Practice focused topic-wise MCQs with answers and explanations for quick revision and exam preparation.
Difficulty select karke sirf is topic ke focused MCQs practice karein. Score, timer aur explanations student-friendly flow me milenge.
Showing 411-420 of 649 questions.
A. (0,-5)
(\sqrt{5}+\(-\sqrt{5}\)=0) और (\sqrt{5}\cdot\(-\sqrt{5}\)=-5) है। विपरीत अपरिमेयों का योग शून्य हो सकता है। / (\sqrt{5}+\(-\sqrt{5}\)=0) and (\sqrt{5}\cdot\(-\sqrt{5}\)=-5). Opposite irrationals can have zero sum.
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Which polynomial has zeroes \(1+\sqrt{10}\) and \(1-\sqrt{10}\)?
A. \(x^2-2x-9\)
योग (2) और गुणनफल (1-10=-9) है। इसलिए बहुपद \(x^2-2x-9\) है। / The sum is (2) and the product is (1-10=-9). So the polynomial is \(x^2-2x-9\).
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If (p(x)=3x-2-12x+6), what are its zeroes?
A. \(2\pm\sqrt{2}\)
(3x-2-12x+6=3\(x^2-4x+2\)), इसलिए शून्यक \(2\pm\sqrt{2}\) हैं। पहले सामान्य गुणनखंड हटाना गणना आसान करता है। / Since (3x-2-12x+6=3\(x^2-4x+2\)), the zeroes are \(2\pm\sqrt{2}\). Removing a common factor first makes calculation easier.
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If (p(x)=x-2+4x+1), what are the sum and type of its zeroes?
A. योग (-4), दोनों अपरिमेय वास्तविकSum (-4), both irrational real
योग \(-\frac{b}{a}=-4\) और (D=16-4=12) है, जो पूर्ण वर्ग नहीं है। इसलिए दोनों शून्यक अपरिमेय वास्तविक हैं। / The sum is \(-\frac{b}{a}=-4\) and (D=16-4=12), not a perfect square. Hence both zeroes are irrational real.
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If the zeroes are \(\frac{1+\sqrt{3}}{2}\) and \(\frac{1-\sqrt{3}}{2}\), what is their product?
A. \(-\frac{1}{2}\)
गुणनफल (\frac{\(1+\sqrt{3}\)\(1-\sqrt{3}\)}{4}=\frac{1-3}{4}=-\frac{1}{2}) है। संयुग्मी गुणनफल में \(a^2-b\) प्रयोग करें। / The product is (\frac{\(1+\sqrt{3}\)\(1-\sqrt{3}\)}{4}=\frac{1-3}{4}=-\frac{1}{2}). Use \(a^2-b\) for conjugate products.
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For which value of (k) will the zeroes of \(x^2-2x+k\) be real and irrational?
C. (k=-1)
यहाँ (D=4-4k) है। (k=-1) पर (D=8), जो धनात्मक पूर्ण वर्ग नहीं है। / Here (D=4-4k). For (k=-1), (D=8), which is positive and not a perfect square.
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If (p(x)=x-2-8x+7), how do its zeroes compare with those of \(x^2-8x+10\)?
A. पहले के परिमेय, दूसरे के अपरिमेय वास्तविकFirst are rational, second are irrational real
पहले में (D=64-28=36) पूर्ण वर्ग है, दूसरे में (D=64-40=24) पूर्ण वर्ग नहीं है। विविक्तकर से शून्यकों का प्रकार तुरंत पता चलता है। / For the first, (D=64-28=36) is a perfect square; for the second, (D=64-40=24) is not. The discriminant quickly tells the type of zeroes.
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A. \(a^2-b\)
एकक बहुपद में स्थिर पद शून्यकों का गुणनफल होता है। यहाँ गुणनफल (\(a+\sqrt{b}\)\(a-\sqrt{b}\)=a-2-b) है। / In a monic polynomial, the constant term is the product of zeroes. Here the product is (\(a+\sqrt{b}\)\(a-\sqrt{b}\)=a-2-b).
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Which polynomial has real zeroes but not rational zeroes?
C. \(x^2-8\)
\(x^2-8=0\) से \(x=\pm2\sqrt{2}\), जो अपरिमेय वास्तविक हैं। पूर्ण वर्ग और धनात्मकता दोनों जाँचें। / From \(x^2-8=0\), \(x=\pm2\sqrt{2}\), which are irrational real. Check both perfect-square status and positivity.
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B. कम से कम एक गुणांक अपरिमेय होगाAt least one coefficient will be irrational
योग \(\sqrt{2}+\sqrt{3}\) अपरिमेय है, इसलिए एकक बहुपद में (x) का गुणांक अपरिमेय होगा। परिमेय गुणांक के लिए ऐसे शून्यक संयुग्मी रूप में होने चाहिए। / The sum \(\sqrt{2}+\sqrt{3}\) is irrational, so the coefficient of (x) in the monic polynomial is irrational. For rational coefficients, such zeroes must occur as conjugates.
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