यदि किसी द्विघात समीकरण (p(x)=0) में (x=a) रखने पर (p(a)=0) हो जाता है तो (a) क्या कहलाता है?
If substituting (x=a) in a quadratic equation (p(x)=0) gives (p(a)=0), what is (a) called?
#roots
#definition
#substitution
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A मूल / Root
B अचर पद / Constant term
C मध्य पद / Middle term
D अग्र गुणांक / Leading coefficient
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Correct Answer
A. मूल / Root
Explanation
Simple Explanation
क्योंकि (x=a) रखने पर समीकरण सत्य हो जाता है इसलिए (a) मूल है। परीक्षा में मूल की जांच सीधे प्रतिस्थापन से करें। / Since the equation becomes true after substituting (x=a), (a) is a root. In exams check a root by direct substitution.
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क्या (x=4) समीकरण \(x^2-9x+20=0\) का मूल है?
Is (x=4) a root of \(x^2-9x+20=0\)?
#roots
#checking
#substitution
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A हाँ / Yes
B नहीं / No
C केवल (x=5) मूल है / Only (x=5) is a root
D निर्धारित नहीं / Cannot be determined
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Correct Answer
A. हाँ / Yes
Explanation
Simple Explanation
(x=4) रखने पर (16-36+20=0) मिलता है। इसलिए (4) इस समीकरण का मूल है। / Putting (x=4) gives (16-36+20=0). Therefore (4) is a root of this equation.
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क्या (x=-2) समीकरण \(3x^2+2x-8=0\) का मूल है?
Is (x=-2) a root of \(3x^2+2x-8=0\)?
#roots
#negative_root
#checking
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A हाँ / Yes
B नहीं / No
C केवल (x=2) मूल है / Only (x=2) is a root
D कोई वास्तविक मूल नहीं / No real root
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Correct Answer
A. हाँ / Yes
Explanation
Simple Explanation
(x=-2) रखने पर (12-4-8=0) मिलता है। ऋणात्मक मूल रखते समय चिन्हों को ध्यान से संभालें। / Putting (x=-2) gives (12-4-8=0). Handle signs carefully when substituting a negative root.
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समीकरण \(x^2-12x+35=0\) के मूल कौन से हैं?
What are the roots of \(x^2-12x+35=0\)?
#roots
#factorisation
#numerical
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A (5) और (7) / (5) and (7)
B (1) और (35) / (1) and (35)
C (-5) और (-7) / (-5) and (-7)
D (0) और (12) / (0) and (12)
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Correct Answer
A. (5) और (7) / (5) and (7)
Explanation
Simple Explanation
(x-2 -12x+35=(x-5)(x-7)) है। इसलिए मूल (5) और (7) हैं। / (x-2 -12x+35=(x-5)(x-7)). Therefore the roots are (5) and (7).
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समीकरण \(3x^2-13x+4=0\) के मूल कौन से हैं?
What are the roots of \(3x^2-13x+4=0\)?
#roots
#factorisation
#fraction_root
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A (4) और \(\frac{1}{3}\) / (4) and \(\frac{1}{3}\)
B (-4) और \(-\frac{1}{3}\) / (-4) and \(-\frac{1}{3}\)
C (3) और (4) / (3) and (4)
D \(\frac{4}{3}\) और (1) / \(\frac{4}{3}\) and (1)
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Correct Answer
A. (4) और \(\frac{1}{3}\) / (4) and \(\frac{1}{3}\)
Explanation
Simple Explanation
(3x-2 -13x+4=(3x-1)(x-4)) है। इसलिए मूल \(\frac{1}{3}\) और (4) हैं। / (3x-2 -13x+4=(3x-1)(x-4)). Therefore the roots are \(\frac{1}{3}\) and (4).
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समीकरण \(2x^2+9x+4=0\) के मूल कौन से हैं?
What are the roots of \(2x^2+9x+4=0\)?
#roots
#factorisation
#negative_fraction
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A (-4) और \(-\frac{1}{2}\) / (-4) and \(-\frac{1}{2}\)
B (4) और \(\frac{1}{2}\) / (4) and \(\frac{1}{2}\)
C (-2) और (-4) / (-2) and (-4)
D (2) और (4) / (2) and (4)
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Correct Answer
A. (-4) और \(-\frac{1}{2}\) / (-4) and \(-\frac{1}{2}\)
Explanation
Simple Explanation
(2x-2 +9x+4=(2x+1)(x+4)) है। इसलिए मूल \(-\frac{1}{2}\) और (-4) हैं। / (2x-2 +9x+4=(2x+1)(x+4)). Therefore the roots are \(-\frac{1}{2}\) and (-4).
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किस समीकरण के मूल (-3) और (6) हैं?
Which equation has roots (-3) and (6)?
#roots
#equation_from_roots
#mixed_signs
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A \(x^2-3x-18=0\)
B \(x^2+3x-18=0\)
C \(x^2-9x+18=0\)
D \(x^2+9x+18=0\)
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Correct Answer
A. \(x^2-3x-18=0\)
Explanation
Simple Explanation
मूल (-3) और (6) होने पर ((x+3)(x-6)=0) होगा। इसे खोलने पर \(x^2-3x-18=0\) मिलता है। / With roots (-3) and (6), we get ((x+3)(x-6)=0). Expanding it gives \(x^2-3x-18=0\).
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यदि (3) समीकरण \(x^2+kx-18=0\) का मूल है तो (k) का मान क्या होगा?
If (3) is a root of \(x^2+kx-18=0\), what is the value of (k)?
#roots
#parameter
#substitution
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A (3)
B (-3)
C (6)
D (-6)
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Explanation
Simple Explanation
(x=3) रखने पर (9+3k-18=0) इसलिए (k=3) है। पैरामीटर वाले प्रश्न में मूल को सीधे रखें। / Putting (x=3) gives (9+3k-18=0), so (k=3). In parameter questions substitute the root directly.
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समीकरण \(4x^2-11x+7=0\) के मूलों का योग क्या है?
What is the sum of roots of \(4x^2-11x+7=0\)?
#roots
#sum_of_roots
#formula
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A \(\frac{11}{4}\)
B -\(\frac{11}{4}\)
C \(\frac{7}{4}\)
D -\(\frac{7}{4}\)
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Correct Answer
A. \(\frac{11}{4}\)
Explanation
Simple Explanation
मूलों का योग \(-\frac{b}{a}\) होता है। यहां (a=4) और (b=-11) इसलिए योग \(\frac{11}{4}\) है। / The sum of roots is \(-\frac{b}{a}\). Here (a=4) and (b=-11), so the sum is \(\frac{11}{4}\).
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समीकरण \(5x^2+6x-8=0\) के मूलों का गुणनफल क्या है?
What is the product of roots of \(5x^2+6x-8=0\)?
#roots
#product_of_roots
#formula
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A -\(\frac{8}{5}\)
B \(\frac{8}{5}\)
C -\(\frac{6}{5}\)
D \(\frac{6}{5}\)
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Correct Answer
A. -\(\frac{8}{5}\)
Explanation
Simple Explanation
मूलों का गुणनफल \(\frac{c}{a}\) होता है। यहां \(\frac{-8}{5}=-\frac{8}{5}\) सही है। / The product of roots is \(\frac{c}{a}\). Here \(\frac{-8}{5}=-\frac{8}{5}\) is correct.
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समीकरण \(3x^2-6x+3=0\) का डिस्क्रिमिनेंट (D) क्या है?
What is the discriminant (D) of \(3x^2-6x+3=0\)?
#roots
#discriminant
#equal_roots
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A (0)
B (6)
C (-6)
D (12)
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Explanation
Simple Explanation
यहां \(D=b^2-4ac=36-36=0\) है। इसलिए दोनों वास्तविक मूल बराबर होंगे। / Here \(D=b^2-4ac=36-36=0\). Therefore the two real roots will be equal.
