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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: B. Irrational because it equals (\sqrt{5})
Explanation: Step 1: (\sqrt{45}=3\sqrt{5}). Step 2: (\frac{\sqrt{45}}{3}=\sqrt{5}) which is irrational. Step 3: Even after division check the remaining radical.
Correct answer: A. (\frac{3\sqrt{2}}{2}) and irrational
Explanation: Step 1: (\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}). Step 2: (\sqrt{2}+\frac{\sqrt{2}}{2}=\frac{3\sqrt{2}}{2}) which is irrational. Step 3: Rationalize the denominator before combining terms.
Correct answer: A. (\sqrt{2})
Explanation: Step 1: (1<2<4). Step 2: Hence (1<\sqrt{2}<2) and (\sqrt{2}) is irrational. Step 3: To locate an irrational number compare squares.
Correct answer: A. (\sqrt{2}=-\frac{a}{b}) would be rational which is impossible
Explanation: Step 1: The equation gives (\sqrt{2}=-\frac{a}{b}). Step 2: (-\frac{a}{b}) is rational but (\sqrt{2}) is irrational. Step 3: Such questions use irrationality to create a contradiction.
Correct answer: A. It is (6\sqrt{2}) and irrational
Explanation: Step 1: (\sqrt{32}=4\sqrt{2}), (\sqrt{50}=5\sqrt{2}), and (\sqrt{18}=3\sqrt{2}). Step 2: The result is (6\sqrt{2}) which is irrational. Step 3: Add and subtract coefficients of like radicals.
Correct answer: B. (\sqrt{5}+(2-\sqrt{5})=2)
Explanation: Step 1: (\sqrt{5}) is irrational and (2-\sqrt{5}) is also irrational. Step 2: Their sum is (2) which is rational. Step 3: There is no single always rule for the sum of two irrational numbers.
Correct answer: B. (\sqrt{11})
Explanation: Step 1: (\sqrt{11}) is irrational because (11) is not a perfect square. Step 2: ((\sqrt{11})^2=11) which is rational. Step 3: Squaring may remove the radical.
Correct answer: A. (10)
Explanation: Step 1: (\sqrt{2}\sqrt{8}=\sqrt{16}=4). Step 2: (\sqrt{3}\sqrt{12}=\sqrt{36}=6), so the sum is (10). Step 3: In products combine radicals and check for perfect squares.
Correct answer: B. (x) is irrational
Explanation: Step 1: Suppose (\sqrt{3}-\sqrt{2}) is rational. Step 2: Squaring gives (5-2\sqrt{6}), forcing (\sqrt{6}) to be rational which is false. Step 3: The difference of unlike radicals is not directly an integer.
Correct answer: A. (\frac{\sqrt{5}-1}{2})
Explanation: Step 1: Multiply by the conjugate (\sqrt{5}-1). Step 2: (\frac{2(\sqrt{5}-1)}{5-1}=\frac{\sqrt{5}-1}{2}). Step 3: Multiplying by the conjugate makes the denominator rational.
Correct answer: B. Rational because (\sqrt{0.04}=0.2)
Explanation: Step 1: (0.04=\frac{4}{100}). Step 2: (\sqrt{0.04}=\frac{2}{10}=0.2) which is rational. Step 3: Every square root is not irrational.
Correct answer: B. Irrational because (\sqrt{\frac{2}{9}}=\frac{\sqrt{2}}{3})
Explanation: Step 1: (\sqrt{\frac{2}{9}}=\frac{\sqrt{2}}{3}). Step 2: (\sqrt{2}) is irrational and remains irrational after division by (3). Step 3: Check both numerator and denominator of the fraction.
Correct answer: B. (2.12112111211112\ldots) with no repeating block
Explanation: Step 1: Terminating or recurring decimals are rational. Step 2: The given decimal is non-terminating and has no fixed repeating block, so it is irrational. Step 3: Decide by checking repetition, not just by seeing many digits.
Correct answer: A. (x) is not a perfect square
Explanation: Step 1: The square root of a perfect square is an integer and rational. Step 2: If (\sqrt{x}) is irrational then (x) cannot be a perfect square. Step 3: For irrational square roots first check perfect-square status.
Correct answer: A. Make a right triangle with legs (1) and (1) and use the hypotenuse
Explanation: Step 1: (1^2+1^2=2). Step 2: So the hypotenuse of that right triangle is (\sqrt{2}). Step 3: The Pythagoras theorem helps locate irrational numbers on a number line.
Correct answer: B. It is (\sqrt{6}) and irrational
Explanation: Step 1: The product of radicals is (\sqrt{2}\cdot \sqrt{3}=\sqrt{6}). Step 2: Since (6) is not a perfect square (\sqrt{6}) is irrational. Step 3: In multiplication the numbers inside radicals multiply, not add.
Correct answer: B. Rational because (4+\sqrt{9}=7)
Explanation: Step 1: (\sqrt{9}=3). Step 2: (4+3=7), a rational number. Step 3: Do not call every expression with a square root irrational immediately.
Correct answer: A. When (r=0)
Explanation: Step 1: From (r+s=s), subtract (s) from both sides to get (r=0). Step 2: (0) is rational, so the condition is possible. Step 3: Form a simple equation before judging number types.
Correct answer: A. (2\sqrt{2}) and irrational
Explanation: Step 1: (\sqrt{98}=7\sqrt{2}) and (\sqrt{50}=5\sqrt{2}). Step 2: The difference is (2\sqrt{2}), which is irrational. Step 3: Directly subtracting numbers inside radicals is wrong.
Correct answer: A. The first is greater
Explanation: Step 1: (\sqrt{2}+\sqrt{18}=4\sqrt{2}). Step 2: (\sqrt{8}+\sqrt{12}=2\sqrt{2}+2\sqrt{3}). Since (\sqrt{3}>\sqrt{2}), the second expression is greater. Step 3: Simplify first and compare carefully.
Correct answer: A. Whether it becomes the square root of a perfect square
Explanation: Step 1: In square-root questions first check whether the number inside is a perfect square. Step 2: A perfect square may give a rational square root while a non-perfect square often gives an irrational value. Step 3: Simplifying is the safest first step in identification questions.
Correct answer: A. (3\sqrt{7}) and irrational
Explanation: Step 1: Since (28=4\cdot 7), (\sqrt{28}=2\sqrt{7}). Step 2: Now (\sqrt{7}+2\sqrt{7}=3\sqrt{7}), and (\sqrt{7}) is irrational. Step 3: In exams, combine like radicals by adding their coefficients.
Correct answer: B. Because adding an irrational number to a rational number gives an irrational result
Explanation: Step 1: (\frac{3}{2}) is rational and (\sqrt{5}) is irrational. Step 2: If their sum were rational, then (\sqrt{5}) would become the difference of two rational numbers, which is impossible. Step 3: For rational-plus-irrational questions, contradiction is a very useful method.
Correct answer: A. ((\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2}))
Explanation: Step 1: The first option is a product of conjugate terms. Step 2: ((\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})=6-2=4), which is rational. Step 3: Identifying conjugates helps remove radicals quickly.
Correct answer: A. When (a) can be written as a ratio of two perfect squares
Explanation: Step 1: The square root of a rational fraction is rational when both numerator and denominator can be perfect squares. Step 2: For example, (\sqrt{\frac{4}{9}}=\frac{2}{3}), so a ratio of two perfect squares is a safe condition. Step 3: Being positive or less than (1) does not guarantee a rational square root.
