Muft Shiksha™ एक 100% Free Education Portal है 🇮🇳, जिसका उद्देश्य Class 9–12 के हर विद्यार्थी तक High-Quality Education को पूरी तरह मुफ्त पहुँचाना है। 🇮🇳 हम मानते हैं कि अच्छी शिक्षा किसी student की आर्थिक स्थिति पर निर्भर नहीं होनी चाहिए। 🇮🇳 हर विद्यार्थी को वही Quality Study Material, MCQs, Quizzes, Exam Preparation, Concept-Based Learning और Bilingual Support मिलना चाहिए, जो आमतौर पर महंगी Coaching या Premium Platforms में मिलता है। Muft Shiksha™ 🇮🇳 इसी सोच के साथ बनाया गया है
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. It is always irrational
Explanation: Step 1: (a) is rational and non-zero. Step 2: If (a\sqrt{3}) were rational then (\sqrt{3}) would also become rational which is false. Step 3: In exams always check the special case of multiplication by zero.
Correct answer: C. (\sqrt{2}+\sqrt{8})
Explanation: Step 1: Simplify (\sqrt{8}=2\sqrt{2}). Step 2: (\sqrt{2}+\sqrt{8}=3\sqrt{2}) and (\sqrt{2}) is irrational. Step 3: Do not choose the answer before simplifying square roots.
Correct answer: C. (n) is 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: For example (\sqrt{25}=5). Step 3: The square root of a prime number is usually irrational.
Correct answer: B. (x) is irrational
Explanation: Step 1: Suppose (\sqrt{2}+\sqrt{3}) is rational. Step 2: Squaring gives (5+2\sqrt{6}) so (\sqrt{6}) would be rational which is false. Step 3: Do not decide the sum of two different irrational numbers without reasoning.
Correct answer: C. (\sqrt{49})
Explanation: Step 1: The square root of a perfect square is rational. Step 2: (\sqrt{49}=7) so it is rational. Step 3: First look for perfect-square factors inside the radical.
Correct answer: B. Both (p) and (q) become even
Explanation: Step 1: From (\sqrt{2}=\frac{p}{q}) we get (p^2=2q^2). Step 2: This makes (p) even and then (q) even. Step 3: Coprime numbers cannot both be even so the assumption is false.
Correct answer: C. It is non-recurring and non-terminating decimal
Explanation: Step 1: (\sqrt{5}) is not the square root of a perfect square. Step 2: So it is irrational and its decimal expansion is non-terminating and non-recurring. Step 3: While using decimal form check whether repetition exists.
Correct answer: C. (\frac{3}{5}\cdot \sqrt{13})
Explanation: Step 1: (\frac{3}{5}) is a non-zero rational number. Step 2: Multiplying it by (\sqrt{13}) gives an irrational number. Step 3: In products check first whether square roots combine to a perfect square.
Correct answer: B. Both are irrational and (\sqrt{8}=2\sqrt{2})
Explanation: Step 1: Since (8=4\cdot 2) we have (\sqrt{8}=2\sqrt{2}). Step 2: (\sqrt{2}) is irrational and its double is also irrational. Step 3: Compare like radicals after simplifying them.
Correct answer: B. Always irrational
Explanation: Step 1: Dividing by a non-zero rational number is the same as multiplying by its reciprocal. Step 2: An irrational number multiplied by a non-zero rational number remains irrational. Step 3: Convert division questions into multiplication for easier reasoning.
Correct answer: B. Irrational because the answer is (\sqrt{2})
Explanation: Step 1: (\sqrt{18}=3\sqrt{2}) and (\sqrt{8}=2\sqrt{2}). Step 2: The difference is (\sqrt{2}) which is irrational. Step 3: Do not subtract the numbers inside square roots directly.
Correct answer: C. Non-terminating non-recurring decimal
Explanation: Step 1: Rational numbers have terminating or recurring decimals. Step 2: Irrational numbers have non-terminating and non-recurring decimals. Step 3: If a repeating block is visible the number may be rational.
Correct answer: B. (\sqrt{5}\cdot \sqrt{5})
Explanation: Step 1: (\sqrt{5}) is irrational. Step 2: (\sqrt{5}\cdot \sqrt{5}=5) which is rational. Step 3: The product of two identical irrational square roots becomes the number inside.
Correct answer: B. (x) is irrational
Explanation: Step 1: (3) is rational and (\sqrt{2}) is irrational. Step 2: The sum of a rational and an irrational number is irrational. Step 3: Adding an irrational number to an integer does not give an integer.
Correct answer: B. It is irrational
Explanation: Step 1: If (\frac{1}{\sqrt{3}}) were rational then its reciprocal (\sqrt{3}) would be rational. Step 2: (\sqrt{3}) is irrational so the given number is irrational. Step 3: A denominator with an irrational radical does not make the value rational automatically.
Correct answer: B. Irrational because (\sqrt{48}=4\sqrt{3})
Explanation: Step 1: (48=16\cdot 3). Step 2: (\sqrt{48}=4\sqrt{3}) and (\sqrt{3}) is irrational. Step 3: The square root of an even number need not be rational.
Correct answer: B. (\sqrt{18}) and (\sqrt{2})
Explanation: Step 1: (\sqrt{18}) and (\sqrt{2}) are both irrational. Step 2: (\frac{\sqrt{18}}{\sqrt{2}}=\sqrt{9}=3) which is rational. Step 3: In quotients check whether the ratio inside the radical becomes a perfect square.
Correct answer: B. (\sqrt{6}) is rational
Explanation: Step 1: Suppose (5-\sqrt{6}) is rational. Step 2: Then (\sqrt{6}=5-) that rational number so (\sqrt{6}) would be rational. Step 3: In contradiction proofs identify the result that clashes with a known fact.
Correct answer: B. (\sqrt{5})
Explanation: Step 1: (4<5<9). Step 2: Therefore (2<\sqrt{5}<3) and since (5) is not a perfect square (\sqrt{5}) is irrational. Step 3: Use squares to locate irrational square roots between integers.
Correct answer: B. It equals (2\sqrt{a}) and is irrational
Explanation: Step 1: Like terms give (\sqrt{a}+\sqrt{a}=2\sqrt{a}). Step 2: Since (a) is not a perfect square (\sqrt{a}) is irrational and its double is irrational. Step 3: Add like radicals like algebraic terms.
Correct answer: A. (2) and rational
Explanation: Step 1: This is a difference of squares form. Step 2: ((\sqrt{3}+1)(\sqrt{3}-1)=3-1=2) which is rational. Step 3: Product of conjugates often removes the radical.
Correct answer: B. Always irrational
Explanation: Step 1: Let (r) be rational and (s) be irrational. Step 2: If (r-s) were rational then (s=r-(r-s)) would be rational which is false. Step 3: Adding or subtracting a rational and an irrational number gives an irrational number.
Correct answer: A. (7\sqrt{3}) and irrational
Explanation: Step 1: (\sqrt{75}=5\sqrt{3}) and (\sqrt{12}=2\sqrt{3}). Step 2: The sum is (7\sqrt{3}) which is irrational. Step 3: Simplify radicals before adding them.
Correct answer: A. Because (p) has no square factor except (1)
Explanation: Step 1: A prime number (p) is not a perfect square. Step 2: If it is not a perfect square then (\sqrt{p}) cannot be rational. Step 3: Assuming the square root of a prime to be rational leads to a factor contradiction.
Correct answer: B. Rational
Explanation: Step 1: (x=\sqrt{10}) is irrational. Step 2: (x^2=(\sqrt{10})^2=10) which is rational. Step 3: The square of an irrational number can be rational.