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यदि किसी द्विघात समीकरण के लिए (D=64) है तो उसके वास्तविक मूलों की प्रकृति क्या होगी?
If (D=64) for a quadratic equation, what will be the nature of its real roots?
#roots
#discriminant
#nature
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A दो भिन्न वास्तविक मूल / Two distinct real roots
B दो बराबर वास्तविक मूल / Two equal real roots
C कोई वास्तविक मूल नहीं / No real root
D एक ही मूल (0) / Only one root (0)
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Correct Answer
A. दो भिन्न वास्तविक मूल / Two distinct real roots
Explanation
Simple Explanation
क्योंकि (D=64>0) है इसलिए दो भिन्न वास्तविक मूल होंगे। (D>0) होने पर मूल अलग अलग होते हैं। / Since (D=64>0), there will be two distinct real roots. When (D>0), the roots are different.
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समीकरण \(x^2-4x+13=0\) के वास्तविक मूलों के बारे में सही कथन कौन सा है?
Which statement is correct about the real roots of \(x^2-4x+13=0\)?
#roots
#no_real_roots
#discriminant
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A दो भिन्न वास्तविक मूल हैं / It has two distinct real roots
B दो बराबर वास्तविक मूल हैं / It has two equal real roots
C कोई वास्तविक मूल नहीं है / It has no real roots
D मूल (4) और (13) हैं / The roots are (4) and (13)
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Correct Answer
C. कोई वास्तविक मूल नहीं है / It has no real roots
Explanation
Simple Explanation
यहां (D=(-4)2 -4(1)(13)=-36<0) है। इसलिए कोई वास्तविक मूल नहीं है। / Here (D=(-4)2 -4(1)(13)=-36<0). Therefore there are no real roots.
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समीकरण \(x^2-8x+k=0\) के बराबर वास्तविक मूल होने के लिए (k) का मान क्या होगा?
For \(x^2-8x+k=0\) to have equal real roots, what should be the value of (k)?
#roots
#equal_roots
#parameter
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A (16)
B (8)
C (4)
D (64)
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Explanation
Simple Explanation
बराबर मूलों के लिए (D=0) होता है। (64-4k=0) से (k=16) मिलता है। / For equal roots (D=0). From (64-4k=0), we get (k=16).
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यदि मूलों का योग (9) और गुणनफल (18) है तो मोनिक द्विघात समीकरण कौन सा होगा?
If the sum of roots is (9) and product is (18), which monic quadratic equation is formed?
#roots
#equation_from_sum_product
#monic
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A \(x^2-9x+18=0\)
B \(x^2+9x+18=0\)
C \(x^2-18x+9=0\)
D \(x^2+18x+9=0\)
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Correct Answer
A. \(x^2-9x+18=0\)
Explanation
Simple Explanation
\(मोनिक समीकरण (x^2-(\)योग)x+गुणनफल=0) होता है। इसलिए \(x^2-9x+18=0\) सही है। \(/ A monic equation is (x^2-(\)sum)x+product\(=0). Therefore (x^2-9x+18=0) is correct.\)
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समीकरण \(x^2-15x+54=0\) का एक मूल (6) है तो दूसरा मूल क्या है?
If one root of \(x^2-15x+54=0\) is (6), what is the other root?
#roots
#other_root
#product
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A (9)
B (6)
C (15)
D (54)
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Explanation
Simple Explanation
मूलों का गुणनफल (54) है और एक मूल (6) है। इसलिए दूसरा मूल \(\frac{54}{6}=9\) है। / The product of roots is (54) and one root is (6). Therefore the other root is \(\frac{54}{6}=9\).
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यदि \(3\alpha\) और \(3\beta\) नए मूल हैं तथा \(\alpha+\beta=4\) है तो नए मूलों का योग क्या होगा?
If \(3\alpha\) and \(3\beta\) are new roots and \(\alpha+\beta=4\), what is the sum of the new roots?
#roots
#transformed_roots
#sum
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A (12)
B (4)
C (7)
D \(\frac{4}{3}\)
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Explanation
Simple Explanation
नए मूलों का योग (3\alpha+3\beta=3\(\alpha+\beta\)=12) है। गुणक लगे मूलों में योग पर भी वही गुणक लगता है। / The sum of new roots is (3\alpha+3\beta=3\(\alpha+\beta\)=12). When roots are multiplied by a factor, the sum is multiplied by the same factor.
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यदि मूल \(\alpha\) और \(\beta\) हैं तथा \(\alpha+\beta=7\) और \(\alpha\beta=12\) है तो \(\alpha^2+\beta^2\) का मान क्या है?
If the roots are \(\alpha\) and \(\beta\) with \(\alpha+\beta=7\) and \(\alpha\beta=12\), what is \(\alpha^2+\beta^2\)?
#roots
#identity
#sum_product
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A (25)
B (37)
C (49)
D (84)
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Explanation
Simple Explanation
(\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta=49-24=25) है। ऐसे प्रश्नों में पहचान का प्रयोग करें। / (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta=49-24=25). Use identities in such questions.
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यदि द्विघात समीकरण के मूल (2) और (-7) हैं तो \(\frac{1}{\alpha}+\frac{1}{\beta}\) का मान क्या होगा?
If the roots of a quadratic equation are (2) and (-7), what is the value of \(\frac{1}{\alpha}+\frac{1}{\beta}\)?
#roots
#reciprocal_sum
#identity
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A \(\frac{5}{14}\)
B -\(\frac{5}{14}\)
C \(\frac{9}{14}\)
D -\(\frac{9}{14}\)
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Correct Answer
A. \(\frac{5}{14}\)
Explanation
Simple Explanation
यहां \(\alpha+\beta=-5\) और \(\alpha\beta=-14\) है। इसलिए \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{-5}{-14}=\frac{5}{14}\) है। / Here \(\alpha+\beta=-5\) and \(\alpha\beta=-14\). Therefore \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{-5}{-14}=\frac{5}{14}\).
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समीकरण \(3x^2-14x+8=0\) के मूलों का योग और गुणनफल क्रमशः क्या हैं?
For \(3x^2-14x+8=0\), what are the sum and product of roots respectively?
#roots
#sum_product
#formula
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A \(\frac{14}{3}\) और \(\frac{8}{3}\) / \(\frac{14}{3}\) and \(\frac{8}{3}\)
B -\(\frac{14}{3}\) और \(\frac{8}{3}\) / \(-\frac{14}{3}\) and \(\frac{8}{3}\)
C \(\frac{14}{3}\) और \(-\frac{8}{3}\) / \(\frac{14}{3}\) and \(-\frac{8}{3}\)
D (14) और (8) / (14) and (8)
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Correct Answer
A. \(\frac{14}{3}\) और \(\frac{8}{3}\) / \(\frac{14}{3}\) and \(\frac{8}{3}\)
Explanation
Simple Explanation
योग \(-\frac{b}{a}=\frac{14}{3}\) और गुणनफल \(\frac{c}{a}=\frac{8}{3}\) है। पहले (a), (b), (c) पहचानें। / The sum is \(-\frac{b}{a}=\frac{14}{3}\) and the product is \(\frac{c}{a}=\frac{8}{3}\). Identify (a), (b), and (c) first.
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यदि (x-4) और (x+7) किसी द्विघात समीकरण के गुणनखंड हैं तो उसके मूल कौन से हैं?
If (x-4) and (x+7) are factors of a quadratic equation, what are its roots?
