Which option correctly gives the position of (4-\sqrt{18}) on the number line?
Correct answer: A
( \sqrt{18}\approx4.243 ), so (4-\sqrt{18}\approx-0.243). The sign can change when subtracting a root.
View question detailsMathematics
Representing real numbers on the number line
संख्या रेखा पर वास्तविक संख्याओं का निरूपण
In this Class 10 Mathematics topic, students learn how real numbers—including rational numbers, irrational numbers and square roots—are located and represented accurately on the number line. They explore scale, order, distance and interval placement, and connect numerical values with geometric positions. This understanding supports the Polynomials chapter by helping students interpret and visualise real zeroes as points on the number line.
TOPIC PRACTICE
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Hard · Level 51 · number-line,root-expression,interval
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- Between ( -1 ) and (0)
- Between (0) and (1)
- Between (1) and (2)
- Between (2) and (3)
Hard · Level 51 · number-line,fraction-square-root,exact-value
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- ( \frac{9}{14} )
- ( \frac{81}{14} )
- ( \frac{14}{9} )
- ( \frac{9}{196} )
Hard · Level 51 · number line,polynomial radicals,square root simplification,real numbers,perfect square factors
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- \(6\sqrt{3}\)
- \(3\sqrt{6}\)
- \(12\sqrt{3}\)
- \(9\sqrt{3}\)
Hard · Level 51 · number line,decimal to fraction,negative rational numbers,terminating decimals,real numbers
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- \(-\frac{5}{16}\)
- \(-\frac{3}{16}\)
- \(-\frac{7}{16}\)
- \(-\frac{1}{16}\)
Hard · Level 51 · number-line,between-values,root-fraction
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- (3.2)
- (3.4)
- (3.33)
- (3.5)
Hard · Level 51 · number-line,distance,difference
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- (1.5)
- (2.7)
- (4.2)
- (6.9)
Hard · Level 51 · number-line,decimal-roots,between-values
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- (1.4)
- (1.55)
- (1.7)
- (1.2)
Hard · Level 51 · number-line,ordering,root-fraction-decimal
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- ( \frac{13}{3},4.34,\sqrt{19} )
- (4.34,\frac{13}{3},\sqrt{19} )
- ( \sqrt{19},4.34,\frac{13}{3} )
- ( \frac{13}{3},\sqrt{19},4.34 )
Hard · Level 51 · polynomials,real numbers,number line,square root,decimal calculation
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- 20.09
- 21.09
- 22.09
- 23.09
Hard · Level 51 · number-line,negative-comparison,between-values
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- ( -3.5 )
- ( -3.9 )
- ( -3.65 )
- ( -4.0 )
Hard · Level 51 · number-line,radical-addition,simplification
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- (4\sqrt{3})
- (5\sqrt{3})
- ( \sqrt{39} )
- (9\sqrt{3})
Hard · Level 51 · number line,real numbers,interval estimation,negative numbers
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- Between \(-3\) and \(-2\)
- Between \(-4\) and \(-3\)
- Between \(-5\) and \(-4\)
- Between \(2\) and \(3\)
Hard · Level 51 · number-line,irrational-expression,estimation
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- (1) and (2)
- (2) and (3)
- (3) and (4)
- (4) and (5)
Expert · Level 50 · number-line,irrational-difference,estimation
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- (1) and (2)
- (2) and (3)
- (0) and (1)
- (3) and (4)
Expert · Level 50 · number-line,absolute-value,distance
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- (1.25) and (-6.25)
- (6.25) and (-1.25)
- (3.75) and (-2.5)
- (2.5) and (-3.75)
Expert · Level 50 · number-line,ordering,mixed-values
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- ( \sqrt{13},3.61,\frac{29}{8} )
- (3.61,\sqrt{13},\frac{29}{8} )
- ( \sqrt{13},\frac{29}{8},3.61 )
- ( \frac{29}{8},3.61,\sqrt{13} )
Expert · Level 50 · number-line,negative-root,comparison
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- (P)
- (Q)
- Both are equal
- Both are right of (0)
Expert · Level 50 · polynomials,number-line,decimal-square-root,principal-square-root
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- 2.2
- 2.3
- 2.4
- 2.6
Expert · Level 50 · number-line,distance,fractions
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- \( \frac{47}{12} \)
- \( \frac{19}{12} \)
- \( \frac{23}{10} \)
- \( \frac{13}{6} \)
Expert · Level 50 · number-line,between-roots,estimation
View options
- (5.70)
- (5.75)
- (5.90)
- (5.60)
On the number line, ( \sqrt{\frac{81}{196}} ) is equal to which point?
