roots se related questions ko ek jagah revise karein. Har question me bilingual content, answer feedback aur explanation available hai.
632 questions tagged with roots.
A. \(x^2-\frac{5}{6}x+\frac{1}{6}=0\)
पुराने योग (5) और गुणनफल (6) हैं। उलटे मूलों का योग \(\frac{5}{6}\) और गुणनफल \(\frac{1}{6}\) होगा। / The old sum is (5) and product is (6). The reciprocal roots have sum \(\frac{5}{6}\) and product \(\frac{1}{6}\).
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If roots of \(2x^2+px+8=0\) are (-1) and (-4), what is the value of (p)?
A. (10)
मूलों का योग (-5) है और \(-\frac{p}{2}=-5\) होगा। इसलिए (p=10) है। / The sum of roots is (-5), and \(-\frac{p}{2}=-5\). Therefore (p=10).
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A. \(\frac{5}{2}\)
यहां \(\alpha+\beta=12\) और \(\alpha\beta=32\) है। \(\alpha^2+\beta^2=144-64=80\) और \(\frac{80}{32}=\frac{5}{2}\) है। / Here \(\alpha+\beta=12\) and \(\alpha\beta=32\). \(\alpha^2+\beta^2=144-64=80\), and \(\frac{80}{32}=\frac{5}{2}\).
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If roots of \(x^2+kx+36=0\) are equal and negative, what is (k)?
A. (12)
बराबर ऋणात्मक मूल (-6) और (-6) हैं। योग (-12) है इसलिए (-k=-12) से (k=12) है। / The equal negative roots are (-6) and (-6). Their sum is (-12), so (-k=-12) gives (k=12).
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A. (7)
यहां \(\alpha+\beta=2\) और \(\alpha\beta=-8\) है। (\(\alpha+3\)\(\beta+3\)=-8+3(2)+9=7) है। / Here \(\alpha+\beta=2\) and \(\alpha\beta=-8\). (\(\alpha+3\)\(\beta+3\)=-8+3(2)+9=7).
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B. \(\frac{7}{3}\) या \(-\frac{7}{3}\)\(\frac{7}{3}\) or \(-\frac{7}{3}\)
मूल (5) और (2) हैं इसलिए योग (7) और अंतर \(\pm3\) हो सकता है। इसलिए मान \(\frac{7}{3}\) या \(-\frac{7}{3}\) है। / The roots are (5) and (2), so the sum is (7) and the difference can be \(\pm3\). Hence the value is \(\frac{7}{3}\) or \(-\frac{7}{3}\).
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If \(\alpha\) and \(\beta\) are roots of \(x^2-6x+8=0\), what is the value of \(\alpha^2\beta^2\)?
A. (64)
(\alpha-2\beta-2=\(\alpha\beta\)2) होता है। यहां \(\alpha\beta=8\) इसलिए मान (64) है। / (\alpha-2\beta-2=\(\alpha\beta\)2). Here \(\alpha\beta=8\), so the value is (64).
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A. (1)
समीकरण के मूल (5) और (6) हैं। इसलिए छोटा मूल (5) और बड़ा मूल (6) है तथा \(\beta-\alpha=1\) है। / The roots are (5) and (6). Thus the smaller root is (5) and the larger root is (6), so \(\beta-\alpha=1\).
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A. (8)
यहां \(\alpha+\beta=8\) और \(\alpha\beta=15\) है। (\(\alpha-1\)\(\beta-1\)=15-8+1=8) है। / Here \(\alpha+\beta=8\) and \(\alpha\beta=15\). (\(\alpha-1\)\(\beta-1\)=15-8+1=8).
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A. \(\frac{17}{8}\)
मूल (3) और (5) हैं इसलिए \(\alpha-1\) और \(\beta-1\) (2) और (4) होंगे। मान \(\frac{2}{4}+\frac{4}{2}=\frac{5}{2}\) है। / The roots are (3) and (5), so \(\alpha-1\) and \(\beta-1\) are (2) and (4). The value is \(\frac{2}{4}+\frac{4}{2}=\frac{5}{2}\).
