\(5.4^2=29.16\) and \(5.5^2=30.25\), so \( \sqrt{30} \) lies between them. Check squares for decimal bounds.
Step 2
Why this answer is correct
The correct answer is B. \(5.4<\sqrt{30}<5.5\). \(5.4^2=29.16\) and \(5.5^2=30.25\), so \( \sqrt{30} \) lies between them. Check squares for decimal bounds.
Step 3
Exam Tip
\(5.4^2=29.16\) और \(5.5^2=30.25\), इसलिए \( \sqrt{30} \) इनके बीच है। दशमलव सीमा के लिए वर्ग जाँचें।
\( \frac{355}{113}\approx3.14159292 \) and \( \pi\approx3.14159265 \), so the fraction is slightly larger. Do not treat approximations as exactly equal.
Step 2
Why this answer is correct
The correct answer is C. \( \frac{355}{113}>\pi \). \( \frac{355}{113}\approx3.14159292 \) and \( \pi\approx3.14159265 \), so the fraction is slightly larger. Do not treat approximations as exactly equal.
Step 3
Exam Tip
\( \frac{355}{113}\approx3.14159292 \) और \( \pi\approx3.14159265 \), इसलिए भिन्न थोड़ा बड़ा है। अनुमानों को ठीक बराबर न मानें।
\( \sqrt{7}\approx2.646 \), so it is slightly less than (2.65). Compare close values to more decimal places.
Step 2
Why this answer is correct
The correct answer is A. \( \sqrt{7}<2.65 \). \( \sqrt{7}\approx2.646 \), so it is slightly less than (2.65). Compare close values to more decimal places.
Step 3
Exam Tip
\( \sqrt{7}\approx2.646 \), इसलिए यह (2.65) से थोड़ा छोटा है। निकट मानों में अधिक दशमलव स्थान तक तुलना करें।
This decimal is non-terminating and non-repeating, so it is irrational. Every irrational number also has a place on the number line.
Step 2
Why this answer is correct
The correct answer is B. अपरिमेय संख्या / Irrational number. This decimal is non-terminating and non-repeating, so it is irrational. Every irrational number also has a place on the number line.
Step 3
Exam Tip
यह दशमलव असांत और अनावर्ती है, इसलिए अपरिमेय है। संख्या रेखा पर हर अपरिमेय संख्या का भी स्थान होता है।
By Pythagoras, the hypotenuse is \( \sqrt{2^2+1^2}=\sqrt{5} \). Right triangles help in square-root construction.
Step 2
Why this answer is correct
The correct answer is C. \( \sqrt{5} \). By Pythagoras, the hypotenuse is \( \sqrt{2^2+1^2}=\sqrt{5} \). Right triangles help in square-root construction.
Step 3
Exam Tip
पायथागोरस से कर्ण \( \sqrt{2^2+1^2}=\sqrt{5} \) होगा। वर्गमूल निर्माण में समकोण त्रिभुज उपयोगी है।
Both (2.499) and (2.501) are at distance (0.001) from (2.5). Points at equal distance are equally close.
Step 2
Why this answer is correct
The correct answer is C. (2.499) और (2.501) / (2.499) and (2.501). Both (2.499) and (2.501) are at distance (0.001) from (2.5). Points at equal distance are equally close.
Step 3
Exam Tip
(2.499) और (2.501) दोनों की (2.5) से दूरी (0.001) है। समान दूरी पर दोनों समान रूप से निकट होते हैं।
Moving left gives \( \frac{3}{2}-\frac{11}{6}=-\frac{1}{3} \). Subtract the distance according to direction.
Step 2
Why this answer is correct
The correct answer is B. \( -\frac{1}{3} \). Moving left gives \( \frac{3}{2}-\frac{11}{6}=-\frac{1}{3} \). Subtract the distance according to direction.
Step 3
Exam Tip
बाईं ओर जाने पर \( \frac{3}{2}-\frac{11}{6}=-\frac{1}{3} \) मिलता है। दिशा के अनुसार दूरी घटाएँ।
\( \frac{4}{9}\approx0.444 \), \( \frac{1}{2}=0.5 \), and \( \frac{5}{9}\approx0.556 \). Hence \( \frac{1}{2} \) lies between them.
Step 2
Why this answer is correct
The correct answer is B. \( \frac{1}{2} \). \( \frac{4}{9}\approx0.444 \), \( \frac{1}{2}=0.5 \), and \( \frac{5}{9}\approx0.556 \). Hence \( \frac{1}{2} \) lies between them.
Step 3
Exam Tip
\( \frac{4}{9}\approx0.444 \), \( \frac{1}{2}=0.5 \), और \( \frac{5}{9}\approx0.556 \)। इसलिए \( \frac{1}{2} \) बीच में है।
\( \sqrt{5}\approx2.236 \) and \( \frac{9}{4}=2.25 \), so \( \frac{9}{4} \) is greater. The greater number lies farther right.
Step 2
Why this answer is correct
The correct answer is B. \( \frac{9}{4} \). \( \sqrt{5}\approx2.236 \) and \( \frac{9}{4}=2.25 \), so \( \frac{9}{4} \) is greater. The greater number lies farther right.
