Practice focused topic-wise MCQs with answers and explanations for quick revision and exam preparation.
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If the roots of (x-2-(m+1)x+m=0) are reciprocals of each other, what is (m)?
A. (1)
व्युत्क्रम जड़ों के लिए \(\alpha\beta=1\) होता है। यहाँ \(\alpha\beta=m\), इसलिए (m=1)। / For reciprocal roots, \(\alpha\beta=1\). Here \(\alpha\beta=m\), so (m=1).
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If \(2x^2+kx+2=0\) has equal and negative roots, what is (k)?
A. (4)
समान जड़ों के लिए \(k^2-16=0\), इसलिए \(k=\pm4\)। समान जड़ \(-\frac{k}{4}\) है, जो ऋणात्मक तभी होगी जब (k=4)। / For equal roots, \(k^2-16=0\), so \(k=\pm4\). The equal root is \(-\frac{k}{4}\), which is negative only when (k=4).
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If \(\alpha,\beta\) are roots of \(x^2-4x+2=0\), what is (\(\alpha+2\)\(\beta+2\))?
C. (14)
(\(\alpha+2\)\(\beta+2\)=\alpha\beta+2\(\alpha+\beta\)+4) है। \(\alpha+\beta=4\) और \(\alpha\beta=2\), इसलिए मान (14) है। / (\(\alpha+2\)\(\beta+2\)=\alpha\beta+2\(\alpha+\beta\)+4). Since \(\alpha+\beta=4\) and \(\alpha\beta=2\), the value is (14).
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If the difference between the roots of \(x^2+6x+r=0\) is \(2\sqrt{5}\), what is (r)?
A. (4)
(\(\alpha-\beta\)2=\(\alpha+\beta\)2-4\alpha\beta) से (20=36-4r) मिलता है। इसलिए (r=4)। / Using (\(\alpha-\beta\)2=\(\alpha+\beta\)2-4\alpha\beta), we get (20=36-4r). Hence (r=4).
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If (x=0) is a root of \(ax^2+bx+c=0\), which condition must be true?
A. (c=0)
(x=0) रखने पर समीकरण (c=0) बनता है। इसलिए शून्य जड़ होने की सीधी शर्त (c=0) है। / Putting (x=0) gives (c=0). Thus the direct condition for zero to be a root is (c=0).
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What is the value of both roots of \(4x^2-4x+1=0\)?
A. \(\frac{1}{2}\)
(4x-2-4x+1=(2x-1)2) है। इसलिए समान जड़ \(x=\frac{1}{2}\) है। / (4x-2-4x+1=(2x-1)2). Therefore the equal root is \(x=\frac{1}{2}\).
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Which is the correct pair of roots of \(x^2+3x-18=0\)?
A. (3) और (-6)(3) and (-6)
गुणनफल (-18) और योग (-3) वाली संख्याएँ (3) और (-6) हैं। इसलिए यही जड़ों का सही युग्म है। / The numbers with product (-18) and sum (-3) are (3) and (-6). Hence this is the correct pair of roots.
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If \(x^2-sx+9=0\) has equal and positive roots, what is (s)?
A. (6)
समान जड़ों के लिए \(s^2-36=0\), इसलिए \(s=\pm6\)। समान जड़ \(\frac{s}{2}\) है, जो धनात्मक होने पर (s=6) देता है। / For equal roots, \(s^2-36=0\), so \(s=\pm6\). The equal root is \(\frac{s}{2}\), which is positive when (s=6).
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If ((k-2)x-2+4x+1=0) has equal roots, what is (k)?
A. (6)
समान जड़ों के लिए (D=0) रखें। (16-4(k-2)=0) से (k=6) मिलता है। / For equal roots, put (D=0). From (16-4(k-2)=0), we get (k=6).
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If \(\alpha,\beta\) are roots of \(x^2-6x+8=0\), what is (\(\alpha-2\)\(\beta-2\))?
A. (0)
(\(\alpha-2\)\(\beta-2\)=\alpha\beta-2\(\alpha+\beta\)+4) है। \(\alpha+\beta=6\) और \(\alpha\beta=8\), इसलिए मान (0) है। / (\(\alpha-2\)\(\beta-2\)=\alpha\beta-2\(\alpha+\beta\)+4). Since \(\alpha+\beta=6\) and \(\alpha\beta=8\), the value is (0).
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