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SubjectsMathematics
अपरिमेय संख्याएँ
In this Class 9 Mathematics topic from the Number Systems chapter, students learn that irrational numbers cannot be written in the form p/q, where p and q are integers and q is not zero. They explore familiar examples such as √2 and π, understand their non-terminating, non-repeating decimal expansions, and distinguish them from rational numbers. The topic also develops skills for representing irrational numbers on the number line and understanding their place within the real number system.
Correct answer: A. \(rx\) is irrational
Explanation: If \(rx\) were rational, then \(x=(rx)/r\) would be rational, a contradiction. Hence \(rx\) is irrational. But \(x^2\) need not be irrational: for \(x=\sqrt{2}\), \(x^2=2\). Exam tip: use contradiction by dividing by a non-zero rational number.
Correct answer: A. (2\sqrt{17})
Explanation: (\frac{1}{\sqrt{17}+4}=\sqrt{17}-4) because the denominator becomes (17-16=1). So the sum is (2\sqrt{17}).
Correct answer: A. Its decimal expansion is non-terminating and non-repeating.
Explanation: An irrational number has a non-terminating, non-repeating decimal expansion, so A is correct. B fails since \((\sqrt{2})^2=2\) is rational. Exam tip: repeating decimals are rational.
Correct answer: D. \(10+\sqrt{21}\)
Explanation: The square of a square root gives the number inside. So \(\left(\sqrt{10+\sqrt{21}}\right)^2=10+\sqrt{21}\).
Correct answer: B. (9)
Explanation: Area is ((\sqrt{14}+\sqrt{5})(\sqrt{14}-\sqrt{5})=14-5=9). Conjugate dimensions can give a rational area.
Correct answer: D. The decimal expansion is non-terminating and non-repeating.
Explanation: An irrational number has an endless decimal expansion with no repeating block. In contrast, \(1/8=0.125\) terminates, so it is rational. Exam tip: choose non-terminating, non-repeating decimals.
Correct answer: B. \(\sqrt{12},\,2-\sqrt{12}\)
Explanation: \(\sqrt{12}\) and \(2-\sqrt{12}\) are irrational; otherwise subtracting from 2 would make \(\sqrt{12}\) rational. Their sum is \(2\), which is rational. Option A gives \(5\sqrt{3}\). Check both conditions in such questions.
Correct answer: A. \(x+r\)
Explanation: \(x+r\) is irrational: if it were rational, subtracting the rational number \(r\) would make \(x\) rational, a contradiction. Also, \(x/x=1\) is rational. Exam tip: adding a rational number does not change irrationality.
Correct answer: C. (\sqrt{7.5})
Explanation: Since (7<7.5<8), (\sqrt{7.5}) lies between them. Compare square roots using the numbers inside.
Correct answer: A. \(\sqrt{3}\times\sqrt{3}=3\)
Explanation: \(\sqrt{3}\) is irrational because 3 is not a perfect square, yet \(\sqrt{3}\times\sqrt{3}=3\), a rational number. Option B still gives irrational \(\sqrt6\). Exam tip: one valid counterexample disproves an “always” statement.
Correct answer: C. Irrational number
Explanation: Since \(m\) is not a perfect square, \(\sqrt{m}\) is irrational. A rational \(a+b\sqrt{m}\) would give \((a+b\sqrt{m}-a)/b=\sqrt{m}\) rational, a contradiction. Tip: check \(b\ne0\).
Correct answer: A. (9\sqrt{3})
Explanation: (\sqrt{300}=10\sqrt{3}), (\sqrt{108}=6\sqrt{3}), and (\sqrt{75}=5\sqrt{3}). Therefore the result is (9\sqrt{3}).
Correct answer: A. \(rx\)
Explanation: If \(rx\) were rational, then \(x=\frac{rx}{r}\) would also be rational because \(r\neq0\) is rational. This is a contradiction. However, \(x^2\) can be rational, for example when \(x=\sqrt{2}\). Exam tip: multiplying by a non-zero rational preserves irrationality.
