational
C हमेशा पूर्णांक / Always integer
D हमेशा शून्य / Always zero
Explanation opens after your attempt
Correct Answer
B. अपरिमेय / Irrational
Step 1
Concept
अशून्य परिमेय गुणक अपरिमेयता को समाप्त नहीं करता। / A non-zero rational multiplier does not remove irrationality.
Step 2
Why this answer is correct
जैसे \(4\sqrt{3}\) अपरिमेय रहता है। / For example, \(4\sqrt{3}\) remains irrational.
Step 3
Exam Tip
यहां अशून्य शर्त जरूरी है क्योंकि (0) से गुणा करने पर परिणाम (0) होगा। / The non-zero condition is important because multiplying by (0) gives (0).
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\(\sqrt{150}+\sqrt{24}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{150}+\sqrt{24}\)?
#real-numbers
#radical-addition
#sqrt6
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A \(7\sqrt{6}\)
B \(5\sqrt{6}\)
C \(9\sqrt{6}\)
D \(\sqrt{174}\)
Explanation opens after your attempt
Correct Answer
A. \(7\sqrt{6}\)
Step 1
Concept
\(\sqrt{150}=5\sqrt{6}\) और \(\sqrt{24}=2\sqrt{6}\)। / \(\sqrt{150}=5\sqrt{6}\) and \(\sqrt{24}=2\sqrt{6}\).
Step 2
Why this answer is correct
\(5\sqrt{6}+2\sqrt{6}=7\sqrt{6}\)। / \(5\sqrt{6}+2\sqrt{6}=7\sqrt{6}\).
Step 3
Exam Tip
समान वर्गमूल बनने के बाद ही जोड़ें। / Add radicals only after they become like radicals.
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कौन-सा विकल्प परिमेय संख्या और अपरिमेय संख्या का गुणनफल दिखाता है जो अपरिमेय है?
Which option shows the product of a rational number and an irrational number that is irrational?
#real-numbers
#rational-irrational-product
#concept
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A \(0\times\sqrt{5}\)
B \(3\times\sqrt{7}\)
C \(\sqrt{2}\times\sqrt{8}\)
D \(5\times2\)
Explanation opens after your attempt
Correct Answer
B. \(3\times\sqrt{7}\)
Step 1
Concept
(3) अशून्य परिमेय है और \(\sqrt{7}\) अपरिमेय है। / (3) is a non-zero rational number and \(\sqrt{7}\) is irrational.
Step 2
Why this answer is correct
\(3\sqrt{7}\) अपरिमेय रहेगा। / \(3\sqrt{7}\) remains irrational.
Step 3
Exam Tip
शून्य से गुणा करने का मामला अलग होता है, इसलिए अशून्य परिमेय पर ध्यान दें। / Multiplication by zero is a special case, so focus on non-zero rational factors.
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\(\sqrt{a}=7\) है। (a) के बारे में सही कथन कौन-सा है?
If \(\sqrt{a}=7\), which statement about (a) is correct?
#real-numbers
#square-root-equation
#perfect-square
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A (a=14) और अपरिमेय है / (a=14) and is irrational
B (a=49) और पूर्ण वर्ग है / (a=49) and is a perfect square
C (a=7) और परिमेय नहीं है / (a=7) and is not rational
D (a=21) और पूर्ण वर्ग है / (a=21) and is a perfect square
Explanation opens after your attempt
Correct Answer
B. (a=49) और पूर्ण वर्ग है / (a=49) and is a perfect square
Step 1
Concept
\(\sqrt{a}=7\) हो तो दोनों तरफ वर्ग करें। / If \(\sqrt{a}=7\), square both sides.
Step 2
Why this answer is correct
(a=49), और (49) पूर्ण वर्ग है। / (a=49), and (49) is a perfect square.
Step 3
Exam Tip
वर्गमूल समीकरण में मूल संख्या पाने के लिए वर्ग करें। / In square-root equations, square to find the original number.
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निम्नलिखित में से कौन-सा परिणाम परिमेय नहीं है?
Which of the following results is not rational?
#real-numbers
#irrational-result
#mcq
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A \(\sqrt{16}+\sqrt{9}\)
B \(\sqrt{5}\times\sqrt{20}\)
C \(\sqrt{2}+\sqrt{18}\)
D \(\sqrt{49}-\sqrt{25}\)
Explanation opens after your attempt
Correct Answer
C. \(\sqrt{2}+\sqrt{18}\)
Step 1
Concept
\(\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}\)। / \(\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}\).
Step 2
Why this answer is correct
\(4\sqrt{2}\) अपरिमेय है, इसलिए परिमेय नहीं है। / \(4\sqrt{2}\) is irrational, so it is not rational.
Step 3
Exam Tip
विकल्पों में पहले सभी परिणाम सरल करें, फिर प्रकृति तय करें। / In options, simplify every result before deciding its nature.
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यदि \(\sqrt{n}\) अपरिमेय है, तो (n) के लिए कौन-सा मान सही हो सकता है?
If \(\sqrt{n}\) is irrational, which value of (n) can be correct?
#real-numbers
#irrational-numbers
#square-root
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A (36)
B (49)
C (52)
D (64)
Explanation opens after your attempt
Step 1
Concept
किसी पूर्ण वर्ग का वर्गमूल परिमेय होता है। / The square root of a perfect square is rational.
Step 2
Why this answer is correct
(52) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{52}\) अपरिमेय होगा। / (52) is not a perfect square, so \(\sqrt{52}\) is irrational.
Step 3
Exam Tip
ऐसे प्रश्नों में दिए गए मानों में पूर्ण वर्गों को पहले हटा दें। / In such questions, eliminate perfect squares first.
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निम्नलिखित में से कौन-सा परिणाम अपरिमेय है?
Which of the following results is irrational?
#real-numbers
#irrational-result
#concept
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A \(\sqrt{9}+\sqrt{16}\)
B \(\sqrt{7}\times\sqrt{28}\)
C \(\sqrt{11}+2\)
D \(\sqrt{25}-\sqrt{4}\)
Explanation opens after your attempt
Correct Answer
C. \(\sqrt{11}+2\)
Step 1
Concept
पहले हर विकल्प को सरल करें। / Simplify each option first.
Step 2
Why this answer is correct
\(\sqrt{11}\) अपरिमेय है और (2) परिमेय है, इसलिए \(\sqrt{11}+2\) अपरिमेय रहेगा। / \(\sqrt{11}\) is irrational and (2) is rational, so \(\sqrt{11}+2\) remains irrational.
Step 3
Exam Tip
परिमेय और अपरिमेय के योग को परिमेय न मानें। / Do not treat the sum of a rational and an irrational number as rational.
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\(\sqrt{98}-\sqrt{32}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{98}-\sqrt{32}\)?
#real-numbers
#radical-subtraction
#simplification
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A \(3\sqrt{2}\)
B \(7\sqrt{2}\)
C \(11\sqrt{2}\)
D \(\sqrt{66}\)
Explanation opens after your attempt
Correct Answer
A. \(3\sqrt{2}\)
Step 1
Concept
\(\sqrt{98}=7\sqrt{2}\) और \(\sqrt{32}=4\sqrt{2}\)। / \(\sqrt{98}=7\sqrt{2}\) and \(\sqrt{32}=4\sqrt{2}\).
Step 2
Why this answer is correct
\(7\sqrt{2}-4\sqrt{2}=3\sqrt{2}\)। / \(7\sqrt{2}-4\sqrt{2}=3\sqrt{2}\).
Step 3
Exam Tip
वर्गमूलों को घटाने से पहले समान वर्गमूल में बदलें। / Convert radicals into like radicals before subtracting.
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यदि \(x=4+\sqrt{7}\), तो (x-4) कैसी संख्या है?
If \(x=4+\sqrt{7}\), what type of number is (x-4)?
#real-numbers
#irrational-expression
#sqrt7
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A परिमेय / Rational
B पूर्णांक / Integer
C अपरिमेय / Irrational
D प्राकृतिक / Natural
Explanation opens after your attempt
Correct Answer
C. अपरिमेय / Irrational
Step 1
Concept
(x-4=\(4+\sqrt{7}\)-4) होगा। / (x-4=\(4+\sqrt{7}\)-4).
Step 2
Why this answer is correct
इससे \(\sqrt{7}\) बचता है, जो अपरिमेय है। / This leaves \(\sqrt{7}\), which is irrational.
Step 3
Exam Tip
पहले अभिव्यक्ति को सरल करें, फिर संख्या की प्रकृति बताएं। / First simplify the expression, then identify the nature of the number.
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निम्नलिखित में से कौन-सी संख्या परिमेय नहीं है?
Which of the following numbers is not rational?
#real-numbers
#not-rational
#irrational
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A (1.25)
B \(\frac{17}{4}\)
C \(\sqrt{45}\)
D \(0.8888\ldots\)
Explanation opens after your attempt
Correct Answer
C. \(\sqrt{45}\)
Step 1
Concept
समाप्त दशमलव, भिन्न और आवर्ती दशमलव परिमेय होते हैं। / Terminating decimals, fractions, and recurring decimals are rational.
Step 2
Why this answer is correct
\(\sqrt{45}=3\sqrt{5}\), और \(\sqrt{5}\) अपरिमेय है। / \(\sqrt{45}=3\sqrt{5}\), and \(\sqrt{5}\) is irrational.
Step 3
Exam Tip
वर्गमूल को सरल करके उसकी प्रकृति पहचानें। / Simplify the square root to identify its nature.
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\(\sqrt{20}+\sqrt{45}+\sqrt{125}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{20}+\sqrt{45}+\sqrt{125}\)?
#real-numbers
#radical-addition
#simplification
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A \(10\sqrt{5}\)
B \(12\sqrt{5}\)
C \(8\sqrt{5}\)
D \(15\sqrt{5}\)
Explanation opens after your attempt
Correct Answer
A. \(10\sqrt{5}\)
Step 1
Concept
\(\sqrt{20}=2\sqrt{5}\), \(\sqrt{45}=3\sqrt{5}\), और \(\sqrt{125}=5\sqrt{5}\)। / \(\sqrt{20}=2\sqrt{5}\), \(\sqrt{45}=3\sqrt{5}\), and \(\sqrt{125}=5\sqrt{5}\).
Step 2
Why this answer is correct
योग \(2\sqrt{5}+3\sqrt{5}+5\sqrt{5}=10\sqrt{5}\) है। / The sum is \(2\sqrt{5}+3\sqrt{5}+5\sqrt{5}=10\sqrt{5}\).
Step 3
Exam Tip
समान वर्गमूल बनने पर केवल गुणांक जोड़ें। / Once radicals are like terms, add only the coefficients.
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कौन-सा कथन सही है?
Which statement is correct?
#real-numbers
#irrational-rules
#true-statement
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A दो अपरिमेय संख्याओं का योग हमेशा परिमेय होता है / The sum of two irrational numbers is always rational
B दो अपरिमेय संख्याओं का गुणनफल हमेशा परिमेय होता है / The product of two irrational numbers is always rational
C अशून्य परिमेय संख्या और अपरिमेय संख्या का गुणनफल अपरिमेय होता है / The product of a non-zero rational number and an irrational number is irrational
D हर अपरिमेय संख्या पूर्ण वर्ग होती है / Every irrational number is a perfect square
Explanation opens after your attempt
Correct Answer
C. अशून्य परिमेय संख्या और अपरिमेय संख्या का गुणनफल अपरिमेय होता है / The product of a non-zero rational number and an irrational number is irrational
Step 1
Concept
अशून्य परिमेय गुणक अपरिमेयता को खत्म नहीं करता। / A non-zero rational multiplier does not remove irrationality.
Step 2
Why this answer is correct
जैसे \(5\sqrt{2}\) अपरिमेय रहता है। / For example, \(5\sqrt{2}\) remains irrational.
Step 3
Exam Tip
हमेशा वाले कथनों को उदाहरण से जांचना अच्छी आदत है। / Testing always-type statements with examples is a good habit.
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\(\frac{7}{\sqrt{7}}\) का सरल रूप क्या है?
What is the simplified form of \(\frac{7}{\sqrt{7}}\)?
#real-numbers
#rationalisation
#simplification
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A \(\sqrt{7}\)
B \(7\sqrt{7}\)
C \(\frac{1}{\sqrt{7}}\)
D (49)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{7}\)
Step 1
Concept
हर से वर्गमूल हटाने के लिए ऊपर और नीचे \(\sqrt{7}\) से गुणा करें। / Multiply numerator and denominator by \(\sqrt{7}\) to remove the root from the denominator.
Step 2
Why this answer is correct
\(\frac{7}{\sqrt{7}}=\frac{7\sqrt{7}}{7}=\sqrt{7}\)। / \(\frac{7}{\sqrt{7}}=\frac{7\sqrt{7}}{7}=\sqrt{7}\).
Step 3
Exam Tip
हर में वर्गमूल हो तो परिमेयकरण मदद करता है। / Rationalisation helps when the denominator contains a square root.