Step 3
Exam Tip
\( \sqrt{5}\approx2.236 \) और \( \frac{9}{4}=2.25 \), इसलिए \( \frac{9}{4} \) बड़ा है। बड़ी संख्या संख्या रेखा पर दाईं ओर होती है।
The midpoint is \( \frac{-\frac{5}{3}+\frac{7}{3}}{2}=\frac{1}{3} \). Take the average to find the midpoint.
Step 2
Why this answer is correct
The correct answer is A. \( \frac{1}{3} \). The midpoint is \( \frac{-\frac{5}{3}+\frac{7}{3}}{2}=\frac{1}{3} \). Take the average to find the midpoint.
Step 3
Exam Tip
मध्य बिंदु \( \frac{-\frac{5}{3}+\frac{7}{3}}{2}=\frac{1}{3} \) है। मध्य बिंदु के लिए औसत लें।
A. \( \frac{1}{2} \) और \( \frac{15}{2} \)/\( \frac{1}{2} \) and \( \frac{15}{2} \)
Step 1
Concept
\( |x-4|=\frac{7}{2} \) means (x) is at distance \( \frac{7}{2} \) from (4). Moving both directions gives \( \frac{1}{2} \) and \( \frac{15}{2} \).
Step 2
Why this answer is correct
The correct answer is A. \( \frac{1}{2} \) और \( \frac{15}{2} \) / \( \frac{1}{2} \) and \( \frac{15}{2} \). \( |x-4|=\frac{7}{2} \) means (x) is at distance \( \frac{7}{2} \) from (4). Moving both directions gives \( \frac{1}{2} \) and \( \frac{15}{2} \).
Step 3
Exam Tip
\( |x-4|=\frac{7}{2} \) का अर्थ (x) की (4) से दूरी \( \frac{7}{2} \) है। दोनों दिशाओं में \( \frac{1}{2} \) और \( \frac{15}{2} \) मिलते हैं।
B. (0) से (1) तक (12) बराबर भागों में (11)वाँ बिंदु/The (11)th point among (12) equal parts from (0) to (1)
Step 1
Concept
\( \frac{11}{12} \) means (11) parts out of (12) equal parts. The denominator gives the number of equal parts.
Step 2
Why this answer is correct
The correct answer is B. (0) से (1) तक (12) बराबर भागों में (11)वाँ बिंदु / The (11)th point among (12) equal parts from (0) to (1). \( \frac{11}{12} \) means (11) parts out of (12) equal parts. The denominator gives the number of equal parts.
Step 3
Exam Tip
\( \frac{11}{12} \) का अर्थ (12) बराबर भागों में (11) भाग है। हर बराबर भागों की संख्या बताता है।
\( \sqrt{21}\approx4.583 \) and \( \sqrt{22}\approx4.690 \), so (4.65) lies between them. Estimate accurately for close roots.
Step 2
Why this answer is correct
The correct answer is C. (4.65). \( \sqrt{21}\approx4.583 \) and \( \sqrt{22}\approx4.690 \), so (4.65) lies between them. Estimate accurately for close roots.
Step 3
Exam Tip
\( \sqrt{21}\approx4.583 \) और \( \sqrt{22}\approx4.690 \), इसलिए (4.65) इनके बीच है। पास-पास मूलों में सटीक अनुमान करें।
C. यह (-4) और (-3) के बीच है/It lies between (-4) and (-3)
Step 1
Concept
\( -\frac{19}{6}\approx-3.167 \), so it lies between (-4) and (-3). Converting a negative fraction to decimal is useful.
Step 2
Why this answer is correct
The correct answer is C. यह (-4) और (-3) के बीच है / It lies between (-4) and (-3). \( -\frac{19}{6}\approx-3.167 \), so it lies between (-4) and (-3). Converting a negative fraction to decimal is useful.
Step 3
Exam Tip
\( -\frac{19}{6}\approx-3.167 \), इसलिए यह (-4) और (-3) के बीच है। ऋणात्मक भिन्न को दशमलव में बदलना उपयोगी है।
Moving right means adding the distance, so the coordinate is \( -3+\sqrt{14} \). Choose addition or subtraction by direction.
Step 2
Why this answer is correct
The correct answer is C. \( -3+\sqrt{14} \). Moving right means adding the distance, so the coordinate is \( -3+\sqrt{14} \). Choose addition or subtraction by direction.
Step 3
Exam Tip
दाईं ओर जाने पर दूरी जोड़ी जाती है, इसलिए निर्देशांक \( -3+\sqrt{14} \) होगा। दिशा देखकर जोड़ या घटाव चुनें।
B. ( -6 ) और ( -5 ) के बीच/Between ( -6 ) and ( -5 )
Step 1
Concept
Since \(5<\sqrt{29}<6\), \(-6<-\sqrt{29}<-5\). For negative roots, write the reversed interval carefully.
Step 2
Why this answer is correct
The correct answer is B. ( -6 ) और ( -5 ) के बीच / Between ( -6 ) and ( -5 ). Since \(5<\sqrt{29}<6\), \(-6<-\sqrt{29}<-5\). For negative roots, write the reversed interval carefully.
Step 3
Exam Tip
क्योंकि \(5<\sqrt{29}<6\), इसलिए \(-6<-\sqrt{29}<-5\)। ऋणात्मक मूलों में क्रम उलटकर लिखें।
