यदि (m) पूर्णांक है और \(m<\sqrt{145}<m+1\), तो (m) का मान क्या है?
If (m) is an integer and \(m<\sqrt{145}<m+1\), what is the value of (m)?
#number-line
#integer-bounds
#square-roots
A (11)
B (12)
C (13)
D (14)
Explanation opens after your attempt
Step 1
Concept
Since (144<145<169), \(12<\sqrt{145}<13\). Perfect squares give (m) quickly.
Step 2
Why this answer is correct
The correct answer is B. (12). Since (144<145<169), \(12<\sqrt{145}<13\). Perfect squares give (m) quickly.
Step 3
Exam Tip
क्योंकि (144<145<169), इसलिए \(12<\sqrt{145}<13\)। पूर्ण वर्गों से (m) तुरंत मिलता है।
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संख्या रेखा पर \( \sqrt{65} \) का सबसे अच्छा एक दशमलव स्थान तक अनुमान कौन सा है?
What is the best estimate of \( \sqrt{65} \) to one decimal place on the number line?
#number-line
#estimation
#square-roots
A (7.9)
B (8.1)
C (8.5)
D (7.5)
Explanation opens after your attempt
Step 1
Concept
\(8^2=64\) and \(8.1^2=65.61\), so \( \sqrt{65}\approx8.1 \). Estimate using nearby squares.
Step 2
Why this answer is correct
The correct answer is B. (8.1). \(8^2=64\) and \(8.1^2=65.61\), so \( \sqrt{65}\approx8.1 \). Estimate using nearby squares.
Step 3
Exam Tip
\(8^2=64\) और \(8.1^2=65.61\), इसलिए \( \sqrt{65}\approx8.1 \)। निकट वर्ग से अनुमान लगाएँ।
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संख्या रेखा पर \( \sqrt{52} \) किस दो लगातार पूर्णांकों के बीच स्थित है?
Between which two consecutive integers is \( \sqrt{52} \) located on the number line?
#number-line
#square-roots
#integer-bounds
A (6) और (7) / (6) and (7)
B (7) और (8) / (7) and (8)
C (5) और (6) / (5) and (6)
D (8) और (9) / (8) and (9)
Explanation opens after your attempt
Correct Answer
B. (7) और (8) / (7) and (8)
Step 1
Concept
Since (49<52<64), \(7<\sqrt{52}<8\). First identify the nearest perfect squares.
Step 2
Why this answer is correct
The correct answer is B. (7) और (8) / (7) and (8). Since (49<52<64), \(7<\sqrt{52}<8\). First identify the nearest perfect squares.
Step 3
Exam Tip
क्योंकि (49<52<64), इसलिए \(7<\sqrt{52}<8\)। पहले निकटतम पूर्ण वर्गों को पहचानें।
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यदि (m) ऐसा पूर्णांक है कि \(m<\sqrt{57}<m+1\), तो (m) का मान क्या है?
If (m) is an integer such that \(m<\sqrt{57}<m+1\), what is the value of (m)?
#number-line
#integer-bounds
#square-roots
A (7)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
Since (49<57<64), \(7<\sqrt{57}<8\). Therefore (m=7).
Step 2
Why this answer is correct
The correct answer is A. (7). Since (49<57<64), \(7<\sqrt{57}<8\). Therefore (m=7).
Step 3
Exam Tip
क्योंकि (49<57<64), इसलिए \(7<\sqrt{57}<8\)। अतः (m=7) होगा।
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यदि (x) संख्या रेखा पर \( \sqrt{72} \) है, तो (x) के लिए सही सरल रूप कौन सा है?
If (x) is \( \sqrt{72} \) on the number line, what is the correct simplified form of (x)?
#number-line
#simplification
#square-roots
A \(6\sqrt{2}\)
B \(8\sqrt{2}\)
C \(3\sqrt{8}\)
D \(12\sqrt{2}\)
Explanation opens after your attempt
Correct Answer
A. \(6\sqrt{2}\)
Step 1
Concept
\( \sqrt{72}=\sqrt{36\cdot2}=6\sqrt{2}\). To simplify a root, factor out the largest perfect square.
