\(5-\sqrt{11}\approx1.683\) and \( \frac{17}{10}=1.7 \). Therefore the first value is slightly smaller.
Step 2
Why this answer is correct
The correct answer is A. \(5-\sqrt{11}<\frac{17}{10}\). \(5-\sqrt{11}\approx1.683\) and \( \frac{17}{10}=1.7 \). Therefore the first value is slightly smaller.
Step 3
Exam Tip
\(5-\sqrt{11}\approx1.683\) और \( \frac{17}{10}=1.7 \) है। इसलिए पहला मान थोड़ा छोटा है।
\( \sqrt{39}\approx6.245 \), so \(6-\sqrt{39}\approx-0.245\). Always check the sign in root subtraction.
Step 2
Why this answer is correct
The correct answer is A. ( -1 ) और (0) के बीच / Between ( -1 ) and (0). \( \sqrt{39}\approx6.245 \), so \(6-\sqrt{39}\approx-0.245\). Always check the sign in root subtraction.
Step 3
Exam Tip
\( \sqrt{39}\approx6.245 \), इसलिए \(6-\sqrt{39}\approx-0.245\) है। घटाव वाले मूल में चिह्न जरूर जाँचें।
\(4-\sqrt{6}\approx1.551\) and \( \frac{31}{20}=1.55 \), so the first value is slightly greater. Estimate close values accurately.
Step 2
Why this answer is correct
The correct answer is B. \(4-\sqrt{6}>\frac{31}{20}\). \(4-\sqrt{6}\approx1.551\) and \( \frac{31}{20}=1.55 \), so the first value is slightly greater. Estimate close values accurately.
Step 3
Exam Tip
\(4-\sqrt{6}\approx1.551\) और \( \frac{31}{20}=1.55 \), इसलिए पहला मान थोड़ा बड़ा है। निकट मानों में सटीक अनुमान करें।
\( \sqrt{31}\approx5.568 \), so \(5-\sqrt{31}\approx-0.568\). Always check the sign in root subtraction.
Step 2
Why this answer is correct
The correct answer is B. ( -1 ) और (0) के बीच / Between ( -1 ) and (0). \( \sqrt{31}\approx5.568 \), so \(5-\sqrt{31}\approx-0.568\). Always check the sign in root subtraction.
Step 3
Exam Tip
\( \sqrt{31}\approx5.568 \), इसलिए \(5-\sqrt{31}\approx-0.568\) है। घटाव वाले मूलों में चिह्न अवश्य जाँचें।
\( \sqrt{18}\approx4.243 \), so \(4-\sqrt{18}\approx-0.243\). The sign can change when subtracting a root.
Step 2
Why this answer is correct
The correct answer is A. ( -1 ) और (0) के बीच / Between ( -1 ) and (0). \( \sqrt{18}\approx4.243 \), so \(4-\sqrt{18}\approx-0.243\). The sign can change when subtracting a root.
Step 3
Exam Tip
\( \sqrt{18}\approx4.243 \), इसलिए \(4-\sqrt{18}\approx-0.243\)। मूल घटाने पर चिह्न बदल सकता है।
Since \(4<\sqrt{22}<5\), \(-1<-5+\sqrt{22}<0\). Add bounds carefully in mixed expressions.
Step 2
Why this answer is correct
The correct answer is C. ( -1 ) और (0) के बीच / Between ( -1 ) and (0). Since \(4<\sqrt{22}<5\), \(-1<-5+\sqrt{22}<0\). Add bounds carefully in mixed expressions.
Step 3
Exam Tip
\(4<\sqrt{22}<5\), इसलिए \(-1<-5+\sqrt{22}<0\)। मिश्रित अभिव्यक्ति में सीमा जोड़ें।
\(3-\sqrt{2}\approx1.586\) and \( \frac{8}{5}=1.6 \). Therefore the first value is slightly smaller.
Step 2
Why this answer is correct
The correct answer is A. \(3-\sqrt{2}<\frac{8}{5}\). \(3-\sqrt{2}\approx1.586\) and \( \frac{8}{5}=1.6 \). Therefore the first value is slightly smaller.
Step 3
Exam Tip
\(3-\sqrt{2}\approx1.586\) और \( \frac{8}{5}=1.6 \) है। इसलिए पहला मान थोड़ा छोटा है।
\( \sqrt{10}\approx3.162 \), so \(2-\sqrt{10}\approx-1.162\). Estimation is important in root subtraction.
Step 2
Why this answer is correct
The correct answer is A. (-2) और (-1) के बीच / Between (-2) and (-1). \( \sqrt{10}\approx3.162 \), so \(2-\sqrt{10}\approx-1.162\). Estimation is important in root subtraction.
Step 3
Exam Tip
\( \sqrt{10}\approx3.162 \) इसलिए \(2-\sqrt{10}\approx-1.162\) है। घटाव वाले मूल में अनुमान जरूरी है।
\( \sqrt{5}\approx2.236\), so \(2-\sqrt{5}\approx-0.236\). Estimation is fastest for subtraction with roots.
