root-expression se related questions ko ek jagah revise karein. Har question me bilingual content, answer feedback aur explanation available hai.
34 questions tagged with root-expression.
A. \(5-\sqrt{11}<\frac{17}{10}\)
\(5-\sqrt{11}\approx1.683\) और \( \frac{17}{10}=1.7 \) है। इसलिए पहला मान थोड़ा छोटा है। / \(5-\sqrt{11}\approx1.683\) and \( \frac{17}{10}=1.7 \). Therefore the first value is slightly smaller.
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In which interval is \(6-\sqrt{39}\) correctly located on the number line?
A. ( -1 ) और (0) के बीचBetween ( -1 ) and (0)
\( \sqrt{39}\approx6.245 \), इसलिए \(6-\sqrt{39}\approx-0.245\) है। घटाव वाले मूल में चिह्न जरूर जाँचें। / \( \sqrt{39}\approx6.245 \), so \(6-\sqrt{39}\approx-0.245\). Always check the sign in root subtraction.
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Which statement is correct when comparing \(4-\sqrt{6}\) and \( \frac{31}{20} \) on the number line?
B. \(4-\sqrt{6}>\frac{31}{20}\)
\(4-\sqrt{6}\approx1.551\) और \( \frac{31}{20}=1.55 \), इसलिए पहला मान थोड़ा बड़ा है। निकट मानों में सटीक अनुमान करें। / \(4-\sqrt{6}\approx1.551\) and \( \frac{31}{20}=1.55 \), so the first value is slightly greater. Estimate close values accurately.
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Between which two consecutive integers does \(\sqrt{89}+1\) lie on the number line?
B. 10 और 1110 and 11
क्योंकि \(9^2=81<89<100=10^2\), इसलिए \(9<\sqrt{89}<10\)। दोनों पक्षों में 1 जोड़ने पर \(10<\sqrt{89}+1<11\), अतः यह 10 और 11 के बीच है। विकल्प A केवल \(\sqrt{89}\) की सीमा बताता है, पूरे व्यंजक की नहीं। परीक्षा-युक्ति: वर्गमूल की सीमा ज्ञात करने के लिए दी गई संख्या के निकटतम पूर्ण वर्गों की तुलना करें। / Since \(9^2=81<89<100=10^2\), we get \(9<\sqrt{89}<10\). Adding 1 to all parts gives \(10<\sqrt{89}+1<11\), so the expression lies between 10 and 11. Option A bounds only \(\sqrt{89}\), not the complete expression. Exam tip: compare the number with the nearest consecutive perfect squares to bound a square root.
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In which interval is \(5-\sqrt{31}\) correctly located on the number line?
B. ( -1 ) और (0) के बीचBetween ( -1 ) and (0)
\( \sqrt{31}\approx5.568 \), इसलिए \(5-\sqrt{31}\approx-0.568\) है। घटाव वाले मूलों में चिह्न अवश्य जाँचें। / \( \sqrt{31}\approx5.568 \), so \(5-\sqrt{31}\approx-0.568\). Always check the sign in root subtraction.
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Which option correctly gives the position of \(4-\sqrt{18}\) on the number line?
A. ( -1 ) और (0) के बीचBetween ( -1 ) and (0)
\( \sqrt{18}\approx4.243 \), इसलिए \(4-\sqrt{18}\approx-0.243\)। मूल घटाने पर चिह्न बदल सकता है। / \( \sqrt{18}\approx4.243 \), so \(4-\sqrt{18}\approx-0.243\). The sign can change when subtracting a root.
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In which interval is \(2-\sqrt{10}\) correctly located on the number line?
A. (-2) और (-1) के बीचBetween (-2) and (-1)
\( \sqrt{10}\approx3.162 \) इसलिए \(2-\sqrt{10}\approx-1.162\) है। घटाव वाले मूल में अनुमान जरूरी है। / \( \sqrt{10}\approx3.162 \), so \(2-\sqrt{10}\approx-1.162\). Estimation is important in root subtraction.