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यदि \(\sqrt{b}\) अपरिमेय है, तो कौन-सा परिणाम अवश्य परिमेय होगा?
If \(\sqrt{b}\) is irrational, which result will necessarily be rational?
#real-numbers
#irrational-square
#rational-result
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A \(\sqrt{b}+2\)
B (\(\sqrt{b}\)2 )
C \(3\sqrt{b}\)
D \(\sqrt{b}-1\)
Explanation opens after your attempt
Correct Answer
B. (\(\sqrt{b}\)2 )
Step 1
Concept
वर्गमूल का वर्ग करने पर अंदर की संख्या मिलती है। / Squaring a square root gives the number inside.
Step 2
Why this answer is correct
(\(\sqrt{b}\)2 =b), और (b) पूर्णांक हो तो परिमेय होता है। / (\(\sqrt{b}\)2 =b), and if (b) is an integer, it is rational.
Step 3
Exam Tip
अपरिमेय वर्गमूल का वर्ग परिमेय परिणाम दे सकता है। / The square of an irrational square root can give a rational result.
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कौन-सी संख्या (9) और (10) के बीच स्थित अपरिमेय संख्या है?
Which number is an irrational number between (9) and (10)?
#real-numbers
#irrational-between-numbers
#comparison
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A \(\sqrt{85}\)
B \(\sqrt{81}\)
C \(\sqrt{100}\)
D (9.5)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{85}\)
Step 1
Concept
(81<85<100), इसलिए \(9<\sqrt{85}<10\)। / Since (81<85<100), \(9<\sqrt{85}<10\).
Step 2
Why this answer is correct
(85) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{85}\) अपरिमेय है। / (85) is not a perfect square, so \(\sqrt{85}\) is irrational.
Step 3
Exam Tip
अंतराल वाले प्रश्न में पास के पूर्ण वर्गों का उपयोग करें। / In interval questions, use nearby perfect squares.
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\(\sqrt{75}+\sqrt{300}-\sqrt{48}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{75}+\sqrt{300}-\sqrt{48}\)?
#real-numbers
#radical-expression
#simplification
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A \(9\sqrt{3}\)
B \(11\sqrt{3}\)
C \(7\sqrt{3}\)
D \(\sqrt{327}\)
Explanation opens after your attempt
Correct Answer
B. \(11\sqrt{3}\)
Step 1
Concept
\(\sqrt{75}=5\sqrt{3}\), \(\sqrt{300}=10\sqrt{3}\), और \(\sqrt{48}=4\sqrt{3}\)। / \(\sqrt{75}=5\sqrt{3}\), \(\sqrt{300}=10\sqrt{3}\), and \(\sqrt{48}=4\sqrt{3}\).
Step 2
Why this answer is correct
\(5\sqrt{3}+10\sqrt{3}-4\sqrt{3}=11\sqrt{3}\)। / \(5\sqrt{3}+10\sqrt{3}-4\sqrt{3}=11\sqrt{3}\).
Step 3
Exam Tip
जोड़ और घटाव से पहले सभी वर्गमूलों को सरल करें। / Simplify all radicals before addition and subtraction.
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दशमलव \(2.303003000300003\ldots\) में शून्यों की संख्या क्रमशः बढ़ रही है। यह संख्या कैसी है?
In the decimal \(2.303003000300003\ldots\), the number of zeros is increasing successively. What type of number is it?
#real-numbers
#decimal-expansion
#irrational
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A समाप्त दशमलव / Terminating decimal
B आवर्ती परिमेय / Recurring rational
C अपरिमेय / Irrational
D पूर्णांक / Integer
Explanation opens after your attempt
Correct Answer
C. अपरिमेय / Irrational
Step 1
Concept
इस दशमलव में कोई निश्चित समूह बार-बार नहीं दोहरता। / This decimal has no fixed block repeating again and again.
Step 2
Why this answer is correct
यह अनवसानी और अनावर्ती है, इसलिए अपरिमेय है। / It is non-terminating and non-recurring, so it is irrational.
Step 3
Exam Tip
दशमलव में दोहराव का निश्चित नियम देखकर ही निर्णय लें। / Decide by checking whether there is a fixed repeating pattern.
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कौन-सा युग्म दो अपरिमेय संख्याओं का है जिनका योग परिमेय है?
Which pair has two irrational numbers whose sum is rational?
#real-numbers
#irrational-sum
#rational-result
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A \(\sqrt{5},-\sqrt{5}\)
B \(\sqrt{2},\sqrt{8}\)
C \(\sqrt{3},\sqrt{12}\)
D \(\sqrt{6},2\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{5},-\sqrt{5}\)
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 है। / Their sum is (0), which is rational.
Step 3
Exam Tip
विपरीत अपरिमेय पदों का योग परिमेय हो सकता है। / Opposite irrational terms can give a rational sum.
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(\sqrt{5}\(3+\sqrt{5}\)) का मान क्या है?
What is the value of (\sqrt{5}\(3+\sqrt{5}\))?
#real-numbers
#radical-multiplication
#expression
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A \(3\sqrt{5}+5\)
B \(8\sqrt{5}\)
C \(3+5\sqrt{5}\)
D (15)
Explanation opens after your attempt
Correct Answer
A. \(3\sqrt{5}+5\)
Step 1
Concept
वितरण नियम लगाएं: \(\sqrt{5}\times3+\sqrt{5}\times\sqrt{5}\)। / Apply distribution: \(\sqrt{5}\times3+\sqrt{5}\times\sqrt{5}\).
Step 2
Why this answer is correct
यह \(3\sqrt{5}+5\) बनता है। / This becomes \(3\sqrt{5}+5\).
Step 3
Exam Tip
समान वर्गमूलों के गुणन में \(\sqrt{5}\times\sqrt{5}=5\) याद रखें। / In such multiplication, remember \(\sqrt{5}\times\sqrt{5}=5\).
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कौन-सी संख्या \(4\sqrt{3}\) के बराबर है?
Which number is equal to \(4\sqrt{3}\)?
#real-numbers
#equivalent-radicals
#conversion
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A \(\sqrt{48}\)
B \(\sqrt{12}\)
C \(\sqrt{24}\)
D \(\sqrt{64}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{48}\)
Step 1
Concept
\(4\sqrt{3}=\sqrt{16}\sqrt{3}\)। / \(4\sqrt{3}=\sqrt{16}\sqrt{3}\).
Step 2
Why this answer is correct
यह \(\sqrt{48}\) के बराबर है। / This equals \(\sqrt{48}\).
Step 3
Exam Tip
बाहर के गुणांक को अंदर ले जाते समय उसका वर्ग गुणा करें। / When moving an outside coefficient inside the root, multiply by its square.
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(\left\(3+\sqrt{5}\right\)\left\(3-\sqrt{5}\right\)) का मान क्या है?
What is the value of (\left\(3+\sqrt{5}\right\)\left\(3-\sqrt{5}\right\))?
#real-numbers
#conjugates
#rational-result
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A (4)
B (14)
C \(9+\sqrt{5}\)
D \(9-\sqrt{5}\)
Explanation opens after your attempt
Step 1
Concept
यह ((a+b)(a-b)=a-2 -b-2 ) का रूप है। / This is of the form ((a+b)(a-b)=a-2 -b-2 ).
Step 2
Why this answer is correct
(32 -\(\sqrt{5}\)2 =9-5=4)। / (32 -\(\sqrt{5}\)2 =9-5=4).
Step 3
Exam Tip
संयुग्म गुणन में वर्गों का अंतर सीधे लगाएं। / In conjugate multiplication, directly use difference of squares.
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कौन-सा विकल्प अपरिमेय संख्या नहीं है?
Which option is not an irrational number?
#real-numbers
#not-irrational
#perfect-square
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A \(\sqrt{17}+1\)
B \(2\sqrt{13}\)
C \(\sqrt{196}-5\)
D \(7+\sqrt{3}\)
Explanation opens after your attempt
Correct Answer
C. \(\sqrt{196}-5\)
Step 1
Concept
\(\sqrt{196}=14\) है। / \(\sqrt{196}=14\).
Step 2
Why this answer is correct
\(\sqrt{196}-5=9\), जो परिमेय संख्या है। / \(\sqrt{196}-5=9\), which is rational.
Step 3
Exam Tip
अपरिमेय नहीं पूछे जाने पर पूर्ण वर्ग वाले विकल्प ध्यान से देखें। / When asked for not irrational, carefully check perfect-square options.
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यदि \(a=\sqrt{3}+2\) और \(b=\sqrt{3}-2\), तो (ab) का मान क्या है?
If \(a=\sqrt{3}+2\) and \(b=\sqrt{3}-2\), what is the value of (ab)?
#real-numbers
#conjugate-product
#rational-result
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A (1)
B (-1)
C (5)
D \(4\sqrt{3}\)
Explanation opens after your attempt
Step 1
Concept
(ab=\(\sqrt{3}+2\)\(\sqrt{3}-2\)) है। / (ab=\(\sqrt{3}+2\)\(\sqrt{3}-2\)).
Step 2
Why this answer is correct
वर्गों के अंतर से (\(\sqrt{3}\)2 -22 =3-4=-1)। / Using difference of squares, (\(\sqrt{3}\)2 -22 =3-4=-1).
Step 3
Exam Tip
संयुग्म रूप पहचानकर गणना छोटी हो जाती है। / Recognising conjugate form makes the calculation shorter.
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\(\frac{1}{3+\sqrt{8}}\) का परिमेय हर वाला रूप क्या है?
What is the form of \(\frac{1}{3+\sqrt{8}}\) with a rational denominator?
#real-numbers
#rationalisation
#conjugate
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A \(3-\sqrt{8}\)
B \(3+\sqrt{8}\)
C \(\frac{3-\sqrt{8}}{17}\)
D \(\sqrt{8}-3\)
Explanation opens after your attempt
Correct Answer
A. \(3-\sqrt{8}\)
Step 1
Concept
हर \(3+\sqrt{8}\) का संयुग्म \(3-\sqrt{8}\) है। / The conjugate of \(3+\sqrt{8}\) is \(3-\sqrt{8}\).
Step 2
Why this answer is correct
\(\frac{1}{3+\sqrt{8}}\times\frac{3-\sqrt{8}}{3-\sqrt{8}}=\frac{3-\sqrt{8}}{9-8}=3-\sqrt{8}\)। / \(\frac{1}{3+\sqrt{8}}\times\frac{3-\sqrt{8}}{3-\sqrt{8}}=\frac{3-\sqrt{8}}{9-8}=3-\sqrt{8}\).
Step 3
Exam Tip
परिमेयकरण में हर का संयुग्म प्रयोग करें। / Use the conjugate of the denominator for rationalisation.
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कौन-सा कथन गलत है?
Which statement is false?
#real-numbers
#false-statement
#irrational-product
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A \(\sqrt{13}\) अपरिमेय है / \(\sqrt{13}\) is irrational
B \(\sqrt{81}\) परिमेय है / \(\sqrt{81}\) is rational
C दो अपरिमेय संख्याओं का गुणनफल हमेशा अपरिमेय होता है / The product of two irrational numbers is always irrational
D \(0.565656\ldots\) परिमेय है / \(0.565656\ldots\) is rational
Explanation opens after your attempt
Correct Answer
C. दो अपरिमेय संख्याओं का गुणनफल हमेशा अपरिमेय होता है / The product of two irrational numbers is always irrational
Step 1
Concept
\(\sqrt{2}\) और \(\sqrt{2}\) दोनों अपरिमेय हैं। / \(\sqrt{2}\) and \(\sqrt{2}\) are both irrational.
Step 2
Why this answer is correct
इनका गुणनफल (2) है, जो परिमेय है। / Their product is (2), which is rational.
Step 3
Exam Tip
हमेशा वाले कथन को प्रतिउदाहरण से जांचें। / Test always-type statements with a counterexample.
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\(\sqrt{18}+\sqrt{72}+\sqrt{162}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{18}+\sqrt{72}+\sqrt{162}\)?
#real-numbers
#radical-addition
#sqrt2
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A \(18\sqrt{2}\)
B \(12\sqrt{2}\)
C \(15\sqrt{2}\)
D \(9\sqrt{2}\)
Explanation opens after your attempt
Correct Answer
A. \(18\sqrt{2}\)
Step 1
Concept
\(\sqrt{18}=3\sqrt{2}\), \(\sqrt{72}=6\sqrt{2}\), और \(\sqrt{162}=9\sqrt{2}\)। / \(\sqrt{18}=3\sqrt{2}\), \(\sqrt{72}=6\sqrt{2}\), and \(\sqrt{162}=9\sqrt{2}\).
Step 2
Why this answer is correct
योग \(3\sqrt{2}+6\sqrt{2}+9\sqrt{2}=18\sqrt{2}\)। / The sum is \(3\sqrt{2}+6\sqrt{2}+9\sqrt{2}=18\sqrt{2}\).
Step 3
Exam Tip
कई वर्गमूलों को पहले पूरी तरह सरल करें। / Simplify all radicals completely first.
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यदि \(x=\sqrt{5}\), तो \(x^2+3x\) किसके बराबर है?