Step 2
Why this answer is correct
The correct answer is A. \(6\sqrt{2}\). \( \sqrt{72}=\sqrt{36\cdot2}=6\sqrt{2}\). To simplify a root, factor out the largest perfect square.
Step 3
Exam Tip
\( \sqrt{72}=\sqrt{36\cdot2}=6\sqrt{2}\)। मूल सरल करने के लिए सबसे बड़ा पूर्ण वर्ग निकालें।
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किस विकल्प में \( \sqrt{15} \) की संख्या रेखा पर स्थिति सबसे सही बताई गई है?
Which option states the position of \( \sqrt{15} \) on the number line most correctly?
#number-line
#decimal-estimation
#square-roots
A (3.8) और (3.9) के बीच / Between (3.8) and (3.9)
B (3.5) और (3.6) के बीच / Between (3.5) and (3.6)
C (4.1) और (4.2) के बीच / Between (4.1) and (4.2)
D (2.9) और (3.0) के बीच / Between (2.9) and (3.0)
Explanation opens after your attempt
Correct Answer
A. (3.8) और (3.9) के बीच / Between (3.8) and (3.9)
Step 1
Concept
\(3.8^2=14.44\) and \(3.9^2=15.21\), so \( \sqrt{15}\) lies between them. Check squares for decimal bounds.
Step 2
Why this answer is correct
The correct answer is A. (3.8) और (3.9) के बीच / Between (3.8) and (3.9). \(3.8^2=14.44\) and \(3.9^2=15.21\), so \( \sqrt{15}\) lies between them. Check squares for decimal bounds.
Step 3
Exam Tip
\(3.8^2=14.44\) और \(3.9^2=15.21\), इसलिए \( \sqrt{15}\) इनके बीच है। दशमलव सीमा के लिए वर्ग जाँचें।
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यदि \( \sqrt{n} \) संख्या रेखा पर (8) और (9) के बीच है, तो (n) के लिए कौन सा मान संभव है?
If \( \sqrt{n} \) lies between (8) and (9) on the number line, which value of (n) is possible?
#number-line
#inequality
#square-roots
A (70)
B (60)
C (81)
D (100)
Explanation opens after your attempt
Step 1
Concept
From \(8<\sqrt{n}<9\), we get (64<n<81), so (70) is possible. Square positive bounds carefully.
Step 2
Why this answer is correct
The correct answer is A. (70). From \(8<\sqrt{n}<9\), we get (64<n<81), so (70) is possible. Square positive bounds carefully.
Step 3
Exam Tip
\(8<\sqrt{n}<9\) से (64<n<81), इसलिए (70) संभव है। असमानता को वर्ग करते समय धनात्मक सीमा का उपयोग करें।
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कौन सी संख्या \( \sqrt{8} \) और (3) के बीच संख्या रेखा पर स्थित है?
Which number lies between \( \sqrt{8} \) and (3) on the number line?
#number-line
#between-numbers
#square-roots
A (2.9)
B (2.7)
C (3.1)
D (2)
Explanation opens after your attempt
Step 1
Concept
\( \sqrt{8}\approx2.828\), so (2.9) lies between it and (3). First estimate the irrational number.
Step 2
Why this answer is correct
The correct answer is A. (2.9). \( \sqrt{8}\approx2.828\), so (2.9) lies between it and (3). First estimate the irrational number.
Step 3
Exam Tip
\( \sqrt{8}\approx2.828\), इसलिए (2.9) उसके और (3) के बीच है। पहले अपरिमेय संख्या का अनुमान करें।
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संख्या रेखा पर \( \sqrt{99} \) का सबसे निकटतम पूर्णांक कौन सा है?
Which integer is closest to \( \sqrt{99} \) on the number line?
#number-line
#nearest-integer
#square-roots
A (10)
B (9)
C (11)
D (8)
Explanation opens after your attempt
Step 1
Concept
(99) is very close to (100), so \( \sqrt{99}\) is about (10). The nearest perfect square gives a quick answer.
