Practice focused topic-wise MCQs with answers and explanations for quick revision and exam preparation.
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If (p(x)=x-2+2\sqrt{7}x+6), what is the discriminant?
A. (4)
(D=\(2\sqrt{7}\)2-4\cdot1\cdot6=28-24=4)। अपरिमेय गुणांक होने पर भी विविक्तकर परिमेय हो सकता है। / (D=\(2\sqrt{7}\)2-4\cdot1\cdot6=28-24=4). Even with an irrational coefficient, the discriminant can be rational.
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If (p(x)=x-2+2\sqrt{7}x+6), what are its zeroes?
A. \(-\sqrt{7}+1\) और \(-\sqrt{7}-1\)\(-\sqrt{7}+1\) and \(-\sqrt{7}-1\)
सूत्र से \(x=\frac{-2\sqrt{7}\pm2}{2}=-\sqrt{7}\pm1\)। पहले विविक्तकर सरल करने से उत्तर साफ मिलता है। / Using the formula, \(x=\frac{-2\sqrt{7}\pm2}{2}=-\sqrt{7}\pm1\). Simplifying the discriminant first gives a clean answer.
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If the zeroes of \(x^2+px-3\) are \(2+\sqrt{7}\) and \(2-\sqrt{7}\), what is (p)?
A. (-4)
योग (4) है और एकक बहुपद में (p=-) योग होता है। गुणनफल (4-7=-3) भी स्थिर पद से मेल खाता है। / The sum is (4), and in a monic polynomial, (p=-) sum. The product (4-7=-3) also matches the constant term.
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If (p(x)=x-2-2x+n) has equal and irrational zeroes, which statement about (n) is correct?
A. ऐसा कोई वास्तविक (n) नहीं हैNo such real (n) exists
समान शून्यकों के लिए (D=0), यानी (4-4n=0), इसलिए (n=1)। तब शून्यक (1) है, जो अपरिमेय नहीं है। / For equal zeroes, (D=0), so (4-4n=0) and (n=1). Then the zero is (1), which is not irrational.
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A. \(\frac{3-\sqrt{5}}{2}\)
परिमेय गुणांकों में अपरिमेय भाग का संयुग्मी भी शून्यक होता है। इसलिए \(\frac{3-\sqrt{5}}{2}\) दूसरा शून्यक है। / With rational coefficients, the conjugate of the irrational part is also a zero. Hence \(\frac{3-\sqrt{5}}{2}\) is the other zero.
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A. \(x^2-3x+1\)
योग (3) और गुणनफल \(\frac{9-5}{4}=1\) है। इसलिए बहुपद \(x^2-3x+1\) है। / The sum is (3) and the product is \(\frac{9-5}{4}=1\). Therefore the polynomial is \(x^2-3x+1\).
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If (p(x)=x-2-2\sqrt{2}x+1), what is the type of its zeroes?
A. दो भिन्न वास्तविक अपरिमेयTwo distinct real irrational
(D=\(2\sqrt{2}\)2-4=8-4=4) है और शून्यक \(\sqrt{2}\pm1\) हैं। ये वास्तविक और अपरिमेय हैं। / (D=\(2\sqrt{2}\)2-4=8-4=4), and the zeroes are \(\sqrt{2}\pm1\). They are real and irrational.
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If (p(x)=x-2-\(\sqrt{2}+\sqrt{8}\)x+4), what is the sum of its zeroes?
A. \(3\sqrt{2}\)
योग \(\sqrt{2}+\sqrt{8}=\sqrt{2}+2\sqrt{2}=3\sqrt{2}\) है। मूलों को सरल करके ही अंतिम उत्तर दें। / The sum is \(\sqrt{2}+\sqrt{8}=\sqrt{2}+2\sqrt{2}=3\sqrt{2}\). Simplify radicals before giving the final answer.
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If (p(x)=x-2-11x+24) and (q(x)=x-2-11x+23), which statement is correct?
A. (p(x)) के शून्यक परिमेय और (q(x)) के अपरिमेय वास्तविक हैं(p(x)) has rational zeroes and (q(x)) has irrational real zeroes
(p(x)) के लिए (D=121-96=25) पूर्ण वर्ग है। (q(x)) के लिए (D=121-92=29) धनात्मक अपूर्ण वर्ग है। / For (p(x)), (D=121-96=25), a perfect square. For (q(x)), (D=121-92=29), positive and not a perfect square.
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If (p(x)=x-2-2x-3\sqrt{2}), what does the constant term tell about the zeroes?
A. शून्यकों का गुणनफल \(-3\sqrt{2}\) हैThe product of zeroes is \(-3\sqrt{2}\)
एकक द्विघात में स्थिर पद शून्यकों का गुणनफल होता है। यहाँ \(\alpha\beta=-3\sqrt{2}\) है। / In a monic quadratic, the constant term is the product of zeroes. Here \(\alpha\beta=-3\sqrt{2}\).
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