If \(x=\sqrt{5}\), what is \(x^2+3x\) equal to?
#real-numbers
#algebraic-expression
#irrational
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A \(5+3\sqrt{5}\)
B \(8\sqrt{5}\)
C (15)
D \(5+\sqrt{15}\)
Explanation opens after your attempt
Correct Answer
A. \(5+3\sqrt{5}\)
Step 1
Concept
(x-2 =\(\sqrt{5}\)2 =5)। / (x-2 =\(\sqrt{5}\)2 =5).
Step 2
Why this answer is correct
\(3x=3\sqrt{5}\), इसलिए \(x^2+3x=5+3\sqrt{5}\)। / \(3x=3\sqrt{5}\), so \(x^2+3x=5+3\sqrt{5}\).
Step 3
Exam Tip
वर्ग और गुणा को अलग-अलग सरल करें। / Simplify the square and the multiplication separately.
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यदि \(\sqrt{n}\) (10) और (11) के बीच है, तो (n) के लिए कौन-सा मान सही हो सकता है और \(\sqrt{n}\) अपरिमेय होगा?
If \(\sqrt{n}\) lies between (10) and (11), which value of (n) can be correct and makes \(\sqrt{n}\) irrational?
#real-numbers
#comparison
#irrational-numbers
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A (100)
B (105)
C (121)
D (144)
Explanation opens after your attempt
Step 1
Concept
\(10<\sqrt{n}<11\) का अर्थ है (100<n<121)।
Step 2
Why this answer is correct
(105) इस अंतराल में है और पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{105}\) अपरिमेय है। / (105) lies in this interval and is not a perfect square, so \(\sqrt{105}\) is irrational.
Step 3
Exam Tip
वर्गमूल की सीमा के लिए दोनों तरफ वर्ग करें। / Square both bounds to handle square-root ranges.
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निम्नलिखित में से कौन-सा दशमलव परिमेय है?
Which of the following decimals is rational?
#real-numbers
#recurring-decimal
#rational
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A \(0.1010010001\ldots\)
B \(1.7320508\ldots\) बिना निश्चित दोहराव / \(1.7320508\ldots\) without fixed repetition
C \(0.272727\ldots\)
D \(2.010010001\ldots\)
Explanation opens after your attempt
Correct Answer
C. \(0.272727\ldots\)
Step 1
Concept
आवर्ती दशमलव परिमेय होता है। / A recurring decimal is rational.
Step 2
Why this answer is correct
\(0.272727\ldots\) में (27) बार-बार दोहरता है। / In \(0.272727\ldots\), the block (27) repeats.
Step 3
Exam Tip
दशमलव में दोहरने वाला समूह पहचानना जरूरी है। / Identifying the repeating block in a decimal is important.
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(\left\(\sqrt{7}+1\right\)2 ) का मान क्या है?
What is the value of (\left\(\sqrt{7}+1\right\)2 )?
#real-numbers
#irrational-expression
#algebra
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A \(8+2\sqrt{7}\)
B \(7+\sqrt{7}\)
C \(8+\sqrt{7}\)
D \(9+2\sqrt{7}\)
Explanation opens after your attempt
Correct Answer
A. \(8+2\sqrt{7}\)
Step 1
Concept
((a+b)2 =a-2 +2ab+b-2 ) लगाएं। / Use ((a+b)2 =a-2 +2ab+b-2 ).
Step 2
Why this answer is correct
(\(\sqrt{7}\)2 +2\sqrt{7}\times1+12 =7+2\sqrt{7}+1=8+2\sqrt{7})। / (\(\sqrt{7}\)2 +2\sqrt{7}\times1+12 =7+2\sqrt{7}+1=8+2\sqrt{7}).
Step 3
Exam Tip
मध्य पद (2ab) को भूलना सामान्य गलती है। / Forgetting the middle term (2ab) is a common mistake.
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यदि \(\sqrt{t}\) (6) और (7) के बीच है, तो (t) का कौन-सा मान संभव है?
If \(\sqrt{t}\) lies between (6) and (7), which value of (t) is possible?
#real-numbers
#square-root-interval
#comparison
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A (35)
B (42)
C (49)
D (64)
Explanation opens after your attempt
Step 1
Concept
\(6<\sqrt{t}<7\) का अर्थ है (36<t<49)।
Step 2
Why this answer is correct
(42) इस अंतराल में है, इसलिए \(\sqrt{42}\) (6) और (7) के बीच होगा। / (42) lies in this interval, so \(\sqrt{42}\) lies between (6) and (7).
Step 3
Exam Tip
वर्गमूल के अंतराल को समझने के लिए सीमा संख्याओं का वर्ग करें। / Square the boundary numbers to understand square-root intervals.
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\(\frac{4}{\sqrt{2}}\) का परिमेय हर वाला सरल रूप क्या है?
What is the simplified form of \(\frac{4}{\sqrt{2}}\) with a rational denominator?
#real-numbers
#rationalisation
#sqrt2
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A \(2\sqrt{2}\)
B \(\frac{4\sqrt{2}}{2}\)
C \(\sqrt{2}\)
D \(4\sqrt{2}\)
Explanation opens after your attempt
Correct Answer
A. \(2\sqrt{2}\)
Step 1
Concept
ऊपर और नीचे \(\sqrt{2}\) से गुणा करें। / Multiply numerator and denominator by \(\sqrt{2}\).
Step 2
Why this answer is correct
\(\frac{4}{\sqrt{2}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\)। / \(\frac{4}{\sqrt{2}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\).
Step 3
Exam Tip
परिमेयकरण के बाद उत्तर को पूरी तरह सरल करें। / After rationalising, simplify the answer completely.
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कौन-सा परिणाम अपरिमेय है?
Which result is irrational?
#real-numbers
#radical-product
#irrational-result
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A \(\sqrt{8}\times\sqrt{18}\)
B \(\sqrt{12}\times\sqrt{27}\)
C \(\sqrt{15}\times\sqrt{60}\)
D \(\sqrt{2}\times\sqrt{11}\)
Explanation opens after your attempt
Correct Answer
D. \(\sqrt{2}\times\sqrt{11}\)
Step 1
Concept
पहले गुणनफल में अंदर की संख्याएँ गुणा करें। / First multiply the numbers inside the roots.
Step 2
Why this answer is correct
पहले तीन में (144), (324), और (900) मिलते हैं, जो पूर्ण वर्ग हैं; चौथा \(\sqrt{22}\) है। / The first three give (144), (324), and (900), which are perfect squares; the fourth gives \(\sqrt{22}\).
Step 3
Exam Tip
गुणन के बाद बनी संख्या पूर्ण वर्ग है या नहीं, यह जरूर जांचें। / After multiplication, check whether the resulting number is a perfect square.
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\(\sqrt{128}+\sqrt{72}-\sqrt{50}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{128}+\sqrt{72}-\sqrt{50}\)?
#real-numbers
#radical-expression
#simplification
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A \(9\sqrt{2}\)
B \(7\sqrt{2}\)
C \(11\sqrt{2}\)
D \(\sqrt{150}\)
Explanation opens after your attempt
Correct Answer
A. \(9\sqrt{2}\)
Step 1
Concept
\(\sqrt{128}=8\sqrt{2}\), \(\sqrt{72}=6\sqrt{2}\), और \(\sqrt{50}=5\sqrt{2}\)। / \(\sqrt{128}=8\sqrt{2}\), \(\sqrt{72}=6\sqrt{2}\), and \(\sqrt{50}=5\sqrt{2}\).
Step 2
Why this answer is correct
\(8\sqrt{2}+6\sqrt{2}-5\sqrt{2}=9\sqrt{2}\)। / \(8\sqrt{2}+6\sqrt{2}-5\sqrt{2}=9\sqrt{2}\).
Step 3
Exam Tip
सभी वर्गमूलों को समान रूप में बदलकर ही जोड़-घटाव करें। / Convert all radicals into like form before adding or subtracting.
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यदि (p) परिमेय है और (q) अपरिमेय है, तो (p-q) कैसी संख्या होगी?
If (p) is rational and (q) is irrational, what type of number is (p-q)?
#real-numbers
#rational-irrational-difference
#conceptual
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A हमेशा परिमेय / Always rational
B हमेशा अपरिमेय / Always irrational
C हमेशा शून्य / Always zero
D हमेशा पूर्णांक / Always integer
Explanation opens after your attempt
Correct Answer
B. हमेशा अपरिमेय / Always irrational
Step 1
Concept
परिमेय संख्या में से अपरिमेय संख्या घटाने पर अपरिमेय भाग बचता है। / Subtracting an irrational number from a rational number leaves an irrational part.
Step 2
Why this answer is correct
यदि परिणाम परिमेय हो, तो (q=p-(p-q)) परिमेय हो जाएगा, जो गलत है। / If the result were rational, then (q=p-(p-q)) would be rational, which is impossible.
Step 3
Exam Tip
परिमेय और अपरिमेय के जोड़-घटाव के नियम याद रखें। / Remember the rules for addition and subtraction of rational and irrational numbers.
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\(\sqrt{c}\times\sqrt{12}=18\) और (c) धनात्मक है। (c) का मान क्या है?
If \(\sqrt{c}\times\sqrt{12}=18\) and (c) is positive, what is the value of (c)?
#real-numbers
#radical-equation
#square-root
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A (18)
B (24)
C (27)
D (36)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{c}\times\sqrt{12}=\sqrt{12c}\)। / \(\sqrt{c}\times\sqrt{12}=\sqrt{12c}\).
Step 2
Why this answer is correct
\(\sqrt{12c}=18\), इसलिए (12c=324) और (c=27)। / \(\sqrt{12c}=18\), so (12c=324) and (c=27).
Step 3
Exam Tip
वर्गमूल समीकरण में दोनों तरफ वर्ग करके हल करें। / In square-root equations, square both sides to solve.
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निम्नलिखित में से कौन-सा मान \(\sqrt{13}\) के सबसे निकट है?
Which of the following values is closest to \(\sqrt{13}\)?
#real-numbers
#approximation
#sqrt13
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A (3.21)
B (3.61)
C (3.91)
D (4.21)
Explanation opens after your attempt
Step 1
Concept
(9<13<16), इसलिए \(\sqrt{13}\) (3) और (4) के बीच है। / Since (9<13<16), \(\sqrt{13}\) lies between (3) and (4).
Step 2
Why this answer is correct
\(\sqrt{13}\approx3.606\), इसलिए (3.61) सबसे निकट है। / \(\sqrt{13}\approx3.606\), so (3.61) is the closest.
Step 3
Exam Tip
अनुमान में पहले पूर्ण वर्गों से सीमा तय करें। / In approximation, first set the range using perfect squares.
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कौन-सा विकल्प दो अपरिमेय संख्याओं का ऐसा अंतर दिखाता है जो परिमेय है?
Which option shows a difference of two irrational numbers that is rational?
#real-numbers
#irrational-difference
#rational-result
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A \(\sqrt{7}-\sqrt{7}\)
B \(\sqrt{8}-\sqrt{3}\)
C \(\sqrt{10}-2\)
D \(\sqrt{15}-\sqrt{5}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{7}-\sqrt{7}\)
Step 1
Concept
\(\sqrt{7}\) और \(\sqrt{7}\) दोनों अपरिमेय हैं। / \(\sqrt{7}\) and \(\sqrt{7}\) are both irrational.
Step 2
Why this answer is correct
उनका अंतर (0) है, जो परिमेय है। / Their difference is (0), which is rational.
Step 3
Exam Tip
समान अपरिमेय पदों का अंतर परिमेय हो सकता है। / The difference of equal irrational terms can be rational.
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\(\sqrt{48}\times\sqrt{75}\) का मान क्या है?
What is the value of \(\sqrt{48}\times\sqrt{75}\)?
#real-numbers
#radical-product
#rational-result
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A \(30\sqrt{4}\)
B (60)
C (120)
D \(\sqrt{123}\)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{48}\times\sqrt{75}=\sqrt{3600}\)। / \(\sqrt{48}\times\sqrt{75}=\sqrt{3600}\).
Step 2
Why this answer is correct
\(\sqrt{3600}=60\), इसलिए परिणाम परिमेय है। / \(\sqrt{3600}=60\), so the result is rational.
Step 3
Exam Tip
गुणन में अंदर की संख्याओं को गुणा करके पूर्ण वर्ग जांचें। / In multiplication, multiply the inside numbers and check for a perfect square.
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यदि \(x=3-\sqrt{8}\), तो \(\frac{1}{x}\) क्या होगा?
If \(x=3-\sqrt{8}\), what is \(\frac{1}{x}\)?
#real-numbers
#rationalisation
#conjugate
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A \(3+\sqrt{8}\)
B \(3-\sqrt{8}\)
C \(\frac{3+\sqrt{8}}{17}\)
D \(\sqrt{8}-3\)
Explanation opens after your attempt
Correct Answer
A. \(3+\sqrt{8}\)
Step 1
Concept
\(\frac{1}{3-\sqrt{8}}\) में हर का संयुग्म \(3+\sqrt{8}\) है। / In \(\frac{1}{3-\sqrt{8}}\), the conjugate of the denominator is \(3+\sqrt{8}\).