Step 2
Why this answer is correct
The correct answer is A. (10). (99) is very close to (100), so \( \sqrt{99}\) is about (10). The nearest perfect square gives a quick answer.
Step 3
Exam Tip
(99) संख्या (100) के बहुत निकट है, इसलिए \( \sqrt{99}\) लगभग (10) है। निकटतम पूर्ण वर्ग तेजी से उत्तर देता है।
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संख्या रेखा पर \( \sqrt{12}-1 \) किस दो पूर्णांकों के बीच आएगा?
Between which two integers will \( \sqrt{12}-1 \) lie on the number line?
#number-line
#compound-expression
#square-roots
A (2) और (3) / (2) and (3)
B (3) और (4) / (3) and (4)
C (1) और (2) / (1) and (2)
D (4) और (5) / (4) and (5)
Explanation opens after your attempt
Correct Answer
A. (2) और (3) / (2) and (3)
Step 1
Concept
Since \(3<\sqrt{12}<4\), \(2<\sqrt{12}-1<3\). First find the root interval, then subtract.
Step 2
Why this answer is correct
The correct answer is A. (2) और (3) / (2) and (3). Since \(3<\sqrt{12}<4\), \(2<\sqrt{12}-1<3\). First find the root interval, then subtract.
Step 3
Exam Tip
\(3<\sqrt{12}<4\), इसलिए \(2<\sqrt{12}-1<3\)। पहले मूल का अंतराल निकालें, फिर घटाएँ।
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यदि \(a=\sqrt{2}\), \(b=\sqrt{3}\), और \(c=\sqrt{5}\), तो संख्या रेखा पर बाएँ से दाएँ सही क्रम कौन सा है?
If \(a=\sqrt{2}\), \(b=\sqrt{3}\), and \(c=\sqrt{5}\), what is the correct left-to-right order on the number line?
#number-line
#ordering
#square-roots
A (a,b,c)
B (c,b,a)
C (b,a,c)
D (a,c,b)
Explanation opens after your attempt
Correct Answer
A. (a,b,c)
Step 1
Concept
Since (2<3<5), \( \sqrt{2}<\sqrt{3}<\sqrt{5}\). For positive square roots, order follows the radicand.
Step 2
Why this answer is correct
The correct answer is A. (a,b,c). Since (2<3<5), \( \sqrt{2}<\sqrt{3}<\sqrt{5}\). For positive square roots, order follows the radicand.
Step 3
Exam Tip
क्योंकि (2<3<5), इसलिए \( \sqrt{2}<\sqrt{3}<\sqrt{5}\)। धनात्मक वर्गमूल में अंदर की संख्या से क्रम तय होता है।
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संख्या रेखा पर \( \sqrt{18} \) को सबसे सही किस दो लगातार पूर्णांकों के बीच दिखाया जाएगा?
Between which two consecutive integers should \( \sqrt{18} \) be shown most accurately on the number line?
#number-line
#real-numbers
#square-roots
A (4) और (5) / (4) and (5)
B (3) और (4) / (3) and (4)
C (5) और (6) / (5) and (6)
D (2) और (3) / (2) and (3)
Explanation opens after your attempt
Correct Answer
A. (4) और (5) / (4) and (5)
Step 1
Concept
Since (16<18<25), we get \(4<\sqrt{18}<5\). In exams, compare perfect squares first.
Step 2
Why this answer is correct
The correct answer is A. (4) और (5) / (4) and (5). Since (16<18<25), we get \(4<\sqrt{18}<5\). In exams, compare perfect squares first.
Step 3
Exam Tip
क्योंकि (16<18<25), इसलिए \(4<\sqrt{18}<5\)। परीक्षा में वर्गों की तुलना पहले करें।
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किस विकल्प में \(\sqrt{11}\) और \(\sqrt{15}\) के बीच संख्या रेखा पर स्थित संख्या है?
