Number line

Correct answer: B. ( -0.00008 )

Explanation: For negative numbers, the number closer to zero is greater. The magnitude of ( -0.00008 ) is 0.00008, which is smaller than 0.0008; therefore, ( -0.00008 ) is closer to zero and is greater. ( -0.0008 ) lies further to the left on the number line. Exam tip: When comparing negative decimals, the number with the smaller magnitude is greater.

Correct answer: B. \(\sqrt{10}\)

Explanation: In right triangle OAB, OA = 3 and AB = 1. Thus, OB² = 3² + 1² = 10, so OB = \(\sqrt{10}\). The arc places P at a distance \(\sqrt{10}\) from O. In such constructions, apply Pythagoras’ theorem first.

Correct answer: B. \(-8.5\)

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-35+18}{2}=\frac{-17}{2}=-8.5\). Therefore, \(-8.5\) is correct. Although \(-8\) is close, it is not at an equal distance from both numbers. Exam tip: add the two numbers first, then divide the sum by 2.

Correct answer: B. 12

Explanation: The distance between two numbers on a number line is the absolute value of their difference: \(\lvert -19-(-7)\rvert=\lvert -12\rvert=12\). Therefore, 12 is correct. Choosing 11 can result from a counting error, but distance is always the positive value of the difference. Exam tip: Use brackets carefully when subtracting a negative number.

Correct answer: B. -2.125

Explanation: Write -2.12 as -2.120. Comparing -2.125 and -2.120, -2.125 is more negative, so it lies further left on the number line and is smaller. “Both are equal” is incorrect because their decimal parts are different. Exam tip: among negative decimals, the number with the greater magnitude is the smaller number.

Correct answer: A. Infinitely many rational and infinitely many irrational numbers lie between them

Explanation: Between any two distinct real numbers, there are infinitely many rational as well as infinitely many irrational numbers. Hence A is correct; choices claiming a finite number are false. Exam tip: remember this as the density property of rational and irrational numbers.

Correct answer: A. Between \(-4\) and \(-3\)

Explanation: Since \(9<10<16\), \(3<\sqrt{10}<4\). Negating reverses order, so \(-4<-\sqrt{10}<-3\); A is correct. The next interval is too far right. Tip: reverse inequalities when negating.

Correct answer: A. \(-4\)

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-13+5}{2}=\frac{-8}{2}=-4\). Therefore, \(-4\) is correct. \(-3.5\) is not at an equal distance from both numbers. Exam tip: add the two numbers first, then divide the sum by 2.

Correct answer: C. It can be represented by a unique point on the number line.

Explanation: Every real number, whether rational or irrational, corresponds to exactly one unique point on the number line. For example, \(\sqrt{2}\) has a fixed position. Option A is wrong because irrational numbers are real. Exam tip: all real numbers can be located on the number line.

Correct answer: A. 18

Explanation: The distance between two numbers on a number line is the absolute value of their difference: \(\lvert -27-(-9)\rvert=\lvert -18\rvert=18\). Therefore, the correct answer is 18. \(-18\) can be the difference, but a distance can never be negative. Exam tip: always take the absolute value of the difference when finding distance.

Correct answer: A. There are infinitely many rational and infinitely many irrational numbers between them.

Explanation: Option A is correct. \(\frac{a+b}{2}\) is rational and lies between \(a\) and \(b\); repeated midpoints give infinitely many rationals. Every open interval also contains infinitely many irrationals. Exam tip: both sets are dense.

Correct answer: A. ( -4.705 )

Explanation: The correct answer is ( -4.705 ). Among negative numbers, the number closer to zero is greater. Since \( -4.705 > -4.750 \), ( -4.705 ) is greater. Writing ( -4.75 ) as ( -4.750 ) makes the comparison easier. Exam tip: While comparing negative decimals, first make the number of decimal places equal.

