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1536 questions tagged with hard.
Which quadratic equation has sum of roots (0) and product (-81)?
A. \(x^2-81=0\)
\(मोनिक समीकरण (x^2-(\)योग)x+गुणनफल=0) है। योग (0) और गुणनफल (-81) रखने पर \(x^2-81=0\) मिलता है। \(/ The monic equation is (x^2-(\)sum)x+product\(=0). Using sum (0) and product (-81) gives (x^2-81=0).\)
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If (x=-1) is not a root of \(kx^2+3x+2=0\), what condition must (k) satisfy?
A. \(k\neq1\)
(x=-1) रखने पर बायां पक्ष (k-3+2=k-1) होता है। मूल न होने के लिए \(k-1\neq0\), इसलिए \(k\neq1\)। / Putting (x=-1), the left side becomes (k-3+2=k-1). For it not to be a root, \(k-1\neq0\), so \(k\neq1\).
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The roots of \(x^2+px+16=0\) are equal and positive. What is the value of (p)?
A. (-8)
समान मूलों के लिए \(p^2-64=0\) से \(p=\pm8\) मिलता है। धनात्मक समान मूल के लिए \(-\frac{p}{2}>0\), इसलिए (p=-8)। / For equal roots, \(p^2-64=0\) gives \(p=\pm8\). For equal positive roots, \(-\frac{p}{2}>0\), so (p=-8).
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If one root of \(x^2+mx+25=0\) is the reciprocal of the other, what can be said about (m)?
A. ऐसा संभव नहीं हैIt is not possible
एक मूल दूसरे का व्युत्क्रम हो तो मूलों का गुणनफल (1) होना चाहिए। यहाँ गुणनफल (25) है, इसलिए ऐसा संभव नहीं है। / If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (25), so it is not possible.
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What is the nature of roots of \(9x^2+12kx+4k^2=0\)?
A. दो समान वास्तविक मूलTwo equal real roots
यह ((3x+2k)2=0) के रूप में लिखा जा सकता है। इसलिए दोनों मूल समान वास्तविक होते हैं। / It can be written as ((3x+2k)2=0). Therefore both roots are equal and real.
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If ((x+a)2=x-2-12x+36), what is the value of (a)?
A. (-6)
((x+a)2=x-2+2ax+a-2) होता है। (2a=-12) से (a=-6) मिलता है। / ((x+a)2=x-2+2ax+a-2). From (2a=-12), we get (a=-6).
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Which equation will have equal and negative roots?
A. \(x^2+16x+64=0\)
(x-2+16x+64=(x+8)2), इसलिए दोनों मूल (-8) हैं। दोनों मूल बराबर और ऋणात्मक हैं। / (x-2+16x+64=(x+8)2), so both roots are (-8). Both roots are equal and negative.
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If \(\alpha,\beta\) are roots of \(x^2-7x+12=0\), what is \(\frac{1}{\alpha}+\frac{1}{\beta}\)?
A. \( \frac{7}{12} \)
\(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}\) होता है। यहाँ मान \(\frac{7}{12}\) है। / \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}\). Here the value is \(\frac{7}{12}\).
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Two numbers have sum (13) and product (40). They can be roots of which quadratic equation?
A. \(x^2-13x+40=0\)
यदि मूलों का योग (13) और गुणनफल (40) है, तो समीकरण \(x^2-13x+40=0\) होगा। मोनिक रूप का सूत्र याद रखें। / If the sum of roots is (13) and product is (40), the equation is \(x^2-13x+40=0\). Remember the monic form formula.
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A rectangle has length (x+7) and breadth (x-3). If its area is (90), which equation is correct?
A. \(x^2+4x-111=0\)
क्षेत्रफल ((x+7)(x-3)=90) होगा। विस्तार से \(x^2+4x-21=90\) और \(x^2+4x-111=0\) मिलता है। / The area is ((x+7)(x-3)=90). Expanding gives \(x^2+4x-21=90\) and \(x^2+4x-111=0\).
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In which option will the sum of roots be negative and the product of roots be positive?