Which option lies between \(\sqrt{11}\) and \(\sqrt{15}\) on the number line?
#polynomials
#number-line
#ordering
#square-roots
A \(\sqrt{13}\)
B \(\sqrt{10}\)
C \(\sqrt{16}\)
D \(\sqrt{18}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{13}\)
Step 1
Concept
Because (11<13<15), \(\sqrt{11}<\sqrt{13}<\sqrt{15}\). For positive square roots, the order of radicands is preserved.
Step 2
Why this answer is correct
The correct answer is A. \(\sqrt{13}\). Because (11<13<15), \(\sqrt{11}<\sqrt{13}<\sqrt{15}\). For positive square roots, the order of radicands is preserved.
Step 3
Exam Tip
क्योंकि (11<13<15), इसलिए \(\sqrt{11}<\sqrt{13}<\sqrt{15}\)। धनात्मक वर्गमूल में मूल संख्या का क्रम बना रहता है।
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संख्या रेखा पर \(\sqrt{15}\) किस दो पूर्णांकों के बीच स्थित होगा?
Between which two integers will \(\sqrt{15}\) lie on the number line?
#polynomials
#number-line
#square-roots
#class-10
A (2) और (3) / (2) and (3)
B (3) और (4) / (3) and (4)
C (4) और (5) / (4) and (5)
D (5) और (6) / (5) and (6)
Explanation opens after your attempt
Correct Answer
B. (3) और (4) / (3) and (4)
Step 1
Concept
Since \(3^2=9\) and \(4^2=16\), \(\sqrt{15}\) lies between (3) and (4). In exams, use nearby perfect squares to locate square roots.
Step 2
Why this answer is correct
The correct answer is B. (3) और (4) / (3) and (4). Since \(3^2=9\) and \(4^2=16\), \(\sqrt{15}\) lies between (3) and (4). In exams, use nearby perfect squares to locate square roots.
Step 3
Exam Tip
क्योंकि \(3^2=9\) और \(4^2=16\), इसलिए \(\sqrt{15}\) (3) और (4) के बीच है। परीक्षा में वर्गमूल की स्थिति के लिए नजदीकी पूर्ण वर्ग देखें।
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संख्या रेखा पर \(-\sqrt{16}\) किस संख्या के बराबर स्थान पर होगा?
On the number line, \(-\sqrt{16}\) will be at the same position as which number?
#polynomials
#number-line
#square-roots
#class-10
A (4)
B (16)
C (-4)
D (-16)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{16}=4\), so \(-\sqrt{16}=-4\). In exams, apply the outside negative sign after finding the square root.
Step 2
Why this answer is correct
The correct answer is C. (-4). \(\sqrt{16}=4\), so \(-\sqrt{16}=-4\). In exams, apply the outside negative sign after finding the square root.
Step 3
Exam Tip
\(\sqrt{16}=4\), इसलिए \(-\sqrt{16}=-4\) है। परीक्षा में वर्गमूल निकालने के बाद बाहर का ऋण चिह्न लगाएं।
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संख्या रेखा पर \(\sqrt{9}\) किस संख्या के बराबर स्थान पर होगा?
On the number line, \(\sqrt{9}\) will be at the same position as which number?
#polynomials
#number-line
#square-roots
#class-10
A (2)
B (3)
C (9)
D (-3)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{9}=3\), so its position is at (3). In exams, identify square roots of perfect squares quickly.
Step 2
Why this answer is correct
The correct answer is B. (3). \(\sqrt{9}=3\), so its position is at (3). In exams, identify square roots of perfect squares quickly.
Step 3
Exam Tip
\(\sqrt{9}=3\), इसलिए इसका स्थान (3) पर होगा। परीक्षा में पूर्ण वर्गों के वर्गमूल तुरंत पहचानें।
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संख्या रेखा पर \(\sqrt{13}\) किस दो पूर्णांकों के बीच स्थित होगा?
Between which two integers will \(\sqrt{13}\) lie on the number line?
