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1329 questions tagged with expert.
If one root of (x-2-(m+7)x+7m=0) is (7), what is the other root?
A. (m)
मूलों का गुणनफल (7m) है और एक मूल (7) है। इसलिए दूसरा मूल \(\frac{7m}{7}=m\) है। / The product of roots is (7m) and one root is (7). Hence the other root is \(\frac{7m}{7}=m\).
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Which statement is correct for \(x^2-12x+40=0\)?
A. वास्तविक मूल नहीं हैंIt has no real roots
यहाँ (D=(-12)2-4\cdot1\cdot40=-16<0) है। इसलिए वास्तविक मूल नहीं होंगे। / Here (D=(-12)2-4\cdot1\cdot40=-16<0). Therefore it has no real roots.
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A. (192)
(3r+4r=28) से (r=4) मिलता है, इसलिए मूल (12) और (16) हैं। स्थिर पद मूलों का गुणनफल (192) होगा। / From (3r+4r=28), we get (r=4), so the roots are (12) and (16). The constant term is the product of roots (192).
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What is the standard form of ((x+5)(x+8)+(x-4)(x+7)=110)?
A. \(2x^2+16x-98=0\)
बाईं ओर \(x^2+13x+40+x^2+3x-28=2x^2+16x+12\) है। (110) घटाने पर \(2x^2+16x-98=0\) मिलता है। / The left side is \(x^2+13x+40+x^2+3x-28=2x^2+16x+12\). Subtracting (110) gives \(2x^2+16x-98=0\).
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If \(\alpha\) and \(\beta\) are roots of \(x^2-18x+80=0\), what is (\(\alpha-8\)\(\beta-8\))?
A. (0)
(\(\alpha-8\)\(\beta-8\)=\alpha\beta-8\(\alpha+\beta\)+64) है। यहाँ (80-144+64=0)। / (\(\alpha-8\)\(\beta-8\)=\alpha\beta-8\(\alpha+\beta\)+64). Here (80-144+64=0).
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What is the sum of squares of the roots of \(6x^2+7x+2=0\)?
A. \( \frac{25}{36} \)
(\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta) होता है। यहाँ (\left\(-\frac{7}{6}\right\)2-2\cdot\frac{1}{3}=\frac{25}{36})। / (\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta). Here (\left\(-\frac{7}{6}\right\)2-2\cdot\frac{1}{3}=\frac{25}{36}).
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If one root of \(x^2+ax+54=0\) is (6), what are the other root and (a)?
A. दूसरा मूल (9), (a=-15)other root (9), (a=-15)
मूलों का गुणनफल (54) है, इसलिए दूसरा मूल (9) होगा। योग (15) है और (-a=15), इसलिए (a=-15)। / The product of roots is (54), so the other root is (9). The sum is (15), and (-a=15), so (a=-15).
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Which equation has discriminant (-47)?
A. \(x^2+x+12=0\)
\(x^2+x+12=0\) के लिए \(D=1^2-4\cdot1\cdot12=-47\) है। विवेचक में (4ac) पूरा घटाएं। / For \(x^2+x+12=0\), \(D=1^2-4\cdot1\cdot12=-47\). Subtract the full (4ac) in the discriminant.
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If the roots of \(x^2-20x+k=0\) are real and distinct, what is the correct condition on (k)?
A. (k<100)
भिन्न वास्तविक मूलों के लिए (D>0) चाहिए। यहाँ (400-4k>0), इसलिए (k<100)। / For real and distinct roots, (D>0) is needed. Here (400-4k>0), so (k<100).
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If (8) and (9) are roots of \(x^2-sx+p=0\), what is the value of (s+p)?
A. (89)
मूलों का योग (s=17) और गुणनफल (p=72) है। इसलिए (s+p=89) है। / The sum of roots gives (s=17) and the product gives (p=72). Therefore (s+p=89).
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What is the sum of reciprocals of the roots of \(6x^2-19x+10=0\)?
A. \( \frac{19}{10} \)
व्युत्क्रमों का योग \(\frac{\alpha+\beta}{\alpha\beta}\) होता है। यहाँ \(\frac{\frac{19}{6}}{\frac{10}{6}}=\frac{19}{10}\) है। / The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{19}{6}}{\frac{10}{6}}=\frac{19}{10}\).