A. \(x^2+7x+12=0\)
पहले विकल्प में योग \(-\frac{b}{a}=-7\) और गुणनफल \(\frac{c}{a}=12\) है। इसलिए योग ऋणात्मक और गुणनफल धनात्मक है। / In the first option, the sum is \(-\frac{b}{a}=-7\) and the product is \(\frac{c}{a}=12\). So the sum is negative and the product is positive.
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If both roots of \(x^2+bx+49=0\) are equal and their value is (-7), what is (b)?
A. (14)
दोनों मूल (-7) हैं, इसलिए योग (-14) है। यहाँ मूलों का योग (-b) है, अतः (b=14)। / Both roots are (-7), so their sum is (-14). Here the sum of roots is (-b), so (b=14).
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If \(x^2-14x+c=0\) is a perfect square quadratic equation, what is the value of (c)?
A. (49)
पूर्ण वर्ग के लिए (x-2-14x+c=(x-7)2) होना चाहिए। इसलिए (c=49) होगा। / For a perfect square, (x-2-14x+c=(x-7)2) is needed. Hence (c=49).
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What is the difference between the roots of \(2x^2-9x+10=0\)?
A. \( \frac{1}{2} \)
मूल (2) और \(\frac{5}{2}\) हैं। उनका अंतर \(\frac{5}{2}-2=\frac{1}{2}\) है। / The roots are (2) and \(\frac{5}{2}\). Their difference is \(\frac{5}{2}-2=\frac{1}{2}\).
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If one root of (x-2-(m+4)x+4m=0) is (4), what is the other root?
A. (m)
गुणनफल (4m) है और एक मूल (4) है। इसलिए दूसरा मूल \(\frac{4m}{4}=m\) है। / The product of roots is (4m) and one root is (4). Hence the other root is \(\frac{4m}{4}=m\).
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Which statement is correct for \(x^2-6x+11=0\)?
A. वास्तविक मूल नहीं हैंIt has no real roots
यहाँ (D=(-6)2-4\cdot1\cdot11=-8<0) है। इसलिए वास्तविक मूल नहीं होंगे। / Here (D=(-6)2-4\cdot1\cdot11=-8<0). Therefore it has no real roots.
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A. (18)
(r+2r=9) से (r=3) मिलता है, इसलिए मूल (3) और (6) हैं। स्थिर पद मूलों का गुणनफल (18) होगा। / From (r+2r=9), we get (r=3), so the roots are (3) and (6). The constant term will be the product of roots (18).
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What is the standard form of ((x+2)(x+5)+(x-1)(x+4)=60)?
A. \(2x^2+10x-54=0\)
बाईं ओर \(x^2+7x+10+x^2+3x-4=2x^2+10x+6\) है। (60) घटाने पर \(2x^2+10x-54=0\) मिलता है। / The left side is \(x^2+7x+10+x^2+3x-4=2x^2+10x+6\). Subtracting (60) gives \(2x^2+10x-54=0\).
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If \(\alpha\) and \(\beta\) are roots of \(x^2+7x+10=0\), what is (\(\alpha+2\)\(\beta+2\))?
A. (0)
(\(\alpha+2\)\(\beta+2\)=\alpha\beta+2\(\alpha+\beta\)+4) होता है। यहाँ (10+2(-7)+4=0)। / (\(\alpha+2\)\(\beta+2\)=\alpha\beta+2\(\alpha+\beta\)+4). Here (10+2(-7)+4=0).
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What is the sum of squares of the roots of \(3x^2+8x+4=0\)?
A. \( \frac{40}{9} \)
यदि मूल \(\alpha,\beta\) हैं, तो (\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta)। यहाँ (\left\(-\frac{8}{3}\right\)2-2\cdot\frac{4}{3}=\frac{40}{9})। / If the roots are \(\alpha,\beta\), then (\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta). Here (\left\(-\frac{8}{3}\right\)2-2\cdot\frac{4}{3}=\frac{40}{9}).
