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SubjectsMathematics
संख्या रेखा
The Number Line topic in Class 9 Mathematics, within Number Systems, helps students visualise numbers as points on a continuous line. They learn to locate and compare integers, rational numbers, irrational numbers and real numbers, understand their order and relative position, and interpret distance using intervals. The topic also supports the geometric representation of irrational numbers such as √2, making the connection between numerical expressions and their positions on the real number line clear.
Correct answer: A. \(\sqrt{2}\)
Explanation: \(\sqrt{2}\) cannot be expressed as \(\frac{p}{q}\), so it is irrational. \(-\frac{7}{4}\), \(0.125\), and \(0.\overline{3}\) are rational. Exam tip: terminating or recurring decimals are rational.
Correct answer: B. -9.99
Explanation: For negative numbers, the number closer to zero is greater. Here, \(-9.99\) is closer to zero than \(-10\), so \(-9.99 > -10\). “Both are equal” is incorrect because the two decimal values are different. Exam tip: On a number line, the number farther to the right is greater.
Correct answer: A. 3
Explanation: The midpoint of two numbers on a number line is their average: \(\frac{11+(-5)}{2}=\frac{6}{2}=3\). Therefore, 3 is correct. The number 2 is not at an equal distance from 11 and -5. Exam tip: While finding a midpoint, add a negative number using brackets to avoid a sign error.
Correct answer: B. ( -1.75 )
Explanation: Write -1.75 as -1.750. On the number line, -1.750 lies to the left of -1.705, so it is smaller. Among negative numbers, the number with the greater magnitude is smaller. Option A is a close distractor, but -1.705 is less negative and therefore greater. Exam tip: Add trailing zeros to make the decimal places equal before comparing.
Correct answer: B. 16
Explanation: On a number line, the distance between two numbers is the absolute value of their difference: \(\lvert -24-(-8)\rvert=\lvert -16\rvert=16\). Therefore, the correct answer is 16. The value 24 is the magnitude of one number, not the distance between the two numbers. Exam tip: A distance is always non-negative.
Correct answer: B. \( -54 \)
Explanation: The successor of an integer is 1 greater than that integer. Thus, \( -55+1=-54 \), so \( -54 \) is correct. \( -56 \) is 1 less than \( -55 \), so it is the predecessor. Exam tip: To find a successor, add 1 even for negative integers.
Correct answer: B. ( -0.00001 )
Explanation: For negative numbers, the number closer to zero is greater. \(-0.00001\) is closer to zero than \(-0.0001\), so \(-0.00001 > -0.0001\). Option A is smaller because it has a greater negative magnitude. Exam tip: When comparing negative decimals, the negative number with the smaller magnitude is greater.
Correct answer: B. \(2.1\)
Explanation: \(2.1\) can be written as \(2.10\). Comparing \(2.10\) and \(1.99\), their whole-number parts are 2 and 1 respectively; therefore, \(2.10>1.99\). Hence, \(2.1\) is greater. Although \(1.99\) is very close to 2, it is still less than 2. Exam tip: When comparing decimals, add trailing zeroes if needed to make the number of decimal places equal.
Correct answer: A. \(-8\)
Explanation: The midpoint of two numbers is their average: \(\frac{-16+0}{2}=\frac{-16}{2}=-8\). Thus, \(-8\) is equally distant from \(-16\) and \(0\). \(-7\) is not the midpoint because it is not at the same distance from both numbers. Exam tip: use \(\frac{a+b}{2}\) to find the midpoint on a number line.
Correct answer: B. -5.555
Explanation: Write the numbers to the same number of decimal places: \(-5.55=-5.550\). Comparing \(-5.555\) and \(-5.550\), \(-5.555\) is more negative, so it lies farther left on the number line and is smaller. Thus, \(-5.550\) is greater. Exam tip: among negative decimals, the number with the greater magnitude is the smaller number.
Correct answer: B. 100
Explanation: On a number line, the distance between two numbers is the absolute value of their difference: \(|-100-0|=|-100|=100\). Therefore, the correct answer is 100. The value 99 would be the distance from -100 to -1, not from -100 to 0. Exam tip: distance is always non-negative.
Correct answer: C. Infinitely many rational and infinitely many irrational numbers exist
Explanation: Both sets are dense. Choose rational \(r\) between \(a\) and \(b\); for large \(n\), \(r+\sqrt{2}/n\) stays in the interval and is irrational. Thus C is correct. Exam tip: dense means infinitely many numbers in every interval.
