Muft Shiksha™ एक 100% Free Education Portal है 🇮🇳, जिसका उद्देश्य Class 9–12 के हर विद्यार्थी तक High-Quality Education को पूरी तरह मुफ्त पहुँचाना है। 🇮🇳 हम मानते हैं कि अच्छी शिक्षा किसी student की आर्थिक स्थिति पर निर्भर नहीं होनी चाहिए। 🇮🇳 हर विद्यार्थी को वही Quality Study Material, MCQs, Quizzes, Exam Preparation, Concept-Based Learning और Bilingual Support मिलना चाहिए, जो आमतौर पर महंगी Coaching या Premium Platforms में मिलता है। Muft Shiksha™ 🇮🇳 इसी सोच के साथ बनाया गया है
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. There are infinitely many rational and infinitely many irrational numbers between them
Explanation: The real number line is dense. If \(x<y\), there are infinitely many rational as well as infinitely many irrational numbers between \(x\) and \(y\). Exam tip: the phrase “between two real numbers” often tests density.
Correct answer: A. \(\frac{5}{2}\)
Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-3+8}{2}=\frac{5}{2}\). Hence, \(\frac{5}{2}\) is correct. The option 3 is not correct because the sum of the two numbers is 5, not 6. Exam tip: add the two endpoints first and then divide by 2.
Correct answer: A. 12
Explanation: On a number line, the distance between two numbers is the absolute value of their difference. Thus, \(\lvert -18-(-6)\rvert=\lvert -12\rvert=12\). Therefore, 12 is correct. A value such as 13 may result from an error while handling negative signs. Exam tip: distance is always non-negative.
Correct answer: A. \(\sqrt{2}\)
Explanation: The distance from the origin is \(|x|\). Here \(|x|=\sqrt{2}\), and the point lies in the positive direction, so \(x=\sqrt{2}\). \(-\sqrt{2}\) is equally far from the origin but lies to the left. Exam tip: use the direction to choose the sign.
Correct answer: A. Between any two rational numbers, there is at least one irrational number.
Explanation: For rational \(r<s\), \(r+\frac{s-r}{\sqrt2}\) is irrational and lies between them, so A is correct and B is false. Exam tip: rational and irrational numbers are both dense on the number line.
Correct answer: A. \(5.5\)
Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-4+15}{2}=\frac{11}{2}=5.5\). Therefore, \(5.5\) is correct. \(4.5\) is not half of the sum, so it cannot be the midpoint. Exam tip: while finding a midpoint, add the numbers with their signs before dividing by 2.
Correct answer: A. 12
Explanation: On a number line, the distance between two numbers is the absolute value of their difference. Thus, \(\lvert -9-(-21)\rvert=\lvert 12\rvert=12\). Therefore, the correct answer is 12. Choosing 11 may result from incorrectly counting the integers between the numbers rather than finding their distance. Exam tip: always take the absolute value of the difference when finding distance.
Correct answer: B. -0.009
Explanation: For negative numbers, the number closer to zero is greater. Here, -0.009 = -9/1000 and -0.01 = -10/1000; therefore, -9/1000 > -10/1000. Hence, -0.009 is greater. “Both are equal” is incorrect because the decimal values are different. Exam tip: Write negative decimals to the same number of decimal places before comparing them.
Correct answer: A. \(a+b=0\)
Explanation: The midpoint coordinate is \(\frac{a+b}{2}\). For it to be the origin, \(\frac{a+b}{2}=0\), so \(a+b=0\). In contrast, \(a-b=0\) gives the same coordinate for both points. Exam tip: use the average of coordinates for a midpoint.
Correct answer: B. P is \(\sqrt{2}\) and is irrational; irrational numbers are also represented on the number line
Explanation: Since \(P^2=2\) and \(P>0\), \(P=\sqrt{2}\). Its decimal expansion \(1.414\ldots\) is non-terminating and non-repeating, so it is irrational. Every real number has a point on the number line. Exam tip: being between two integers does not make a number rational.
Correct answer: A. The hypotenuse has length \(\sqrt{13}\), not \(\sqrt{10}\).
Explanation: By Pythagoras’ theorem, hypotenuse² = \(3^2+2^2=13\), so the length obtained is \(\sqrt{13}\). For \(\sqrt{10}\), use perpendicular sides 3 and 1. Exam tip: always add the squares of the two legs first.
Correct answer: A. 2
Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-6+10}{2}=\frac{4}{2}=2\). Therefore, the correct answer is 2. Option 3 is not the average of -6 and 10. Exam tip: Add the two numbers and divide by 2 to find their midpoint.