Step 2
Why this answer is correct
The correct answer is A. (-1) और (0) के बीच / Between (-1) and (0). \( \sqrt{5}\approx2.236\), so \(2-\sqrt{5}\approx-0.236\). Estimation is fastest for subtraction with roots.
Step 3
Exam Tip
\( \sqrt{5}\approx2.236\), इसलिए \(2-\sqrt{5}\approx-0.236\)। घटाव वाले मूलों में अनुमान सबसे तेज होता है।
\(x=-4+\sqrt{13}\), and \(3<\sqrt{13}<4\), so (-1<x<0). Add bounds in combined expressions.
Step 2
Why this answer is correct
The correct answer is A. (-1) और (0) के बीच / Between (-1) and (0). \(x=-4+\sqrt{13}\), and \(3<\sqrt{13}<4\), so (-1<x<0). Add bounds in combined expressions.
Step 3
Exam Tip
\(x=-4+\sqrt{13}\) और \(3<\sqrt{13}<4\), इसलिए (-1<x<0)। संयुक्त अभिव्यक्ति में सीमा जोड़ें।
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.
Step 2
Why this answer is correct
The correct answer is B. (-11). 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.
Step 3
Exam Tip
\(\alpha+\beta=7\) और \(\alpha\beta=10\) है। \(\alpha^2+\beta^2=49-20=29\), इसलिए (29-6(7)=-13), अतः विकल्पों में कोई सही नहीं है।
We use (\(\alpha-5\)\(\beta-5\)=\alpha\beta-5\(\alpha+\beta\)+25). Since \(\alpha+\beta=4\) and \(\alpha\beta=-12\), the value is (-7).
Step 2
Why this answer is correct
The correct answer is A. (-7). We use (\(\alpha-5\)\(\beta-5\)=\alpha\beta-5\(\alpha+\beta\)+25). Since \(\alpha+\beta=4\) and \(\alpha\beta=-12\), the value is (-7).
Step 3
Exam Tip
(\(\alpha-5\)\(\beta-5\)=\alpha\beta-5\(\alpha+\beta\)+25) है। \(\alpha+\beta=4\) और \(\alpha\beta=-12\), इसलिए मान (-7) है।
Here \(\alpha+\beta=3\) and \(\alpha\beta=\frac{5}{4}\). Thus \(\alpha^3+\beta^3=27-\frac{45}{4}=\frac{63}{4}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{63}{4}\). Here \(\alpha+\beta=3\) and \(\alpha\beta=\frac{5}{4}\). Thus \(\alpha^3+\beta^3=27-\frac{45}{4}=\frac{63}{4}\).
Step 3
Exam Tip
यहाँ \(\alpha+\beta=3\) और \(\alpha\beta=\frac{5}{4}\) है। \(\alpha^3+\beta^3=27-\frac{45}{4}=\frac{63}{4}\)।
Here \(\alpha^2+\beta^2=64-4=60\) and (\(\alpha\beta\)2=4). Thus \(\frac{1}{\alpha^2}+\frac{1}{\beta^2}=\frac{60}{4}=15\).
Step 2
Why this answer is correct
The correct answer is A. (15). Here \(\alpha^2+\beta^2=64-4=60\) and (\(\alpha\beta\)2=4). Thus \(\frac{1}{\alpha^2}+\frac{1}{\beta^2}=\frac{60}{4}=15\).
Step 3
Exam Tip
\(\alpha^2+\beta^2=64-4=60\) और (\(\alpha\beta\)2=4) है। इसलिए \(\frac{1}{\alpha^2}+\frac{1}{\beta^2}=\frac{60}{4}=15\)।
We use (\(\alpha-4\)\(\beta-4\)=\alpha\beta-4\(\alpha+\beta\)+16). Since \(\alpha+\beta=3\) and \(\alpha\beta=-10\), the value is (-6).
Step 2
Why this answer is correct
The correct answer is C. (-6). We use (\(\alpha-4\)\(\beta-4\)=\alpha\beta-4\(\alpha+\beta\)+16). Since \(\alpha+\beta=3\) and \(\alpha\beta=-10\), the value is (-6).
Step 3
Exam Tip
(\(\alpha-4\)\(\beta-4\)=\alpha\beta-4\(\alpha+\beta\)+16) है। \(\alpha+\beta=3\) और \(\alpha\beta=-10\), इसलिए मान (-6) 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 \(\frac{224}{27}\).
Step 2
Why this answer is correct
The correct answer is B. \(\frac{224}{27}\). 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 \(\frac{224}{27}\).
Step 3
Exam Tip
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।
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 \(\frac{10}{3}\).
Step 2
Why this answer is correct
The correct answer is B. \(\frac{10}{3}\). 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 \(\frac{10}{3}\).
Step 3
Exam Tip
(\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।
Here \(\alpha+\beta=5\) and \(\alpha\beta=6\). Since \(\alpha^2+\beta^2=13\), the value is (13-4\(\alpha+\beta\)=13-20=-7).
Step 2
Why this answer is correct
The correct answer is A. (-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).
Step 3
Exam Tip
\(\alpha+\beta=5\) और \(\alpha\beta=6\) है। \(\alpha^2+\beta^2=13\), इसलिए (13-4\(\alpha+\beta\)=13-20=-7)।
