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.
What is the product of roots of \(7x^2+4x-9=0\)?
A. -\(\frac{9}{7}\)
मूलों का गुणनफल \(\frac{c}{a}\) होता है। यहां \(\frac{-9}{7}=-\frac{9}{7}\) सही है। / The product of roots is \(\frac{c}{a}\). Here \(\frac{-9}{7}=-\frac{9}{7}\) is correct.
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What is the sum of roots of \(6x^2-13x+5=0\)?
A. \(\frac{13}{6}\)
मूलों का योग \(-\frac{b}{a}\) होता है। यहां (a=6) और (b=-13) हैं इसलिए योग \(\frac{13}{6}\) है। / The sum of roots is \(-\frac{b}{a}\). Here (a=6) and (b=-13), so the sum is \(\frac{13}{6}\).
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If (4) is a root of \(x^2+kx-32=0\), what is the value of (k)?
A. (4)
(x=4) रखने पर (16+4k-32=0) इसलिए (k=4) है। पैरामीटर के लिए दिए गए मूल को सीधे रखें। / Putting (x=4) gives (16+4k-32=0), so (k=4). Substitute the given root directly for a parameter.
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Which equation has roots (3) and (-8)?
A. \(x^2+5x-24=0\)
मूल (3) और (-8) होने पर ((x-3)(x+8)=0) होगा। खोलने पर \(x^2+5x-24=0\) मिलता है। / With roots (3) and (-8), we get ((x-3)(x+8)=0). Expanding gives \(x^2+5x-24=0\).
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What are the roots of \(5x^2+16x+3=0\)?
A. (-3) और \(-\frac{1}{5}\)(-3) and \(-\frac{1}{5}\)
(5x-2+16x+3=(5x+1)(x+3)) है। इसलिए मूल \(-\frac{1}{5}\) और (-3) हैं। / (5x-2+16x+3=(5x+1)(x+3)). Therefore the roots are \(-\frac{1}{5}\) and (-3).
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What are the roots of \(4x^2-17x+4=0\)?
A. (4) और \(\frac{1}{4}\)(4) and \(\frac{1}{4}\)
(4x-2-17x+4=(4x-1)(x-4)) है। इसलिए मूल \(\frac{1}{4}\) और (4) हैं। / (4x-2-17x+4=(4x-1)(x-4)). Therefore the roots are \(\frac{1}{4}\) and (4).
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What are the roots of \(x^2-14x+45=0\)?
A. (5) और (9)(5) and (9)
(x-2-14x+45=(x-5)(x-9)) है। इसलिए मूल (5) और (9) हैं। / (x-2-14x+45=(x-5)(x-9)). Therefore the roots are (5) and (9).
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Is (x=-3) a root of \(2x^2+7x+3=0\)?
A. हाँYes
(x=-3) रखने पर (18-21+3=0) मिलता है। ऋणात्मक मूल रखते समय चिन्हों को ध्यान से देखें। / Putting (x=-3) gives (18-21+3=0). Watch the signs carefully when substituting a negative root.
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Is (x=5) a root of \(x^2-10x+25=0\)?
A. हाँYes
(x=5) रखने पर (25-50+25=0) मिलता है। इसलिए (5) इसका मूल है। / Putting (x=5) gives (25-50+25=0). Therefore (5) is its root.
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If (q(t)=0) in a quadratic equation (q(x)=0), what is (t) called?
A. मूलRoot
(q(t)=0) का अर्थ है कि (x=t) रखने पर समीकरण सत्य है। इसलिए (t) उस समीकरण का मूल है। / (q(t)=0) means the equation is true when (x=t). Therefore (t) is a root of the equation.
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A. (2)
यहां \(\alpha+\beta=\frac{7}{2}\) और \(\alpha\beta=\frac{3}{2}\) है। इसलिए \(\alpha+\beta-\alpha\beta=\frac{7}{2}-\frac{3}{2}=2\) होगा। / Here \(\alpha+\beta=\frac{7}{2}\) and \(\alpha\beta=\frac{3}{2}\). Therefore \(\alpha+\beta-\alpha\beta=\frac{7}{2}-\frac{3}{2}=2\).
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For \(x^2-18x+81=0\), what are the sum and product of roots respectively?
A. (18) और (81)(18) and (81)
यह ((x-9)2=0) है इसलिए दोनों मूल (9) हैं। योग (18) और गुणनफल (81) होगा। / It is ((x-9)2=0), so both roots are (9). The sum is (18) and the product is (81).
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If (x=3) is a root of \(ax^2-8x+6=0\), what is the value of (a)?
A. (2)
(x=3) रखने पर (9a-24+6=0) इसलिए (9a=18) और (a=2)। अज्ञात गुणांक वाले प्रश्न में प्रतिस्थापन करें। / Putting (x=3) gives (9a-24+6=0), so (9a=18) and (a=2). Substitute in questions with an unknown coefficient.
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What is the sum of squares of the roots of \(x^2-7x+10=0\)?
A. (29)
यहां \(\alpha+\beta=7\) और \(\alpha\beta=10\) है। \(\alpha^2+\beta^2=49-20=29\) होगा। / Here \(\alpha+\beta=7\) and \(\alpha\beta=10\). Thus \(\alpha^2+\beta^2=49-20=29\).
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A. \(x^2+5x-14=0\)
मोनिक समीकरण (x-2-\(\alpha+\beta\)x+\alpha\beta=0) होता है। इसलिए \(x^2+5x-14=0\) सही है। / The monic equation is (x-2-\(\alpha+\beta\)x+\alpha\beta=0). Therefore \(x^2+5x-14=0\) is correct.
