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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: B. \(-1.11\)
Explanation: \(-1.11\) can be written as \(-1.110\) up to three decimal places. Comparing \(-1.110\) and \(-1.101\), \(-1.110\) is more negative, so it lies further left on the number line and is smaller. \(-1.101\) is closer to zero and is therefore greater. Exam tip: for negative decimals, the number with the greater magnitude is the smaller number.
Correct answer: B. 12
Explanation: The distance between two numbers on a number line is their absolute difference: \(\left|-13-(-1)\right|=\left|-12\right|=12\). Therefore, the correct answer is 12. Choosing 11 misses one unit. Exam tip: a distance is always non-negative.
Correct answer: C. Both are equal
Explanation: The correct answer is that both are equal. A zero added at the end of a decimal does not change its value, so \(-0.70=-0.7\). Therefore, neither number is greater; they represent the same point on the number line. Exam tip: Adding or removing trailing zeros in a decimal does not change its value.
Correct answer: A. 3.5
Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-3+10}{2}=\frac{7}{2}=3.5\). Therefore, 3.5 is correct. Although 4 is close, it is not at an equal distance from both numbers. Exam tip: add the two numbers and divide by 2 to find their midpoint.
Correct answer: C. 8
Explanation: On a number line, the distance between two numbers is the positive value of their difference. Thus, \(|-16-(-8)|=|-8|=8\). Therefore, 8 is the correct answer. Choosing 7 would result from an incorrect subtraction. Exam tip: a distance is always non-negative.
Correct answer: A. ( -2.01 )
Explanation: Among negative numbers, the number with the greater magnitude is smaller. Here, \(2.01=2.010\), and \(2.010>2.001\). Therefore, \(-2.010<-2.001\), so \((-2.01)\) is smaller. \((-2.001)\) is closer to zero and is therefore greater. Exam tip: Add trailing zeros when needed to compare decimal places.
Correct answer: A. 5
Explanation: The midpoint of two numbers on a number line is their average:
\(\frac{-5+15}{2}=\frac{10}{2}=5\). Therefore, 5 is correct. The value 10 is the distance between the numbers, not their midpoint. Exam tip: add the two numbers first, then divide by 2.
Correct answer: A. 15
Explanation: The distance between two numbers on a number line is the absolute value of their difference. Thus, \(|6-(-9)|=|15|=15\). Therefore, 15 is correct. A result such as 14 may come from handling the negative sign incorrectly. Exam tip: always take the absolute value of the difference, because distance cannot be negative.
Correct answer: A. Infinitely many irrational numbers lie between any two distinct rational numbers.
Explanation: If \(a<b\) are rational, then \(a+(b-a)\sqrt2/n\), for \(n\ge2\), is an irrational number between them. Different values of \(n\) give infinitely many such numbers. Exam tip: both rational and irrational numbers are dense on the number line.
Correct answer: A. \(OB^2=OA^2+AB^2=2^2+1^2=5\), so \(OB=\sqrt{5}\)
Explanation: Triangle \(OAB\) is right-angled. By Pythagoras, \(OB^2=2^2+1^2=5\), hence \(OB=\sqrt{5}\). Transferring this radius from \(O\) to the number line marks the required point. Exam tip: verify the sum of squares, not the sum of sides.
Correct answer: B. \(-7\)
Explanation: For negative numbers, the number closer to zero is greater. \(-7\) is closer to zero than \(-7.7\), so \(-7 > -7.7\). Choosing \(-7.7\) is incorrect because it lies to the left of \(-7\) on the number line. Exam tip: on a number line, the number farther to the right is always greater.
Correct answer: A. \(-4\)
Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-12+4}{2}=\frac{-8}{2}=-4\). Hence, \(-4\) is correct. \(-8\) is the sum of the two numbers, not their midpoint. Exam tip: Add the two numbers first and then divide by 2 to find the midpoint.
Correct answer: C. 16
Explanation: On a number line, the distance between two numbers is the positive value of their difference. Thus, \(\lvert -18-(-2)\rvert=\lvert -16\rvert=16\). Therefore, 16 is correct. Choosing 18 results from not subtracting the negative number correctly. Exam tip: always take the absolute value of the difference for distance.