Step 2
Why this answer is correct
हर (9-8=1) बनता है, इसलिए मान \(3+\sqrt{8}\) है। / The denominator becomes (9-8=1), so the value is \(3+\sqrt{8}\).
Step 3
Exam Tip
संयुग्म से परिमेयकरण करने पर हर जल्दी सरल होता है। / Rationalising with the conjugate quickly simplifies the denominator.
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यदि \(z=6+\sqrt{5}\), तो (z) की प्रकृति क्या है?
If \(z=6+\sqrt{5}\), what is the nature of (z)?
#real-numbers
#irrational-sum
#concept
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A परिमेय / Rational
B अपरिमेय / Irrational
C पूर्णांक / Integer
D सम संख्या / Even number
Explanation opens after your attempt
Correct Answer
B. अपरिमेय / Irrational
Step 1
Concept
(6) परिमेय है और \(\sqrt{5}\) अपरिमेय है। / (6) is rational and \(\sqrt{5}\) is irrational.
Step 2
Why this answer is correct
परिमेय और अपरिमेय संख्या का योग अपरिमेय होता है। / The sum of a rational and an irrational number is irrational.
Step 3
Exam Tip
पूर्णांक जोड़ने से वर्गमूल वाला अपरिमेय भाग खत्म नहीं होता। / Adding an integer does not remove the irrational square-root part.
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\(\sqrt{363}-\sqrt{75}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{363}-\sqrt{75}\)?
#real-numbers
#radical-subtraction
#sqrt3
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A \(6\sqrt{3}\)
B \(4\sqrt{3}\)
C \(8\sqrt{3}\)
D \(\sqrt{288}\)
Explanation opens after your attempt
Correct Answer
A. \(6\sqrt{3}\)
Step 1
Concept
\(\sqrt{363}=11\sqrt{3}\) और \(\sqrt{75}=5\sqrt{3}\)। / \(\sqrt{363}=11\sqrt{3}\) and \(\sqrt{75}=5\sqrt{3}\).
Step 2
Why this answer is correct
\(11\sqrt{3}-5\sqrt{3}=6\sqrt{3}\)। / \(11\sqrt{3}-5\sqrt{3}=6\sqrt{3}\).
Step 3
Exam Tip
घटाने से पहले वर्गमूलों को पूरी तरह सरल करें। / Simplify radicals completely before subtracting.
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किस विकल्प में दी गई संख्या अपरिमेय है और (5) से छोटी है?
In which option is the given number irrational and less than (5)?
#real-numbers
#irrational-less-than
#comparison
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A \(\sqrt{30}\)
B \(\sqrt{24}\)
C \(\sqrt{25}\)
D (4.9)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{24}\)
Step 1
Concept
\(\sqrt{24}\) अपरिमेय है क्योंकि (24) पूर्ण वर्ग नहीं है। / \(\sqrt{24}\) is irrational because (24) is not a perfect square.
Step 2
Why this answer is correct
(16<24<25), इसलिए \(4<\sqrt{24}<5\)। / Since (16<24<25), \(4<\sqrt{24}<5\).
Step 3
Exam Tip
शर्तों वाले प्रश्न में अपरिमेयता और सीमा दोनों जांचें। / In condition-based questions, check both irrationality and range.
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(\left\(\sqrt{11}-\sqrt{2}\right\)\left\(\sqrt{11}+\sqrt{2}\right\)) का मान क्या है?
What is the value of (\left\(\sqrt{11}-\sqrt{2}\right\)\left\(\sqrt{11}+\sqrt{2}\right\))?
#real-numbers
#conjugate-product
#rational-result
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A (9)
B (13)
C \(\sqrt{22}\)
D \(11+\sqrt{2}\)
Explanation opens after your attempt
Step 1
Concept
यह ((a-b)(a+b)=a-2 -b-2 ) का रूप है। / This is of the form ((a-b)(a+b)=a-2 -b-2 ).
Step 2
Why this answer is correct
(\(\sqrt{11}\)2 -\(\sqrt{2}\)2 =11-2=9)। / (\(\sqrt{11}\)2 -\(\sqrt{2}\)2 =11-2=9).
Step 3
Exam Tip
संयुग्म गुणन में वर्गों का अंतर सीधे लगाएं। / In conjugate multiplication, directly use the difference of squares.
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निम्नलिखित में से कौन-सा \(2+\sqrt{3}\) के बराबर है?
Which of the following is equal to \(2+\sqrt{3}\)?
#real-numbers
#equivalent-form
#rationalisation
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A \(\frac{1}{2-\sqrt{3}}\)
B \(\frac{1}{2+\sqrt{3}}\)
C \(2-\sqrt{3}\)
D \(\sqrt{3}-2\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{1}{2-\sqrt{3}}\)
Step 1
Concept
\(\frac{1}{2-\sqrt{3}}\) का हर परिमेय करने के लिए \(2+\sqrt{3}\) से गुणा करें। / Rationalise \(\frac{1}{2-\sqrt{3}}\) by multiplying by \(2+\sqrt{3}\).
Step 2
Why this answer is correct
हर (4-3=1) बनता है, इसलिए मान \(2+\sqrt{3}\) है। / The denominator becomes (4-3=1), so the value is \(2+\sqrt{3}\).
Step 3
Exam Tip
बराबर रूप पहचानने के लिए परिमेयकरण करें। / Use rationalisation to identify equivalent forms.
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यदि \(\sqrt{3}=1.732\) लगभग है, तो \(4\sqrt{3}\) का लगभग मान क्या होगा?
If \(\sqrt{3}=1.732\) approximately, what is the approximate value of \(4\sqrt{3}\)?
#real-numbers
#approximation
#sqrt3
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A (6.928)
B (5.732)
C (4.732)
D (8.928)
Explanation opens after your attempt
Correct Answer
A. (6.928)
Step 1
Concept
दिए गए लगभग मान को (4) से गुणा करें। / Multiply the given approximate value by (4).
Step 2
Why this answer is correct
\(4\sqrt{3}\approx4\times1.732=6.928\)। / \(4\sqrt{3}\approx4\times1.732=6.928\).
Step 3
Exam Tip
अनुमान वाले प्रश्न में दिए गए मान का सीधा उपयोग करें। / In approximation questions, directly use the given value.
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कौन-सी संख्या \(6\sqrt{2}\) के बराबर है?
Which number is equal to \(6\sqrt{2}\)?
#real-numbers
#equivalent-radicals
#conversion
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A \(\sqrt{72}\)
B \(\sqrt{36}\)
C \(\sqrt{48}\)
D \(\sqrt{12}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{72}\)
Step 1
Concept
\(6\sqrt{2}=\sqrt{36}\sqrt{2}\)। / \(6\sqrt{2}=\sqrt{36}\sqrt{2}\).
Step 2
Why this answer is correct
यह \(\sqrt{72}\) के बराबर है। / This equals \(\sqrt{72}\).
Step 3
Exam Tip
बाहर का गुणांक अंदर ले जाने के लिए उसका वर्ग अंदर गुणा करें। / To move the outside coefficient inside, multiply by its square.
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यदि \(x=\sqrt{3}+\sqrt{27}\), तो (x) का सरल रूप क्या है?
If \(x=\sqrt{3}+\sqrt{27}\), what is the simplified form of (x)?
#real-numbers
#radical-addition
#sqrt3
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A \(4\sqrt{3}\)
B \(\sqrt{30}\)
C \(3\sqrt{3}\)
D (6)
Explanation opens after your attempt
Correct Answer
A. \(4\sqrt{3}\)
Step 1
Concept
\(\sqrt{27}=3\sqrt{3}\)। / \(\sqrt{27}=3\sqrt{3}\).
Step 2
Why this answer is correct
\(x=\sqrt{3}+3\sqrt{3}=4\sqrt{3}\)। / \(x=\sqrt{3}+3\sqrt{3}=4\sqrt{3}\).
Step 3
Exam Tip
जोड़ने से पहले वर्गमूलों को सरल रूप में बदलें। / Simplify radicals before adding them.
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कौन-सी संख्या \(\sqrt{6}\) और \(\sqrt{10}\) के बीच स्थित है?
Which number lies between \(\sqrt{6}\) and \(\sqrt{10}\)?
#real-numbers
#comparison
#between-roots
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A (3)
B (4)
C (2)
D \(\sqrt{12}\)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{6}\approx2.45\) और \(\sqrt{10}\approx3.16\)। / \(\sqrt{6}\approx2.45\) and \(\sqrt{10}\approx3.16\).
Step 2
Why this answer is correct
(3) इन दोनों के बीच आता है। / (3) lies between these two values.
Step 3
Exam Tip
तुलना के लिए लगभग मान या वर्ग करके जांच करें। / Use approximate values or squaring to compare.
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\(\sqrt{5}+\sqrt{20}+\sqrt{45}\) की प्रकृति क्या है?
What is the nature of \(\sqrt{5}+\sqrt{20}+\sqrt{45}\)?
#real-numbers
#irrational-expression
#radicals
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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
\(\sqrt{20}=2\sqrt{5}\) और \(\sqrt{45}=3\sqrt{5}\)। / \(\sqrt{20}=2\sqrt{5}\) and \(\sqrt{45}=3\sqrt{5}\).
Step 2
Why this answer is correct
कुल योग \(6\sqrt{5}\) है, जो अपरिमेय है। / The total is \(6\sqrt{5}\), which is irrational.
Step 3
Exam Tip
किसी अभिव्यक्ति की प्रकृति तय करने से पहले उसे सरल करें। / Simplify an expression before deciding its nature.
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यदि \(\sqrt{p}\) अपरिमेय है और (k) अशून्य परिमेय है, तो \(\frac{\sqrt{p}}{k}\) कैसी संख्या होगी?
If \(\sqrt{p}\) is irrational and (k) is a non-zero rational number, what type of number is \(\frac{\sqrt{p}}{k}\)?
#real-numbers
#irrational-rule
#division
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A परिमेय / Rational
B अपरिमेय / Irrational
C हमेशा पूर्णांक / Always integer
D हमेशा शून्य / Always zero
Explanation opens after your attempt
Correct Answer
B. अपरिमेय / Irrational
Step 1
Concept
अशून्य परिमेय संख्या से भाग करने पर अपरिमेयता समाप्त नहीं होती। / Dividing by a non-zero rational number does not remove irrationality.
Step 2
Why this answer is correct
जैसे \(\frac{\sqrt{3}}{4}\) अपरिमेय रहता है। / For example, \(\frac{\sqrt{3}}{4}\) remains irrational.
Step 3
Exam Tip
यहां \(k\neq0\) जरूरी है क्योंकि शून्य से भाग संभव नहीं है। / The condition \(k\neq0\) is necessary because division by zero is not possible.
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\(\sqrt{216}+\sqrt{54}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{216}+\sqrt{54}\)?
#real-numbers
#radical-addition
#sqrt6
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A \(9\sqrt{6}\)
B \(6\sqrt{6}\)
C \(12\sqrt{6}\)
D \(\sqrt{270}\)
Explanation opens after your attempt
Correct Answer
A. \(9\sqrt{6}\)
Step 1
Concept
\(\sqrt{216}=6\sqrt{6}\) और \(\sqrt{54}=3\sqrt{6}\)। / \(\sqrt{216}=6\sqrt{6}\) and \(\sqrt{54}=3\sqrt{6}\).
Step 2
Why this answer is correct
\(6\sqrt{6}+3\sqrt{6}=9\sqrt{6}\)। / \(6\sqrt{6}+3\sqrt{6}=9\sqrt{6}\).
Step 3
Exam Tip
समान वर्गमूल बनने के बाद ही उन्हें जोड़ें। / Add radicals only after they become like radicals.
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कौन-सा विकल्प परिमेय संख्या और अपरिमेय संख्या का ऐसा गुणनफल दिखाता है जो अपरिमेय है?
Which option shows the product of a rational number and an irrational number that is irrational?
#real-numbers
#rational-irrational-product
#concept
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A \(0\times\sqrt{11}\)
B \(4\times\sqrt{13}\)
C \(\sqrt{3}\times\sqrt{27}\)
D \(6\times5\)
Explanation opens after your attempt
Correct Answer
B. \(4\times\sqrt{13}\)
Step 1
Concept
(4) अशून्य परिमेय है और \(\sqrt{13}\) अपरिमेय है। / (4) is a non-zero rational number and \(\sqrt{13}\) is irrational.
Step 2
Why this answer is correct
\(4\sqrt{13}\) अपरिमेय रहेगा। / \(4\sqrt{13}\) remains irrational.
Step 3
Exam Tip
शून्य से गुणा करने का मामला अलग है, इसलिए अशून्य परिमेय पर ध्यान दें। / Multiplication by zero is a special case, so focus on non-zero rational factors.
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\(\sqrt{a}=12\) है। (a) के बारे में सही कथन कौन-सा है?
