Level 28 • 21/50 questions • 35 seconds per question.
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What is the discriminant (D) of \(4x^2+4x+1=0\)?
A. (0)
यहाँ \(D=b^2-4ac=4^2-4\cdot4\cdot1=0\) है। विवेचक में (a,b,c) सही पहचानना जरूरी है। / Here \(D=b^2-4ac=4^2-4\cdot4\cdot1=0\). Correctly identifying (a,b,c) is necessary for the discriminant.
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If (D=25), what is correct about the real roots of the quadratic equation?
A. दो भिन्न वास्तविक मूल होंगेThere will be two distinct real roots
(D=25>0), इसलिए दो भिन्न वास्तविक मूल होंगे। (D) का चिन्ह मूलों की प्रकृति बताता है। / Since (D=25>0), there will be two distinct real roots. The sign of (D) tells the nature of roots.
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What is the nature of roots of \(x^2+6x+9=0\)?
B. दो समान वास्तविक मूलTwo equal real roots
यहाँ \(D=6^2-4\cdot1\cdot9=0\), इसलिए मूल समान हैं। (D=0) का अर्थ दो समान वास्तविक मूल होता है। / Here \(D=6^2-4\cdot1\cdot9=0\), so the roots are equal. (D=0) means two equal real roots.
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Which option has roots (x=1) and (x=4)?
A. \(x^2-5x+4=0\)
मूल (1) और (4) हों तो गुणनखंड ((x-1)) और ((x-4)) होंगे। इनका गुणन \(x^2-5x+4=0\) देता है। / If the roots are (1) and (4), the factors are ((x-1)) and ((x-4)). Their product gives \(x^2-5x+4=0\).
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If the roots are (-2) and (5), what will be the monic quadratic equation?
A. \(x^2-3x-10=0\)
गुणनखंड ((x+2)) और ((x-5)) होंगे, जिनसे \(x^2-3x-10=0\) मिलता है। मूल के विपरीत चिन्ह वाले गुणनखंड बनाएं। / The factors are ((x+2)) and ((x-5)), giving \(x^2-3x-10=0\). Make factors with signs opposite to the roots.
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What are the roots of \(x^2-8x=0\)?
A. (0,8)
(x-2-8x=x(x-8)), इसलिए मूल (0) और (8) हैं। समान गुणनखंड बाहर निकालना आसान तरीका है। / (x-2-8x=x(x-8)), so the roots are (0) and (8). Taking the common factor is an easy method.
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What are the roots of \(9x^2-25=0\)?
A. \(x=\pm \frac{5}{3}\)
\(9x^2=25\), इसलिए \(x^2=\frac{25}{9}\) और \(x=\pm \frac{5}{3}\)। वर्गमूल लेते समय दोनों चिन्ह लें। / \(9x^2=25\), so \(x^2=\frac{25}{9}\) and \(x=\pm \frac{5}{3}\). Take both signs while taking square roots.
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If (x=2) and (x=-3) are roots of a quadratic equation, what is the sum of the roots?
B. (-1)
मूलों का योग (2+(-3)=-1) है। अलग-अलग चिन्ह वाले पूर्णांकों को जोड़ते समय बड़े परिमाण का चिन्ह लें। / The sum of roots is (2+(-3)=-1). When adding integers with unlike signs, take the sign of the larger magnitude.
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If the roots are (4) and (6), what is the product of the roots?
B. (24)
मूलों का गुणनफल \(4\cdot6=24\) है। मूलों से समीकरण बनाते समय योग और गुणनफल दोनों उपयोगी होते हैं। / The product of roots is \(4\cdot6=24\). Sum and product are both useful while forming an equation from roots.
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Which quadratic equation has sum of roots (5) and product (6)?
A. \(x^2-5x+6=0\)
\(मोनिक समीकरण (x^2-(\)योग)x+गुणनफल=0) होता है। इसलिए \(x^2-5x+6=0\) सही है। \(/ A monic equation is (x^2-(\)sum)x+product\(=0). Hence (x^2-5x+6=0) is correct.\)
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What is the sum of roots of \(2x^2-5x+3=0\)?
A. \(\frac{5}{2}\)
मूलों का योग \(-\frac{b}{a}=-\frac{-5}{2}=\frac{5}{2}\) है। सूत्र में (b) का चिन्ह ध्यान से रखें। / The sum of roots is \(-\frac{b}{a}=-\frac{-5}{2}=\frac{5}{2}\). Pay attention to the sign of (b) in the formula.
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What is the product of roots of \(3x^2+2x-8=0\)?
A. \(-\frac{8}{3}\)
मूलों का गुणनफल \(\frac{c}{a}=\frac{-8}{3}\) है। (c) का चिन्ह सीधे सूत्र में रखें। / The product of roots is \(\frac{c}{a}=\frac{-8}{3}\). Use the sign of (c) directly in the formula.
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If the sum of roots of \(2x^2+mx+8=0\) is (-3), what is (m)?
A. (6)
मूलों का योग \(-\frac{m}{2}=-3\), इसलिए (m=6) है। योग के सूत्र में ऋण चिन्ह पर ध्यान दें। / The sum of roots is \(-\frac{m}{2}=-3\), so (m=6). Notice the negative sign in the sum formula.
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If the product of roots of \(kx^2-4x+2=0\) is (1), what is the value of (k)?
B. (2)
मूलों का गुणनफल \(\frac{2}{k}=1\), इसलिए (k=2) है। गुणनफल के सूत्र में (c/a) प्रयोग करें। / The product of roots is \(\frac{2}{k}=1\), so (k=2). Use (c/a) for the product formula.
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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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If the roots of \(x^2+px+16=0\) are (4) and (4), what is the value of (p)?
B. (-8)
मूलों का योग (8) है और \(x^2+px+16=0\) में योग (-p) होता है। इसलिए (p=-8) है। / The sum of roots is (8), and in \(x^2+px+16=0\) the sum is (-p). Therefore (p=-8).
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Which equation will have roots that are opposite numbers?
A. \(x^2-16=0\)
\(x^2-16=0\) से \(x=\pm4\) मिलता है, जो विपरीत संख्याएँ हैं। शुद्ध द्विघात में अक्सर विपरीत मूल मिलते हैं। / \(x^2-16=0\) gives \(x=\pm4\), which are opposite numbers. Pure quadratics often give opposite roots.
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If the roots of \(x^2-6x+k=0\) are equal, what is (k)?
B. (9)
समान मूलों के लिए (D=0), इसलिए (36-4k=0) और (k=9)। समान मूल की शर्त में विवेचक शून्य रखें। / For equal roots, (D=0), so (36-4k=0) and (k=9). Set the discriminant to zero for equal roots.
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In \(x^2+kx+25=0\), if the roots are equal and negative, which possible value of (k) is correct?
A. (10)
समान मूलों के लिए (D=0) से \(k^2=100\), और ऋणात्मक समान मूल के लिए \(-\frac{k}{2}<0\) चाहिए। इसलिए (k=10) सही है। / For equal roots, (D=0) gives \(k^2=100\), and for equal negative roots \(-\frac{k}{2}<0\) is needed. Hence (k=10) is correct.
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Which pair of numbers is useful to identify the roots of \(x^2-11x+30=0\)?
A. (5) और (6)(5) and (6)
\(5\cdot6=30\) और (5+6=11) होता है। ऋण मध्य पद के कारण गुणनखंड ((x-5)(x-6)) बनते हैं। / \(5\cdot6=30\) and (5+6=11). Because the middle term is negative, the factors are ((x-5)(x-6)).
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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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