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Which option correctly gives the position of \(2-\sqrt{5}\) on the number line?
A. (-1) और (0) के बीचBetween (-1) and (0)
\( \sqrt{5}\approx2.236\), इसलिए \(2-\sqrt{5}\approx-0.236\)। घटाव वाले मूलों में अनुमान सबसे तेज होता है। / \( \sqrt{5}\approx2.236\), so \(2-\sqrt{5}\approx-0.236\). Estimation is fastest for subtraction with roots.
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A. (-13)
(\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta=49-20=29) है। इसलिए मान (29-6\(\alpha+\beta\)=29-42=-13) है। / (\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta=49-20=29). Therefore the value is (29-6\(\alpha+\beta\)=29-42=-13).
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If \(\alpha,\beta\) are roots of \(x^2-7x+10=0\), what is \(\alpha^2-6\alpha+\beta^2-6\beta\)?
B. (-11)
\(\alpha+\beta=7\) और \(\alpha\beta=10\) है। \(\alpha^2+\beta^2=49-20=29\), इसलिए (29-6(7)=-13), अतः विकल्पों में कोई सही नहीं है। / Here \(\alpha+\beta=7\) and \(\alpha\beta=10\). Since \(\alpha^2+\beta^2=49-20=29\), the value is (29-6(7)=-13), so none of the options is correct.
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If \(\alpha,\beta\) are roots of \(x^2-5x+1=0\), what is \(\alpha^4+\beta^4\)?
A. (527)
\(\alpha+\beta=5\) और \(\alpha\beta=1\) है। पहले \(\alpha^2+\beta^2=23\), फिर \(\alpha^4+\beta^4=23^2-2=527\)। / Here \(\alpha+\beta=5\) and \(\alpha\beta=1\). First \(\alpha^2+\beta^2=23\), then \(\alpha^4+\beta^4=23^2-2=527\).
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If \(\alpha,\beta\) are roots of \(x^2-7x+r=0\) and \(\alpha^2\beta+\alpha\beta^2=84\), what is (r)?
B. (12)
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)) होता है। यहाँ \(\alpha+\beta=7\), इसलिए (7r=84) और (r=12)। / We use (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)). Here \(\alpha+\beta=7\), so (7r=84) and (r=12).
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If \(\alpha,\beta\) are roots of \(x^2-4x-12=0\), what is (\(\alpha-5\)\(\beta-5\))?
A. (-7)
(\(\alpha-5\)\(\beta-5\)=\alpha\beta-5\(\alpha+\beta\)+25) है। \(\alpha+\beta=4\) और \(\alpha\beta=-12\), इसलिए मान (-7) है। / We use (\(\alpha-5\)\(\beta-5\)=\alpha\beta-5\(\alpha+\beta\)+25). Since \(\alpha+\beta=4\) and \(\alpha\beta=-12\), the value is (-7).
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A. \(\frac{63}{4}\)
यहाँ \(\alpha+\beta=3\) और \(\alpha\beta=\frac{5}{4}\) है। \(\alpha^3+\beta^3=27-\frac{45}{4}=\frac{63}{4}\)। / Here \(\alpha+\beta=3\) and \(\alpha\beta=\frac{5}{4}\). Thus \(\alpha^3+\beta^3=27-\frac{45}{4}=\frac{63}{4}\).
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A. (15)
\(\alpha^2+\beta^2=64-4=60\) और (\(\alpha\beta\)2=4) है। इसलिए \(\frac{1}{\alpha^2}+\frac{1}{\beta^2}=\frac{60}{4}=15\)। / Here \(\alpha^2+\beta^2=64-4=60\) and (\(\alpha\beta\)2=4). Thus \(\frac{1}{\alpha^2}+\frac{1}{\beta^2}=\frac{60}{4}=15\).
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If \(\alpha,\beta\) are roots of \(x^2-6x+5=0\), what is \(\alpha^2-5\alpha+\beta^2-5\beta\)?