Correct answer: B. (14)
Explanation: (\sqrt{72}=6\sqrt{2}) and (\sqrt{128}=8\sqrt{2}), so the numerator is (14\sqrt{2}). Dividing gives (14).
Correct answer: A. (4\sqrt{14})
Explanation: The direct answer is A, 4√14. Use the identity r²−s²=(r−s)(r+s). With r=√7+√2 and s=√7−√2, r−s = (√7+√2)−(√7−√2)=2√2. Also, r+s=(√7+√2)+(√7−√2)=2√7. Multiplying gives (2√2)(2√7)=4√14, so option A is correct. Option B, 9, is not obtained and is too small for the exact expression. Option C, 2√14, misses a factor of 2. Option D, 14, incorrectly removes the radical and treats the result as a whole number. The difference-of-squares identity is much safer than expanding both squares separately. Memory cue: in r²−s², first find r−s and r+s; the middle terms then disappear automatically.
Correct answer: B. (9\sqrt{2})
Explanation: (\sqrt{242}=11\sqrt{2}), (\sqrt{50}=5\sqrt{2}), and (\sqrt{98}=7\sqrt{2}). Therefore (P=9\sqrt{2}).
Correct answer: A. (175)
Explanation: The direct answer is A, 175. Simplify first: √28 = √(4×7)=2√7 and √63=√(9×7)=3√7. Their sum is 5√7. Therefore (5√7)^2=25×7=175, so option A is correct. Option B, 91, is not the result of squaring the simplified sum. Option C, 25√7, stops after squaring the coefficient and forgets that (√7)^2=7; the radical must disappear. Option D, 700, is four times the correct value and usually comes from an incorrect treatment of the cross term. A direct check gives 28+63+2√(28×63)=91+2√1764=91+84=175. The key idea is to take perfect-square factors out of radicals before adding like radical terms.
Correct answer: A. \(x+r\)
Explanation: If \(x+r\) were rational, subtracting the rational number \(r\) would make \(x\) rational, which is a contradiction. \(x^2\) is not always irrational; for \(x=\sqrt{2}\), it equals 2. Exam tip: rational ± irrational is irrational.
Correct answer: A. \(x+3\)
Explanation: If \(x+3\) were rational, subtracting the rational number 3 would make \(x\) rational, a contradiction. However, \(x^2\) can be rational, for example when \(x=\sqrt{2}\). Exam tip: rational ± irrational is always irrational.
Correct answer: A. \(\sqrt{2}\times\sqrt{8}=4\)
Explanation: Both \(\sqrt{2}\) and \(\sqrt{8}\) are irrational, but their product is \(\sqrt{16}=4\), a rational number. Thus the word “always” makes the claim false. Exam tip: one valid counterexample is enough to disprove a universal statement.
Correct answer: A. (\frac{5(\sqrt{17}-\sqrt{8})}{9})
Explanation: Multiplying by the conjugate makes the denominator (17-8=9). So the form is (\frac{5(\sqrt{17}-\sqrt{8})}{9}).
Correct answer: A. (21-14\sqrt{2})
Explanation: ((\sqrt{14}-\sqrt{7})^2=14+7-2\sqrt{98}). Since (\sqrt{98}=7\sqrt{2}), the answer is (21-14\sqrt{2}).
Correct answer: C. (7)
Explanation: (\sqrt{45}=3\sqrt{5}) and (\sqrt{80}=4\sqrt{5}), so the numerator is (7\sqrt{5}). Dividing by (\sqrt{5}) gives (7).
Correct answer: D. \(x^2\)
Explanation: \(x^2\) need not be irrational. For example, \(x=\sqrt{2}\) is irrational, but \(x^2=2\) is rational. Adding a rational number to, or multiplying/dividing an irrational number by a non-zero rational, keeps it irrational. Exam tip: test such claims using \(\sqrt{2}\).
Correct answer: A. Always irrational
Explanation: Option A is correct. If \(qx\) were rational, then \(x=(qx)/q\) would also be rational because \(q\ne0\) is rational. This contradicts that \(x\) is irrational. Exam tip: always check the non-zero condition.
