roots se related questions ko ek jagah revise karein. Har question me bilingual content, answer feedback aur explanation available hai.
673 questions tagged with roots.
If the roots of \(x^2+px+q=0\) are (1) and (p), which relation is correct for (q)?
A. (q=p)
मूलों का गुणनफल (q) होता है। दिए मूल (1) और (p) हैं, इसलिए \(q=1\cdot p=p\)। / The product of roots is (q). The given roots are (1) and (p), so \(q=1\cdot p=p\).
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The roots of \(x^2-4x+1=0\) are \(\alpha,\beta\). What is \(\alpha+\beta+\alpha\beta\)?
A. (5)
मूलों का योग (4) और गुणनफल (1) है। इसलिए \(\alpha+\beta+\alpha\beta=5\)। / The sum of roots is (4) and the product is (1). Therefore \(\alpha+\beta+\alpha\beta=5\).
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If the roots of \(x^2+bx+c=0\) are (-2) and (7), what is the value of (b+c)?
A. (-19)
मूलों का योग (5) है, इसलिए (b=-5), और गुणनफल (-14) है, इसलिए (c=-14)। अतः (b+c=-19)। / The sum of roots is (5), so (b=-5), and the product is (-14), so (c=-14). Therefore (b+c=-19).
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What is the sum of cubes of the roots of \(x^2-2x-3=0\)?
A. (26)
मूलों का योग (2) और गुणनफल (-3) है। (\alpha-3+\beta-3=\(\alpha+\beta\)3-3\alpha\beta\(\alpha+\beta\)=8+18=26)। / The sum of roots is (2) and the product is (-3). (\alpha-3+\beta-3=\(\alpha+\beta\)3-3\alpha\beta\(\alpha+\beta\)=8+18=26).
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If in \(2x^2+kx+18=0\), the product of roots is (-3) times the sum of roots, what is (k)?
A. (6)
गुणनफल (9) और योग \(-\frac{k}{2}\) है। (9=-3\left\(-\frac{k}{2}\right\)) से (k=6) मिलता है। / The product is (9) and the sum is \(-\frac{k}{2}\). From (9=-3\left\(-\frac{k}{2}\right\)), we get (k=6).
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What is the difference between the roots of \(3x^2-10x+8=0\)?
A. \( \frac{2}{3} \)
मूल (2) और \(\frac{4}{3}\) हैं। उनका अंतर \(2-\frac{4}{3}=\frac{2}{3}\) है। / The roots are (2) and \(\frac{4}{3}\). Their difference is \(2-\frac{4}{3}=\frac{2}{3}\).
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If one root of (x-2-(m+2)x+2m=0) is (2), what is the other root?
A. (m)
गुणनफल (2m) है और एक मूल (2) है। इसलिए दूसरा मूल \(\frac{2m}{2}=m\) है। / The product of roots is (2m) and one root is (2). Hence the other root is \(\frac{2m}{2}=m\).
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If the roots of \(x^2+px+q=0\) are (p) and (q) themselves, which relation is correct?
A. (p+q=-p) और (pq=q)(p+q=-p) and (pq=q)
मूलों का योग (-p) और गुणनफल (q) होता है। दिए मूल (p,q) हैं, इसलिए (p+q=-p) और (pq=q)। / The sum of roots is (-p) and the product is (q). Since the roots are (p,q), we get (p+q=-p) and (pq=q).
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If \(\alpha\) and \(\beta\) are roots of \(x^2-8x+15=0\), what is (\(\alpha-1\)\(\beta-1\))?
A. (8)
(\(\alpha-1\)\(\beta-1\)=\alpha\beta-\(\alpha+\beta\)+1) होता है। यहाँ (15-8+1=8)। / (\(\alpha-1\)\(\beta-1\)=\alpha\beta-\(\alpha+\beta\)+1). Here (15-8+1=8).
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What is the sum of squares of the roots of \(2x^2+7x+3=0\)?
A. \( \frac{37}{4} \)
यदि मूल \(\alpha,\beta\) हैं, तो (\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta)। यहाँ (\left\(-\frac{7}{2}\right\)2-2\cdot\frac{3}{2}=\frac{37}{4})। / If the roots are \(\alpha,\beta\), then (\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta). Here (\left\(-\frac{7}{2}\right\)2-2\cdot\frac{3}{2}=\frac{37}{4}).
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If (2) and (3) are roots of \(x^2-sx+p=0\), what is the value of (s+p)?
A. (11)
मूलों का योग (s=5) और गुणनफल (p=6) है। इसलिए (s+p=11) है। / The sum of roots gives (s=5) and the product gives (p=6). Therefore (s+p=11).
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What is the sum of reciprocals of the roots of \(2x^2-5x+3=0\)?
A. \( \frac{5}{3} \)
व्युत्क्रमों का योग \(\frac{\alpha+\beta}{\alpha\beta}\) होता है। यहाँ \(\frac{\frac{5}{2}}{\frac{3}{2}}=\frac{5}{3}\) है। / The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{5}{2}}{\frac{3}{2}}=\frac{5}{3}\).
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If the roots of \(x^2+vx+28=0\) are (-4) and (-7), what is the value of (v)?
A. (11)
मूलों का योग (-11) है और \(x^2+vx+28=0\) में योग (-v) होता है। इसलिए (v=11) है। / The sum of roots is (-11), and in \(x^2+vx+28=0\), the sum is (-v). Therefore (v=11).
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Which pair is the correct roots of \(x^2-3x-40=0\)?
A. (8,-5)
((x-8)(x+5)=0) से मूल (8) और (-5) मिलते हैं। गुणनफल और योग दोनों मिलाकर जांचें। / From ((x-8)(x+5)=0), the roots are (8) and (-5). Check both product and sum.