Correct answer: B. -0.405
Explanation: The correct answer is -0.405. Write -0.45 as -0.450 for comparison. Since -0.405 is closer to zero than -0.450, -0.405 is greater. -0.45 is smaller because among negative numbers, the number with the greater magnitude is smaller. Exam tip: Write negative decimals to the same number of decimal places before comparing them.
Correct answer: A. कम-से-कम एक परिमेय संख्या
Explanation: Between any two distinct real numbers, there is at least one rational number; in fact, infinitely many exist. An integer is not necessary: there is no integer between 0.2 and 0.3. Exam tip: the average of two numbers often gives a number between them.
Correct answer: A. Between any two distinct rational numbers, there is at least one irrational number.
Explanation: For rational numbers \(a<b\), \(a+\frac{b-a}{\sqrt2}\) is an irrational number between them since \(0<\frac1{\sqrt2}<1\). Exam tip: every interval on the number line contains both rational and irrational numbers.
Correct answer: A. -1.5
Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-10+7}{2}=\frac{-3}{2}=-1.5\). Therefore, \(-1.5\) is correct. Although \(-2\) is close, it is not equally distant from both numbers. Exam tip: add the two numbers and divide by 2 to find their midpoint.
Correct answer: A. 16
Explanation: The distance between two numbers on a number line is the absolute value of their difference. Thus, \(\lvert -20-(-4)\rvert=\lvert -16\rvert=16\). Therefore, 16 is correct. A result of 15 can arise from an arithmetic error. Exam tip: always express distance as a positive value.
Correct answer: B. ( -0.00009 )
Explanation: ( -0.00009 ) is closer to zero, whereas ( -0.0009 ) is a negative number farther from zero. On the number line, the negative number farther to the right is greater; therefore, ( -0.00009 ) is greater. Choosing ( -0.0009 ) is a common mistake because, among negative numbers, the one with the larger magnitude is smaller. Exam tip: When comparing negative decimals, first identify which number is closer to zero.
Correct answer: B. ( -3.6 )
Explanation: We can write -3.6 as -3.60. Since -3.60 is more negative than -3.06, it lies farther to the left on the number line and is therefore smaller. The close distractor, -3.06, has a smaller negative magnitude. Exam tip: while comparing negative decimals, first write them with the same number of decimal places.
Correct answer: A. There is at least one rational and one irrational number
Explanation: Every interval between two distinct real numbers contains infinitely many rational as well as irrational numbers. Thus, the “only rational” and “only irrational” statements are false. Exam tip: pay close attention to the word “always”.
Correct answer: A. P lies between 3 and 4 because the hypotenuse has length \(\sqrt{3^2+2^2}=\sqrt{13}\).
Explanation: By the Pythagorean theorem, the hypotenuse is \(\sqrt{3^2+2^2}=\sqrt{13}\). Since \(9<13<16\), we get \(3<\sqrt{13}<4\). Adding 3 and 2 is incorrect. Exam tip: bracket the number between perfect squares to locate its square root.
Correct answer: A. 20
Explanation: On a number line, the distance between two numbers is the absolute value of their difference: \(\lvert -30-(-10)\rvert=\lvert -20\rvert=20\). Therefore, the correct answer is 20. \(-20\) can be a signed difference, but distance can never be negative. Exam tip: always take the absolute value of the difference when finding distance.
Correct answer: A. It lies between 2 and 3 and is irrational
Explanation: Since \(2^2<5<3^2\), we get \(2<\sqrt{5}<3\). As 5 is not a perfect square, \(\sqrt{5}\) is irrational; it is not 2.5. Exam tip: compare with nearby perfect squares first.
Correct answer: A. 24
Explanation: The distance between two numbers on a number line is the absolute value of their difference: \(\lvert -40-(-16)\rvert=\lvert -24\rvert=24\). Therefore, the correct answer is 24. \(-24\) can be the signed difference, but a distance can never be negative. Exam tip: always take the absolute value when finding distance.
Correct answer: B. ( -0.333 )
Explanation: On a number line, the number farther to the left is smaller. Here, \(-0.3=-0.300\), and \(-0.333\) is more negative; therefore, \(-0.333 < -0.300\). Hence, \((-0.333)\) is the correct answer. Although \(-0.3\) is close, it is nearer to zero and is therefore greater. Exam tip: While comparing negative decimals, write equal decimal places; the number with the greater magnitude is smaller.