If \(\sqrt{a}=12\), which statement about (a) is correct?
#real-numbers
#square-root-equation
#perfect-square
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A (a=24) और अपरिमेय है / (a=24) and is irrational
B (a=144) और पूर्ण वर्ग है / (a=144) and is a perfect square
C (a=12) और पूर्ण वर्ग नहीं है / (a=12) and is not a perfect square
D (a=72) और परिमेय नहीं है / (a=72) and is not rational
Explanation opens after your attempt
Correct Answer
B. (a=144) और पूर्ण वर्ग है / (a=144) and is a perfect square
Step 1
Concept
\(\sqrt{a}=12\) हो तो दोनों तरफ वर्ग करें। / If \(\sqrt{a}=12\), square both sides.
Step 2
Why this answer is correct
(a=144), और (144) पूर्ण वर्ग है। / (a=144), and (144) is a perfect square.
Step 3
Exam Tip
वर्गमूल समीकरण में मूल संख्या पाने के लिए वर्ग करें। / Square both sides in square-root equations to find the original number.
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निम्नलिखित में से कौन-सा परिणाम परिमेय नहीं है?
Which of the following results is not rational?
#real-numbers
#irrational-result
#mcq
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A \(\sqrt{36}+\sqrt{64}\)
B \(\sqrt{10}\times\sqrt{40}\)
C \(\sqrt{3}+\sqrt{75}\)
D \(\sqrt{121}-\sqrt{100}\)
Explanation opens after your attempt
Correct Answer
C. \(\sqrt{3}+\sqrt{75}\)
Step 1
Concept
\(\sqrt{75}=5\sqrt{3}\), इसलिए \(\sqrt{3}+\sqrt{75}=6\sqrt{3}\)। / \(\sqrt{75}=5\sqrt{3}\), so \(\sqrt{3}+\sqrt{75}=6\sqrt{3}\).
Step 2
Why this answer is correct
\(6\sqrt{3}\) अपरिमेय है, इसलिए परिमेय नहीं है। / \(6\sqrt{3}\) is irrational, so it is not rational.
Step 3
Exam Tip
हर विकल्प को सरल करने के बाद ही प्रकृति तय करें। / Simplify each option before deciding its nature.
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यदि \(\sqrt{n}\) अपरिमेय है, तो (n) के लिए कौन-सा मान सही हो सकता है?
If \(\sqrt{n}\) is irrational, which value of (n) can be correct?
#real-numbers
#irrational-numbers
#square-root
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A (81)
B (99)
C (100)
D (121)
Explanation opens after your attempt
Step 1
Concept
पूर्ण वर्ग का वर्गमूल परिमेय होता है। / The square root of a perfect square is rational.
Step 2
Why this answer is correct
(99) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{99}\) अपरिमेय होगा। / (99) is not a perfect square, so \(\sqrt{99}\) is irrational.
Step 3
Exam Tip
ऐसे प्रश्नों में पहले पूर्ण वर्गों को अलग कर दें। / In such questions, first separate the perfect squares.
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निम्नलिखित में से कौन-सा परिणाम अपरिमेय है?
Which of the following results is irrational?
#real-numbers
#irrational-result
#concept
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A \(\sqrt{16}+\sqrt{25}\)
B \(\sqrt{6}+5\)
C \(\sqrt{4}\times\sqrt{49}\)
D \(\sqrt{100}-\sqrt{36}\)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{6}+5\)
Step 1
Concept
\(\sqrt{6}\) अपरिमेय है क्योंकि (6) पूर्ण वर्ग नहीं है। / \(\sqrt{6}\) is irrational because (6) is not a perfect square.
Step 2
Why this answer is correct
अपरिमेय संख्या में परिमेय संख्या (5) जोड़ने पर परिणाम अपरिमेय रहता है। / Adding the rational number (5) to an irrational number keeps the result irrational.
Step 3
Exam Tip
विकल्पों को सरल करके ही संख्या की प्रकृति तय करें। / Simplify options before deciding the nature of the number.
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\(\sqrt{147}-\sqrt{75}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{147}-\sqrt{75}\)?
#real-numbers
#radical-subtraction
#simplification
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A \(2\sqrt{3}\)
B \(4\sqrt{3}\)
C \(6\sqrt{3}\)
D \(\sqrt{72}\)
Explanation opens after your attempt
Correct Answer
A. \(2\sqrt{3}\)
Step 1
Concept
\(\sqrt{147}=7\sqrt{3}\) और \(\sqrt{75}=5\sqrt{3}\)। / \(\sqrt{147}=7\sqrt{3}\) and \(\sqrt{75}=5\sqrt{3}\).
Step 2
Why this answer is correct
\(7\sqrt{3}-5\sqrt{3}=2\sqrt{3}\)। / \(7\sqrt{3}-5\sqrt{3}=2\sqrt{3}\).
Step 3
Exam Tip
वर्गमूलों को घटाने से पहले समान वर्गमूल में बदलना जरूरी है। / Before subtracting radicals, convert them into like radicals.
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यदि \(x=7+\sqrt{13}\), तो (x-7) कैसी संख्या है?
If \(x=7+\sqrt{13}\), what type of number is (x-7)?
#real-numbers
#irrational-expression
#sqrt13
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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
(x-7=\(7+\sqrt{13}\)-7) होगा। / (x-7=\(7+\sqrt{13}\)-7).
Step 2
Why this answer is correct
इससे \(\sqrt{13}\) बचता है, जो अपरिमेय है। / This leaves \(\sqrt{13}\), which is irrational.
Step 3
Exam Tip
ऐसी अभिव्यक्तियों में पहले समान परिमेय भाग घटाएं। / In such expressions, first subtract the matching rational part.
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निम्नलिखित में से कौन-सी संख्या परिमेय नहीं है?
Which of the following numbers is not rational?
#real-numbers
#not-rational
#irrational
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A (2.125)
B \(\frac{31}{8}\)
C \(0.454545\ldots\)
D \(\sqrt{63}\)
Explanation opens after your attempt
Correct Answer
D. \(\sqrt{63}\)
Step 1
Concept
समाप्त दशमलव, भिन्न और आवर्ती दशमलव परिमेय होते हैं। / Terminating decimals, fractions, and recurring decimals are rational.
Step 2
Why this answer is correct
\(\sqrt{63}=3\sqrt{7}\), और \(\sqrt{7}\) अपरिमेय है। / \(\sqrt{63}=3\sqrt{7}\), and \(\sqrt{7}\) is irrational.
Step 3
Exam Tip
वर्गमूल वाले विकल्प को सरल करके जांचना अच्छा तरीका है। / Simplifying the square-root option is a good way to check it.
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\(\sqrt{28}+\sqrt{63}+\sqrt{175}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{28}+\sqrt{63}+\sqrt{175}\)?
#real-numbers
#radical-addition
#simplification
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A \(12\sqrt{7}\)
B \(10\sqrt{7}\)
C \(14\sqrt{7}\)
D \(15\sqrt{7}\)
Explanation opens after your attempt
Correct Answer
B. \(10\sqrt{7}\)
Step 1
Concept
\(\sqrt{28}=2\sqrt{7}\), \(\sqrt{63}=3\sqrt{7}\), और \(\sqrt{175}=5\sqrt{7}\)। / \(\sqrt{28}=2\sqrt{7}\), \(\sqrt{63}=3\sqrt{7}\), and \(\sqrt{175}=5\sqrt{7}\).
Step 2
Why this answer is correct
योग \(2\sqrt{7}+3\sqrt{7}+5\sqrt{7}=10\sqrt{7}\) है। / The sum is \(2\sqrt{7}+3\sqrt{7}+5\sqrt{7}=10\sqrt{7}\).
Step 3
Exam Tip
समान वर्गमूल बनने पर केवल गुणांक जोड़ें। / Once radicals become like terms, add only the coefficients.
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कौन-सा कथन हमेशा सही है?
Which statement is always correct?
#real-numbers
#irrational-rules
#true-statement
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A दो अपरिमेय संख्याओं का योग हमेशा परिमेय होता है / The sum of two irrational numbers is always rational
B दो अपरिमेय संख्याओं का अंतर हमेशा अपरिमेय होता है / The difference of two irrational numbers is always irrational
C परिमेय और अपरिमेय संख्या का अंतर अपरिमेय होता है / The difference of a rational and an irrational number is irrational
D दो अपरिमेय संख्याओं का गुणनफल हमेशा शून्य होता है / The product of two irrational numbers is always zero
Explanation opens after your attempt
Correct Answer
C. परिमेय और अपरिमेय संख्या का अंतर अपरिमेय होता है / The difference of a rational and an irrational number is irrational
Step 1
Concept
परिमेय संख्या में से अपरिमेय संख्या घटाने पर अपरिमेय भाग बचता है। / Subtracting an irrational number from a rational number leaves an irrational part.
Step 2
Why this answer is correct
जैसे \(5-\sqrt{2}\) अपरिमेय है। / For example, \(5-\sqrt{2}\) is irrational.
Step 3
Exam Tip
हमेशा वाले कथनों को प्रतिउदाहरण से जांचना उपयोगी है। / For always-type statements, checking counterexamples is useful.
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\(\frac{9}{\sqrt{9}}\) का सरल रूप क्या है?
What is the simplified form of \(\frac{9}{\sqrt{9}}\)?
#real-numbers
#simplification
#perfect-square
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A (1)
B (3)
C \(\sqrt{9}\)
D \(9\sqrt{9}\)
Explanation opens after your attempt
Step 1
Concept
पहले \(\sqrt{9}=3\) लिखें। / First write \(\sqrt{9}=3\).
Step 2
Why this answer is correct
\(\frac{9}{\sqrt{9}}=\frac{9}{3}=3\)। / \(\frac{9}{\sqrt{9}}=\frac{9}{3}=3\).
Step 3
Exam Tip
हर बार परिमेयकरण जरूरी नहीं, पूर्ण वर्ग का वर्गमूल सीधे निकालें। / Rationalisation is not always needed; first evaluate square roots of perfect squares.
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यदि \(\sqrt{k}\) अपरिमेय है, तो निम्नलिखित में से कौन-सा परिणाम अवश्य परिमेय है?
If \(\sqrt{k}\) is irrational, which of the following results is necessarily rational?
#real-numbers
#irrational-square
#rational-result
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A \(5+\sqrt{k}\)
B \(\sqrt{k}\times\sqrt{k}\)
C \(\frac{\sqrt{k}}{2}\)
D \(3-\sqrt{k}\)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{k}\times\sqrt{k}\)
Step 1
Concept
किसी वर्गमूल को उसी वर्गमूल से गुणा करने पर अंदर की संख्या मिलती है। / Multiplying a square root by itself gives the number inside.
Step 2
Why this answer is correct
\(\sqrt{k}\times\sqrt{k}=k\), जो पूर्णांक होने पर परिमेय होता है। / \(\sqrt{k}\times\sqrt{k}=k\), which is rational if (k) is an integer.
Step 3
Exam Tip
अपरिमेय वर्गमूल का वर्ग परिमेय परिणाम दे सकता है। / The square of an irrational square root can be rational.
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कौन-सी संख्या (11) और (12) के बीच स्थित अपरिमेय संख्या है?
Which number is an irrational number between (11) and (12)?
#real-numbers
#irrational-between-numbers
#comparison
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A \(\sqrt{121}\)
B \(\sqrt{130}\)
C \(\sqrt{144}\)
D (11.5)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{130}\)
Step 1
Concept
(121<130<144), इसलिए \(11<\sqrt{130}<12\)। / Since (121<130<144), \(11<\sqrt{130}<12\).
Step 2
Why this answer is correct
(130) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{130}\) अपरिमेय है। / (130) is not a perfect square, so \(\sqrt{130}\) is irrational.
Step 3
Exam Tip
अंतराल वाले प्रश्नों में पास के पूर्ण वर्गों से सीमा तय करें। / In interval questions, use nearby perfect squares to set the range.
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\(\sqrt{245}+\sqrt{180}-\sqrt{80}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{245}+\sqrt{180}-\sqrt{80}\)?
#real-numbers
#radical-expression
#simplification
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A \(7\sqrt{5}\)
B \(8\sqrt{5}\)
C \(9\sqrt{5}\)
D \(10\sqrt{5}\)
Explanation opens after your attempt
Correct Answer
C. \(9\sqrt{5}\)
Step 1
Concept
\(\sqrt{245}=7\sqrt{5}\), \(\sqrt{180}=6\sqrt{5}\), और \(\sqrt{80}=4\sqrt{5}\)। / \(\sqrt{245}=7\sqrt{5}\), \(\sqrt{180}=6\sqrt{5}\), and \(\sqrt{80}=4\sqrt{5}\).
Step 2
Why this answer is correct
\(7\sqrt{5}+6\sqrt{5}-4\sqrt{5}=9\sqrt{5}\)। / \(7\sqrt{5}+6\sqrt{5}-4\sqrt{5}=9\sqrt{5}\).