Correct answer: A
The expression is \\(\sqrt{\frac{81}{196}}\\). Since both 81 and 196 are perfect squares, take their positive square roots: \\(\sqrt{81}=9\\) and \\(\sqrt{196}=14\\). Therefore \\(\sqrt{\frac{81}{196}}=\frac{\sqrt{81}}{\sqrt{196}}=\frac{9}{14}\\). The principal square root is nonnegative, so the result is positive. Hence option A represents the correct point on the number line. Options such as \\(\frac{81}{14}\\) or \\(\frac{9}{196}\\) do not result from taking the square root of numerator and denominator.
The fraction is already in lowest terms because 9 and 14 have no common factor greater than 1. Squaring the answer also verifies it: \\(\left(\frac{9}{14}\right)^2=\frac{81}{196}\\). Thus the point corresponding to the given radical is exactly \\(\frac{9}{14}\\), not a negative value or an unreduced expression. The supplied answer A is mathematically consistent and follows from the square-root property for positive numerator and denominator.
If the coordinate of a point on the number line is \\(x=\\sqrt{108}\\), what is the simplest form of \(x\)?
Correct answer: A
Since \(108=36\times3\) and \(36\) is the greatest perfect-square factor of \(108\), \(\sqrt{108}=\sqrt{36\times3}=6\sqrt{3}\). Thus, the point’s coordinate on the number line is \(6\sqrt{3}\). Exam tip: extract the greatest perfect-square factor from under the radical. Option B does not represent the correct simplification, while options C and D give larger values than \(\sqrt{108}\).
View question detailsHow can the number \(-0.3125\), represented on the number line, be written in its simplest fractional form?
Correct answer: A
\(-0.3125=-\frac{3125}{10000}\). Dividing the numerator and denominator by 625 gives \(-\frac{3125}{10000}=-\frac{5}{16}\), so option A is correct. The closest distractor, \(-\frac{3}{16}\), equals \(-0.1875\), not \(-0.3125\). Exam tip: for a terminating decimal, place it over the corresponding power of 10 and then reduce the fraction completely.
View question detailsWhich value is greater than ( \sqrt{11} ) and less than ( \frac{10}{3} )?
Correct answer: C
( \sqrt{11}\approx3.316 ) and ( \frac{10}{3}\approx3.333 ), so (3.33) lies between them. Compare very close values accurately.
View question detailsIf (A=2.4), (B=-1.8), and (C=5.1), what is the difference between lengths (AB) and (AC)?
Correct answer: A
(AB=|2.4-(-1.8)|=4.2) and (AC=|2.4-5.1|=2.7), so the difference is (1.5). Use absolute value for distance.
View question detailsWhich value lies between ( \sqrt{2.25} ) and ( \sqrt{2.56} ) on the number line?
Correct answer: B
( \sqrt{2.25}=1.5 ) and ( \sqrt{2.56}=1.6 ), so (1.55) lies between them. Remember decimal square roots.
View question detailsWhat is the correct increasing order of ( \sqrt{19} ), ( \frac{13}{3} ), and (4.34) on the number line?
Correct answer: D
( \frac{13}{3}\approx4.333 ), ( \sqrt{19}\approx4.359 ), and (4.34) lies between them. The correct order is ( \frac{13}{3},4.34,\sqrt{19} ).