#square-roots
#estimation
#number-line
A (3) और (4) / (3) and (4)
B (2) और (3) / (2) and (3)
C (4) और (5) / (4) and (5)
D (1) और (2) / (1) and (2)
Explanation opens after your attempt
Correct Answer
A. (3) और (4) / (3) and (4)
Step 1
Concept
Since \(3^2=9\) and \(4^2=16\), \(\sqrt{13}\) lies between (3) and (4). Use nearby perfect squares to locate square roots.
Step 2
Why this answer is correct
The correct answer is A. (3) और (4) / (3) and (4). Since \(3^2=9\) and \(4^2=16\), \(\sqrt{13}\) lies between (3) and (4). Use nearby perfect squares to locate square roots.
Step 3
Exam Tip
क्योंकि \(3^2=9\) और \(4^2=16\), इसलिए \(\sqrt{13}\) (3) और (4) के बीच है। वर्गमूल की स्थिति के लिए निकट पूर्ण वर्ग देखें।
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संख्या रेखा पर \(\sqrt{7}\) को किसके बीच रखा जाएगा?
Between which numbers will \(\sqrt{7}\) be placed on the number line?
#square-roots
#estimation
#number-line
A (2) और (3) / (2) and (3)
B (3) और (4) / (3) and (4)
C (1) और (2) / (1) and (2)
D (7) और (8) / (7) and (8)
Explanation opens after your attempt
Correct Answer
A. (2) और (3) / (2) and (3)
Step 1
Concept
Since \(2^2=4\) and \(3^2=9\), \(\sqrt{7}\) lies between (2) and (3). Bound the number using nearby squares.
Step 2
Why this answer is correct
The correct answer is A. (2) और (3) / (2) and (3). Since \(2^2=4\) and \(3^2=9\), \(\sqrt{7}\) lies between (2) and (3). Bound the number using nearby squares.
Step 3
Exam Tip
क्योंकि \(2^2=4\) और \(3^2=9\), इसलिए \(\sqrt{7}\) (2) और (3) के बीच है। संख्या को निकट वर्गों से घेरें।
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संख्या रेखा पर \(\sqrt{16}\) और (4) के बारे में कौन-सा कथन सही है?
Which statement about \(\sqrt{16}\) and (4) on the number line is correct?
#perfect-squares
#square-roots
#number-line
A दोनों एक ही बिंदु पर हैं / both are at the same point
B \(\sqrt{16}\), (4) के बाईं ओर है / \(\sqrt{16}\) is left of (4)
C \(\sqrt{16}\), (4) के दाईं ओर है / \(\sqrt{16}\) is right of (4)
D दोनों ऋणात्मक हैं / both are negative
Explanation opens after your attempt
Correct Answer
A. दोनों एक ही बिंदु पर हैं / both are at the same point
Step 1
Concept
\(\sqrt{16}=4\), so both have the same position. Identify square roots of perfect squares directly.
Step 2
Why this answer is correct
The correct answer is A. दोनों एक ही बिंदु पर हैं / both are at the same point. \(\sqrt{16}=4\), so both have the same position. Identify square roots of perfect squares directly.
Step 3
Exam Tip
\(\sqrt{16}=4\), इसलिए दोनों की स्थिति समान है। पूर्ण वर्ग का वर्गमूल सीधे पहचानें।
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\(\sqrt{10}\) को संख्या रेखा पर रखने के लिए उसका मान किसके बीच होगा?
To place \(\sqrt{10}\) on the number line, its value will be between which numbers?
#square-roots
#estimation
#number-line
A (3) और (4) / (3) and (4)
B (2) और (3) / (2) and (3)
C (4) और (5) / (4) and (5)
D (1) और (2) / (1) and (2)
Explanation opens after your attempt
Correct Answer
A. (3) और (4) / (3) and (4)
Step 1
Concept
Since \(3^2=9\) and \(4^2=16\), \(\sqrt{10}\) lies between (3) and (4). Use perfect squares to locate square roots.
