Irrational numbers

Correct answer: B. It is always irrational

Explanation: Step 1: Dividing by a non-zero rational number is the same as multiplying by its reciprocal. Step 2: (\frac{1}{r}) is also a non-zero rational number, so (\frac{x}{r}) remains irrational. Step 3: In division questions, always check that the denominator is not zero.

Correct answer: A. (4\sqrt{2}), irrational

Explanation: Step 1: (\sqrt{18}=3\sqrt{2}). Step 2: (\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}), which is irrational. Step 3: For like surds, add only the outside coefficients.

Correct answer: C. (1.01001000100001\ldots)

Explanation: Step 1: Terminating and recurring decimals are rational. Step 2: (1.01001000100001\ldots) is non-terminating and has no fixed repeating block. Step 3: To identify an irrational decimal, check both non-termination and non-repetition.

Correct answer: A. (2), rational

Explanation: Step 1: Substitute the given value of (x). Step 2: (x-\sqrt{3}=(\sqrt{3}+2)-\sqrt{3}=2), which is rational. Step 3: Like irrational terms may cancel, so decide the nature only after simplifying.

Correct answer: B. (\frac{\sqrt{45}}{3})

Explanation: Step 1: (\frac{\sqrt{45}}{3}=\frac{3\sqrt{5}}{3}=\sqrt{5}). Step 2: Since (5) is not a perfect square, (\sqrt{5}) is irrational. Step 3: Do not choose an answer in multiplication or division of surds without simplifying.

Correct answer: B. (\sqrt{72}) is irrational because (72) is not a perfect square

Explanation: Step 1: The square root of a positive integer is rational only when the integer is a perfect square. Step 2: (72) is not a perfect square, so (\sqrt{72}) is irrational. Step 3: Being even does not make a square root rational.

Correct answer: A. (18)

Explanation: Step 1: This is multiplication of conjugates. Step 2: (ab=5^2-(\sqrt{7})^2=25-7=18). Step 3: In conjugate multiplication, the middle irrational terms cancel.

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

Explanation: Step 1: (3=\sqrt{9}) and (4=\sqrt{16}). Step 2: (10) is not a perfect square and (9<10<16), so (\sqrt{10}) is an irrational number between (3) and (4). Step 3: A non-perfect square between two square numbers helps find an irrational number between two integers.

Correct answer: B. (x) must be rational

Explanation: Step 1: If (\frac{1}{x}) is rational and non-zero, then its reciprocal is also rational. Step 2: Therefore (x) is rational, not irrational. Step 3: In reciprocal questions, always check the non-zero condition.

Correct answer: B. (3(\sqrt{5}-2))

Explanation: Step 1: The conjugate of the denominator is (2-\sqrt{5}). Step 2: (\frac{3}{2+\sqrt{5}}\times\frac{2-\sqrt{5}}{2-\sqrt{5}}=\frac{3(2-\sqrt{5})}{4-5}=3(\sqrt{5}-2)). Step 3: Use the difference of squares in the denominator when multiplying by a conjugate.

Correct answer: B. (45)

Explanation: Step 1: (\sqrt{20}=2\sqrt{5}), so (x=3\sqrt{5}). Step 2: (x^2=(3\sqrt{5})^2=9\times5=45). Step 3: Simplify surd terms before squaring.

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

Explanation: Step 1: (\sqrt{98}=7\sqrt{2}) and (\sqrt{50}=5\sqrt{2}). Step 2: The difference is (7\sqrt{2}-5\sqrt{2}=2\sqrt{2}), which is irrational. Step 3: For like surds, subtract only the coefficients.

Correct answer: A. Both (p) and (q) turn out divisible by (3)

Explanation: Step 1: Assuming (\sqrt{3}=\frac{p}{q}) gives (p^2=3q^2). Step 2: This makes both (p) and (q) divisible by (3), contradicting that they are coprime. Step 3: In such proofs, finding a common factor creates the contradiction.

Correct answer: B. (\sqrt{12}) and (\sqrt{3})

Explanation: Step 1: (\sqrt{12}=2\sqrt{3}) and (\sqrt{3}) are both irrational. Step 2: (\frac{\sqrt{12}}{\sqrt{3}}=\sqrt{4}=2), which is rational. Step 3: In quotients, check whether the value inside the root becomes a perfect square.

Correct answer: C. Every non-terminating decimal is irrational

Explanation: Step 1: A non-terminating decimal can also be recurring. Step 2: For example, (0.\overline{6}) is non-terminating but rational. Step 3: For irrational decimals, non-repetition is also necessary.

Correct answer: A. (6)

Explanation: Step 1: (\sqrt{8}=2\sqrt{2}) and (\sqrt{32}=4\sqrt{2}). Step 2: (y=6\sqrt{2}), so (\frac{y}{\sqrt{2}}=6). Step 3: First add like surds, then divide.

Correct answer: B. (\sqrt{60})

Explanation: Step 1: (2\sqrt{15}=\sqrt{4}\sqrt{15}). Step 2: Therefore (2\sqrt{15}=\sqrt{60}). Step 3: When moving a coefficient inside a square root, its square goes inside.

Correct answer: A. (-\sqrt{13})

Explanation: Step 1: (\sqrt{52}=2\sqrt{13}). Step 2: (a=\sqrt{13}-2\sqrt{13}=-\sqrt{13}), which is irrational. Step 3: A negative sign does not change irrationality.

Correct answer: B. ((\sqrt{11}+1)(\sqrt{11}-1))

Explanation: Step 1: ((\sqrt{11}+1)(\sqrt{11}-1)) is a conjugate product. Step 2: Its value is (11-1=10), which is rational. Step 3: In conjugate forms, irrational terms can cancel.

Correct answer: B. It is irrational

Explanation: Step 1: (0) is rational. Step 2: (x+0=x), so the nature remains the same and it is irrational. Step 3: Adding zero does not change either the value or the type of a number.

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

Explanation: Step 1: (\sqrt{45}=3\sqrt{5}) and (\sqrt{20}=2\sqrt{5}). Step 2: (\sqrt{5}+3\sqrt{5}-2\sqrt{5}=2\sqrt{5}). Step 3: In questions with many radicals, first convert all terms to like surds when possible.

Correct answer: A. (x=\sqrt{19})

Explanation: Step 1: (\sqrt{19}) is irrational because (19) is not a perfect square. Step 2: ((\sqrt{19})^2=19), which is rational. Step 3: The square of an irrational number can sometimes be rational.

Correct answer: C. Changing the length of digit groups without a fixed repetition

Explanation: Step 1: An irrational decimal neither terminates nor has a fixed repeating block. Step 2: Digit groups with changing lengths do not form a fixed repetition. Step 3: Once a fixed repetition appears, the decimal becomes rational.

Correct answer: B. (3\sqrt{6})

Explanation: Step 1: (\sqrt{24}=2\sqrt{6}). Step 2: So (x=\sqrt{6}+2\sqrt{6}=3\sqrt{6}), which is irrational. Step 3: Simplify radicals to like terms before adding.

Correct answer: C. Non-terminating non-recurring irrational

Explanation: Step 1: This decimal does not terminate. Step 2: The number of (5)'s keeps increasing, so no fixed repeating block is formed. Step 3: A non-terminating non-recurring decimal is irrational.