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If one root of \(x^2+ax+18=0\) is (2), what are the other root and (a)?
A. दूसरा मूल (9), (a=-11)other root (9), (a=-11)
मूलों का गुणनफल (18) है, इसलिए दूसरा मूल (9) होगा। योग (11) है और (-a=11), इसलिए (a=-11)। / The product of roots is (18), so the other root is (9). The sum is (11), and (-a=11), so (a=-11).
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Which equation has discriminant (-11)?
A. \(x^2+x+3=0\)
\(x^2+x+3=0\) के लिए \(D=1^2-4\cdot1\cdot3=-11\) है। विवेचक निकालते समय (4ac) पूरा घटाएं। / For \(x^2+x+3=0\), \(D=1^2-4\cdot1\cdot3=-11\). Subtract the full (4ac) while finding the discriminant.
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If the roots of \(x^2-8x+k=0\) are real and distinct, what is the correct condition on (k)?
A. (k<16)
भिन्न वास्तविक मूलों के लिए (D>0) चाहिए। यहाँ (64-4k>0), इसलिए (k<16)। / For real and distinct roots, (D>0) is needed. Here (64-4k>0), so (k<16).
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If (4) and (5) are roots of \(x^2-sx+p=0\), what is the value of (s+p)?
A. (29)
मूलों का योग (s=9) और गुणनफल (p=20) है। इसलिए (s+p=29) है। / The sum of roots gives (s=9) and the product gives (p=20). Therefore (s+p=29).
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What is the sum of reciprocals of the roots of \(3x^2-10x+7=0\)?
A. \( \frac{10}{7} \)
व्युत्क्रमों का योग \(\frac{\alpha+\beta}{\alpha\beta}\) होता है। यहाँ \(\frac{\frac{10}{3}}{\frac{7}{3}}=\frac{10}{7}\) है। / The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{10}{3}}{\frac{7}{3}}=\frac{10}{7}\).
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What is the condition for \(x^2+2kx+16=0\) to have real roots?
A. \(k\leq -4\) या \(k\geq 4\)\(k\leq -4\) or \(k\geq 4\)
वास्तविक मूलों के लिए \(D\geq0\) चाहिए। यहाँ \(4k^2-64\geq0\), इसलिए \(k\leq-4\) या \(k\geq4\)। / For real roots, \(D\geq0\) is needed. Here \(4k^2-64\geq0\), so \(k\leq-4\) or \(k\geq4\).
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If the roots of \(x^2-2kx+25=0\) are equal, what are the possible values of (k)?
A. \(k=\pm5\)
समान मूलों के लिए (D=0), इसलिए ((-2k)2-100=0) मिलता है। इससे \(k^2=25\) और \(k=\pm5\) है। / For equal roots, (D=0), so ((-2k)2-100=0). This gives \(k^2=25\) and \(k=\pm5\).
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Which monic quadratic equation has sum of roots (-9) and product (20)?
A. \(x^2+9x+20=0\)
\(मोनिक समीकरण (x^2-(\)योग)x+गुणनफल=0) होता है। योग (-9) रखने पर \(x^2+9x+20=0\) मिलता है। \(/ A monic equation is (x^2-(\)sum)x+product\(=0). Substituting sum (-9) gives (x^2+9x+20=0).\)
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If (x=-4) is a root of \(2x^2+px-12=0\), what is the value of (p)?
A. (5)
(x=-4) रखने पर (32-4p-12=0) मिलता है। इससे (p=5) है। / Putting (x=-4) gives (32-4p-12=0). Hence (p=5).
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Write \(\frac{x^2+1}{3}-\frac{x-2}{2}=5\) in standard form with integer coefficients.
A. \(2x^2-3x-22=0\)
पूरे समीकरण को (6) से गुणा करने पर \(2x^2+2-3x+6=30\) मिलता है। इसलिए मानक रूप \(2x^2-3x-22=0\) है। / Multiplying the whole equation by (6) gives \(2x^2+2-3x+6=30\). Therefore the standard form is \(2x^2-3x-22=0\).
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