Step 2
Why this answer is correct
The correct answer is A. (3) और (4) / (3) and (4). Since \(3^2=9\) and \(4^2=16\), \(\sqrt{10}\) lies between (3) and (4). Use perfect squares to locate square roots.
Step 3
Exam Tip
क्योंकि \(3^2=9\) और \(4^2=16\), इसलिए \(\sqrt{10}\) (3) और (4) के बीच है। वर्गमूल की स्थिति पूर्ण वर्गों से पहचानें।
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\(\sqrt{5}\) संख्या रेखा पर किन दो पूर्णांकों के बीच होगा?
Between which two integers will \(\sqrt{5}\) lie on the number line?
#irrational-numbers
#square-roots
#number-line
A (2) और (3) / (2) and (3)
B (1) और (2) / (1) and (2)
C (5) और (6) / (5) and (6)
D (0) और (1) / (0) and (1)
Explanation opens after your attempt
Correct Answer
A. (2) और (3) / (2) and (3)
Step 1
Concept
Since \(2^2=4\) and \(3^2=9\), \(\sqrt{5}\) lies between (2) and (3). Look at the nearest perfect squares.
Step 2
Why this answer is correct
The correct answer is A. (2) और (3) / (2) and (3). Since \(2^2=4\) and \(3^2=9\), \(\sqrt{5}\) lies between (2) and (3). Look at the nearest perfect squares.
Step 3
Exam Tip
क्योंकि \(2^2=4\) और \(3^2=9\), इसलिए \(\sqrt{5}\) (2) और (3) के बीच है। निकटतम पूर्ण वर्ग देखें।
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संख्या रेखा पर \(\sqrt{9}\) की स्थिति क्या होगी?
What is the position of \(\sqrt{9}\) on the number line?
#square-roots
#perfect-squares
#number-line
A (3)
B (9)
C (-3)
D \(\frac{1}{3}\)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{9}=3\), so it is shown at (3). The principal square root is always non-negative.
Step 2
Why this answer is correct
The correct answer is A. (3). \(\sqrt{9}=3\), so it is shown at (3). The principal square root is always non-negative.
Step 3
Exam Tip
\(\sqrt{9}=3\) है, इसलिए इसे (3) पर दिखाते हैं। मुख्य वर्गमूल हमेशा अऋणात्मक लिया जाता है।
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संख्या रेखा पर \(\sqrt{4}\) किस बिंदु पर होगा?
At which point will \(\sqrt{4}\) lie on the number line?
#square-roots
#number-line
#real-numbers
A (2)
B (4)
C \(\frac{1}{2}\)
D (-2)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{4}=2\), so the point is (2). Remember square roots of perfect squares.
Step 2
Why this answer is correct
The correct answer is A. (2). \(\sqrt{4}=2\), so the point is (2). Remember square roots of perfect squares.
Step 3
Exam Tip
\(\sqrt{4}=2\) होता है, इसलिए बिंदु (2) पर होगा। पूर्ण वर्गों के वर्गमूल याद रखें।
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समीकरण \(x^2=9\) के मूल कौन-से हैं?
What are the roots of \(x^2=9\)?
#quadratic-equations
#square-roots
#roots
A (0,9)
B (3,9)
C (-3,3)
D (-9,9)
Explanation opens after your attempt
Step 1
Concept
From \(x^2=9\), we get \(x=\pm3\). While taking square roots, take both positive and negative values.
Step 2
Why this answer is correct
The correct answer is C. (-3,3). From \(x^2=9\), we get \(x=\pm3\). While taking square roots, take both positive and negative values.
Step 3
Exam Tip
\(x^2=9\) से \(x=\pm3\) मिलता है। वर्गमूल लेते समय धन और ऋण दोनों मान लें।
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कौन सा विकल्प \(\sqrt{2}\), \(\sqrt{3}\), और \(\sqrt{5}\) के प्रमाणों में सबसे बड़ा सामान्य भ्रम है?
Which option is the biggest common misconception in the proofs of \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{5}\)?