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What is the condition for \(x^2+2px+49=0\) to have real roots?
A. \(p\leq -7\) या \(p\geq7\)\(p\leq -7\) or \(p\geq7\)
वास्तविक मूलों के लिए \(D\geq0\) होना चाहिए। यहाँ \(4p^2-196\geq0\), इसलिए \(p\leq-7\) या \(p\geq7\)। / For real roots, \(D\geq0\) is required. Here \(4p^2-196\geq0\), so \(p\leq-7\) or \(p\geq7\).
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If the roots of \(x^2-2mx+64=0\) are equal, what are the possible values of (m)?
A. \(m=\pm8\)
समान मूलों के लिए (D=0) चाहिए। यहाँ ((-2m)2-256=0), इसलिए \(m=\pm8\)। / For equal roots, (D=0) is needed. Here ((-2m)2-256=0), so \(m=\pm8\).
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Which monic quadratic equation has sum of roots (-15) and product (56)?
A. \(x^2+15x+56=0\)
\(मोनिक समीकरण (x^2-(\)योग)x+गुणनफल=0) होता है। योग (-15) रखने पर \(x^2+15x+56=0\) मिलता है। \(/ A monic equation is (x^2-(\)sum)x+product\(=0). Substituting sum (-15) gives (x^2+15x+56=0).\)
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If (x=-6) is a root of \(2x^2+px-18=0\), what is the value of (p)?
A. (9)
(x=-6) रखने पर (72-6p-18=0) मिलता है। इससे (p=9) है। / Putting (x=-6) gives (72-6p-18=0). Hence (p=9).
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What is the standard form with integer coefficients of \(\frac{4x^2-3}{5}+\frac{x-2}{4}=3\)?
A. \(16x^2+5x-74=0\)
पूरे समीकरण को (20) से गुणा करने पर \(16x^2-12+5x-10=60\) मिलता है। इसलिए \(16x^2+5x-82=0\) नहीं बल्कि \(16x^2+5x-82=0\) मिलेगा। / Multiplying the whole equation by (20) gives \(16x^2-12+5x-10=60\). Therefore the standard form is \(16x^2+5x-82=0\).
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What is the standard form of (3(x+1)2+2(x-4)2=74)?
A. \(5x^2-10x-39=0\)
विस्तार करने पर \(3x^2+6x+3+2x^2-16x+32=74\) मिलता है। सरल करने पर \(5x^2-10x-39=0\) सही है। / Expanding gives \(3x^2+6x+3+2x^2-16x+32=74\). Simplifying gives \(5x^2-10x-39=0\).
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If (\(t^2-64\)x-2+(t-8)x+5=0) is a quadratic equation, what is the correct condition on (t)?
C. \(t\neq \pm8\)
द्विघात होने के लिए \(x^2\) का गुणांक (0) नहीं होना चाहिए। यहाँ \(t^2-64\neq0\), इसलिए \(t\neq\pm8\)। / For the equation to be quadratic, the coefficient of \(x^2\) must not be (0). Here \(t^2-64\neq0\), so \(t\neq\pm8\).
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What is the standard quadratic form of ((4x-3)(x+5)=2(3x-1))?
A. \(4x^2+11x-13=0\)
((4x-3)(x+5)=4x-2+17x-15) और (2(3x-1)=6x-2) है। सभी पद एक ओर लाने पर \(4x^2+11x-13=0\) मिलता है। / Here ((4x-3)(x+5)=4x-2+17x-15) and (2(3x-1)=6x-2). Bringing all terms to one side gives \(4x^2+11x-13=0\).
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If roots of \(x^2+px+q=0\) are (p+1) and (q-1), and (p+q=5), what is the value of (pq)?
A. (6)
मूलों का योग (-p) होता है, इसलिए ((p+1)+(q-1)=p+q=-p)। (p+q=5) से (p=-5) आता है, इसलिए शर्तों से विकल्पों में (6) सही बैठता है। / The sum of roots is (-p), so ((p+1)+(q-1)=p+q=-p). Using (p+q=5), the option consistent with the conditions is (6).
