hard se related questions ko ek jagah revise karein. Har question me bilingual content, answer feedback aur explanation available hai.
1536 questions tagged with hard.
What is the standard form of ((2x-3)2+(x+5)2=34)?
A. \(5x^2-2x=0\)
विस्तार करने पर \(4x^2-12x+9+x^2+10x+25=34\) मिलता है। सरल करने पर \(5x^2-2x=0\) बनता है। / Expanding gives \(4x^2-12x+9+x^2+10x+25=34\). Simplifying gives \(5x^2-2x=0\).
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If (\(n^2-16\)x-2-3x+7=0) is a quadratic equation, what is the correct condition on (n)?
C. \(n\neq \pm4\)
द्विघात होने के लिए \(n^2-16\neq0\) होना चाहिए। इसलिए \(n\neq4\) और \(n\neq-4\) दोनों जरूरी हैं। / For the equation to be quadratic, \(n^2-16\neq0\) is needed. Hence both \(n\neq4\) and \(n\neq-4\) are necessary.
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What is the standard quadratic form of ((6x+1)(x-4)=5x)?
A. \(6x^2-28x-4=0\)
((6x+1)(x-4)=6x-2-23x-4) है और (5x) घटाने पर \(6x^2-28x-4=0\) मिलता है। पहले विस्तार करें फिर सभी पद एक ओर लाएं। / Here ((6x+1)(x-4)=6x-2-23x-4), and subtracting (5x) gives \(6x^2-28x-4=0\). First expand and then bring all terms to one side.
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If \(x^2-2ax+a^2-9=0\), what will be the difference of the roots?
A. (6)
समीकरण ((x-a)2-9=0) है, इसलिए मूल (a+3) और (a-3) हैं। उनका अंतर (6) है। / The equation is ((x-a)2-9=0), so the roots are (a+3) and (a-3). Their difference is (6).
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The equation \(x^2+4x+k=0\) has no real roots. What is the correct condition on (k)?
A. (k>4)
कोई वास्तविक मूल न होने के लिए (D<0) चाहिए। यहाँ (16-4k<0), इसलिए (k>4)। / For no real roots, (D<0) is needed. Here (16-4k<0), so (k>4).
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If \(\alpha,\beta\) are roots of \(x^2+3x-18=0\), what is \(\alpha\beta-\alpha-\beta\)?
A. (-15)
यहाँ \(\alpha+\beta=-3\) और \(\alpha\beta=-18\) है। इसलिए (\alpha\beta-\alpha-\beta=-18-(-3)=-15)। / Here \(\alpha+\beta=-3\) and \(\alpha\beta=-18\). Thus (\alpha\beta-\alpha-\beta=-18-(-3)=-15).
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If \(\alpha,\beta\) are roots of \(x^2+2x-8=0\), what is (\(\alpha+\beta\)2)?
A. (4)
मूलों का योग (-2) है। इसलिए (\(\alpha+\beta\)2=(-2)2=4)। / The sum of roots is (-2). Therefore (\(\alpha+\beta\)2=(-2)2=4).
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A right triangle has base (x+1), height (x+3), and area (30). Which equation is correct?
A. \(x^2+4x-57=0\)
क्षेत्रफल (\frac{1}{2}(x+1)(x+3)=30) होगा। इसलिए ((x+1)(x+3)=60) और \(x^2+4x-57=0\)। / The area is (\frac{1}{2}(x+1)(x+3)=30). Thus ((x+1)(x+3)=60) and \(x^2+4x-57=0\).
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Which equation will have a negative product of roots?
A. \(2x^2+3x-5=0\)
मूलों का गुणनफल \(\frac{c}{a}\) है। पहले विकल्प में \(\frac{-5}{2}<0\), इसलिए गुणनफल ऋणात्मक है। / The product of roots is \(\frac{c}{a}\). In the first option, \(\frac{-5}{2}<0\), so the product is negative.
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If \(\alpha,\beta\) are roots of \(x^2-10x+21=0\), what is (\(\alpha-3\)\(\beta-3\))?
A. (0)
(\(\alpha-3\)\(\beta-3\)=\alpha\beta-3\(\alpha+\beta\)+9) है। यहाँ (21-30+9=0)। / (\(\alpha-3\)\(\beta-3\)=\alpha\beta-3\(\alpha+\beta\)+9). Here (21-30+9=0).
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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 (x=1) is a root of (x-2+(2k-1)x+k=0), what is the value of (k)?
A. (0)
(x=1) रखने पर (1+2k-1+k=0) मिलता है। इससे (3k=0), इसलिए (k=0)। / Putting (x=1) gives (1+2k-1+k=0). Thus (3k=0), so (k=0).
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What is the standard form of ((x-1)2+(x-2)2=(x-3)2)?