#common misconception
#square roots
#hard
#class 10
A वर्गमूल को उसके अंदर की संख्या के बराबर मान लेना / Treating the square root as equal to the number inside it
B परिमेय मानकर शुरुआत करना / Beginning by assuming rationality
C दोनों ओर वर्ग करना / Squaring both sides
D सहअभाज्य शर्त लिखना / Writing the coprime condition
Explanation opens after your attempt
Correct Answer
A. वर्गमूल को उसके अंदर की संख्या के बराबर मान लेना / Treating the square root as equal to the number inside it
Step 1
Concept
Writing \(\sqrt{2}=2\), \(\sqrt{3}=3\), or \(\sqrt{5}=5\) is wrong.
Step 2
Why this answer is correct
In the correct proof, the square root is assumed as a fraction and then squared.
Step 3
Exam Tip
Do not confuse the square root with the number inside it. चरण 1: \(\sqrt{2}=2\), \(\sqrt{3}=3\), या \(\sqrt{5}=5\) लिखना गलत है। चरण 2: सही प्रमाण में वर्गमूल को भिन्न के रूप में मानकर वर्ग किया जाता है। चरण 3: वर्गमूल और अंदर की संख्या को समान न समझें।
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कौन-सा विकल्प \(\sqrt{a}+\sqrt{b}=\sqrt{a+b}\) जैसी गलत सोच को खंडित करता है?
Which option disproves the wrong idea \(\sqrt{a}+\sqrt{b}=\sqrt{a+b}\)?
#common mistake
#square roots
#class 10
A (a=4,b=9)
B (a=0,b=9)
C (a=1,b=0)
D (a=0,b=0)
Explanation opens after your attempt
Correct Answer
A. (a=4,b=9)
Step 1
Concept
For (a=4,b=9), the left side is (2+3=5).
Step 2
Why this answer is correct
The right side is \(\sqrt{13}\), which is not (5).
Step 3
Exam Tip
When adding square roots, the numbers inside the roots are not added directly. चरण 1: (a=4,b=9) रखने पर बायाँ पक्ष (2+3=5) है। चरण 2: दायाँ पक्ष \(\sqrt{13}\) है, जो (5) नहीं है। चरण 3: वर्गमूलों को जोड़ते समय अंदर की संख्याएँ सीधे नहीं जोड़ी जातीं।
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यदि (a) और (b) धनात्मक पूर्णांक हैं तथा \(\sqrt{a}+\sqrt{b}\) परिमेय है, जबकि (a) पूर्ण वर्ग नहीं है, तो (b) के बारे में कौन-सा निष्कर्ष निश्चित रूप से सही हो सकता है?
If (a) and (b) are positive integers and \(\sqrt{a}+\sqrt{b}\) is rational, while (a) is not a perfect square, which conclusion about (b) can definitely be true?
#irrational numbers
#square roots
#expert
#class 10
A (b) भी (a) जैसा समान अपूर्ण वर्ग भाग रखता है / (b) also has the same non-square part as (a)
B (b) हमेशा पूर्ण वर्ग होगा / (b) will always be a perfect square
C ऐसा होना संभव नहीं है / This is not possible
D (b) अवश्य अभाज्य होगा / (b) must be prime
Explanation opens after your attempt
Correct Answer
C. ऐसा होना संभव नहीं है / This is not possible
Step 1
Concept
Since (a) is not a perfect square, \(\sqrt{a}\) is irrational.
Step 2
Why this answer is correct
A sum of two positive square roots could become rational only if irrational parts cancel, but both terms are positive here.
Step 3
Exam Tip
Without opposite signs, irrational surd parts remain in the sum. चरण 1: (a) पूर्ण वर्ग नहीं है, इसलिए \(\sqrt{a}\) अपरिमेय है। चरण 2: दो धनात्मक वर्गमूलों का योग परिमेय तभी हो सकता है जब अपरिमेय भाग कटे, पर यहाँ दोनों पद धनात्मक हैं इसलिए कटना संभव नहीं है। चरण 3: धनात्मक मूलों के योग में विपरीत चिह्न न होने पर अपरिमेय भाग बचता है।
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