Real Numbers

Correct answer: A. Rational real number

Explanation: ( \frac{22}{7} ) is a ratio of two integers with non-zero denominator. So it is a rational real number.

Correct answer: A. ( \frac{1}{2} ), ( \sqrt{3} ), ( -5 )

Explanation: ( \frac{1}{2} ), ( \sqrt{3} ), and ( -5 ) are all real numbers. A fraction with zero denominator is not real.

Correct answer: B. −2

Explanation: The governing concept is the principal square root and the effect of an outside negative sign. The symbol √4 denotes the non-negative principal square root of 4, so √4 = 2. The minus sign is written before the radical and therefore changes the result after the square root is evaluated: −√4 = −(2) = −2. Thus option B is correct. Option A would be correct for √4 without the outside negative sign. Option C is the radicand, not its square root, while option D incorrectly treats the square root of 4 as 4 and also applies a negative sign. It is important not to confuse −√4 with √(−4), which is not a real number.

Correct answer: C. 7

Explanation: Absolute value represents a number’s distance from zero, so it is never negative. The distance of −7 from zero is 7; therefore, \(|-7|=7\). Option A is incorrect because it gives the signed value, not the absolute value. Exam tip: To find the absolute value of a negative number, remove its minus sign.

Correct answer: D. Infinitely many

Explanation: There are infinitely many real numbers between any two distinct real numbers. This is an important property of the number line.

Correct answer: C. \(\sqrt{36}<\sqrt{64}\)

Explanation: \(\sqrt{36}=6\) and \(\sqrt{64}=8\), so \(6<8\), which gives \(\sqrt{36}<\sqrt{64}\). Therefore, option C is correct. Option A reverses the inequality, while option B incorrectly treats the two values as equal. Option D is also incorrect because 6 and 8 are both real and rational numbers. Exam tip: For perfect squares, calculate their square roots first and then compare the results.

Correct answer: C. 0

Explanation: The governing concept is the identity property of addition. An additive identity is a number that leaves every number unchanged when it is added to that number. For any real number x, the defining relation is x + 0 = 0 + x = x. Therefore, zero is the additive identity in the set of real numbers, and option C is correct. Option A is not an additive identity because x + 1 is generally greater than x. Similarly, adding −1 changes most numbers, so option B cannot be correct. Adding 2 also changes the original number. Notice that 1 is the multiplicative identity because x × 1 = x; this distinction helps avoid confusing the identities for addition and multiplication.

Correct answer: B. (1)

Explanation: Multiplying any real number by (1) gives the same number. Hence (1) is the multiplicative identity.

Correct answer: D. 9

Explanation: The additive inverse of a number is the number that gives a sum of 0 when added to the original number. Here, \((-9)+9=0\), so the additive inverse of \((-9)\) is 9. \(\frac{1}{9}\) is its multiplicative inverse, not its additive inverse. Exam tip: to find an additive inverse, change the sign of the number.

Correct answer: A. ( \frac{7}{4} )

Explanation: The multiplicative inverse of a non-zero number is the number that gives product 1 when multiplied by the original number. For a non-zero fraction \\(\frac{a}{b}\\), the inverse is \\(\frac{b}{a}\\), because \\(\frac{a}{b}\times\frac{b}{a}=1\\). Here the numerator 4 and denominator 7 are interchanged, so the inverse of \\(\frac{4}{7}\\) is \\(\frac{7}{4}\\). Therefore, option A is correct.

We can verify the result directly: \\(\frac{4}{7}\times\frac{7}{4}=\frac{28}{28}=1\\). The negative choices are not correct because multiplying \\(\frac{4}{7}\\) by a negative version gives a negative product, not 1. The number \\(\frac{11}{7}\\) also does not produce 1. The fraction is non-zero, so its reciprocal exists.

Correct answer: A. 0

Explanation: Zero is neither positive nor negative. Positive numbers are greater than 0, such as 1 and \(\frac{1}{2}\), whereas negative numbers are less than 0, such as -1. Therefore, 0 is the correct answer. Exam tip: On the number line, numbers to the right of 0 are positive and those to the left are negative.

Correct answer: B. 11

Explanation: \(\sqrt{121}\) is the positive number whose square is 121. Since \(11^2=121\), we get \(\sqrt{121}=11\). Option 121 is the radicand, not its square root. Exam tip: for a positive number \(a\), \(\sqrt{a^2}=a\).

Correct answer: B. 2√3

Explanation: The governing concept is extracting a perfect-square factor from a radical. Factor 12 as 4 × 3, where 4 is a perfect square. Using √(ab) = √a × √b for non-negative factors, √12 = √(4 × 3) = √4 × √3 = 2√3. Therefore option B is correct. Option A, 6√2, would square to 72 and is too large. Option C, 3√2, would square to 18, not 12. Option D, 4√3, would square to 48. The expression 2√3 cannot be reduced further because 3 has no factor greater than 1 that is a perfect square. This method is preferable to using a decimal approximation because it gives the exact simplified radical.