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If the roots are (-4) and (-9), what is the product of the roots?
A. (36)
मूलों का गुणनफल ((-4)(-9)=36) है। दो ऋणात्मक संख्याओं का गुणनफल धनात्मक होता है। / The product of roots is ((-4)(-9)=36). The product of two negative numbers is positive.
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If (x=-6) and (x=1) are roots, what is the sum of the roots?
A. (-5)
मूलों का योग (-6+1=-5) है। विपरीत चिन्ह वाली संख्याओं में बड़े परिमाण का चिन्ह आता है। / The sum of roots is (-6+1=-5). For numbers with opposite signs, the sign of the larger magnitude is used.
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What are the roots of \(49x^2-64=0\)?
A. \(x=\pm \frac{8}{7}\)
\(49x^2=64\), इसलिए \(x^2=\frac{64}{49}\) और \(x=\pm\frac{8}{7}\) है। वर्गमूल लेते समय दोनों चिन्ह लें। / \(49x^2=64\), so \(x^2=\frac{64}{49}\) and \(x=\pm\frac{8}{7}\). Take both signs while taking square roots.
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What are the roots of \(x^2-10x=0\)?
A. (0,10)
(x-2-10x=x(x-10)), इसलिए मूल (0) और (10) हैं। समान गुणनखंड निकालना तेज तरीका है। / (x-2-10x=x(x-10)), so the roots are (0) and (10). Taking the common factor is a quick method.
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If the roots are (-5) and (4), what will be the monic quadratic equation?
A. \(x^2+x-20=0\)
गुणनखंड ((x+5)) और ((x-4)) होंगे। इन्हें गुणा करने पर \(x^2+x-20=0\) मिलता है। / The factors are ((x+5)) and ((x-4)). Multiplying them gives \(x^2+x-20=0\).
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Which option has roots (x=3) and (x=8)?
A. \(x^2-11x+24=0\)
मूल (3) और (8) हों तो गुणनखंड ((x-3)) और ((x-8)) होंगे। इनका गुणन \(x^2-11x+24=0\) देता है। / If the roots are (3) and (8), the factors are ((x-3)) and ((x-8)). Their product gives \(x^2-11x+24=0\).
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If the roots of \(x^2+sx+18=0\) are (-3) and (-6), what is the value of (s)?
A. (9)
मूलों का योग (-9) है और \(x^2+sx+18=0\) में योग (-s) होता है। इसलिए (s=9) है। / The sum of roots is (-9), and in \(x^2+sx+18=0\) the sum is (-s). Therefore (s=9).
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Which pair is the correct roots of \(x^2+2x-35=0\)?
A. (5,-7)
((x+7)(x-5)=0) से मूल (-7) और (5) मिलते हैं। गुणनफल और योग दोनों मिलाकर जांचें। / From ((x+7)(x-5)=0), the roots are (-7) and (5). Check both product and sum.
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If the roots are (-3) and (-8), what is the product of the roots?
A. (24)
मूलों का गुणनफल ((-3)(-8)=24) है। दो ऋणात्मक संख्याओं का गुणनफल धनात्मक होता है। / The product of roots is ((-3)(-8)=24). The product of two negative numbers is positive.
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If (x=-5) and (x=2) are roots, what is the sum of the roots?
A. (-3)
मूलों का योग (-5+2=-3) है। विपरीत चिन्ह वाली संख्याओं में बड़े परिमाण का चिन्ह आता है। / The sum of roots is (-5+2=-3). For numbers with opposite signs, the sign of the larger magnitude is used.
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What are the roots of \(16x^2-81=0\)?
A. \(x=\pm \frac{9}{4}\)
\(16x^2=81\), इसलिए \(x^2=\frac{81}{16}\) और \(x=\pm\frac{9}{4}\) है। वर्गमूल लेते समय दोनों चिन्ह लें। / \(16x^2=81\), so \(x^2=\frac{81}{16}\) and \(x=\pm\frac{9}{4}\). Take both signs while taking square roots.
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What are the roots of \(x^2+9x=0\)?
A. (0,-9)
(x-2+9x=x(x+9)), इसलिए मूल (0) और (-9) हैं। समान गुणनखंड निकालना जल्दी तरीका है। / (x-2+9x=x(x+9)), so the roots are (0) and (-9). Taking the common factor is a quick method.
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If the roots are (-4) and (3), what will be the monic quadratic equation?
A. \(x^2+x-12=0\)
गुणनखंड ((x+4)) और ((x-3)) होंगे। इनसे \(x^2+x-12=0\) मिलता है। / The factors will be ((x+4)) and ((x-3)). These give \(x^2+x-12=0\).
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Which option has roots (x=2) and (x=7)?
A. \(x^2-9x+14=0\)
मूल (2) और (7) हों तो गुणनखंड ((x-2)) और ((x-7)) होंगे। इनका गुणन \(x^2-9x+14=0\) देता है। / If the roots are (2) and (7), the factors are ((x-2)) and ((x-7)). Their product gives \(x^2-9x+14=0\).
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If the roots of \(x^2+ax+10=0\) are (-2) and (-5), what is the value of (a)?
A. (7)
मूलों का योग (-7) है और \(x^2+ax+10=0\) में योग (-a) होता है। इसलिए (a=7) है। / The sum of roots is (-7), and in \(x^2+ax+10=0\) the sum is (-a). Therefore (a=7).
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Which pair is the correct roots of \(x^2-2x-15=0\)?
B. (5,-3)
((x-5)(x+3)=0) से मूल (5) और (-3) मिलते हैं। ऐसे प्रश्नों में गुणनफल और योग दोनों मिलाएं। / From ((x-5)(x+3)=0), the roots are (5) and (-3). Match both product and sum in such questions.
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