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A. \(-b^2\)
\(\frac{c}{a}\) मूलों का गुणनफल होता है। यहां (b(-b)=-b-2) है। / \(\frac{c}{a}\) is the product of roots. Here (b(-b)=-b-2).
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What are the roots of \(8x^2-10x+3=0\)?
A. \(\frac{1}{2}\) और \(\frac{3}{4}\)\(\frac{1}{2}\) and \(\frac{3}{4}\)
(8x-2-10x+3=(2x-1)(4x-3)) है। इसलिए मूल \(\frac{1}{2}\) और \(\frac{3}{4}\) हैं। / (8x-2-10x+3=(2x-1)(4x-3)). Therefore the roots are \(\frac{1}{2}\) and \(\frac{3}{4}\).
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If the roots of \(x^2+ax+16=0\) are (4) and (4), what is the value of (a)?
A. (-8)
मूलों का योग (8) है और (-a=8) होगा। इसलिए (a=-8) है। / The sum of roots is (8), and (-a=8). Therefore (a=-8).
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What will be the signs of both roots of \(x^2+9x+18=0\)?
A. दोनों ऋणात्मकBoth negative
मूल (-3) और (-6) हैं क्योंकि ((x+3)(x+6)=0)। योग ऋणात्मक और गुणनफल धनात्मक होने पर दोनों मूल ऋणात्मक होते हैं। / The roots are (-3) and (-6) because ((x+3)(x+6)=0). When the sum is negative and product is positive, both roots are negative.
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For \(x^2+4x+k=0\) to have real roots, which condition on (k) is correct?
A. \(k\le4\)
वास्तविक मूलों के लिए \(D\ge0\) चाहिए। \(16-4k\ge0\) से \(k\le4\) मिलता है। / For real roots \(D\ge0\) is required. From \(16-4k\ge0\), we get \(k\le4\).
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If \(\alpha\) and \(\beta\) are roots of \(5x^2+3x-2=0\), what is \(\alpha\beta\)?
A. -\(\frac{2}{5}\)
मूलों का गुणनफल \(\frac{c}{a}=\frac{-2}{5}\) होता है। (c) का चिन्ह न भूलें। / The product of roots is \(\frac{c}{a}=\frac{-2}{5}\). Do not forget the sign of (c).
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If \(\alpha\) and \(\beta\) are roots of \(3x^2-12x+5=0\), what is \(\alpha+\beta\)?
A. (4)
मूलों का योग \(-\frac{b}{a}=-\frac{-12}{3}=4\) है। सूत्र में (b) का चिन्ह सही रखें। / The sum of roots is \(-\frac{b}{a}=-\frac{-12}{3}=4\). Keep the sign of (b) correct in the formula.
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What is the difference between the roots of \(x^2-6x-7=0\)?
A. (8)
(x-2-6x-7=(x-7)(x+1)) है। मूल (7) और (-1) हैं इसलिए अंतर (8) है। / (x-2-6x-7=(x-7)(x+1)). The roots are (7) and (-1), so the difference is (8).
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If (D=0) and the sum of roots is (12), what will each root be?
A. (6)
(D=0) होने पर दोनों मूल बराबर होते हैं। इसलिए प्रत्येक मूल \(\frac{12}{2}=6\) होगा। / When (D=0), both roots are equal. Therefore each root is \(\frac{12}{2}=6\).
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What is the positive root of \(x^2-5x-24=0\)?
A. (8)
(x-2-5x-24=(x-8)(x+3)) है। इसलिए मूल (8) और (-3) हैं तथा धनात्मक मूल (8) है। / (x-2-5x-24=(x-8)(x+3)). Therefore the roots are (8) and (-3), and the positive root is (8).
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A. वे एक दूसरे के व्युत्क्रम हैंThey are reciprocals of each other
\(5\cdot\frac{1}{5}=1\) है इसलिए दोनों व्युत्क्रम मूल हैं। व्युत्क्रम मूलों का गुणनफल (1) होता है। / \(5\cdot\frac{1}{5}=1\), so the roots are reciprocals. Reciprocal roots have product (1).
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If the roots of \(x^2+mx+20=0\) are (-4) and (-5), what is the value of (m)?
A. (9)
मूलों का योग (-9) है और (-m=-9) होगा। इसलिए (m=9) है। / The sum of roots is (-9), and (-m=-9). Therefore (m=9).
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What are the roots of \(3x^2+2x-8=0\)?
A. \(\frac{4}{3}\) और (-2)\(\frac{4}{3}\) and (-2)
(3x-2+2x-8=(3x-4)(x+2)) है। इसलिए मूल \(\frac{4}{3}\) और (-2) हैं। / (3x-2+2x-8=(3x-4)(x+2)). Therefore the roots are \(\frac{4}{3}\) and (-2).
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If (x=2) and (x=7) are roots of a monic quadratic equation, what will be the coefficient of (x)?
A. (-9)
मूलों का योग (2+7=9) है। मोनिक समीकरण में (x) का गुणांक (-(योग)=-9) होता है। \(/ The sum of roots is (2+7=9). In a monic equation the coefficient of (x) is (-(\)sum\()=-9).\)
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If (x=0) is a root of \(ax^2+bx+c=0\) and \(a\ne0\), which conclusion is correct?
A. (c=0)
(x=0) रखने पर समीकरण (c=0) बनता है। इसलिए शून्य मूल के लिए अचर पद शून्य होता है। / Putting (x=0) makes the equation (c=0). Therefore the constant term is zero for a zero root.
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