Correct answer: B. 14
Explanation: The distance between two numbers on a number line is the absolute value of their difference: \(\lvert -28-(-14)\rvert=\lvert -14\rvert=14\). Therefore, the correct answer is 14. Values such as 13 and 15 are one less or one more, so they do not represent the distance. Exam tip: a distance is always non-negative.
Correct answer: B. -0.49
Explanation: The correct answer is -0.49. On a number line, the negative number closer to zero is greater. Since -0.49 is closer to zero than -0.5, we have -0.49 > -0.5. “Both are equal” is incorrect because the decimal values are different. Exam tip: among negative numbers, the number with the smaller magnitude is greater.
Correct answer: A. 3
Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-7+13}{2}=\frac{6}{2}=3\). Therefore, 3 is the correct answer. The number 2 is not the midpoint because it is not equally distant from -7 and 13. Exam tip: add the two numbers and divide by 2 to find their midpoint.
Correct answer: B. -13.5
Explanation: -13.5 can be written as -13.50. On a number line, the more negative number lies farther to the left and is smaller. Since -13.50 < -13.05, -13.5 is the smaller number. Although -13.05 looks similar, it is closer to zero and is therefore greater. Exam tip: write negative decimals with the same number of decimal places before comparing them.
Correct answer: A. \(\displaystyle -\frac{11}{2}\)
Explanation: The midpoint of two numbers is their average: \(\frac{-9+(-2)}{2}=\frac{-11}{2}=-5.5\). Therefore, \(\displaystyle -\frac{11}{2}\) is correct. Although \(-6\) lies between the two numbers, it is not at equal distance from both. Exam tip: add the two coordinates and divide by 2 to find a midpoint on the number line.
Correct answer: B. \(12\)
Explanation: The distance between two numbers on a number line is the absolute value of their difference. Thus, \(\left|-3-(-15)\right|=\left|12\right|=12\). Therefore, \(12\) is correct. \(18\) is obtained by adding the absolute values of the numbers, but distance must be found using the absolute difference. Exam tip: distance can never be negative.
Correct answer: B. -0.101
Explanation: -0.11 can be written as -0.110. On comparing, -0.101 is closer to zero than -0.110, so -0.101 is greater. The numbers are not equal because their decimal digits differ. Exam tip: Among negative numbers, the number closer to zero is greater.
Correct answer: A. \( -4 \)
Explanation: The midpoint of two numbers on a number line is their average: \(\frac{4+(-12)}{2}=\frac{-8}{2}=-4\). Therefore, \( -4 \) is correct. \( -8 \) is the sum of the two numbers, not their midpoint. Exam tip: add the two numbers and divide by 2 to find the midpoint.
Correct answer: B. 80
Explanation: The distance between two numbers on a number line is the absolute value of their difference: \(|20-(-60)|=|80|=80\). Therefore, the correct answer is 80. A value such as 79 can result from handling the negative sign incorrectly. Exam tip: always take the absolute value of the difference when finding distance.
Correct answer: B. 1 और 2 के बीच, 2 के अधिक निकट
Explanation: Since \(1^2=1\) and \(2^2=4\), \(\sqrt{2}\) lies between 1 and 2. Also, \(1.4^2=1.96\) and \(1.5^2=2.25\), so \(\sqrt{2}\approx1.414\), which is closer to 1 than to 2. Therefore the stated option should identify it as nearer 1; check midpoint \(1.5\) in exams.
Correct answer: B. ( -0.03 )
Explanation: -0.03 can be written as -0.0300. On comparison, -0.0300 is greater than -0.0302 because, among negative numbers, the number closer to zero is greater. Therefore, ( -0.03 ) is the correct answer. Exam tip: Add trailing zeros when needed to make the number of decimal places equal before comparing decimals.
Correct answer: A. Every real number has a unique point on the number line.
Explanation: Real numbers include both rational and irrational numbers; for example, \(\sqrt{2}\) has a definite point on the number line. A single point cannot represent two real numbers. Exam tip: remember the one-to-one correspondence between real numbers and points.
Correct answer: B. -0.07
Explanation: For negative numbers, the number closer to zero is greater.
\(-0.07\) is closer to zero than \(-0.7\), so it lies to the right of \(-0.7\) on the number line. Hence, \(-0.07\) is greater. Writing \(-0.7\) as \(-0.70\) makes the comparison clear: \(-0.07>-0.70\). Exam tip: While comparing negative decimals, first write them with the same number of decimal places.