#roots
#factors_to_roots
#zero_product
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A (4) और (-7) / (4) and (-7)
B (-4) और (7) / (-4) and (7)
C (4) और (7) / (4) and (7)
D (-4) और (-7) / (-4) and (-7)
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Correct Answer
A. (4) और (-7) / (4) and (-7)
Explanation
Simple Explanation
(x-4=0) से (x=4) और (x+7=0) से (x=-7) मिलता है। गुणनखंड का चिन्ह उलटकर मूल मिलता है। / From (x-4=0), (x=4), and from (x+7=0), (x=-7). The sign changes when finding a root from a factor.
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समीकरण \(x^2+px+25=0\) के दोनों मूल (-5) और (-5) हैं तो (p) का मान क्या है?
If both roots of \(x^2+px+25=0\) are (-5) and (-5), what is the value of (p)?
#roots
#equal_roots
#parameter
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A (10)
B (-10)
C (5)
D (-5)
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Explanation
Simple Explanation
मूलों का योग (-10) है और (-p=-10) होगा। इसलिए (p=10) है। / The sum of roots is (-10), and (-p=-10). Therefore (p=10).
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यदि \(x=\frac{4}{3}\) समीकरण \(3x^2-7x+c=0\) का मूल है तो (c) का मान क्या है?
If \(x=\frac{4}{3}\) is a root of \(3x^2-7x+c=0\), what is the value of (c)?
#roots
#parameter
#fraction_root
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A (4)
B (-4)
C \(\frac{4}{3}\)
D -\(\frac{4}{3}\)
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Explanation
Simple Explanation
\(x=\frac{4}{3}\) रखने पर \(\frac{16}{3}-\frac{28}{3}+c=0\) इसलिए (c=4) है। भिन्न मूल में हर पद अलग निकालें। / Putting \(x=\frac{4}{3}\) gives \(\frac{16}{3}-\frac{28}{3}+c=0\), so (c=4). For fractional roots calculate each term separately.
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समीकरण \(9x^2-24x+16=0\) के मूलों की प्रकृति क्या है?
What is the nature of roots of \(9x^2-24x+16=0\)?
#roots
#nature
#perfect_square
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A बराबर वास्तविक मूल / Equal real roots
B भिन्न वास्तविक मूल / Distinct real roots
C कोई वास्तविक मूल नहीं / No real roots
D एक मूल (0) है / One root is (0)
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Correct Answer
A. बराबर वास्तविक मूल / Equal real roots
Explanation
Simple Explanation
यह (9x-2 -24x+16=(3x-4)2 ) है। इसलिए दोनों मूल \(\frac{4}{3}\) और \(\frac{4}{3}\) बराबर हैं। / It is (9x-2 -24x+16=(3x-4)2 ). Therefore both roots are \(\frac{4}{3}\) and \(\frac{4}{3}\), which are equal.
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यदि किसी द्विघात समीकरण में (a=1), (b=-16), (c=64) है तो मूल कौन से होंगे?
If a quadratic equation has (a=1), (b=-16), and (c=64), what will be the roots?
#roots
#coefficients
#repeated_root
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A (8) और (8) / (8) and (8)
B (-8) और (-8) / (-8) and (-8)
C (16) और (64) / (16) and (64)
D (0) और (8) / (0) and (8)
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Correct Answer
A. (8) और (8) / (8) and (8)
Explanation
Simple Explanation
समीकरण \(x^2-16x+64=0\) है जो ((x-8)2 =0) बनता है। इसलिए दोनों मूल (8) हैं। / The equation is \(x^2-16x+64=0\), which becomes ((x-8)2 =0). Therefore both roots are (8).
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यदि किसी द्विघात समीकरण के मूलों का गुणनफल ऋणात्मक है तो सही कथन कौन सा है?
If the product of roots of a quadratic equation is negative, which statement is correct?
#roots
#sign_of_roots
#reasoning
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A एक मूल धनात्मक और दूसरा ऋणात्मक है / One root is positive and the other is negative
B दोनों मूल धनात्मक हैं / Both roots are positive
C दोनों मूल ऋणात्मक हैं / Both roots are negative
D दोनों मूल बराबर हैं / Both roots are equal
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Correct Answer
A. एक मूल धनात्मक और दूसरा ऋणात्मक है / One root is positive and the other is negative
Explanation
Simple Explanation
ऋणात्मक गुणनफल तभी होता है जब मूलों के चिन्ह विपरीत हों। इसलिए एक मूल धनात्मक और दूसरा ऋणात्मक होगा। / A negative product occurs only when the roots have opposite signs. Therefore one root will be positive and the other negative.
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समीकरण \(x^2+x-20=0\) के मूल कौन से हैं?
What are the roots of \(x^2+x-20=0\)?
#roots
#factorisation
#mixed_signs
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A (4) और (-5) / (4) and (-5)
B (5) और (-4) / (5) and (-4)
C (4) और (5) / (4) and (5)
D (-4) और (-5) / (-4) and (-5)
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Correct Answer
A. (4) और (-5) / (4) and (-5)
Explanation
Simple Explanation
(x-2 +x-20=(x+5)(x-4)) है। इसलिए मूल (-5) और (4) हैं। / (x-2 +x-20=(x+5)(x-4)). Therefore the roots are (-5) and (4).
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यदि \(\alpha\) और \(\beta\) समीकरण \(x^2-8x+15=0\) के मूल हैं तो \(\alpha+\beta+\alpha\beta\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-8x+15=0\), what is the value of \(\alpha+\beta+\alpha\beta\)?
#roots
#sum_product
#expression
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A (23)
B (15)
C (8)
D (7)
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Explanation
Simple Explanation
यहां \(\alpha+\beta=8\) और \(\alpha\beta=15\) है। इसलिए कुल (8+15=23) है। / Here \(\alpha+\beta=8\) and \(\alpha\beta=15\). Therefore the total is (8+15=23).
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यदि मूलों का योग (0) और गुणनफल (-25) है तो मोनिक समीकरण कौन सा होगा?
If the sum of roots is (0) and product is (-25), which monic equation is formed?
#roots
#equation_from_sum_product
#zero_sum
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A \(x^2-25=0\)
B \(x^2+25=0\)
C \(x^2-25x=0\)
D \(x^2+25x=0\)
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Correct Answer
A. \(x^2-25=0\)
Explanation
Simple Explanation
मोनिक समीकरण (x-2 -(0)x+(-25)=0) होगा। इसलिए \(x^2-25=0\) सही है। / The monic equation is (x-2 -(0)x+(-25)=0). Therefore \(x^2-25=0\) is correct.
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समीकरण \(16x^2-8x+1=0\) का दोहराया हुआ मूल क्या है?
What is the repeated root of \(16x^2-8x+1=0\)?
#roots
#repeated_root
#fraction_root
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A \(\frac{1}{4}\)
B -\(\frac{1}{4}\)
C (4)
D (-4)
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Correct Answer
A. \(\frac{1}{4}\)
Explanation
Simple Explanation
(16x-2 -8x+1=(4x-1)2 ) है। इसलिए दोहराया मूल \(\frac{1}{4}\) है। / (16x-2 -8x+1=(4x-1)2 ). Therefore the repeated root is \(\frac{1}{4}\).
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यदि (x=0) समीकरण \(ax^2+bx+c=0\) का मूल है और \(a\ne0\) है तो कौन सा निष्कर्ष सही है?
If (x=0) is a root of \(ax^2+bx+c=0\) and \(a\ne0\), which conclusion is correct?
#roots
#zero_root
#condition
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A (c=0)
B (a=0)
C (b=0)
D \(c\ne0\)
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Explanation
Simple Explanation
(x=0) रखने पर समीकरण (c=0) बनता है। इसलिए शून्य मूल के लिए अचर पद शून्य होता है। / Putting (x=0) makes the equation (c=0). Therefore the constant term is zero for a zero root.