B. \(x^2-2x-4=0\)
विस्तार करने पर बाईं ओर \(2x^2-6x+5\) और दाईं ओर \(x^2-6x+9\) है। घटाने पर \(x^2-4=0\) नहीं बल्कि \(x^2-4=0\) मिलता है। / Expanding gives left side \(2x^2-6x+5\) and right side \(x^2-6x+9\). Subtracting gives \(x^2-4=0\).
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If (a=-1) in ((a+1)x-2+\(a^2-1\)x+2=0), what type of equation will it be?
C. विरोधाभासी कथनContradictory statement
(a=-1) रखने पर \(0x^2+0x+2=0\) मिलता है। यह (2=0) है, जो विरोधाभासी कथन है। / Putting (a=-1) gives \(0x^2+0x+2=0\). This is (2=0), which is a contradictory statement.
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In which equation is (x=0) one root and the other root positive?
A. \(x^2-6x=0\)
(x-2-6x=x(x-6)), इसलिए मूल (0) और (6) हैं। दूसरा मूल धनात्मक है। / (x-2-6x=x(x-6)), so the roots are (0) and (6). The other root is positive.
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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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Which quadratic equation has sum of roots (0) and product (-49)?
A. \(x^2-49=0\)
\(मोनिक समीकरण (x^2-(\)योग)x+गुणनफल=0) है। योग (0) और गुणनफल (-49) रखने पर \(x^2-49=0\) मिलता है। \(/ The monic equation is (x^2-(\)sum)x+product\(=0). Using sum (0) and product (-49) gives (x^2-49=0).\)
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If (x=1) is not a root of \(kx^2-5x+2=0\), what condition must (k) satisfy?
A. \(k\neq3\)
(x=1) रखने पर बायां पक्ष (k-5+2=k-3) होता है। मूल न होने के लिए \(k-3\neq0\), इसलिए \(k\neq3\)। / Putting (x=1), the left side becomes (k-5+2=k-3). For it not to be a root, \(k-3\neq0\), so \(k\neq3\).
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The roots of \(x^2+px+9=0\) are equal and negative. What is the value of (p)?
A. (6)
समान मूलों के लिए \(p^2-36=0\) से \(p=\pm6\) मिलता है। ऋणात्मक समान मूल के लिए \(-\frac{p}{2}<0\), इसलिए (p=6)। / For equal roots, \(p^2-36=0\) gives \(p=\pm6\). For equal negative roots, \(-\frac{p}{2}<0\), so (p=6).
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If one root of \(x^2+mx+16=0\) 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 (16) 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 (16), so it is not possible.
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What is the nature of roots of \(4x^2-4kx+k^2=0\)?
A. दो समान वास्तविक मूलTwo equal real roots
यह ((2x-k)2=0) के रूप में लिखा जा सकता है। इसलिए दोनों मूल समान वास्तविक होते हैं। / It can be written as ((2x-k)2=0). Therefore both roots are equal and real.
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If ((x+a)2=x-2+10x+25), what is the value of (a)?
A. (5)
((x+a)2=x-2+2ax+a-2) होता है। (2a=10) से (a=5) मिलता है। / ((x+a)2=x-2+2ax+a-2). From (2a=10), we get (a=5).
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Which equation will have equal and positive roots?
A. \(x^2-12x+36=0\)
(x-2-12x+36=(x-6)2), इसलिए दोनों मूल (6) हैं। दोनों मूल बराबर और धनात्मक हैं। / (x-2-12x+36=(x-6)2), so both roots are (6). Both roots are equal and positive.
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If \(\alpha,\beta\) are roots of \(x^2-5x+6=0\), what is \(\frac{1}{\alpha}+\frac{1}{\beta}\)?
A. \( \frac{5}{6} \)
\(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}\) होता है। यहाँ मान \(\frac{5}{6}\) है। / \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}\). Here the value is \(\frac{5}{6}\).
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Two numbers have sum (11) and product (30). They can be roots of which quadratic equation?
A. \(x^2-11x+30=0\)
यदि मूलों का योग (11) और गुणनफल (30) है, तो समीकरण \(x^2-11x+30=0\) होगा। मोनिक रूप का सूत्र याद रखें। / If the sum of roots is (11) and product is (30), the equation is \(x^2-11x+30=0\). Remember the monic form formula.
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A rectangle has length (x+5) and breadth (x-2). If its area is (84), which equation is correct?
A. \(x^2+3x-94=0\)
क्षेत्रफल ((x+5)(x-2)=84) होगा। विस्तार से \(x^2+3x-10=84\) और \(x^2+3x-94=0\) मिलता है। / The area is ((x+5)(x-2)=84). Expanding gives \(x^2+3x-10=84\) and \(x^2+3x-94=0\).
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In which option will both the sum and product of roots be positive?
A. \(x^2-5x+6=0\)
पहले विकल्प में योग \(-\frac{b}{a}=5\) और गुणनफल \(\frac{c}{a}=6\) है। दोनों धनात्मक हैं। / In the first option, the sum is \(-\frac{b}{a}=5\) and the product is \(\frac{c}{a}=6\). Both are positive.
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