Real Numbers

Correct answer: B. (5) and (6)

Explanation: (5^2=25) and (6^2=36), so ( \sqrt{26} ) lies between (5) and (6). Compare nearby perfect squares.

Correct answer: B. ( -4.4 )

Explanation: ( \sqrt{20}\approx4.47 ), so ( -\sqrt{20}\approx-4.47 ). ( -4.4 ) is closer to zero, so it is greater.

Correct answer: C. 5/0 is a real number

Explanation: The governing concepts are the classification of real numbers and the fact that division by zero is undefined. Every rational and every irrational number belongs to the real-number system. Thus √2 is real even though it is irrational. The repeating decimal 0.333… equals 1/3, so it is rational and real. The integer −7 can be written as −7/1, making it rational and therefore real. However, 5/0 is not defined: if 5/0 were a number x, then multiplying by zero would require 0×x = 5, which is impossible because 0×x is always 0. Hence 5/0 is not a real number, and option C is the incorrect statement.

Correct answer: A. The diagonal is rational and the perimeter is irrational

Explanation: The diagonal of a square is side × \(\sqrt{2}\), so it is \(\sqrt{2}\times\sqrt{2}=2\) cm, which is rational. The perimeter is \(4\times\sqrt{2}=4\sqrt{2}\) cm, which is irrational because the product of a non-zero rational number and an irrational number is irrational. Therefore, option A is correct. Exam tip: Do not classify an expression merely by looking at an irrational term; simplify it first.

Correct answer: A. It is terminating

Explanation: A rational number has a terminating decimal expansion when, after simplification, the denominator has no prime factors other than 2 and 5. This happens because powers of 2 and 5 can combine to make a power of 10. For \\(\frac{13}{40}\\), the numerator and denominator have no common factor, and \\(40=2^3\times5\\).

Since the denominator contains only the allowed prime factors 2 and 5, the decimal terminates. In fact, \\(\frac{13}{40}=\frac{13\times25}{40\times25}=\frac{325}{1000}=0.325\\). Therefore, option A is correct. It is not irrational or undefined, and it is not non-terminating non-repeating; those descriptions do not fit this rational fraction.

Correct answer: B. The statement is incorrect; \(\sqrt{18}=3\sqrt{2}\), and \(\sqrt{2}\) is irrational.

Explanation: Factoring 18 gives \(18=9\times2\), so \(\sqrt{18}=3\sqrt{2}\). Since \(\sqrt{2}\) is irrational, multiplying it by the non-zero rational number 3 still gives an irrational number. Therefore, \(\sqrt{18}\) is irrational. Option A is wrong because the square root of every integer is not rational, and 18 is not a perfect square. Exam tip: Before classifying \(\sqrt{n}\), check whether \(n\) is a perfect square.

Correct answer: A. Rational real number

Explanation: \(\sqrt{5}\times\sqrt{20}=\sqrt{5\times20}=\sqrt{100}=10\). The number 10 is an integer and a natural number, so it is also a rational real number. Therefore, option A is correct. It is not irrational because the product simplifies to a whole number. Exam tip: For positive radicands, use \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\), then check whether the resulting square root is a perfect square.

Correct answer: B. ( -9 )

Explanation: The direct answer is B, \\(-9\\). The two expressions are conjugates: \\(2+\\sqrt{13}\\) and \\(2-\\sqrt{13}\\). Use the difference-of-squares identity \\((a+b)(a-b)=a^2-b^2\\). Here, \\(a=2\\) and \\(b=\\sqrt{13}\\). Therefore, \\((2+\\sqrt{13})(2-\\sqrt{13})=2^2-(\\sqrt{13})^2=4-13=-9\\). Option A, 9, has the wrong sign because 4 is smaller than 13. Option B, -9, is correct. Option C, 17, would result from adding 4 and 13, but multiplication of conjugates requires subtraction. Option D, \\(4+\\sqrt{13}\\), is not the result of multiplying the two binomials and leaves an unnecessary radical. You can also check by expansion: \\(4-2\\sqrt{13}+2\\sqrt{13}-13=4-13=-9\\); the middle terms cancel. Memory cue: conjugates multiply to the square of the rational part minus the square of the radical part. The supplied answer is correct.

Correct answer: A. (4)

Explanation: Multiplying by the conjugate ( \sqrt{5}-1 ) makes the denominator (5-1=4). Use difference of squares in the conjugate method.

Correct answer: A. It is irrational because its decimal expansion is non-terminating and non-repeating; the number of zeros between 1s keeps increasing.

Explanation: A rational number has a terminating or recurring decimal expansion. Here the gaps of zeros between 1s are 1, 2, 3, 4, ..., so no fixed repeating block exists. Exam tip: check the repetition pattern, not merely the digits used.

Correct answer: B. It is irrational because its decimal expansion is non-terminating and non-repeating.

Explanation: In 0.1010010001..., the number of zeros between successive 1s is 1, 2, 3, 4..., so no repeating block occurs. A non-terminating, non-repeating decimal is irrational. Exam tip: an infinite decimal can still be rational only if it repeats.

Correct answer: C. ( |a| )

Explanation: In real numbers, ( \sqrt{a^2}=|a| ) because the principal square root is non-negative. Pay attention to the sign.