View question detailsIf \(\sqrt{x}=4.7\), which value does \(x\) represent on the number line?
Correct answer: C
Given \(\sqrt{x}=4.7\), squaring both sides gives \(x=(4.7)^2=22.09\). Therefore, \(x\) represents 22.09 on the number line. The nearby option 21.09 is incorrect because \(4.7\times4.7=22.09\), not 21.09. Exam tip: when squaring a decimal, multiply it carefully and place the decimal point according to the total number of decimal places in the factors.
View question detailsWhich value is less than ( -\sqrt{13} ) and greater than (-3.7)?
Correct answer: C
( -\sqrt{13}\approx-3.606 ), so (-3.65) is less than it and greater than (-3.7). Be careful with direction for negative values.
View question detailsWhat is the simplified form of ( \sqrt{12}+\sqrt{27} ) on the number line?
Correct answer: B
( \sqrt{12}=2\sqrt{3} ) and ( \sqrt{27}=3\sqrt{3} ), so the sum is (5\sqrt{3}). Simplify the radicals first.
View question detailsIf \(u=-2-\sqrt{7}\) is represented on the number line, in which interval does \(u\) lie?
Correct answer: C
Since \(\sqrt{7}\approx 2.646\), we get \(u=-2-2.646\approx -4.646\). Therefore, \(u\) lies between \(-5\) and \(-4\). Note that subtracting a positive number from a negative number makes the result more negative, so it cannot lie between \(-4\) and \(-3\). In an exam, estimate the square root first and then locate the resulting number between consecutive integers.
View question detailsBetween which two consecutive integers is ( \sqrt{6}+\frac{1}{3} ) located on the number line?
Correct answer: B
( \sqrt{6}\approx2.449 ) and ( \frac{1}{3}\approx0.333 ), so the sum is about (2.782). For mixed values, estimate first.
View question detailsBetween which two consecutive integers does ( \sqrt{91}-\sqrt{55} ) lie on the number line?
Correct answer: B
( \sqrt{91}\approx9.54 ) and ( \sqrt{55}\approx7.42 ), so the difference is about (2.12). Estimate both roots first.
View question detailsIf ( |x+2.5|=3.75 ) on the number line, what are the possible values of (x)?
Correct answer: A
( |x+2.5|=3.75 ) means the distance of (x) from (-2.5) is (3.75). Moving both ways gives (1.25) and (-6.25).
View question detailsWhat is the correct increasing order of ( \frac{29}{8} ), ( \sqrt{13} ), and (3.61) on the number line?
Correct answer: A
( \sqrt{13}\approx3.606 ), (3.61), and ( \frac{29}{8}=3.625 ). Compare close values to more decimal places.
View question detailsIf (P=-\sqrt{74}) and (Q=-8.55), which point is farther right on the number line?
Correct answer: B
( -\sqrt{74}\approx-8.602 ) and (-8.55) is greater. The greater number lies to the right on a number line.
View question detailsAt which point on the number line should (sqrt{5.76}) be marked?
Correct answer: C
Since (2.4)^2=5.76, the principal square root (sqrt{5.76}) is 2.4. The negative value (-2.4) is not the principal square root. In an exam, verify the answer quickly by squaring the decimal.
View question detailsIf \(A=-\frac{11}{4}\) and \(B=\frac{7}{6}\) on the number line, what is the length of (AB)?
Correct answer: A
The distance is \( \left|\frac{7}{6}-\left(-\frac{11}{4}\right)\right|=\frac{47}{12} \). Use absolute value while finding distance.
View question detailsWhich value lies between ( \sqrt{33} ) and ( \sqrt{34} ) on the number line?
Correct answer: B
( \sqrt{33}\approx5.745 ) and ( \sqrt{34}\approx5.831 ), so (5.75) lies between them. Keep estimates accurate for close roots.
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