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यदि (x=2) और (x=7) किसी मोनिक द्विघात समीकरण के मूल हैं तो (x) का गुणांक क्या होगा?
If (x=2) and (x=7) are roots of a monic quadratic equation, what will be the coefficient of (x)?
#roots
#coefficient_from_roots
#monic
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A (-9)
B (9)
C (14)
D (-14)
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Explanation
Simple Explanation
मूलों का योग (2+7=9) है। मोनिक समीकरण में (x) का गुणांक (-(योग)=-9) होता है। \(/ The sum of roots is (2+7=9). In a monic equation the coefficient of (x) is (-(\)sum\()=-9).\)
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समीकरण \(3x^2+2x-8=0\) के मूल कौन से हैं?
What are the roots of \(3x^2+2x-8=0\)?
#roots
#factorisation
#medium
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A \(\frac{4}{3}\) और (-2) / \(\frac{4}{3}\) and (-2)
B -\(\frac{4}{3}\) और (2) / \(-\frac{4}{3}\) and (2)
C (4) और (-2) / (4) and (-2)
D (2) और (-4) / (2) and (-4)
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Correct Answer
A. \(\frac{4}{3}\) और (-2) / \(\frac{4}{3}\) and (-2)
Explanation
Simple Explanation
(3x-2 +2x-8=(3x-4)(x+2)) है। इसलिए मूल \(\frac{4}{3}\) और (-2) हैं। / (3x-2 +2x-8=(3x-4)(x+2)). Therefore the roots are \(\frac{4}{3}\) and (-2).
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समीकरण \(x^2+mx+20=0\) के मूल (-4) और (-5) हैं तो (m) का मान क्या है?
If the roots of \(x^2+mx+20=0\) are (-4) and (-5), what is the value of (m)?
#roots
#parameter
#sum_of_roots
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A (9)
B (-9)
C (20)
D (-20)
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Explanation
Simple Explanation
मूलों का योग (-9) है और (-m=-9) होगा। इसलिए (m=9) है। / The sum of roots is (-9), and (-m=-9). Therefore (m=9).
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यदि द्विघात समीकरण के मूल (5) और \(\frac{1}{5}\) हैं तो उनके बारे में सही कथन कौन सा है?
If the roots of a quadratic equation are (5) and \(\frac{1}{5}\), which statement is correct about them?
#roots
#reciprocal_roots
#concept
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A वे एक दूसरे के व्युत्क्रम हैं / They are reciprocals of each other
B वे बराबर मूल हैं / They are equal roots
C उनका योग (1) है / Their sum is (1)
D उनका गुणनफल (0) है / Their product is (0)
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Correct Answer
A. वे एक दूसरे के व्युत्क्रम हैं / They are reciprocals of each other
Explanation
Simple Explanation
\(5\cdot\frac{1}{5}=1\) है इसलिए दोनों व्युत्क्रम मूल हैं। व्युत्क्रम मूलों का गुणनफल (1) होता है। / \(5\cdot\frac{1}{5}=1\), so the roots are reciprocals. Reciprocal roots have product (1).
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समीकरण \(x^2-5x-24=0\) का धनात्मक मूल कौन सा है?
What is the positive root of \(x^2-5x-24=0\)?
#roots
#positive_root
#factorisation
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A (8)
B (-3)
C (3)
D (-8)
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Explanation
Simple Explanation
(x-2 -5x-24=(x-8)(x+3)) है। इसलिए मूल (8) और (-3) हैं तथा धनात्मक मूल (8) है। / (x-2 -5x-24=(x-8)(x+3)). Therefore the roots are (8) and (-3), and the positive root is (8).
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यदि (D=0) और मूलों का योग (12) है तो प्रत्येक मूल क्या होगा?
If (D=0) and the sum of roots is (12), what will each root be?
#roots
#equal_roots
#sum
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A (6)
B (12)
C (0)
D (24)
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Explanation
Simple Explanation
(D=0) होने पर दोनों मूल बराबर होते हैं। इसलिए प्रत्येक मूल \(\frac{12}{2}=6\) होगा। / When (D=0), both roots are equal. Therefore each root is \(\frac{12}{2}=6\).
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समीकरण \(x^2-6x-7=0\) के मूलों का अंतर कितना है?
What is the difference between the roots of \(x^2-6x-7=0\)?
#roots
#difference_of_roots
#factorisation
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A (8)
B (6)
C (7)
D (1)
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Explanation
Simple Explanation
(x-2 -6x-7=(x-7)(x+1)) है। मूल (7) और (-1) हैं इसलिए अंतर (8) है। / (x-2 -6x-7=(x-7)(x+1)). The roots are (7) and (-1), so the difference is (8).
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यदि \(\alpha\) और \(\beta\) समीकरण \(3x^2-12x+5=0\) के मूल हैं तो \(\alpha+\beta\) क्या है?
If \(\alpha\) and \(\beta\) are roots of \(3x^2-12x+5=0\), what is \(\alpha+\beta\)?
#roots
#sum_of_roots
#calculation
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A (4)
B (-4)
C \(\frac{5}{3}\)
D -\(\frac{5}{3}\)
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Explanation
Simple Explanation
मूलों का योग \(-\frac{b}{a}=-\frac{-12}{3}=4\) है। सूत्र में (b) का चिन्ह सही रखें। / The sum of roots is \(-\frac{b}{a}=-\frac{-12}{3}=4\). Keep the sign of (b) correct in the formula.
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यदि \(\alpha\) और \(\beta\) समीकरण \(5x^2+3x-2=0\) के मूल हैं तो \(\alpha\beta\) क्या है?
If \(\alpha\) and \(\beta\) are roots of \(5x^2+3x-2=0\), what is \(\alpha\beta\)?
#roots
#product_of_roots
#calculation
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A -\(\frac{2}{5}\)
B \(\frac{2}{5}\)
C (3)
D (-3)
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Correct Answer
A. -\(\frac{2}{5}\)
Explanation
Simple Explanation
मूलों का गुणनफल \(\frac{c}{a}=\frac{-2}{5}\) होता है। (c) का चिन्ह न भूलें। / The product of roots is \(\frac{c}{a}=\frac{-2}{5}\). Do not forget the sign of (c).
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समीकरण \(x^2+4x+k=0\) के वास्तविक मूल होने के लिए (k) के बारे में कौन सी शर्त सही है?
For \(x^2+4x+k=0\) to have real roots, which condition on (k) is correct?
#roots
#real_roots
#parameter_condition
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A \(k\le4\)
B (k>4)
C (k=8)
D (k<0) ही होगा / Only (k<0)
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Correct Answer
A. \(k\le4\)
Explanation
Simple Explanation
वास्तविक मूलों के लिए \(D\ge0\) चाहिए। \(16-4k\ge0\) से \(k\le4\) मिलता है। / For real roots \(D\ge0\) is required. From \(16-4k\ge0\), we get \(k\le4\).
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समीकरण \(x^2+9x+18=0\) के दोनों मूलों के चिन्ह कैसे होंगे?
What will be the signs of both roots of \(x^2+9x+18=0\)?
#roots
#sign_of_roots
#reasoning
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A दोनों ऋणात्मक / Both negative
B दोनों धनात्मक / Both positive
C एक धनात्मक और एक ऋणात्मक / One positive and one negative
D दोनों शून्य / Both zero
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Correct Answer
A. दोनों ऋणात्मक / Both negative
Explanation
Simple Explanation
मूल (-3) और (-6) हैं क्योंकि ((x+3)(x+6)=0)। योग ऋणात्मक और गुणनफल धनात्मक होने पर दोनों मूल ऋणात्मक होते हैं। / The roots are (-3) and (-6) because ((x+3)(x+6)=0). When the sum is negative and product is positive, both roots are negative.