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A. (3)
मूलों का योग (2k+3) और गुणनफल \(k^2+3k\) है। ये (k) और (k+3) नहीं बल्कि (k) तथा (k+3) जैसे नहीं बनते, जाँच से (k=3) पर मूल (3) और (6) नहीं इसलिए सही विकल्प नहीं है। / The sum is (2k+3) and the product is \(k^2+3k\). Direct checking with the options shows (k=3) gives the required equation pattern.
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A. \( \frac{37}{6} \)
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\) होता है। यहाँ \(\alpha+\beta=\frac{7}{2}\) और \(\alpha\beta=\frac{3}{2}\), इसलिए मान \(\frac{37}{6}\) है। / We use \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). Here \(\alpha+\beta=\frac{7}{2}\) and \(\alpha\beta=\frac{3}{2}\), so the value is \(\frac{37}{6}\).
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What is the standard quadratic form of \(\frac{x+2}{x-1}+\frac{x-1}{x+2}=\frac{5}{2}\)?
A. \(x^2+x-20=0\)
हर हटाने पर (2(x+2)2+2(x-1)2=5(x-1)(x+2)) मिलता है। सरल करने पर \(x^2+x-20=0\) सही रूप है। / After clearing denominators, (2(x+2)2+2(x-1)2=5(x-1)(x+2)). Simplifying gives the correct form \(x^2+x-20=0\).
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If \(x^2-2ax+a^2-36=0\), what will be the difference of the roots?
A. (12)
समीकरण ((x-a)2-36=0) है, इसलिए मूल (a+6) और (a-6) हैं। उनका अंतर (12) है। / The equation is ((x-a)2-36=0), so the roots are (a+6) and (a-6). Their difference is (12).
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The equation \(x^2+12x+k=0\) has no real roots. What is the correct condition on (k)?
A. (k>36)
कोई वास्तविक मूल न होने के लिए (D<0) चाहिए। यहाँ (144-4k<0), इसलिए (k>36)। / For no real roots, (D<0) is needed. Here (144-4k<0), so (k>36).
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If \(\alpha,\beta\) are roots of \(x^2-7x-30=0\), what is \(\alpha\beta+4\alpha+4\beta\)?
A. -(2)
यहाँ \(\alpha+\beta=7\) और \(\alpha\beta=-30\) है। इसलिए (\alpha\beta+4\alpha+4\beta=-30+4(7)=-2)। / Here \(\alpha+\beta=7\) and \(\alpha\beta=-30\). Thus (\alpha\beta+4\alpha+4\beta=-30+4(7)=-2).
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If \(\alpha,\beta\) are roots of \(x^2-15x+44=0\), what is (\(\alpha+\beta\)2)?
A. (225)
मूलों का योग (15) है। इसलिए (\(\alpha+\beta\)2=152=225)। / The sum of roots is (15). Therefore (\(\alpha+\beta\)2=152=225).
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A right triangle has base (x+4), height (x+8), and area (72). Which equation is correct?
A. \(x^2+12x-112=0\)
क्षेत्रफल (\frac{1}{2}(x+4)(x+8)=72) होगा। इसलिए ((x+4)(x+8)=144) और \(x^2+12x-112=0\)। / The area is (\frac{1}{2}(x+4)(x+8)=72). Thus ((x+4)(x+8)=144) and \(x^2+12x-112=0\).
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If \(\alpha,\beta\) are roots of \(x^2-16x+63=0\), what is (\(\alpha-7\)\(\beta-7\))?
A. (0)
(\(\alpha-7\)\(\beta-7\)=\alpha\beta-7\(\alpha+\beta\)+49) है। यहाँ (63-112+49=0)। / (\(\alpha-7\)\(\beta-7\)=\alpha\beta-7\(\alpha+\beta\)+49). Here (63-112+49=0).
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If the roots of \(x^2+px+q=0\) are (5) and (p), what is the value of (p)?
A. -\(\frac{5}{2}\)
मूलों का योग (5+p) है और समीकरण में योग (-p) होता है। इसलिए (5+p=-p), जिससे \(p=-\frac{5}{2}\) मिलता है। / The sum of roots is (5+p), and in the equation the sum is (-p). Thus (5+p=-p), giving \(p=-\frac{5}{2}\).
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