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Which can be the roots of (x-2-(2m+1)x+m(m+1)=0)?
A. (m) और (m+1)(m) and (m+1)
(m+(m+1)=2m+1) और (m(m+1)) गुणनफल है। इसलिए मूल (m) और (m+1) हो सकते हैं। / (m+(m+1)=2m+1) and the product is (m(m+1)). Therefore the roots can be (m) and (m+1).
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A. (2)
यहां \(\alpha+\beta=-1\) और \(\alpha\beta=-1\) है। (\alpha-2+\alpha\beta+\beta-2=\(\alpha+\beta\)2-\alpha\beta=1-(-1)=2) है। / Here \(\alpha+\beta=-1\) and \(\alpha\beta=-1\). (\alpha-2+\alpha\beta+\beta-2=\(\alpha+\beta\)2-\alpha\beta=1-(-1)=2).
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If \(\alpha\) and \(\beta\) are roots of \(x^2-4x+1=0\), what is the value of \(\alpha^2+\beta^2\)?
A. (14)
यहां \(\alpha+\beta=4\) और \(\alpha\beta=1\) है। \(\alpha^2+\beta^2=16-2=14\) है। / Here \(\alpha+\beta=4\) and \(\alpha\beta=1\). \(\alpha^2+\beta^2=16-2=14\).
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If roots of \(x^2+px+q=0\) are (4) and (-1), what is the value of (p-q)?
A. (1)
योग (3) है इसलिए (p=-3) और गुणनफल (-4) है इसलिए (q=-4)। अतः (p-q=-3-(-4)=1) है। / The sum is (3), so (p=-3), and the product is (-4), so (q=-4). Thus (p-q=-3-(-4)=1).
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If (2) and (-3) are roots, what is the value of \(\frac{q}{3}\) in \(3x^2+px+q=0\)?
A. (-6)
मूलों का गुणनफल \(\frac{q}{3}\) होता है। (2\cdot(-3)=-6) इसलिए \(\frac{q}{3}=-6\) है। / The product of roots is \(\frac{q}{3}\). Since (2\cdot(-3)=-6), \(\frac{q}{3}=-6\).
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A. (4)
यहां \(\alpha+\beta=7\) और \(\alpha\beta=10\) है। (\(\alpha-1\)\(\beta-1\)=10-7+1=4) है। / Here \(\alpha+\beta=7\) and \(\alpha\beta=10\). (\(\alpha-1\)\(\beta-1\)=10-7+1=4).
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If roots of (x-2-2(m+1)x+\(m^2+2m+1\)=0) are equal, what is true for (m)?
A. हर वास्तविक (m)Every real (m)
अचर पद ((m+1)2) है और समीकरण ((x-(m+1))2=0) बनता है। इसलिए हर वास्तविक (m) के लिए मूल बराबर हैं। / The constant term is ((m+1)2), and the equation becomes ((x-(m+1))2=0). Hence roots are equal for every real (m).
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If roots of (x-2-2(m+1)x+\(m^2+2m\)=0) are equal, what is the value of (m)?
A. हर वास्तविक (m)Every real (m)
यहां (D=4(m+1)2-4\(m^2+2m\)=4) है। इसलिए मूल बराबर नहीं हो सकते। / Here (D=4(m+1)2-4\(m^2+2m\)=4). Therefore the roots cannot be equal.
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If (x-2-2x+(m+1)=0) has no real roots, which condition on (m) is correct?
A. (m>0)
वास्तविक मूल नहीं होने के लिए (D<0) चाहिए। (4-4(m+1)<0) से (m>0) मिलता है। / For no real roots (D<0) is required. From (4-4(m+1)<0), we get (m>0).
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If the product of roots of \(3x^2-2x+k=0\) equals the sum of roots, what is (k)?