B. (-4)
\(\alpha+\beta=6\) और \(\alpha\beta=5\) है। \(\alpha^2+\beta^2=26\), इसलिए (26-5\(\alpha+\beta\)=-4)। / Here \(\alpha+\beta=6\) and \(\alpha\beta=5\). Since \(\alpha^2+\beta^2=26\), the value is (26-5\(\alpha+\beta\)=-4).
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If \(\alpha,\beta\) are roots of \(x^2-3x+1=0\), what is \(\alpha^4+\beta^4\)?
C. (47)
\(\alpha+\beta=3\) और \(\alpha\beta=1\) है। पहले \(\alpha^2+\beta^2=7\), फिर \(\alpha^4+\beta^4=7^2-2=47\)। / Here \(\alpha+\beta=3\) and \(\alpha\beta=1\). First \(\alpha^2+\beta^2=7\), then \(\alpha^4+\beta^4=7^2-2=47\).
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If \(\alpha,\beta\) are roots of \(x^2-6x+r=0\) and \(\alpha^2\beta+\alpha\beta^2=54\), what is (r)?
C. (9)
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)) होता है। यहाँ \(\alpha+\beta=6\), इसलिए (6r=54) और (r=9)। / We use (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)). Here \(\alpha+\beta=6\), so (6r=54) and (r=9).
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If \(\alpha,\beta\) are the roots of \(x^2-3x-10=0\), what is (\(\alpha-4\)\(\beta-4\))?
C. (-6)
(\(\alpha-4\)\(\beta-4\)=\alpha\beta-4\(\alpha+\beta\)+16) है। \(\alpha+\beta=3\) और \(\alpha\beta=-10\), इसलिए मान (-6) है। / We use (\(\alpha-4\)\(\beta-4\)=\alpha\beta-4\(\alpha+\beta\)+16). Since \(\alpha+\beta=3\) and \(\alpha\beta=-10\), the value is (-6).
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If \(\alpha,\beta\) are the roots of \(3x^2-8x+4=0\), what is \(\alpha^3+\beta^3\)?
B. \(\frac{224}{27}\)
यहाँ \(\alpha+\beta=\frac{8}{3}\) और \(\alpha\beta=\frac{4}{3}\) है। (\alpha-3+\beta-3=\(\alpha+\beta\)3-3\alpha\beta\(\alpha+\beta\)) से \(\frac{224}{27}\) मिलता है। / Here \(\alpha+\beta=\frac{8}{3}\) and \(\alpha\beta=\frac{4}{3}\). Using (\alpha-3+\beta-3=\(\alpha+\beta\)3-3\alpha\beta\(\alpha+\beta\)), we get \(\frac{224}{27}\).
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B. \(\frac{10}{3}\)
(\frac{1}{\alpha-2}+\frac{1}{\beta-2}=\frac{\alpha-2+\beta-2}{\(\alpha\beta\)2}) होता है। \(\alpha^2+\beta^2=30\) और (\(\alpha\beta\)2=9), इसलिए मान \(\frac{10}{3}\) है। / We use (\frac{1}{\alpha-2}+\frac{1}{\beta-2}=\frac{\alpha-2+\beta-2}{\(\alpha\beta\)2}). Since \(\alpha^2+\beta^2=30\) and (\(\alpha\beta\)2=9), the value is \(\frac{10}{3}\).
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A. (-7)
\(\alpha+\beta=5\) और \(\alpha\beta=6\) है। \(\alpha^2+\beta^2=13\), इसलिए (13-4\(\alpha+\beta\)=13-20=-7)। / Here \(\alpha+\beta=5\) and \(\alpha\beta=6\). Since \(\alpha^2+\beta^2=13\), the value is (13-4\(\alpha+\beta\)=13-20=-7).
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If \(\alpha,\beta\) are roots of \(x^2-5x+6=0\), what is \(\alpha^2-4\alpha+\beta^2-4\beta\)?
A. (5)
(\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta=13) है। इसलिए मान (13-4(5)=-7) होगा, अतः कोई विकल्प सही नहीं है। / (\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta=13). Thus the value is (13-4(5)=-7), so none of the options is correct.
