Number line

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.