Correct answer: B. ( \sqrt{6} )

Explanation: For positive numbers, the square root of the greater number is also greater. Since (6>5), ( \sqrt{6}>\sqrt{5} ).

Correct answer: B. Irrational real number

Explanation: Adding the irrational ( \sqrt{2} ) to the rational number (3) generally gives an irrational number. It is also real.

Correct answer: A. A real number

Explanation: Each point on the real number line represents a real number. Understanding the number line helps in comparison.

Correct answer: B. ( \frac{6}{0} )

Explanation: Direct answer: Option B, \(\frac{6}{0}\). Division by zero is undefined, so \(\frac{6}{0}\) is not a real number. A real number must have a definite value on the number line. There is no number \(x\) satisfying \(0\times x=6\), because multiplying zero by any number gives zero. Therefore the expression cannot be assigned an ordinary real value. Option A, -8, is an integer and therefore a real number. Option B is correct because its denominator is zero, making the fraction undefined. Option C, \(\sqrt{19}\), is irrational but still real: 19 is positive, so its square root exists on the real number line. Option D, 0.45, is a terminating decimal and equals \(\frac{45}{100}=\frac9{20}\), so it is rational and real. Do not confuse an irrational number with a non-real number. Irrational numbers are still real. Remember: denominator zero means undefined, not zero and not infinity as an ordinary real answer.

Correct answer: B. 9

Explanation: Since \(81=9^2\), \(\sqrt{81}=9\). The principal square root is the positive number whose square is 81. The squares of 8 and 18 are 64 and 324, respectively, so they are incorrect; 81 itself is the radicand, not its square root. Exam tip: Memorising common perfect squares helps solve such questions quickly.

Correct answer: B. Their decimal expansion is non-terminating and non-repeating.

Explanation: An irrational number has a decimal expansion that is non-terminating and non-repeating, so option B is correct. Option C describes rational numbers, which can be written as the ratio of two integers. Option A applies to rational numbers with terminating decimal expansions. Exam tip: terminating or non-terminating repeating decimals are rational, whereas non-terminating non-repeating decimals are irrational.

Correct answer: B. Rational number

Explanation: The direct answer is B, rational number. A rational number is any number that can be written as \\(p/q\\), where \\(p\\) and \\(q\\) are integers and \\(q\\neq0\\). The decimal 0.125 has three digits after the decimal point, so write it as \\(125/1000\\). Simplifying by dividing numerator and denominator by 125 gives \\(1/8\\). Thus it can be expressed as a ratio of integers and is rational. Option A, irrational number, is wrong because an irrational decimal does not terminate and cannot be written as such a fraction; 0.125 terminates. Option B is correct. Option C, undefined number, is wrong because 0.125 has a clear finite value. Option D, only integer, is wrong because integers have no fractional part, while \\(0.125=1/8\\) is not an integer. The exam cue is that every terminating decimal is rational; repeating decimals are rational too. The supplied answer is correct.

Correct answer: B. ( \frac{-11}{6} )

Explanation: ( \frac{-11}{6} ) is a ratio of two integers with non-zero denominator. So it is a rational real number.

Correct answer: C. \(4\sqrt{2}\)

Explanation: Since \(32=16\times2\) and \(16\) is a perfect square, \(\sqrt{32}=\sqrt{16\times2}=\sqrt{16}\times\sqrt{2}=4\sqrt{2}\). Therefore, option C is correct. Exam tip: To simplify a surd, factor the number using its largest perfect-square factor.

Correct answer: A. \(3\sqrt{5}\)

Explanation: \(45=9\times5=3^2\times5\). Therefore, \(\sqrt{45}=\sqrt{3^2\times5}=3\sqrt{5}\), so option A is correct. Option B results from incorrectly treating 45 as \(5^2\times3\), but \(5^2\times3=75\). Exam tip: factor the number into the largest perfect square multiplied by the remaining factor, then take the square root of the perfect square outside the radical.

Correct answer: B. \(5\sqrt{3}\)

Explanation: Since \(75=25\times3\) and \(25\) is a perfect square, \(\sqrt{75}=\sqrt{25\times3}=\sqrt{25}\times\sqrt{3}=5\sqrt{3}\). Option C, \(3\sqrt{5}\), results from separating the factors incorrectly. Exam tip: take the largest perfect-square factor outside the square root.

Correct answer: A. 6

Explanation: The governing concept is the product property of square roots for non-negative numbers: √a × √b = √(ab). Applying it gives √2 × √18 = √(2 × 18) = √36. The principal square root of 36 is 6 because 6² = 36 and the principal root is non-negative. Therefore option A is correct. Option B, √20, results from an incorrect operation and is not equal to the product. Option C simply rewrites part of the expression without using the correct product property, while option D is incorrect because 9² = 81, not 36. This example also illustrates that multiplying two irrational square roots can produce a rational integer.