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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: C. (\sqrt{169}) is irrational
Explanation: Step 1: (169) is a perfect square. Step 2: (\sqrt{169}=13), which is rational, so calling it irrational is false. Step 3: While choosing a false statement, identify perfect squares carefully.
Correct answer: A. (10\sqrt{3})
Explanation: Step 1: Write (300=100 \times 3). Step 2: (\sqrt{300}=\sqrt{100 \times 3}=10\sqrt{3}). Step 3: When you see a perfect square like (100), take it outside as (10).
Correct answer: A. (14)
Explanation: Step 1: (\sqrt{7}\times\sqrt{28}=\sqrt{196}). Step 2: (\sqrt{196}=14), so the value is rational. Step 3: When multiplying square roots, multiply the numbers inside.
Correct answer: B. (\sqrt{55})
Explanation: Step 1: Since (49<55<64), (7<\sqrt{55}<8). Step 2: (55) is not a perfect square, so (\sqrt{55}) is irrational. Step 3: Nearby perfect squares help locate a square root easily.
Correct answer: A. (12\sqrt{2})
Explanation: Step 1: (288=144 \times 2). Step 2: (\sqrt{288}=\sqrt{144 \times 2}=12\sqrt{2}). Step 3: Using a large perfect square gives the simplified form directly.
Correct answer: B. (3\sqrt{7})
Explanation: Step 1: Both terms contain the same radical (\sqrt{7}). Step 2: (1\sqrt{7}+2\sqrt{7}=3\sqrt{7}). Step 3: For like radicals, add only the outside coefficients.
Correct answer: B. (2.65)
Explanation: Step 1: The value of (\sqrt{7}) is about (2.646). Step 2: Among the options, (2.65) is closest. Step 3: For estimation, use nearby perfect squares (4) and (9) to understand the range.
Correct answer: B. (5) and (6)
Explanation: Step 1: (25<26<36). Step 2: So (\sqrt{25}<\sqrt{26}<\sqrt{36}), meaning (5<\sqrt{26}<6). Step 3: To find the range of a square root, look at nearby perfect squares.
Correct answer: A. (8\sqrt{3})
Explanation: Step 1: (192=64 \times 3). Step 2: (\sqrt{192}=\sqrt{64 \times 3}=8\sqrt{3}). Step 3: To fully simplify the answer, take out the largest perfect square.
Correct answer: C. (17)
Explanation: Step 1: We are given (y=\sqrt{17}). Step 2: (y^2=(\sqrt{17})^2=17). Step 3: Squaring a square root gives the number inside it.
Correct answer: A. (\sqrt{5}\times\sqrt{5})
Explanation: Step 1: (\sqrt{5}) is irrational. Step 2: (\sqrt{5}\times\sqrt{5}=5), which is rational. Step 3: The product of two irrational numbers is not always irrational.
Correct answer: C. (\sqrt{11})
Explanation: Step 1: Adding zero does not change the value of a number. Step 2: (\sqrt{11}+0=\sqrt{11}), which is irrational. Step 3: Even with zero, identify the nature of the original number.
Correct answer: C. Non-terminating and non-recurring
Explanation: Step 1: (21) is not a perfect square. Step 2: So (\sqrt{21}) is irrational and its decimal expansion is non-terminating and non-recurring. Step 3: An irrational number has no fixed recurring block.
Correct answer: B. (3\sqrt{15})
Explanation: Step 1: Write (135=9 \times 15). Step 2: (\sqrt{135}=\sqrt{9 \times 15}=3\sqrt{15}). Step 3: The form is simplified when the remaining number inside has no perfect square factor.
Correct answer: B. (5\sqrt{3})
Explanation: Step 1: (75=25 \times 3). Step 2: (\sqrt{75}=5\sqrt{3}). Step 3: To identify an equivalent form, simplify the square root first.
Correct answer: A. (0)
Explanation: Step 1: The two terms are opposites of each other. Step 2: (\sqrt{13}+(-\sqrt{13})=0). Step 3: The sum of opposite terms is always zero.
Correct answer: B. Irrational
Explanation: Step 1: Dividing by a non-zero rational number does not remove irrationality. Step 2: For example, (\frac{\sqrt{2}}{5}) is irrational. Step 3: The condition (r\neq0) is necessary because division by zero is not possible.
Correct answer: A. (5\sqrt{11})
Explanation: Step 1: (275=25 \times 11). Step 2: (\sqrt{275}=\sqrt{25 \times 11}=5\sqrt{11}). Step 3: Take the perfect square factor outside to simplify the answer.
Correct answer: C. (1.01011011101111\ldots) with no fixed repetition
Explanation: Step 1: Recurring or terminating decimals are rational. Step 2: A non-terminating decimal with no fixed repetition can be irrational. Step 3: If no repeating rule is visible, examine the number carefully.
Correct answer: B. (9\sqrt{3})
Explanation: Step 1: Write (243=81 \times 3). Step 2: (\sqrt{243}=\sqrt{81 \times 3}=9\sqrt{3}). Step 3: Choosing a larger perfect square simplifies the answer in one step.
Correct answer: A. (18)
Explanation: Step 1: (\sqrt{6}\times\sqrt{54}=\sqrt{324}). Step 2: (\sqrt{324}=18), so the result is rational. Step 3: After multiplication, check whether the inside number has become a perfect square.
Correct answer: B. (3.32)
Explanation: Step 1: Since (9<11<16), (\sqrt{11}) lies between (3) and (4). Step 2: Its approximate value is (3.316), so (3.32) is close. Step 3: In estimation, first set the range using perfect squares.
Correct answer: A. (6\sqrt{6})
Explanation: Step 1: (216=36 \times 6). Step 2: (\sqrt{216}=\sqrt{36 \times 6}=6\sqrt{6}). Step 3: The remaining (6) has no perfect square factor, so the form is simplified.
Correct answer: C. (6) and (7)
Explanation: Step 1: (36<39<49). Step 2: Therefore, (6<\sqrt{39}<7). Step 3: Nearby perfect squares give the best clues for comparing square roots.
Correct answer: B. (\sqrt{7}+\sqrt{28})
Explanation: Step 1: (\sqrt{28}=2\sqrt{7}), so (\sqrt{7}+\sqrt{28}=3\sqrt{7}). Step 2: (3\sqrt{7}) is irrational because (\sqrt{7}) is irrational. Step 3: In options, simplify first before deciding the nature of the result.