Correct answer: B. 5

Explanation: Using the identity \(\sqrt{x^2}=|x|\), we get \(\sqrt{(-5)^2}=|-5|=5\). The principal square root is always non-negative, so \(-5\) is not correct. Exam tip: remember that \(\sqrt{x^2}=|x|\), not always \(x\).

Correct answer: C. ( \sqrt{3}-\sqrt{2} )

Explanation: Multiplying by the conjugate ( \sqrt{3}-\sqrt{2} ) makes the denominator (3-2=1). So the rationalised form is ( \sqrt{3}-\sqrt{2} ).

Correct answer: A. \(6\sqrt{5}\)

Explanation: Since \(180=36\times5\) and \(36\) is the largest perfect-square factor, \(\sqrt{180}=\sqrt{36\times5}=\sqrt{36}\times\sqrt{5}=6\sqrt{5}\). Option B is the unsimplified form, while options C and D have incorrect coefficients. Exam tip: factor the radicand using its largest perfect-square factor before simplifying the surd.

Correct answer: B. \(7\sqrt{5}\)

Explanation: \(245=49\times5=7^2\times5\). Therefore, \(\sqrt{245}=\sqrt{7^2\times5}=7\sqrt{5}\), so option B is correct. Option D is incorrect because the square root of the perfect-square factor \(49\) is \(7\), not \(49\). Exam tip: To simplify a radical, factor the radicand into the largest possible perfect square times the remaining factor.

Correct answer: C. \(12\sqrt{2}\)

Explanation: \(288=144\times2=12^2\times2\). Therefore, \(\sqrt{288}=\sqrt{12^2\times2}=12\sqrt{2}\), so option C is correct. In option A, the coefficient is doubled, while options B and D are not equal to \(\sqrt{288}\). Exam tip: factor out the largest perfect-square factor before simplifying a square root.

Correct answer: A. \(12\sqrt{5}\)

Explanation: \(\sqrt{45}=\sqrt{9\times5}=3\sqrt{5}\), so \(2\sqrt{45}=6\sqrt{5}\). Similarly, \(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\), so \(3\sqrt{20}=6\sqrt{5}\). Adding the like surd terms gives \(6\sqrt{5}+6\sqrt{5}=12\sqrt{5}\), so option A is correct. An answer such as option B results from adding the coefficients incorrectly. Exam tip: simplify each surd first, then add only terms with the same surd part.

Correct answer: A. \(2\sqrt{3}\)

Explanation: \(\sqrt{27}=3\sqrt{3}\) and \(\sqrt{75}=5\sqrt{3}\). Therefore, \(4\sqrt{27}-2\sqrt{75}=4(3\sqrt{3})-2(5\sqrt{3})=12\sqrt{3}-10\sqrt{3}=2\sqrt{3}\). Hence, option A is correct. Exam tip: Simplify each surd first, then combine only like surd terms.

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

Explanation: By distributive law, ( \sqrt{5}\times3+\sqrt{5}\times\sqrt{5}=3\sqrt{5}+5 ). Be careful in surd multiplication.

Correct answer: A. \(19+8\sqrt{3}\)

Explanation: Apply the identity \((a+b)^2=a^2+2ab+b^2\). Here, \(a=4\) and \(b=\sqrt{3}\), so \((4+\sqrt{3})^2=4^2+2(4)(\sqrt{3})+(\sqrt{3})^2=16+8\sqrt{3}+3=19+8\sqrt{3}\). Therefore, option A is correct. Remember that \((\sqrt{3})^2=3\), not \(\sqrt{4}\).

Correct answer: B. (44-24\sqrt{2})

Explanation: The direct answer is option B: \(44-24\sqrt{2}\). Start with \((a-b)^2=a^2-2ab+b^2\). Here \(a=6\) and \(b=\sqrt{8}\), so \((6-\sqrt{8})^2=36-12\sqrt{8}+8=44-12\sqrt{8}\). Since \(\sqrt{8}=2\sqrt{2}\), this becomes \(44-24\sqrt{2}\). Option A keeps \(\sqrt{8}\) and has the wrong middle term. Option B is correct after simplification. Option C has an incorrect constant term and radical coefficient. Option D incorrectly treats the cross product as involving \(\sqrt{6}\). Remember: square the first term, subtract twice the product, then square the second term.

Correct answer: A. 8

Explanation: This is a product of conjugate expressions. Using \((a+b)(a-b)=a^2-b^2\), we get \((5+\sqrt{17})(5-\sqrt{17})=5^2-(\sqrt{17})^2=25-17=8\). Therefore, the correct answer is 8. The value 42 results from adding 25 and 17, but this product requires their difference. Exam tip: Whenever you see \((a+b)(a-b)\), apply \(a^2-b^2\) directly.

Correct answer: A. \\(\sqrt{50}=5\sqrt{2}\\), इसलिए यह अपरिमेय संख्या है।

Explanation: Since \\(50=25\times2\\), we get \\(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\\). Because \\(\sqrt{2}\\) is irrational, multiplying it by the non-zero rational number 5 still gives an irrational number. Therefore, option A is correct. Option C has the correct simplification but incorrectly classifies the result as rational. Exam tip: take perfect-square factors outside the square root and check whether a non-perfect-square factor remains inside.

Correct answer: A. ( \frac{5\sqrt{2}}{6} )

Explanation: Multiplying numerator and denominator by ( \sqrt{2} ) gives ( \frac{5\sqrt{2}}{6} ). The radical should be removed from the denominator.