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If \(\alpha,\beta\) are roots of \(x^2-4x+1=0\), what is \(\alpha^4+\beta^4\)?
A. (194)
\(\alpha+\beta=4\) और \(\alpha\beta=1\) है। पहले \(\alpha^2+\beta^2=14\), फिर \(\alpha^4+\beta^4=14^2-2=194\)। / Here \(\alpha+\beta=4\) and \(\alpha\beta=1\). First \(\alpha^2+\beta^2=14\), then \(\alpha^4+\beta^4=14^2-2=194\).
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A. (4)
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)) होता है। यहाँ \(\alpha+\beta=-5\), इसलिए (r(-5)=-20) और (r=4)। / We use (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)). Here \(\alpha+\beta=-5\), so (r(-5)=-20) and (r=4).
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If \(\alpha,\beta\) are the roots of \(x^2-2x-8=0\), what is (\(\alpha+3\)\(\beta+3\))?
A. (7)
(\(\alpha+3\)\(\beta+3\)=\alpha\beta+3\(\alpha+\beta\)+9) है। \(\alpha+\beta=2\) और \(\alpha\beta=-8\), इसलिए मान (7) है। / We use (\(\alpha+3\)\(\beta+3\)=\alpha\beta+3\(\alpha+\beta\)+9). Since \(\alpha+\beta=2\) and \(\alpha\beta=-8\), the value is (7).
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If \(\alpha,\beta\) are the roots of \(2x^2-5x+3=0\), what is \(\alpha^3+\beta^3\)?
A. \(\frac{35}{8}\)
यहाँ \(\alpha+\beta=\frac{5}{2}\) और \(\alpha\beta=\frac{3}{2}\) है। (\alpha-3+\beta-3=\(\alpha+\beta\)3-3\alpha\beta\(\alpha+\beta\)) से \(\frac{35}{8}\) मिलता है। / Here \(\alpha+\beta=\frac{5}{2}\) and \(\alpha\beta=\frac{3}{2}\). Using (\alpha-3+\beta-3=\(\alpha+\beta\)3-3\alpha\beta\(\alpha+\beta\)), we get \(\frac{35}{8}\).
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A. (8)
जड़ें (3) और (5) हैं। सीधे रखने पर \(\frac{5}{1}+\frac{7}{3}=\frac{22}{3}\) आता है, इसलिए विकल्पों में कोई सही नहीं; सही प्रश्न के लिए उत्तर \(\frac{22}{3}\) होना चाहिए। / The roots are (3) and (5). Substitution gives \(\frac{5}{1}+\frac{7}{3}=\frac{22}{3}\), so none of the options is correct; the correct value should be \(\frac{22}{3}\).
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A. \(\frac{21}{4}\)
(\frac{1}{\alpha-2}+\frac{1}{\beta-2}=\frac{\alpha-2+\beta-2}{\(\alpha\beta\)2}) होता है। \(\alpha^2+\beta^2=21\) और (\(\alpha\beta\)2=4), इसलिए मान \(\frac{21}{4}\) है। / We use (\frac{1}{\alpha-2}+\frac{1}{\beta-2}=\frac{\alpha-2+\beta-2}{\(\alpha\beta\)2}). Since \(\alpha^2+\beta^2=21\) and (\(\alpha\beta\)2=4), the value is \(\frac{21}{4}\).
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If \(\alpha,\beta\) are roots of \(x^2-6x+8=0\), what is (\(\alpha-2\)\(\beta-2\))?
A. (0)
(\(\alpha-2\)\(\beta-2\)=\alpha\beta-2\(\alpha+\beta\)+4) है। \(\alpha+\beta=6\) और \(\alpha\beta=8\), इसलिए मान (0) है। / (\(\alpha-2\)\(\beta-2\)=\alpha\beta-2\(\alpha+\beta\)+4). Since \(\alpha+\beta=6\) and \(\alpha\beta=8\), the value is (0).
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