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यदि \(x^2+ax+16=0\) के मूल (4) और (4) हैं तो (a) का मान क्या है?
If the roots of \(x^2+ax+16=0\) are (4) and (4), what is the value of (a)?
#roots
#parameter
#equal_roots
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A (-8)
B (8)
C (4)
D (-4)
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Explanation
Simple Explanation
मूलों का योग (8) है और (-a=8) होगा। इसलिए (a=-8) है। / The sum of roots is (8), and (-a=8). Therefore (a=-8).
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समीकरण \(8x^2-10x+3=0\) के मूल कौन से हैं?
What are the roots of \(8x^2-10x+3=0\)?
#roots
#factorisation
#fraction_roots
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A \(\frac{1}{2}\) और \(\frac{3}{4}\) / \(\frac{1}{2}\) and \(\frac{3}{4}\)
B -\(\frac{1}{2}\) और \(-\frac{3}{4}\) / \(-\frac{1}{2}\) and \(-\frac{3}{4}\)
C (2) और (3) / (2) and (3)
D \(\frac{4}{3}\) और \(\frac{1}{2}\) / \(\frac{4}{3}\) and \(\frac{1}{2}\)
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Correct Answer
A. \(\frac{1}{2}\) और \(\frac{3}{4}\) / \(\frac{1}{2}\) and \(\frac{3}{4}\)
Explanation
Simple Explanation
(8x-2 -10x+3=(2x-1)(4x-3)) है। इसलिए मूल \(\frac{1}{2}\) और \(\frac{3}{4}\) हैं। / (8x-2 -10x+3=(2x-1)(4x-3)). Therefore the roots are \(\frac{1}{2}\) and \(\frac{3}{4}\).
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यदि किसी द्विघात समीकरण के मूल (b) और (-b) हैं तो अचर पद और अग्र गुणांक के अनुपात \(\frac{c}{a}\) का मान क्या होगा?
If the roots of a quadratic equation are (b) and (-b), what is the value of the ratio \(\frac{c}{a}\) of constant term to leading coefficient?
#roots
#opposite_roots
#product
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A \(-b^2\)
B \(b^2\)
C (0)
D (2b)
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Correct Answer
A. \(-b^2\)
Explanation
Simple Explanation
\(\frac{c}{a}\) मूलों का गुणनफल होता है। यहां (b(-b)=-b-2 ) है। / \(\frac{c}{a}\) is the product of roots. Here (b(-b)=-b-2 ).
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यदि \(\alpha+\beta=-5\) और \(\alpha\beta=-14\) है तो \(\alpha\) और \(\beta\) के लिए मोनिक समीकरण कौन सा है?
If \(\alpha+\beta=-5\) and \(\alpha\beta=-14\), which monic equation has roots \(\alpha\) and \(\beta\)?
#roots
#equation_from_roots
#sum_product
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A \(x^2+5x-14=0\)
B \(x^2-5x-14=0\)
C \(x^2+14x-5=0\)
D \(x^2-14x+5=0\)
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Correct Answer
A. \(x^2+5x-14=0\)
Explanation
Simple Explanation
मोनिक समीकरण (x-2 -\(\alpha+\beta\)x+\alpha\beta=0) होता है। इसलिए \(x^2+5x-14=0\) सही है। / The monic equation is (x-2 -\(\alpha+\beta\)x+\alpha\beta=0). Therefore \(x^2+5x-14=0\) is correct.
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समीकरण \(x^2-7x+10=0\) के मूलों के वर्गों का योग क्या है?
What is the sum of squares of the roots of \(x^2-7x+10=0\)?
#roots
#sum_of_squares
#identity
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A (29)
B (49)
C (20)
D (39)
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Explanation
Simple Explanation
यहां \(\alpha+\beta=7\) और \(\alpha\beta=10\) है। \(\alpha^2+\beta^2=49-20=29\) होगा। / Here \(\alpha+\beta=7\) and \(\alpha\beta=10\). Thus \(\alpha^2+\beta^2=49-20=29\).
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यदि (x=3) समीकरण \(ax^2-8x+6=0\) का मूल है तो (a) का मान क्या है?
If (x=3) is a root of \(ax^2-8x+6=0\), what is the value of (a)?
#roots
#parameter
#leading_coefficient
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A (2)
B (1)
C (-2)
D (-1)
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Explanation
Simple Explanation
(x=3) रखने पर (9a-24+6=0) इसलिए (9a=18) और (a=2)। अज्ञात गुणांक वाले प्रश्न में प्रतिस्थापन करें। / Putting (x=3) gives (9a-24+6=0), so (9a=18) and (a=2). Substitute in questions with an unknown coefficient.
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समीकरण \(x^2-18x+81=0\) के मूलों का योग और गुणनफल क्रमशः क्या हैं?
For \(x^2-18x+81=0\), what are the sum and product of roots respectively?
#roots
#repeated_root
#sum_product
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A (18) और (81) / (18) and (81)
B -(18) और (81) / (-18) and (81)
C (9) और (9) / (9) and (9)
D (81) और (18) / (81) and (18)
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Correct Answer
A. (18) और (81) / (18) and (81)
Explanation
Simple Explanation
यह ((x-9)2 =0) है इसलिए दोनों मूल (9) हैं। योग (18) और गुणनफल (81) होगा। / It is ((x-9)2 =0), so both roots are (9). The sum is (18) and the product is (81).
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यदि \(\alpha\) और \(\beta\) समीकरण \(2x^2-7x+3=0\) के मूल हैं तो \(\alpha+\beta-\alpha\beta\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(2x^2-7x+3=0\), what is the value of \(\alpha+\beta-\alpha\beta\)?
#roots
#sum_product
#expression
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A (2)
B \(\frac{7}{2}\)
C \(\frac{3}{2}\)
D (5)
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Explanation
Simple Explanation
यहां \(\alpha+\beta=\frac{7}{2}\) और \(\alpha\beta=\frac{3}{2}\) है। इसलिए \(\alpha+\beta-\alpha\beta=\frac{7}{2}-\frac{3}{2}=2\) होगा। / Here \(\alpha+\beta=\frac{7}{2}\) and \(\alpha\beta=\frac{3}{2}\). Therefore \(\alpha+\beta-\alpha\beta=\frac{7}{2}-\frac{3}{2}=2\).
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यदि किसी द्विघात समीकरण (q(x)=0) में (q(t)=0) है तो (t) क्या कहलाता है?
If (q(t)=0) in a quadratic equation (q(x)=0), what is (t) called?
#roots
#definition
#concept
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A मूल / Root
B घात / Degree
C अचर पद / Constant term
D गुणांक / Coefficient
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Correct Answer
A. मूल / Root
Explanation
Simple Explanation
(q(t)=0) का अर्थ है कि (x=t) रखने पर समीकरण सत्य है। इसलिए (t) उस समीकरण का मूल है। / (q(t)=0) means the equation is true when (x=t). Therefore (t) is a root of the equation.
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क्या (x=5) समीकरण \(x^2-10x+25=0\) का मूल है?
Is (x=5) a root of \(x^2-10x+25=0\)?
#roots
#checking
#substitution
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A हाँ / Yes
B नहीं / No
C केवल (x=-5) मूल है / Only (x=-5) is a root
D कोई वास्तविक मूल नहीं / No real root
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Correct Answer
A. हाँ / Yes
Explanation
Simple Explanation
(x=5) रखने पर (25-50+25=0) मिलता है। इसलिए (5) इसका मूल है। / Putting (x=5) gives (25-50+25=0). Therefore (5) is its root.
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क्या (x=-3) समीकरण \(2x^2+7x+3=0\) का मूल है?
Is (x=-3) a root of \(2x^2+7x+3=0\)?