A. (2)
योग \(-\frac{b}{a}=\frac{2}{3}\) और गुणनफल \(\frac{k}{3}\) है। \(\frac{k}{3}=\frac{2}{3}\) से (k=2) है। / The sum is \(-\frac{b}{a}=\frac{2}{3}\) and the product is \(\frac{k}{3}\). From \(\frac{k}{3}=\frac{2}{3}\), (k=2).
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If roots of \(x^2+bx+9=0\) are equal and positive, what is the value of (b)?
A. (-6)
बराबर धनात्मक मूल (3) और (3) होंगे। योग (6) है इसलिए (-b=6) से (b=-6) है। / The equal positive roots are (3) and (3). The sum is (6), so (-b=6) gives (b=-6).
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If roots of \(x^2+px+12=0\) are in the ratio (1:3) and both are negative, what is (p)?
A. (8)
मूल (-2) और (-6) होंगे क्योंकि गुणनफल (12) है। उनका योग (-8) है इसलिए (-p=-8) से (p=8) है। / The roots are (-2) and (-6) because the product is (12). Their sum is (-8), so (-p=-8) gives (p=8).
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If the roots of \(x^2-6x+c=0\) are in the ratio (1:2), what is the value of (c)?
A. (8)
मूल (t) और (2t) मानें तो योग (3t=6) से (t=2) है। मूल (2) और (4) होंगे इसलिए (c=8) है। / Let the roots be (t) and (2t). From sum (3t=6), (t=2), so roots are (2) and (4), hence (c=8).
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If the sum of roots is (4) and the sum of their squares is (20), what is the product of roots?
A. (-2)
(\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta) है। \(20=16-2\alpha\beta\) से \(\alpha\beta=-2\) मिलता है। / (\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta). From \(20=16-2\alpha\beta\), we get \(\alpha\beta=-2\).
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A. (9)
समीकरण के मूल (3) और (-6) हैं। इसलिए (\left|\alpha-\beta\right|=\left|3-(-6)\right|=9) है। / The roots are (3) and (-6). Therefore (\left|\alpha-\beta\right|=\left|3-(-6)\right|=9).
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If \(\alpha\) and \(\beta\) are roots of \(x^2-10x+25=0\), what is the value of \(\alpha-\beta\)?
A. (0)
यह ((x-5)2=0) है इसलिए दोनों मूल (5) और (5) हैं। अतः \(\alpha-\beta=0\) है। / This is ((x-5)2=0), so both roots are (5) and (5). Hence \(\alpha-\beta=0\).
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A. (9)
पुराने मूलों का योग (5) है। नए योग (\(\alpha+2\)+\(\beta+2\)=\alpha+\beta+4=9) होगा। / The old sum of roots is (5). The new sum is (\(\alpha+2\)+\(\beta+2\)=\alpha+\beta+4=9).
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A. \(x^2-6x+8=0\)
पुराने योग (3) और गुणनफल (2) हैं। नए योग (6) और गुणनफल (8) होंगे इसलिए \(x^2-6x+8=0\) है। / The old sum is (3) and product is (2). The new sum is (6) and product is (8), so the equation is \(x^2-6x+8=0\).
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If each root of \(x^2-4x-12=0\) is decreased by (1), what is the product of the new roots?
A. (-15)
पुराने मूल (6) और (-2) हैं। नए मूल (5) और (-3) होंगे इसलिए गुणनफल (-15) है। / The old roots are (6) and (-2). The new roots are (5) and (-3), so the product is (-15).
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A. \(x^2-6x-15=0\)
पुराने मूल (6) और (-2) हैं। नए मूल (7) और (-1) होंगे इसलिए समीकरण \(x^2-6x-7=0\) नहीं बल्कि \(x^2-6x-7=0\) होता है। / The old roots are (6) and (-2). The new roots are (7) and (-1), so the equation is \(x^2-6x-7=0\).
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