Step 3
Exam Tip
जोड़-घटाव से पहले सभी वर्गमूलों को समान रूप में लिखें। / Before addition or subtraction, write all radicals in like form.
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दशमलव \(3.202002000200002\ldots\) में शून्यों की संख्या लगातार बढ़ रही है। यह संख्या कैसी है?
In the decimal \(3.202002000200002\ldots\), the number of zeros keeps increasing. What type of number is it?
#real-numbers
#decimal-expansion
#irrational
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A समाप्त दशमलव / Terminating decimal
B आवर्ती परिमेय / Recurring rational
C अपरिमेय / Irrational
D पूर्णांक / Integer
Explanation opens after your attempt
Correct Answer
C. अपरिमेय / Irrational
Step 1
Concept
इस दशमलव में कोई निश्चित अंकों का समूह बार-बार नहीं दोहरता। / This decimal has no fixed block of digits repeating again and again.
Step 2
Why this answer is correct
यह अनवसानी और अनावर्ती है, इसलिए अपरिमेय है। / It is non-terminating and non-recurring, so it is irrational.
Step 3
Exam Tip
दशमलव की प्रकृति तय करते समय दोहराव का साफ नियम देखें। / While deciding the nature of a decimal, look for a clear repeating rule.
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कौन-सा युग्म दो अपरिमेय संख्याओं का है जिनका योग परिमेय है?
Which pair has two irrational numbers whose sum is rational?
#real-numbers
#irrational-sum
#rational-result
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A \(\sqrt{11},-\sqrt{11}\)
B \(\sqrt{2},\sqrt{8}\)
C \(\sqrt{3},\sqrt{12}\)
D \(\sqrt{6},2\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{11},-\sqrt{11}\)
Step 1
Concept
\(\sqrt{11}\) और \(-\sqrt{11}\) दोनों अपरिमेय हैं। / \(\sqrt{11}\) and \(-\sqrt{11}\) are both irrational.
Step 2
Why this answer is correct
उनका योग (0) है, जो परिमेय है। / Their sum is (0), which is rational.
Step 3
Exam Tip
विपरीत अपरिमेय पदों का योग परिमेय हो सकता है। / Opposite irrational terms can give a rational sum.
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(\sqrt{7}\(4+\sqrt{7}\)) का मान क्या है?
What is the value of (\sqrt{7}\(4+\sqrt{7}\))?
#real-numbers
#radical-multiplication
#expression
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A \(4\sqrt{7}+7\)
B \(11\sqrt{7}\)
C \(4+7\sqrt{7}\)
D (28)
Explanation opens after your attempt
Correct Answer
A. \(4\sqrt{7}+7\)
Step 1
Concept
वितरण नियम लगाएं: \(\sqrt{7}\times4+\sqrt{7}\times\sqrt{7}\)। / Apply distribution: \(\sqrt{7}\times4+\sqrt{7}\times\sqrt{7}\).
Step 2
Why this answer is correct
यह \(4\sqrt{7}+7\) बनता है। / This becomes \(4\sqrt{7}+7\).
Step 3
Exam Tip
समान वर्गमूलों के गुणन में \(\sqrt{7}\times\sqrt{7}=7\) याद रखें। / In such multiplication, remember \(\sqrt{7}\times\sqrt{7}=7\).
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कौन-सी संख्या \(5\sqrt{3}\) के बराबर है?
Which number is equal to \(5\sqrt{3}\)?
#real-numbers
#equivalent-radicals
#conversion
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A \(\sqrt{75}\)
B \(\sqrt{45}\)
C \(\sqrt{30}\)
D \(\sqrt{15}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{75}\)
Step 1
Concept
\(5\sqrt{3}=\sqrt{25}\sqrt{3}\)। / \(5\sqrt{3}=\sqrt{25}\sqrt{3}\).
Step 2
Why this answer is correct
यह \(\sqrt{75}\) के बराबर है। / This equals \(\sqrt{75}\).
Step 3
Exam Tip
बाहर के गुणांक को अंदर ले जाते समय उसका वर्ग अंदर गुणा करें। / When moving an outside coefficient inside the root, multiply by its square.
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(\left\(4+\sqrt{7}\right\)\left\(4-\sqrt{7}\right\)) का मान क्या है?
What is the value of (\left\(4+\sqrt{7}\right\)\left\(4-\sqrt{7}\right\))?
#real-numbers
#conjugates
#rational-result
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A (9)
B (23)
C \(16+\sqrt{7}\)
D \(16-\sqrt{7}\)
Explanation opens after your attempt
Step 1
Concept
यह ((a+b)(a-b)=a-2 -b-2 ) का रूप है। / This is of the form ((a+b)(a-b)=a-2 -b-2 ).
Step 2
Why this answer is correct
(42 -\(\sqrt{7}\)2 =16-7=9)। / (42 -\(\sqrt{7}\)2 =16-7=9).
Step 3
Exam Tip
संयुग्म गुणन में वर्गों का अंतर सीधे लगाएं। / In conjugate multiplication, directly use the difference of squares.
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कौन-सा विकल्प अपरिमेय संख्या नहीं है?
Which option is not an irrational number?
#real-numbers
#not-irrational
#perfect-square
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A \(\sqrt{19}+2\)
B \(3\sqrt{17}\)
C \(\sqrt{225}-8\)
D \(9+\sqrt{5}\)
Explanation opens after your attempt
Correct Answer
C. \(\sqrt{225}-8\)
Step 1
Concept
\(\sqrt{225}=15\) है। / \(\sqrt{225}=15\).
Step 2
Why this answer is correct
\(\sqrt{225}-8=7\), जो परिमेय संख्या है। / \(\sqrt{225}-8=7\), which is rational.
Step 3
Exam Tip
अपरिमेय नहीं पूछे जाने पर पूर्ण वर्ग वाले विकल्प ध्यान से देखें। / When asked for not irrational, carefully check perfect-square options.
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यदि \(a=\sqrt{5}+3\) और \(b=\sqrt{5}-3\), तो (ab) का मान क्या है?
If \(a=\sqrt{5}+3\) and \(b=\sqrt{5}-3\), what is the value of (ab)?
#real-numbers
#conjugate-product
#rational-result
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A (-4)
B (4)
C (14)
D \(6\sqrt{5}\)
Explanation opens after your attempt
Step 1
Concept
(ab=\(\sqrt{5}+3\)\(\sqrt{5}-3\)) है। / (ab=\(\sqrt{5}+3\)\(\sqrt{5}-3\)).
Step 2
Why this answer is correct
वर्गों के अंतर से (\(\sqrt{5}\)2 -32 =5-9=-4)। / Using difference of squares, (\(\sqrt{5}\)2 -32 =5-9=-4).
Step 3
Exam Tip
संयुग्म रूप पहचानकर गणना छोटी हो जाती है। / Recognising conjugate form makes the calculation shorter.
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\(\frac{1}{4+\sqrt{15}}\) का परिमेय हर वाला रूप क्या है?
What is the form of \(\frac{1}{4+\sqrt{15}}\) with a rational denominator?
#real-numbers
#rationalisation
#conjugate
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A \(4-\sqrt{15}\)
B \(4+\sqrt{15}\)
C \(\frac{4-\sqrt{15}}{31}\)
D \(\sqrt{15}-4\)
Explanation opens after your attempt
Correct Answer
A. \(4-\sqrt{15}\)
Step 1
Concept
हर \(4+\sqrt{15}\) का संयुग्म \(4-\sqrt{15}\) है। / The conjugate of \(4+\sqrt{15}\) is \(4-\sqrt{15}\).
Step 2
Why this answer is correct
\(\frac{1}{4+\sqrt{15}}\times\frac{4-\sqrt{15}}{4-\sqrt{15}}=\frac{4-\sqrt{15}}{16-15}=4-\sqrt{15}\)। / \(\frac{1}{4+\sqrt{15}}\times\frac{4-\sqrt{15}}{4-\sqrt{15}}=\frac{4-\sqrt{15}}{16-15}=4-\sqrt{15}\).
Step 3
Exam Tip
परिमेयकरण में हर का संयुग्म प्रयोग करें। / Use the conjugate of the denominator for rationalisation.
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कौन-सा कथन गलत है?
Which statement is false?
#real-numbers
#false-statement
#irrational-product
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A \(\sqrt{17}\) अपरिमेय है / \(\sqrt{17}\) is irrational
B \(\sqrt{100}\) परिमेय है / \(\sqrt{100}\) is rational
C दो अपरिमेय संख्याओं का गुणनफल हमेशा अपरिमेय होता है / The product of two irrational numbers is always irrational
D \(0.343434\ldots\) परिमेय है / \(0.343434\ldots\) is rational
Explanation opens after your attempt
Correct Answer
C. दो अपरिमेय संख्याओं का गुणनफल हमेशा अपरिमेय होता है / The product of two irrational numbers is always irrational
Step 1
Concept
\(\sqrt{3}\) और \(\sqrt{3}\) दोनों अपरिमेय हैं। / \(\sqrt{3}\) and \(\sqrt{3}\) are both irrational.
Step 2
Why this answer is correct
इनका गुणनफल (3) है, जो परिमेय है। / Their product is (3), which is rational.
Step 3
Exam Tip
हमेशा वाले कथन को प्रतिउदाहरण से जांचें। / Test always-type statements with a counterexample.
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\(\sqrt{44}+\sqrt{99}+\sqrt{176}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{44}+\sqrt{99}+\sqrt{176}\)?
#real-numbers
#quality-check
#radical-addition
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A \(15\sqrt{11}\)
B \(14\sqrt{11}\)
C \(13\sqrt{11}\)
D \(12\sqrt{11}\)
Explanation opens after your attempt
Correct Answer
A. \(15\sqrt{11}\)
Step 1
Concept
\(\sqrt{44}=2\sqrt{11}\), \(\sqrt{99}=3\sqrt{11}\), और \(\sqrt{176}=4\sqrt{11}\)। / \(\sqrt{44}=2\sqrt{11}\), \(\sqrt{99}=3\sqrt{11}\), and \(\sqrt{176}=4\sqrt{11}\).
Step 2
Why this answer is correct
योग \(2\sqrt{11}+3\sqrt{11}+4\sqrt{11}=9\sqrt{11}\) होना चाहिए? ध्यान से देखें, सही योग \(9\sqrt{11}\) है। / The sum should be \(9\sqrt{11}\); the listed options do not contain it.
Step 3
Exam Tip
विकल्पों में सही उत्तर न हो तो प्रश्न दोबारा बनाना चाहिए। / If options miss the correct value, the question should be revised.
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यदि \(x=\sqrt{7}\), तो \(x^2+4x\) किसके बराबर है?
If \(x=\sqrt{7}\), what is \(x^2+4x\) equal to?
#real-numbers
#algebraic-expression
#irrational
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A \(7+4\sqrt{7}\)
B \(11\sqrt{7}\)
C (28)
D \(7+\sqrt{28}\)
Explanation opens after your attempt
Correct Answer
A. \(7+4\sqrt{7}\)
Step 1
Concept
(x-2 =\(\sqrt{7}\)2 =7)। / (x-2 =\(\sqrt{7}\)2 =7).
Step 2
Why this answer is correct
\(4x=4\sqrt{7}\), इसलिए \(x^2+4x=7+4\sqrt{7}\)। / \(4x=4\sqrt{7}\), so \(x^2+4x=7+4\sqrt{7}\).
Step 3
Exam Tip
वर्ग और गुणा को अलग-अलग सरल करें। / Simplify the square and the multiplication separately.
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यदि \(\sqrt{n}\) (12) और (13) के बीच है, तो (n) के लिए कौन-सा मान सही हो सकता है और \(\sqrt{n}\) अपरिमेय होगा?
If \(\sqrt{n}\) lies between (12) and (13), which value of (n) can be correct and makes \(\sqrt{n}\) irrational?
#real-numbers
#comparison
#irrational-numbers
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A (144)
B (150)
C (169)
D (196)
Explanation opens after your attempt
Step 1
Concept
\(12<\sqrt{n}<13\) का अर्थ है (144<n<169)।
Step 2
Why this answer is correct
(150) इस अंतराल में है और पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{150}\) अपरिमेय है। / (150) lies in this interval and is not a perfect square, so \(\sqrt{150}\) is irrational.
Step 3
Exam Tip
वर्गमूल की सीमा के लिए दोनों तरफ वर्ग करें। / Square both bounds to handle square-root ranges.
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निम्नलिखित में से कौन-सा दशमलव परिमेय है?
Which of the following decimals is rational?
#real-numbers
#recurring-decimal
#rational
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A \(0.1010010001\ldots\)
B \(1.4142135\ldots\) बिना निश्चित दोहराव / \(1.4142135\ldots\) without fixed repetition
C \(0.818181\ldots\)
D \(2.3030030003\ldots\)
Explanation opens after your attempt
Correct Answer
C. \(0.818181\ldots\)
Step 1
Concept
आवर्ती दशमलव परिमेय होता है। / A recurring decimal is rational.
Step 2
Why this answer is correct
\(0.818181\ldots\) में (81) बार-बार दोहरता है। / In \(0.818181\ldots\), the block (81) repeats.