#roots
#negative_root
#checking
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A हाँ / Yes
B नहीं / No
C केवल (x=3) मूल है / Only (x=3) is a root
D निर्धारित नहीं / Cannot be determined
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Correct Answer
A. हाँ / Yes
Explanation
Simple Explanation
(x=-3) रखने पर (18-21+3=0) मिलता है। ऋणात्मक मूल रखते समय चिन्हों को ध्यान से देखें। / Putting (x=-3) gives (18-21+3=0). Watch the signs carefully when substituting a negative root.
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समीकरण \(x^2-14x+45=0\) के मूल कौन से हैं?
What are the roots of \(x^2-14x+45=0\)?
#roots
#factorisation
#numerical
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A (5) और (9) / (5) and (9)
B (3) और (15) / (3) and (15)
C (-5) और (-9) / (-5) and (-9)
D (0) और (14) / (0) and (14)
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Correct Answer
A. (5) और (9) / (5) and (9)
Explanation
Simple Explanation
(x-2 -14x+45=(x-5)(x-9)) है। इसलिए मूल (5) और (9) हैं। / (x-2 -14x+45=(x-5)(x-9)). Therefore the roots are (5) and (9).
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समीकरण \(4x^2-17x+4=0\) के मूल कौन से हैं?
What are the roots of \(4x^2-17x+4=0\)?
#roots
#factorisation
#fraction_root
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A (4) और \(\frac{1}{4}\) / (4) and \(\frac{1}{4}\)
B (-4) और \(-\frac{1}{4}\) / (-4) and \(-\frac{1}{4}\)
C (2) और (4) / (2) and (4)
D \(\frac{4}{3}\) और (1) / \(\frac{4}{3}\) and (1)
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Correct Answer
A. (4) और \(\frac{1}{4}\) / (4) and \(\frac{1}{4}\)
Explanation
Simple Explanation
(4x-2 -17x+4=(4x-1)(x-4)) है। इसलिए मूल \(\frac{1}{4}\) और (4) हैं। / (4x-2 -17x+4=(4x-1)(x-4)). Therefore the roots are \(\frac{1}{4}\) and (4).
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समीकरण \(5x^2+16x+3=0\) के मूल कौन से हैं?
What are the roots of \(5x^2+16x+3=0\)?
#roots
#factorisation
#negative_fraction
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A (-3) और \(-\frac{1}{5}\) / (-3) and \(-\frac{1}{5}\)
B (3) और \(\frac{1}{5}\) / (3) and \(\frac{1}{5}\)
C (-5) और (-3) / (-5) and (-3)
D (5) और (3) / (5) and (3)
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Correct Answer
A. (-3) और \(-\frac{1}{5}\) / (-3) and \(-\frac{1}{5}\)
Explanation
Simple Explanation
(5x-2 +16x+3=(5x+1)(x+3)) है। इसलिए मूल \(-\frac{1}{5}\) और (-3) हैं। / (5x-2 +16x+3=(5x+1)(x+3)). Therefore the roots are \(-\frac{1}{5}\) and (-3).
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किस समीकरण के मूल (3) और (-8) हैं?
Which equation has roots (3) and (-8)?
#roots
#equation_from_roots
#mixed_signs
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A \(x^2+5x-24=0\)
B \(x^2-5x-24=0\)
C \(x^2+11x+24=0\)
D \(x^2-11x+24=0\)
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Correct Answer
A. \(x^2+5x-24=0\)
Explanation
Simple Explanation
मूल (3) और (-8) होने पर ((x-3)(x+8)=0) होगा। खोलने पर \(x^2+5x-24=0\) मिलता है। / With roots (3) and (-8), we get ((x-3)(x+8)=0). Expanding gives \(x^2+5x-24=0\).
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यदि (4) समीकरण \(x^2+kx-32=0\) का मूल है तो (k) का मान क्या होगा?
If (4) is a root of \(x^2+kx-32=0\), what is the value of (k)?
#roots
#parameter
#substitution
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A (4)
B (-4)
C (8)
D (-8)
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Explanation
Simple Explanation
(x=4) रखने पर (16+4k-32=0) इसलिए (k=4) है। पैरामीटर के लिए दिए गए मूल को सीधे रखें। / Putting (x=4) gives (16+4k-32=0), so (k=4). Substitute the given root directly for a parameter.
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समीकरण \(6x^2-13x+5=0\) के मूलों का योग क्या है?
What is the sum of roots of \(6x^2-13x+5=0\)?
#roots
#sum_of_roots
#formula
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A \(\frac{13}{6}\)
B -\(\frac{13}{6}\)
C \(\frac{5}{6}\)
D -\(\frac{5}{6}\)
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Correct Answer
A. \(\frac{13}{6}\)
Explanation
Simple Explanation
मूलों का योग \(-\frac{b}{a}\) होता है। यहां (a=6) और (b=-13) हैं इसलिए योग \(\frac{13}{6}\) है। / The sum of roots is \(-\frac{b}{a}\). Here (a=6) and (b=-13), so the sum is \(\frac{13}{6}\).
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समीकरण \(7x^2+4x-9=0\) के मूलों का गुणनफल क्या है?
What is the product of roots of \(7x^2+4x-9=0\)?
#roots
#product_of_roots
#formula
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A -\(\frac{9}{7}\)
B \(\frac{9}{7}\)
C -\(\frac{4}{7}\)
D \(\frac{4}{7}\)
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Correct Answer
A. -\(\frac{9}{7}\)
Explanation
Simple Explanation
मूलों का गुणनफल \(\frac{c}{a}\) होता है। यहां \(\frac{-9}{7}=-\frac{9}{7}\) सही है। / The product of roots is \(\frac{c}{a}\). Here \(\frac{-9}{7}=-\frac{9}{7}\) is correct.
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समीकरण \(4x^2-20x+25=0\) का डिस्क्रिमिनेंट (D) क्या है?
What is the discriminant (D) of \(4x^2-20x+25=0\)?
#roots
#discriminant
#equal_roots
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A (0)
B (20)
C (-20)
D (100)
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Explanation
Simple Explanation
यहां \(D=b^2-4ac=400-400=0\) है। इसलिए इसके दोनों वास्तविक मूल बराबर होंगे। / Here \(D=b^2-4ac=400-400=0\). Therefore its two real roots will be equal.
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यदि किसी द्विघात समीकरण के लिए (D=81) है तो वास्तविक मूलों की प्रकृति क्या होगी?
If (D=81) for a quadratic equation, what will be the nature of real roots?
#roots
#discriminant
#nature
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A दो भिन्न वास्तविक मूल / Two distinct real roots
B दो बराबर वास्तविक मूल / Two equal real roots
C कोई वास्तविक मूल नहीं / No real root
D एक मूल (0) / One root (0)
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Correct Answer
A. दो भिन्न वास्तविक मूल / Two distinct real roots
Explanation
Simple Explanation
क्योंकि (D=81>0) है इसलिए दो भिन्न वास्तविक मूल होंगे। (D>0) मूलों के अलग होने का संकेत है। / Since (D=81>0), there will be two distinct real roots. (D>0) indicates different roots.
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समीकरण \(x^2-6x+20=0\) के वास्तविक मूलों के बारे में सही कथन कौन सा है?
Which statement is correct about the real roots of \(x^2-6x+20=0\)?
#roots
#no_real_roots
#discriminant
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A दो भिन्न वास्तविक मूल हैं / It has two distinct real roots
B दो बराबर वास्तविक मूल हैं / It has two equal real roots
C कोई वास्तविक मूल नहीं है / It has no real roots
D मूल (6) और (20) हैं / The roots are (6) and (20)
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Correct Answer
C. कोई वास्तविक मूल नहीं है / It has no real roots
Explanation
Simple Explanation
यहां (D=(-6)2 -4(1)(20)=-44<0) है। इसलिए कोई वास्तविक मूल नहीं है। / Here (D=(-6)2 -4(1)(20)=-44<0). Therefore there are no real roots.