Correct answer: A. There are infinitely many rational and infinitely many irrational numbers
Explanation: For \(a<b\), both rational and irrational numbers are dense between them, so each type occurs infinitely often. \((a+b)/2\) gives only one midpoint, not the full set. Exam tip: density means that every interval contains such numbers.
Correct answer: B. ( -4 )
Explanation: On a number line, the number farther to the right is greater.
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-4 is closer to zero than -4.4 and lies to its right on the number line. Therefore, ( -4 ) is greater. “Both are equal” is incorrect because -4.0 and -4.4 are different numbers. Exam tip: Among negative numbers, the number with the smaller absolute value is greater.
Correct answer: B. 12
Explanation: On a number line, the distance between two numbers is the absolute value of their difference: \(|2-(-10)|=|12|=12\). Therefore, the correct answer is 12.
-12 may arise as a signed difference, but distance can never be negative. Exam tip: always use the absolute value of the difference when finding distance.
Correct answer: A. \(-\sqrt{3}\)
Explanation: Since \(1<\sqrt{3}<2\), multiplying by \(-1\) reverses the inequalities: \(-2<-\sqrt{3}<-1\). In contrast, \(-\sqrt{5}\) is less than \(-2\). Exam tip: use nearby perfect squares to locate square roots.
Correct answer: A. 4
Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-6+14}{2}=\frac{8}{2}=4\). Therefore, 4 is the correct answer. For example, 5 is not equally distant from -6 and 14. Exam tip: add the two numbers and divide the result by 2 to find the midpoint.
Correct answer: B. 6
Explanation: The distance between two numbers on a number line is their absolute difference: \(|-11-(-5)|=|-6|=6\). Hence, the correct answer is 6. Getting 5 would mean the signs of the negative numbers have not been handled correctly. Exam tip: always take the absolute value of the difference, since distance cannot be negative.
Correct answer: B. -0.11
Explanation: Write the decimals with the same number of places: \(-0.11=-0.110\). Comparing \(-0.110\) and \(-0.101\), \(-0.110\) is more negative, so it lies farther left on the number line and is smaller. Therefore, \(-0.11\) is correct. \(-0.101\) is closer to zero, so it is greater. Exam tip: When comparing negative decimals, add trailing zeros if needed and then compare place values.
Correct answer: A. 0
Explanation: The midpoint of two numbers is found using \(\frac{a+b}{2}\). Here, \(\frac{-8+8}{2}=\frac{0}{2}=0\). Therefore, 0 is at an equal distance from -8 and 8. Option 4 is incorrect because it is not the average of -8 and 8. Exam tip: The midpoint of opposite numbers \(a\) and \(-a\) is always 0.
Correct answer: B. 21
Explanation: The distance between two numbers on a number line is the absolute value of their difference: \(|7-(-14)|=|21|=21\). Therefore, the correct answer is 21. Choosing 20 usually results from missing the sign change while subtracting a negative number. Exam tip: always use \(|a-b|\) to find distance.
Correct answer: C. Both are equal
Explanation: Writing -0.2 as -0.20 only adds a zero to the right of the decimal part; it does not change the value of the number. Hence, -0.2 = -0.20, so both are equal. Options A and B are not correct because neither number is greater than the other. Exam tip: Zeros added at the end of a decimal do not change its value.
Correct answer: B. \(-0.00005\)
Explanation: For negative numbers, the number closer to zero is greater. \(-0.00005\) is closer to zero than \(-0.0005\); therefore, \(-0.00005 > -0.0005\). They are not equal because their decimal values are different. Exam tip: On a number line, the number farther to the right is greater.
Correct answer: A. -2
Explanation: The midpoint of two numbers is their average: \(\frac{-13+9}{2}=\frac{-4}{2}=-2\). Hence, \(-2\) is correct. \(-1\) can result from an error while adding the signed numbers. Exam tip: add the numbers with their signs first, then divide by 2.