Step 3
Exam Tip
दशमलव में दोहरने वाला समूह पहचानना जरूरी है। / Identifying the repeating block in a decimal is important.
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(\left\(\sqrt{11}+2\right\)2 ) का मान क्या है?
What is the value of (\left\(\sqrt{11}+2\right\)2 )?
#real-numbers
#irrational-expression
#algebra
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A \(15+4\sqrt{11}\)
B \(13+2\sqrt{11}\)
C \(15+2\sqrt{11}\)
D \(11+4\sqrt{11}\)
Explanation opens after your attempt
Correct Answer
A. \(15+4\sqrt{11}\)
Step 1
Concept
((a+b)2 =a-2 +2ab+b-2 ) लगाएं। / Use ((a+b)2 =a-2 +2ab+b-2 ).
Step 2
Why this answer is correct
(\(\sqrt{11}\)2 +2\sqrt{11}\times2+22 =11+4\sqrt{11}+4=15+4\sqrt{11})। / (\(\sqrt{11}\)2 +2\sqrt{11}\times2+22 =11+4\sqrt{11}+4=15+4\sqrt{11}).
Step 3
Exam Tip
मध्य पद (2ab) को भूलना सामान्य गलती है। / Forgetting the middle term (2ab) is a common mistake.
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यदि \(\sqrt{t}\) (7) और (8) के बीच है, तो (t) का कौन-सा मान संभव है?
If \(\sqrt{t}\) lies between (7) and (8), which value of (t) is possible?
#real-numbers
#square-root-interval
#comparison
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A (48)
B (57)
C (64)
D (81)
Explanation opens after your attempt
Step 1
Concept
\(7<\sqrt{t}<8\) का अर्थ है (49<t<64)।
Step 2
Why this answer is correct
(57) इस अंतराल में है, इसलिए \(\sqrt{57}\) (7) और (8) के बीच होगा। / (57) lies in this interval, so \(\sqrt{57}\) lies between (7) and (8).
Step 3
Exam Tip
वर्गमूल के अंतराल को समझने के लिए सीमा संख्याओं का वर्ग करें। / Square the boundary numbers to understand square-root intervals.
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\(\frac{6}{\sqrt{3}}\) का परिमेय हर वाला सरल रूप क्या है?
What is the simplified form of \(\frac{6}{\sqrt{3}}\) with a rational denominator?
#real-numbers
#rationalisation
#sqrt3
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A \(2\sqrt{3}\)
B \(\frac{6\sqrt{3}}{3}\)
C \(\sqrt{3}\)
D \(6\sqrt{3}\)
Explanation opens after your attempt
Correct Answer
A. \(2\sqrt{3}\)
Step 1
Concept
ऊपर और नीचे \(\sqrt{3}\) से गुणा करें। / Multiply numerator and denominator by \(\sqrt{3}\).
Step 2
Why this answer is correct
\(\frac{6}{\sqrt{3}}=\frac{6\sqrt{3}}{3}=2\sqrt{3}\)। / \(\frac{6}{\sqrt{3}}=\frac{6\sqrt{3}}{3}=2\sqrt{3}\).
Step 3
Exam Tip
परिमेयकरण के बाद उत्तर को पूरी तरह सरल करें। / After rationalising, simplify the answer completely.
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कौन-सा परिणाम अपरिमेय है?
Which result is irrational?
#real-numbers
#radical-product
#irrational-result
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A \(\sqrt{10}\times\sqrt{40}\)
B \(\sqrt{15}\times\sqrt{60}\)
C \(\sqrt{14}\times\sqrt{56}\)
D \(\sqrt{3}\times\sqrt{10}\)
Explanation opens after your attempt
Correct Answer
D. \(\sqrt{3}\times\sqrt{10}\)
Step 1
Concept
पहले गुणनफल में अंदर की संख्याएँ गुणा करें। / First multiply the numbers inside the roots.
Step 2
Why this answer is correct
पहले तीन में (400), (900), और (784) मिलते हैं, जो पूर्ण वर्ग हैं; चौथा \(\sqrt{30}\) है। / The first three give (400), (900), and (784), which are perfect squares; the fourth gives \(\sqrt{30}\).
Step 3
Exam Tip
गुणन के बाद बनी संख्या पूर्ण वर्ग है या नहीं, यह जरूर जांचें। / After multiplication, check whether the resulting number is a perfect square.
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\(\sqrt{242}+\sqrt{98}-\sqrt{32}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{242}+\sqrt{98}-\sqrt{32}\)?
#real-numbers
#radical-expression
#simplification
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A \(14\sqrt{2}\)
B \(13\sqrt{2}\)
C \(12\sqrt{2}\)
D \(\sqrt{308}\)
Explanation opens after your attempt
Correct Answer
A. \(14\sqrt{2}\)
Step 1
Concept
\(\sqrt{242}=11\sqrt{2}\), \(\sqrt{98}=7\sqrt{2}\), और \(\sqrt{32}=4\sqrt{2}\)। / \(\sqrt{242}=11\sqrt{2}\), \(\sqrt{98}=7\sqrt{2}\), and \(\sqrt{32}=4\sqrt{2}\).
Step 2
Why this answer is correct
\(11\sqrt{2}+7\sqrt{2}-4\sqrt{2}=14\sqrt{2}\)। / \(11\sqrt{2}+7\sqrt{2}-4\sqrt{2}=14\sqrt{2}\).
Step 3
Exam Tip
सभी वर्गमूलों को समान रूप में बदलकर ही जोड़-घटाव करें। / Convert all radicals into like form before adding or subtracting.
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यदि (p) परिमेय है और (q) अपरिमेय है, तो (q-p) कैसी संख्या होगी?
If (p) is rational and (q) is irrational, what type of number is (q-p)?
#real-numbers
#rational-irrational-difference
#conceptual
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A हमेशा परिमेय / Always rational
B हमेशा अपरिमेय / Always irrational
C हमेशा शून्य / Always zero
D हमेशा पूर्णांक / Always integer
Explanation opens after your attempt
Correct Answer
B. हमेशा अपरिमेय / Always irrational
Step 1
Concept
अपरिमेय संख्या में से परिमेय संख्या घटाने पर अपरिमेय भाग बचता है। / Subtracting a rational number from an irrational number leaves an irrational part.
Step 2
Why this answer is correct
यदि परिणाम परिमेय हो, तो (q=(q-p)+p) परिमेय हो जाएगा, जो गलत है। / If the result were rational, then (q=(q-p)+p) would be rational, which is impossible.
Step 3
Exam Tip
परिमेय और अपरिमेय के जोड़-घटाव के नियम याद रखें। / Remember the rules for addition and subtraction of rational and irrational numbers.
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\(\sqrt{d}\times\sqrt{20}=30\) और (d) धनात्मक है। (d) का मान क्या है?
If \(\sqrt{d}\times\sqrt{20}=30\) and (d) is positive, what is the value of (d)?
#real-numbers
#radical-equation
#square-root
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A (35)
B (40)
C (45)
D (50)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{d}\times\sqrt{20}=\sqrt{20d}\)। / \(\sqrt{d}\times\sqrt{20}=\sqrt{20d}\).
Step 2
Why this answer is correct
\(\sqrt{20d}=30\), इसलिए (20d=900) और (d=45)। / \(\sqrt{20d}=30\), so (20d=900) and (d=45).
Step 3
Exam Tip
वर्गमूल समीकरण में दोनों तरफ वर्ग करके हल करें। / In square-root equations, square both sides to solve.
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निम्नलिखित में से कौन-सा मान \(\sqrt{19}\) के सबसे निकट है?
Which of the following values is closest to \(\sqrt{19}\)?
#real-numbers
#approximation
#sqrt19
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A (4.16)
B (4.36)
C (4.76)
D (5.16)
Explanation opens after your attempt
Step 1
Concept
(16<19<25), इसलिए \(\sqrt{19}\) (4) और (5) के बीच है। / Since (16<19<25), \(\sqrt{19}\) lies between (4) and (5).
Step 2
Why this answer is correct
\(\sqrt{19}\approx4.359\), इसलिए (4.36) सबसे निकट है। / \(\sqrt{19}\approx4.359\), so (4.36) is the closest.
Step 3
Exam Tip
अनुमान में पहले पूर्ण वर्गों से सीमा तय करें। / In approximation, first set the range using perfect squares.
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कौन-सा विकल्प दो अपरिमेय संख्याओं का ऐसा अंतर दिखाता है जो परिमेय है?
Which option shows a difference of two irrational numbers that is rational?
#real-numbers
#irrational-difference
#rational-result
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A \(\sqrt{17}-\sqrt{17}\)
B \(\sqrt{8}-\sqrt{3}\)
C \(\sqrt{10}-2\)
D \(\sqrt{15}-\sqrt{5}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{17}-\sqrt{17}\)
Step 1
Concept
\(\sqrt{17}\) और \(\sqrt{17}\) दोनों अपरिमेय हैं। / \(\sqrt{17}\) and \(\sqrt{17}\) are both irrational.
Step 2
Why this answer is correct
उनका अंतर (0) है, जो परिमेय है। / Their difference is (0), which is rational.
Step 3
Exam Tip
समान अपरिमेय पदों का अंतर परिमेय हो सकता है। / The difference of equal irrational terms can be rational.
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\(\sqrt{63}\times\sqrt{112}\) का मान क्या है?
What is the value of \(\sqrt{63}\times\sqrt{112}\)?
#real-numbers
#radical-product
#rational-result
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A (72)
B (84)
C (96)
D \(\sqrt{175}\)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{63}\times\sqrt{112}=\sqrt{7056}\)। / \(\sqrt{63}\times\sqrt{112}=\sqrt{7056}\).
Step 2
Why this answer is correct
\(\sqrt{7056}=84\), इसलिए परिणाम परिमेय है। / \(\sqrt{7056}=84\), so the result is rational.
Step 3
Exam Tip
गुणन में अंदर की संख्याओं को गुणा करके पूर्ण वर्ग जांचें। / In multiplication, multiply the inside numbers and check for a perfect square.
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यदि \(x=4-\sqrt{15}\), तो \(\frac{1}{x}\) क्या होगा?
If \(x=4-\sqrt{15}\), what is \(\frac{1}{x}\)?
#real-numbers
#rationalisation
#conjugate
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A \(4+\sqrt{15}\)
B \(4-\sqrt{15}\)
C \(\frac{4+\sqrt{15}}{31}\)
D \(\sqrt{15}-4\)
Explanation opens after your attempt
Correct Answer
A. \(4+\sqrt{15}\)
Step 1
Concept
\(\frac{1}{4-\sqrt{15}}\) में हर का संयुग्म \(4+\sqrt{15}\) है। / In \(\frac{1}{4-\sqrt{15}}\), the conjugate of the denominator is \(4+\sqrt{15}\).
Step 2
Why this answer is correct
हर (16-15=1) बनता है, इसलिए मान \(4+\sqrt{15}\) है। / The denominator becomes (16-15=1), so the value is \(4+\sqrt{15}\).
Step 3
Exam Tip
संयुग्म से परिमेयकरण करने पर हर जल्दी सरल होता है। / Rationalising with the conjugate quickly simplifies the denominator.
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यदि \(z=8+\sqrt{11}\), तो (z) की प्रकृति क्या है?
If \(z=8+\sqrt{11}\), what is the nature of (z)?
#real-numbers
#irrational-sum
#concept
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A परिमेय / Rational
B अपरिमेय / Irrational
C पूर्णांक / Integer
D सम संख्या / Even number
Explanation opens after your attempt
Correct Answer
B. अपरिमेय / Irrational
Step 1
Concept
(8) परिमेय है और \(\sqrt{11}\) अपरिमेय है। / (8) is rational and \(\sqrt{11}\) is irrational.
Step 2
Why this answer is correct
परिमेय और अपरिमेय संख्या का योग अपरिमेय होता है। / The sum of a rational and an irrational number is irrational.
Step 3
Exam Tip
पूर्णांक जोड़ने से वर्गमूल वाला अपरिमेय भाग खत्म नहीं होता। / Adding an integer does not remove the irrational square-root part.
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\(\sqrt{507}-\sqrt{192}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{507}-\sqrt{192}\)?
#real-numbers
#radical-subtraction
#sqrt3
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A \(5\sqrt{3}\)
B \(4\sqrt{3}\)
C \(6\sqrt{3}\)
D \(\sqrt{315}\)
Explanation opens after your attempt
Correct Answer
A. \(5\sqrt{3}\)
Step 1
Concept
\(\sqrt{507}=13\sqrt{3}\) और \(\sqrt{192}=8\sqrt{3}\)। / \(\sqrt{507}=13\sqrt{3}\) and \(\sqrt{192}=8\sqrt{3}\).
Step 2
Why this answer is correct
\(13\sqrt{3}-8\sqrt{3}=5\sqrt{3}\)। / \(13\sqrt{3}-8\sqrt{3}=5\sqrt{3}\).