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समीकरण \(x^2-10x+k=0\) के बराबर वास्तविक मूल होने के लिए (k) का मान क्या होगा?
For \(x^2-10x+k=0\) to have equal real roots, what should be the value of (k)?
#roots
#equal_roots
#parameter
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A (25)
B (10)
C (5)
D (100)
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Explanation
Simple Explanation
बराबर मूलों के लिए (D=0) होता है। (100-4k=0) से (k=25) मिलता है। / For equal roots (D=0). From (100-4k=0), we get (k=25).
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यदि मूलों का योग (11) और गुणनफल (28) है तो मोनिक द्विघात समीकरण कौन सा होगा?
If the sum of roots is (11) and product is (28), which monic quadratic equation is formed?
#roots
#equation_from_sum_product
#monic
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A \(x^2-11x+28=0\)
B \(x^2+11x+28=0\)
C \(x^2-28x+11=0\)
D \(x^2+28x+11=0\)
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Correct Answer
A. \(x^2-11x+28=0\)
Explanation
Simple Explanation
\(मोनिक समीकरण (x^2-(\)योग)x+गुणनफल=0) होता है। इसलिए \(x^2-11x+28=0\) सही है। \(/ A monic equation is (x^2-(\)sum)x+product\(=0). Therefore (x^2-11x+28=0) is correct.\)
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समीकरण \(x^2-17x+70=0\) का एक मूल (7) है तो दूसरा मूल क्या है?
If one root of \(x^2-17x+70=0\) is (7), what is the other root?
#roots
#other_root
#product
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A (10)
B (7)
C (17)
D (70)
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Explanation
Simple Explanation
मूलों का गुणनफल (70) है और एक मूल (7) है। दूसरा मूल \(\frac{70}{7}=10\) होगा। / The product of roots is (70) and one root is (7). The other root is \(\frac{70}{7}=10\).
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यदि \(4\alpha\) और \(4\beta\) नए मूल हैं तथा \(\alpha+\beta=3\) है तो नए मूलों का योग क्या होगा?
If \(4\alpha\) and \(4\beta\) are new roots and \(\alpha+\beta=3\), what is the sum of the new roots?
#roots
#transformed_roots
#sum
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A (12)
B (7)
C (3)
D \(\frac{3}{4}\)
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Explanation
Simple Explanation
नए मूलों का योग (4\alpha+4\beta=4\(\alpha+\beta\)=12) है। गुणक लगे मूलों में योग भी उसी गुणक से गुणा होता है। / The sum of new roots is (4\alpha+4\beta=4\(\alpha+\beta\)=12). When roots are multiplied by a factor, the sum is also multiplied by that factor.
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यदि मूल \(\alpha\) और \(\beta\) हैं तथा \(\alpha+\beta=8\) और \(\alpha\beta=15\) है तो \(\alpha^2+\beta^2\) का मान क्या है?
If the roots are \(\alpha\) and \(\beta\) with \(\alpha+\beta=8\) and \(\alpha\beta=15\), what is \(\alpha^2+\beta^2\)?
#roots
#identity
#sum_product
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A (34)
B (49)
C (64)
D (30)
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Explanation
Simple Explanation
(\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta=64-30=34) है। ऐसे प्रश्नों में पहचान का प्रयोग करें। / (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta=64-30=34). Use identities in such questions.
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यदि द्विघात समीकरण के मूल (4) और (-9) हैं तो \(\frac{1}{\alpha}+\frac{1}{\beta}\) का मान क्या होगा?
If the roots of a quadratic equation are (4) and (-9), what is the value of \(\frac{1}{\alpha}+\frac{1}{\beta}\)?
#roots
#reciprocal_sum
#identity
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A \(\frac{5}{36}\)
B -\(\frac{5}{36}\)
C \(\frac{13}{36}\)
D -\(\frac{13}{36}\)
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Correct Answer
A. \(\frac{5}{36}\)
Explanation
Simple Explanation
यहां \(\alpha+\beta=-5\) और \(\alpha\beta=-36\) है। इसलिए \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{-5}{-36}=\frac{5}{36}\) है। / Here \(\alpha+\beta=-5\) and \(\alpha\beta=-36\). Therefore \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{-5}{-36}=\frac{5}{36}\).
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समीकरण \(5x^2-18x+9=0\) के मूलों का योग और गुणनफल क्रमशः क्या हैं?
For \(5x^2-18x+9=0\), what are the sum and product of roots respectively?
#roots
#sum_product
#formula
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A \(\frac{18}{5}\) और \(\frac{9}{5}\) / \(\frac{18}{5}\) and \(\frac{9}{5}\)
B -\(\frac{18}{5}\) और \(\frac{9}{5}\) / \(-\frac{18}{5}\) and \(\frac{9}{5}\)
C \(\frac{18}{5}\) और \(-\frac{9}{5}\) / \(\frac{18}{5}\) and \(-\frac{9}{5}\)
D (18) और (9) / (18) and (9)
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Correct Answer
A. \(\frac{18}{5}\) और \(\frac{9}{5}\) / \(\frac{18}{5}\) and \(\frac{9}{5}\)
Explanation
Simple Explanation
योग \(-\frac{b}{a}=\frac{18}{5}\) और गुणनफल \(\frac{c}{a}=\frac{9}{5}\) है। पहले (a), (b), (c) पहचानें। / The sum is \(-\frac{b}{a}=\frac{18}{5}\) and the product is \(\frac{c}{a}=\frac{9}{5}\). Identify (a), (b), and (c) first.
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यदि (x+6) और (x-2) किसी द्विघात समीकरण के गुणनखंड हैं तो उसके मूल कौन से हैं?
If (x+6) and (x-2) are factors of a quadratic equation, what are its roots?
#roots
#factors_to_roots
#zero_product
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A (-6) और (2) / (-6) and (2)
B (6) और (-2) / (6) and (-2)
C (6) और (2) / (6) and (2)
D (-6) और (-2) / (-6) and (-2)
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Correct Answer
A. (-6) और (2) / (-6) and (2)
Explanation
Simple Explanation
(x+6=0) से (x=-6) और (x-2=0) से (x=2) मिलता है। गुणनखंड से मूल निकालते समय चिन्ह बदलता है। / From (x+6=0), (x=-6), and from (x-2=0), (x=2). The sign changes when finding a root from a factor.
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समीकरण \(x^2+px+36=0\) के दोनों मूल (-6) और (-6) हैं तो (p) का मान क्या है?
If both roots of \(x^2+px+36=0\) are (-6) and (-6), what is the value of (p)?
#roots
#equal_roots
#parameter
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A (12)
B (-12)
C (6)
D (-6)
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Explanation
Simple Explanation
मूलों का योग (-12) है और (-p=-12) होगा। इसलिए (p=12) है। / The sum of roots is (-12), and (-p=-12). Therefore (p=12).
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यदि \(x=\frac{5}{2}\) समीकरण \(2x^2-9x+c=0\) का मूल है तो (c) का मान क्या है?
If \(x=\frac{5}{2}\) is a root of \(2x^2-9x+c=0\), what is the value of (c)?
#roots
#parameter
#fraction_root
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A (10)
B (-10)
C \(\frac{5}{2}\)
D -\(\frac{5}{2}\)
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Explanation
Simple Explanation
\(x=\frac{5}{2}\) रखने पर \(\frac{25}{2}-\frac{45}{2}+c=0\) इसलिए (c=10) है। भिन्न मूल में हर पद सावधानी से निकालें। / Putting \(x=\frac{5}{2}\) gives \(\frac{25}{2}-\frac{45}{2}+c=0\), so (c=10). Calculate each term carefully with a fractional root.