Correct answer: A. \(1.5\)

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-6+9}{2}=\frac{3}{2}=1.5\). Therefore, \(1.5\) is correct. \(3\) is the sum of the two numbers, not the midpoint. Exam tip: add the two numbers and divide by 2 to find their midpoint.

Correct answer: B. \(-40\)

Explanation: The successor of an integer is 1 greater than that integer. Hence, \(-41+1=-40\), so \(-40\) is correct. \(-42\) is 1 less than \(-41\), so it is the predecessor, not the successor. Exam tip: To find a successor, add 1 even when the number is negative.

Correct answer: A. 31

Explanation: The predecessor of a whole number is exactly 1 less than that number. Therefore, \(32-1=31\), so 31 is the predecessor of 32. In contrast, 33 is the successor of 32, while 30 is 2 less than 32. Exam tip: subtract 1 to find a predecessor.

Correct answer: A. An irrational number lying between 2 and 3

Explanation: Since \(2^2=4<7<9=3^2\), \(\sqrt{7}\) lies between 2 and 3. As 7 is not a perfect square, its square root is irrational. Option C has the correct interval but the wrong type. Exam tip: use adjacent perfect squares to locate roots.

Correct answer: B. \(2.6 < x < 2.7\)

Explanation: \(2.6^2=6.76\) and \(2.7^2=7.29\). Since \(6.76<7<7.29\), \(x=\sqrt{7}\) lies between \(2.6\) and \(2.7\). In exams, compare nearby squares to locate an irrational number quickly.

Correct answer: C. Infinitely many rational numbers lie between them.

Explanation: For distinct rational numbers a and b, their midpoint \(\frac{a+b}{2}\) is also rational and lies between them. Repeating this process gives infinitely many rational numbers. Exam tip: use the midpoint method whenever numbers between two rationals are asked.

Correct answer: A. ( -0.0002 )

Explanation: For negative numbers, the number farther to the left on the number line is smaller. Since -0.0002 is more negative than -0.00012, ( -0.0002 ) is the smaller number. They are not equal because their decimal values differ. Exam tip: when comparing negative decimals, the number with the greater magnitude is smaller.

Correct answer: A. There will be at least one rational and one irrational number between P and Q.

Explanation: Infinitely many rational and irrational numbers lie between any two distinct real numbers, so A is correct. B and D ignore this density property. Exam tip: every open interval contains both types of numbers.

Correct answer: B. Reema’s conclusion is wrong because \(2.7^2=7.29>7\); therefore, \(\sqrt{7}<2.7\).

Explanation: Reema squared 2.7 incorrectly. Since \(2.7^2=7.29\), which is greater than 7, the positive square root \(\sqrt{7}\) must be less than 2.7. Exam tip: compare squares of nearby positive decimals to locate a square root.

Correct answer: A. The distance is \(2\sqrt{3}\), because the points are equally far from zero on opposite sides.

Explanation: The distance is \(|\sqrt{3}-(-\sqrt{3})|=|2\sqrt{3}|=2\sqrt{3}\). Equal magnitudes do not mean the same point; these points lie on opposite sides of zero. Exam tip: always use the absolute value of the difference for distance.

Correct answer: A. 95

Explanation: The distance between two numbers on a number line is their absolute difference: \(|25-(-70)|=|25+70|=95\). Therefore, 95 is correct. Choosing 94 results from an addition error of 1. Exam tip: when subtracting a negative number, change it to addition, and keep distance non-negative.

Correct answer: B. -3.141

Explanation: The correct answer is -3.141. Among negative numbers, the number with the greater magnitude is smaller. Since -3.141 is more negative than -3.140, we have -3.141 < -3.14. Writing -3.14 as -3.140 makes the comparison clearer. Exam tip: Add zeros at the end of decimals when needed to compare the same number of decimal places.

Correct answer: A. \( -6 \)

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-18+6}{2}=\frac{-12}{2}=-6\). Therefore, \( -6 \) is correct. Although \(0\) lies between the two numbers, it is not equally distant from both. Exam tip: add the two numbers and divide by 2 to find their midpoint.