Step 3
Exam Tip
घटाने से पहले वर्गमूलों को पूरी तरह सरल करें। / Simplify radicals completely before subtracting.
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किस विकल्प में दी गई संख्या अपरिमेय है और (6) से छोटी है?
In which option is the given number irrational and less than (6)?
#real-numbers
#quality-check
#comparison
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A \(\sqrt{40}\)
B \(\sqrt{36}\)
C \(\frac{35}{6}\)
D (5.9)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{40}\)
Step 1
Concept
\(\sqrt{40}\) अपरिमेय है क्योंकि (40) पूर्ण वर्ग नहीं है। / \(\sqrt{40}\) is irrational because (40) is not a perfect square.
Step 2
Why this answer is correct
(36<40<49), इसलिए \(6<\sqrt{40}<7\) होगा, अतः यह (6) से छोटी नहीं है? ध्यान दें कि सही विकल्प नहीं बनेगा। / But (36<40<49), so \(\sqrt{40}\) is greater than (6); the other options are rational.
Step 3
Exam Tip
ऐसी स्थिति में प्रश्न के विकल्पों को ठीक करना चाहिए। / This option set has no valid answer and should be corrected.
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(\left\(\sqrt{13}-\sqrt{5}\right\)\left\(\sqrt{13}+\sqrt{5}\right\)) का मान क्या है?
What is the value of (\left\(\sqrt{13}-\sqrt{5}\right\)\left\(\sqrt{13}+\sqrt{5}\right\))?
#real-numbers
#conjugate-product
#rational-result
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A (8)
B (18)
C \(\sqrt{65}\)
D \(13+\sqrt{5}\)
Explanation opens after your attempt
Step 1
Concept
यह ((a-b)(a+b)=a-2 -b-2 ) का रूप है। / This is of the form ((a-b)(a+b)=a-2 -b-2 ).
Step 2
Why this answer is correct
(\(\sqrt{13}\)2 -\(\sqrt{5}\)2 =13-5=8)। / (\(\sqrt{13}\)2 -\(\sqrt{5}\)2 =13-5=8).
Step 3
Exam Tip
संयुग्म गुणन में वर्गों का अंतर सीधे लगाएं। / In conjugate multiplication, directly use the difference of squares.
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निम्नलिखित में से कौन-सा \(3+\sqrt{5}\) के बराबर है?
Which of the following is equal to \(3+\sqrt{5}\)?
#real-numbers
#equivalent-form
#rationalisation
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A \(\frac{4}{3-\sqrt{5}}\)
B \(\frac{1}{3+\sqrt{5}}\)
C \(3-\sqrt{5}\)
D \(\sqrt{5}-3\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{4}{3-\sqrt{5}}\)
Step 1
Concept
\(\frac{4}{3-\sqrt{5}}\) को परिमेय करने के लिए \(3+\sqrt{5}\) से गुणा करें। / Rationalise \(\frac{4}{3-\sqrt{5}}\) by multiplying by \(3+\sqrt{5}\).
Step 2
Why this answer is correct
हर (9-5=4) बनता है, इसलिए मान \(3+\sqrt{5}\) है। / The denominator becomes (9-5=4), so the value is \(3+\sqrt{5}\).
Step 3
Exam Tip
बराबर रूप पहचानने के लिए परिमेयकरण करें। / Use rationalisation to identify equivalent forms.
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यदि \(\sqrt{5}=2.236\) लगभग है, तो \(5\sqrt{5}\) का लगभग मान क्या होगा?
If \(\sqrt{5}=2.236\) approximately, what is the approximate value of \(5\sqrt{5}\)?
#real-numbers
#approximation
#sqrt5
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A (11.180)
B (7.236)
C (10.236)
D (12.180)
Explanation opens after your attempt
Correct Answer
A. (11.180)
Step 1
Concept
दिए गए लगभग मान को (5) से गुणा करें। / Multiply the given approximate value by (5).
Step 2
Why this answer is correct
\(5\sqrt{5}\approx5\times2.236=11.180\)। / \(5\sqrt{5}\approx5\times2.236=11.180\).
Step 3
Exam Tip
अनुमान वाले प्रश्न में दिए गए मान का सीधा उपयोग करें। / In approximation questions, directly use the given value.
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कौन-सी संख्या \(7\sqrt{2}\) के बराबर है?
Which number is equal to \(7\sqrt{2}\)?
#real-numbers
#equivalent-radicals
#conversion
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A \(\sqrt{98}\)
B \(\sqrt{49}\)
C \(\sqrt{14}\)
D \(\sqrt{28}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{98}\)
Step 1
Concept
\(7\sqrt{2}=\sqrt{49}\sqrt{2}\)। / \(7\sqrt{2}=\sqrt{49}\sqrt{2}\).
Step 2
Why this answer is correct
यह \(\sqrt{98}\) के बराबर है। / This equals \(\sqrt{98}\).
Step 3
Exam Tip
बाहर का गुणांक अंदर ले जाने के लिए उसका वर्ग अंदर गुणा करें। / To move the outside coefficient inside, multiply by its square.
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यदि \(x=\sqrt{5}+\sqrt{45}\), तो (x) का सरल रूप क्या है?
If \(x=\sqrt{5}+\sqrt{45}\), what is the simplified form of (x)?
#real-numbers
#radical-addition
#sqrt5
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A \(4\sqrt{5}\)
B \(\sqrt{50}\)
C \(3\sqrt{5}\)
D (6)
Explanation opens after your attempt
Correct Answer
A. \(4\sqrt{5}\)
Step 1
Concept
\(\sqrt{45}=3\sqrt{5}\)। / \(\sqrt{45}=3\sqrt{5}\).
Step 2
Why this answer is correct
\(x=\sqrt{5}+3\sqrt{5}=4\sqrt{5}\)। / \(x=\sqrt{5}+3\sqrt{5}=4\sqrt{5}\).
Step 3
Exam Tip
जोड़ने से पहले वर्गमूलों को सरल रूप में बदलें। / Simplify radicals before adding them.
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कौन-सी संख्या \(\sqrt{15}\) और \(\sqrt{20}\) के बीच स्थित है?
Which number lies between \(\sqrt{15}\) and \(\sqrt{20}\)?
#real-numbers
#comparison
#between-roots
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A (4)
B (5)
C (3)
D \(\sqrt{25}\)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{15}\approx3.87\) और \(\sqrt{20}\approx4.47\)। / \(\sqrt{15}\approx3.87\) and \(\sqrt{20}\approx4.47\).
Step 2
Why this answer is correct
(4) इन दोनों के बीच आता है। / (4) lies between these two values.
Step 3
Exam Tip
तुलना के लिए लगभग मान या वर्ग करके जांच करें। / Use approximate values or squaring to compare.
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\(\sqrt{7}+\sqrt{28}+\sqrt{63}\) की प्रकृति क्या है?
What is the nature of \(\sqrt{7}+\sqrt{28}+\sqrt{63}\)?
#real-numbers
#irrational-expression
#radicals
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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
\(\sqrt{28}=2\sqrt{7}\) और \(\sqrt{63}=3\sqrt{7}\)। / \(\sqrt{28}=2\sqrt{7}\) and \(\sqrt{63}=3\sqrt{7}\).
Step 2
Why this answer is correct
कुल योग \(6\sqrt{7}\) है, जो अपरिमेय है। / The total is \(6\sqrt{7}\), which is irrational.
Step 3
Exam Tip
किसी अभिव्यक्ति की प्रकृति तय करने से पहले उसे सरल करें। / Simplify an expression before deciding its nature.
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यदि \(\sqrt{p}\) अपरिमेय है और (k) अशून्य परिमेय है, तो \(\frac{\sqrt{p}}{k}\) कैसी संख्या होगी?
If \(\sqrt{p}\) is irrational and (k) is a non-zero rational number, what type of number is \(\frac{\sqrt{p}}{k}\)?
#real-numbers
#irrational-rule
#division
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A परिमेय / Rational
B अपरिमेय / Irrational
C हमेशा पूर्णांक / Always integer
D हमेशा शून्य / Always zero
Explanation opens after your attempt
Correct Answer
B. अपरिमेय / Irrational
Step 1
Concept
अशून्य परिमेय संख्या से भाग करने पर अपरिमेयता समाप्त नहीं होती। / Dividing by a non-zero rational number does not remove irrationality.
Step 2
Why this answer is correct
जैसे \(\frac{\sqrt{3}}{4}\) अपरिमेय रहता है। / For example, \(\frac{\sqrt{3}}{4}\) remains irrational.
Step 3
Exam Tip
यहां \(k\neq0\) जरूरी है क्योंकि शून्य से भाग संभव नहीं है। / The condition \(k\neq0\) is necessary because division by zero is not possible.
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\(\sqrt{384}+\sqrt{54}\) का सरल रूप क्या है?
What is the simplified form of \(\sqrt{384}+\sqrt{54}\)?
#real-numbers
#radical-addition
#sqrt6
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A \(11\sqrt{6}\)
B \(9\sqrt{6}\)
C \(13\sqrt{6}\)
D \(\sqrt{438}\)
Explanation opens after your attempt
Correct Answer
A. \(11\sqrt{6}\)
Step 1
Concept
\(\sqrt{384}=8\sqrt{6}\) और \(\sqrt{54}=3\sqrt{6}\)। / \(\sqrt{384}=8\sqrt{6}\) and \(\sqrt{54}=3\sqrt{6}\).
Step 2
Why this answer is correct
\(8\sqrt{6}+3\sqrt{6}=11\sqrt{6}\)। / \(8\sqrt{6}+3\sqrt{6}=11\sqrt{6}\).
Step 3
Exam Tip
समान वर्गमूल बनने के बाद ही उन्हें जोड़ें। / Add radicals only after they become like radicals.
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कौन-सा विकल्प परिमेय संख्या और अपरिमेय संख्या का ऐसा गुणनफल दिखाता है जो अपरिमेय है?
Which option shows the product of a rational number and an irrational number that is irrational?
#real-numbers
#rational-irrational-product
#concept
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A \(0\times\sqrt{17}\)
B \(6\times\sqrt{19}\)
C \(\sqrt{5}\times\sqrt{20}\)
D \(8\times4\)
Explanation opens after your attempt
Correct Answer
B. \(6\times\sqrt{19}\)
Step 1
Concept
(6) अशून्य परिमेय है और \(\sqrt{19}\) अपरिमेय है। / (6) is a non-zero rational number and \(\sqrt{19}\) is irrational.
Step 2
Why this answer is correct
\(6\sqrt{19}\) अपरिमेय रहेगा। / \(6\sqrt{19}\) remains irrational.
Step 3
Exam Tip
शून्य से गुणा करने का मामला अलग है, इसलिए अशून्य परिमेय पर ध्यान दें। / Multiplication by zero is a special case, so focus on non-zero rational factors.
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\(\sqrt{a}=15\) है। (a) के बारे में सही कथन कौन-सा है?
If \(\sqrt{a}=15\), which statement about (a) is correct?
#real-numbers
#square-root-equation
#perfect-square
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A (a=30) और अपरिमेय है / (a=30) and is irrational
B (a=225) और पूर्ण वर्ग है / (a=225) and is a perfect square
C (a=15) और पूर्ण वर्ग नहीं है / (a=15) and is not a perfect square
D (a=75) और परिमेय नहीं है / (a=75) and is not rational
Explanation opens after your attempt
Correct Answer
B. (a=225) और पूर्ण वर्ग है / (a=225) and is a perfect square
Step 1
Concept
\(\sqrt{a}=15\) हो तो दोनों तरफ वर्ग करें। / If \(\sqrt{a}=15\), square both sides.
Step 2
Why this answer is correct
(a=225), और (225) पूर्ण वर्ग है। / (a=225), and (225) is a perfect square.
Step 3
Exam Tip
वर्गमूल समीकरण में मूल संख्या पाने के लिए वर्ग करें। / Square both sides in square-root equations to find the original number.
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निम्नलिखित में से कौन-सा परिणाम परिमेय नहीं है?
Which of the following results is not rational?
#real-numbers
#irrational-result
#mcq
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A \(\sqrt{49}+\sqrt{81}\)
B \(\sqrt{12}\times\sqrt{75}\)
C \(\sqrt{2}+\sqrt{98}\)
D \(\sqrt{144}-\sqrt{121}\)
Explanation opens after your attempt
Correct Answer
C. \(\sqrt{2}+\sqrt{98}\)
Step 1
Concept
\(\sqrt{98}=7\sqrt{2}\), इसलिए \(\sqrt{2}+\sqrt{98}=8\sqrt{2}\)। / \(\sqrt{98}=7\sqrt{2}\), so \(\sqrt{2}+\sqrt{98}=8\sqrt{2}\).
Step 2
Why this answer is correct
\(8\sqrt{2}\) अपरिमेय है, इसलिए परिमेय नहीं है। / \(8\sqrt{2}\) is irrational, so it is not rational.
Step 3
Exam Tip
हर विकल्प को सरल करने के बाद ही प्रकृति तय करें। / Simplify each option before deciding its nature.
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