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SubjectsMathematics
अपरिमेय संख्याएँ
In Class 10 Mathematics, this topic from the Real Numbers chapter introduces irrational numbers as numbers that cannot be expressed as a ratio of two integers. Students learn to identify them through non-terminating, non-repeating decimal expansions, distinguish them from rational numbers, and locate them on the number line. The topic also develops understanding of examples such as √2 and how irrational numbers fit into the wider system of real numbers.
Correct answer: A. (\sqrt{15})
Explanation: Step 1: An irrational number cannot be written exactly as a fraction. Step 2: (15) is not a perfect square, so (\sqrt{15}) is irrational. Step 3: For square-root options, first check whether the number inside is a perfect square.
Correct answer: B. Rational number
Explanation: Step 1: (81) is a perfect square. Step 2: (\sqrt{81}=9), and (9) can be written as (\frac{9}{1}). Step 3: The square root of a perfect square is always rational.
Correct answer: C. (0.1234567891011\ldots)
Explanation: Step 1: Terminating or recurring decimals are rational. Step 2: (0.1234567891011\ldots) has no fixed repeating pattern, so it is non-terminating and non-recurring. Step 3: To identify an irrational decimal, check for a repeating rule.
Correct answer: A. (2\sqrt{7})
Explanation: Step 1: Write (28=4 \times 7). Step 2: (\sqrt{28}=\sqrt{4 \times 7}=2\sqrt{7}). Step 3: While simplifying a square root, take the perfect square factor outside.
Correct answer: A. (6)
Explanation: Step 1: When multiplying square roots, multiply the numbers inside. Step 2: (\sqrt{2}\times\sqrt{18}=\sqrt{36}=6). Step 3: The product of two irrational numbers can sometimes be rational.
Correct answer: C. (\sqrt{100})
Explanation: Step 1: To find the rational option, look for the square root of a perfect square. Step 2: (100) is a perfect square and (\sqrt{100}=10). Step 3: In (\sqrt{n}), if (n) is a perfect square, the value is rational.
Correct answer: B. It is irrational
Explanation: Step 1: (\sqrt{2}) is irrational and (5) is rational. Step 2: Adding a rational number does not remove the irrational part. Step 3: The sum of a rational and an irrational number is generally irrational.
Correct answer: A. (4\sqrt{3})
Explanation: Step 1: (48=16 \times 3). Step 2: (\sqrt{48}=\sqrt{16 \times 3}=4\sqrt{3}). Step 3: Using the largest perfect square gives the simplest form.
Correct answer: B. (\frac{3}{8})
Explanation: Step 1: A number with a terminating decimal is rational. Step 2: (\frac{3}{8}=0.375), so its decimal terminates. Step 3: If the denominator has only factors (2) and (5), the decimal terminates.
Correct answer: A. (2\sqrt{7})
Explanation: Step 1: Like radicals are added like like terms. Step 2: (\sqrt{7}+\sqrt{7}=2\sqrt{7}). Step 3: In addition, do not add the numbers inside roots to write (\sqrt{14}).
Correct answer: A. (0)
Explanation: Step 1: Subtracting a number from itself gives zero. Step 2: (\sqrt{7}-\sqrt{7}=0), and (0) is rational. Step 3: Normal subtraction rules also apply to irrational numbers.
Correct answer: B. Irrational
Explanation: Step 1: (\sqrt{2}\times\sqrt{5}=\sqrt{10}). Step 2: (10) is not a perfect square, so (\sqrt{10}) is irrational. Step 3: After multiplication, check the number inside the new root.
Correct answer: C. (\sqrt{n}) can be irrational when (n) is not a perfect square
Explanation: Step 1: If (n) is not a perfect square, (\sqrt{n}) is not rational. Step 2: For example, (\sqrt{6}) is irrational. Step 3: Testing a statement with an example makes it easier to judge.
Correct answer: A. (7\sqrt{2})
Explanation: Step 1: (98=49 \times 2). Step 2: (\sqrt{98}=\sqrt{49 \times 2}=7\sqrt{2}). Step 3: Recognising larger perfect squares like (49) helps in simplification.
Correct answer: A. (3\sqrt{3})
Explanation: Step 1: (\sqrt{12}=2\sqrt{3}). Step 2: (\sqrt{3}+2\sqrt{3}=3\sqrt{3}). Step 3: Before adding, convert radicals into like terms if possible.
Correct answer: A. (\sqrt{17})
Explanation: Step 1: Since (16<17<25), (4<\sqrt{17}<5). Step 2: (17) is not a perfect square, so (\sqrt{17}) is irrational. Step 3: Use nearby perfect squares in interval questions.
Correct answer: B. Irrational
Explanation: Step 1: (\sqrt{5}) is irrational. Step 2: Dividing it by the non-zero rational number (5) keeps it irrational. Step 3: A rational denominator alone does not make the whole expression rational.
Correct answer: A. (3,\sqrt{6})
Explanation: Step 1: (3) is rational. Step 2: (\sqrt{6}) is irrational because (6) is not a perfect square. Step 3: In a pair, identify each number separately.
Correct answer: A. (5\sqrt{5})
Explanation: Step 1: Write (125=25 \times 5). Step 2: (\sqrt{125}=\sqrt{25 \times 5}=5\sqrt{5}). Step 3: Take the perfect square (25) outside as (5).
Correct answer: A. (3\sqrt{6})
Explanation: Step 1: Three like irrational terms are being added. Step 2: (\sqrt{6}+\sqrt{6}+\sqrt{6}=3\sqrt{6}). Step 3: Count like radicals as coefficients.
Correct answer: A. (\sqrt{5})
Explanation: Step 1: (\sqrt{45}=3\sqrt{5}) and (\sqrt{20}=2\sqrt{5}). Step 2: (3\sqrt{5}-2\sqrt{5}=\sqrt{5}). Step 3: Simplify both radicals before subtracting.
Correct answer: B. Rational
Explanation: Step 1: ((\sqrt{2})^2=2). Step 2: (2) is rational because it can be written as (\frac{2}{1}). Step 3: The square of an irrational number can sometimes be rational.
Correct answer: B. Irrational
Explanation: Step 1: (x=\sqrt{5}) is irrational. Step 2: (5x=5\sqrt{5}), and (5) is a non-zero rational number. Step 3: Multiplying an irrational number by a non-zero rational number keeps it irrational.
Correct answer: B. (\sqrt{36})
Explanation: Step 1: The question asks for the number that is not irrational, so find the rational one. Step 2: (\sqrt{36}=6), which is rational. Step 3: Read negative wording carefully in MCQs.
Correct answer: A. (5\sqrt{2})
Explanation: Step 1: (\sqrt{8}=2\sqrt{2}) and (\sqrt{18}=3\sqrt{2}). Step 2: (2\sqrt{2}+3\sqrt{2}=5\sqrt{2}). Step 3: Radicals can be added only after they become like radicals.