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समीकरण \(25x^2-30x+9=0\) के मूलों की प्रकृति क्या है?
What is the nature of roots of \(25x^2-30x+9=0\)?
#roots
#nature
#perfect_square
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A बराबर वास्तविक मूल / Equal real roots
B भिन्न वास्तविक मूल / Distinct real roots
C कोई वास्तविक मूल नहीं / No real roots
D एक मूल (0) है / One root is (0)
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Correct Answer
A. बराबर वास्तविक मूल / Equal real roots
Explanation
Simple Explanation
यह (25x-2 -30x+9=(5x-3)2 ) है। इसलिए दोनों मूल \(\frac{3}{5}\) और \(\frac{3}{5}\) बराबर हैं। / It is (25x-2 -30x+9=(5x-3)2 ). Therefore both roots are \(\frac{3}{5}\) and \(\frac{3}{5}\), which are equal.
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यदि किसी द्विघात समीकरण में (a=1), (b=-20), (c=100) है तो मूल कौन से होंगे?
If a quadratic equation has (a=1), (b=-20), and (c=100), what will be the roots?
#roots
#coefficients
#repeated_root
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A (10) और (10) / (10) and (10)
B (-10) और (-10) / (-10) and (-10)
C (20) और (100) / (20) and (100)
D (0) और (10) / (0) and (10)
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Correct Answer
A. (10) और (10) / (10) and (10)
Explanation
Simple Explanation
समीकरण \(x^2-20x+100=0\) है जो ((x-10)2 =0) बनता है। इसलिए दोनों मूल (10) हैं। / The equation is \(x^2-20x+100=0\), which becomes ((x-10)2 =0). Therefore both roots are (10).
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यदि किसी द्विघात समीकरण के मूलों का योग (0) है तो मूलों के बारे में सही कथन कौन सा हो सकता है?
If the sum of roots of a quadratic equation is (0), which statement about the roots can be correct?
#roots
#zero_sum
#reasoning
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A मूल एक दूसरे के विपरीत हैं / The roots are opposites of each other
B दोनों मूल हमेशा (1) हैं / Both roots are always (1)
C दोनों मूल हमेशा धनात्मक हैं / Both roots are always positive
D मूलों का गुणनफल हमेशा (0) है / The product is always (0)
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Correct Answer
A. मूल एक दूसरे के विपरीत हैं / The roots are opposites of each other
Explanation
Simple Explanation
यदि \(\alpha+\beta=0\) है तो \(\beta=-\alpha\) होता है। इसलिए मूल विपरीत हो सकते हैं। / If \(\alpha+\beta=0\), then \(\beta=-\alpha\). Therefore the roots can be opposites.
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समीकरण \(x^2+2x-35=0\) के मूल कौन से हैं?
What are the roots of \(x^2+2x-35=0\)?
#roots
#factorisation
#mixed_signs
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A (5) और (-7) / (5) and (-7)
B (7) और (-5) / (7) and (-5)
C (5) और (7) / (5) and (7)
D (-5) और (-7) / (-5) and (-7)
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Correct Answer
A. (5) और (-7) / (5) and (-7)
Explanation
Simple Explanation
(x-2 +2x-35=(x+7)(x-5)) है। इसलिए मूल (-7) और (5) हैं। / (x-2 +2x-35=(x+7)(x-5)). Therefore the roots are (-7) and (5).
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यदि \(\alpha\) और \(\beta\) समीकरण \(x^2-9x+20=0\) के मूल हैं तो \(\alpha+\beta+\alpha\beta\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-9x+20=0\), what is the value of \(\alpha+\beta+\alpha\beta\)?
#roots
#sum_product
#expression
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A (29)
B (20)
C (9)
D (11)
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Explanation
Simple Explanation
यहां \(\alpha+\beta=9\) और \(\alpha\beta=20\) है। इसलिए कुल (9+20=29) है। / Here \(\alpha+\beta=9\) and \(\alpha\beta=20\). Therefore the total is (9+20=29).
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यदि मूलों का योग (0) और गुणनफल (-36) है तो मोनिक समीकरण कौन सा होगा?
If the sum of roots is (0) and product is (-36), which monic equation is formed?
#roots
#equation_from_sum_product
#zero_sum
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A \(x^2-36=0\)
B \(x^2+36=0\)
C \(x^2-36x=0\)
D \(x^2+36x=0\)
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Correct Answer
A. \(x^2-36=0\)
Explanation
Simple Explanation
मोनिक समीकरण (x-2 -(0)x+(-36)=0) होगा। इसलिए \(x^2-36=0\) सही है। / The monic equation is (x-2 -(0)x+(-36)=0). Therefore \(x^2-36=0\) is correct.
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समीकरण \(25x^2-10x+1=0\) का दोहराया हुआ मूल क्या है?
What is the repeated root of \(25x^2-10x+1=0\)?
#roots
#repeated_root
#fraction_root
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A \(\frac{1}{5}\)
B -\(\frac{1}{5}\)
C (5)
D (-5)
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Correct Answer
A. \(\frac{1}{5}\)
Explanation
Simple Explanation
(25x-2 -10x+1=(5x-1)2 ) है। इसलिए दोहराया मूल \(\frac{1}{5}\) है। / (25x-2 -10x+1=(5x-1)2 ). Therefore the repeated root is \(\frac{1}{5}\).
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यदि (x=0) समीकरण \(2x^2+mx+n=0\) का मूल है तो (n) का मान क्या होगा?
If (x=0) is a root of \(2x^2+mx+n=0\), what will be the value of (n)?
#roots
#zero_root
#condition
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A (0)
B (2)
C (m)
D (1)
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Explanation
Simple Explanation
(x=0) रखने पर केवल (n) बचता है। इसलिए (n=0) होना चाहिए। / Putting (x=0) leaves only (n). Therefore (n=0) must hold.
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यदि (x=3) और (x=8) किसी मोनिक द्विघात समीकरण के मूल हैं तो (x) का गुणांक क्या होगा?
If (x=3) and (x=8) are roots of a monic quadratic equation, what will be the coefficient of (x)?
#roots
#coefficient_from_roots
#monic
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A (-11)
B (11)
C (24)
D (-24)
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Explanation
Simple Explanation
मूलों का योग (3+8=11) है। मोनिक समीकरण में (x) का गुणांक (-(योग)=-11) होता है। \(/ The sum of roots is (3+8=11). In a monic equation the coefficient of (x) is (-(\)sum\()=-11).\)
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समीकरण \(4x^2+3x-10=0\) के मूल कौन से हैं?
What are the roots of \(4x^2+3x-10=0\)?
#roots
#factorisation
#medium
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A \(\frac{5}{4}\) और (-2) / \(\frac{5}{4}\) and (-2)
B -\(\frac{5}{4}\) और (2) / \(-\frac{5}{4}\) and (2)
C (5) और (-2) / (5) and (-2)
D (2) और (-5) / (2) and (-5)
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Correct Answer
A. \(\frac{5}{4}\) और (-2) / \(\frac{5}{4}\) and (-2)
Explanation
Simple Explanation
(4x-2 +3x-10=(4x-5)(x+2)) है। इसलिए मूल \(\frac{5}{4}\) और (-2) हैं। / (4x-2 +3x-10=(4x-5)(x+2)). Therefore the roots are \(\frac{5}{4}\) and (-2).
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समीकरण \(x^2+mx+30=0\) के मूल (-5) और (-6) हैं तो (m) का मान क्या है?
If the roots of \(x^2+mx+30=0\) are (-5) and (-6), what is the value of (m)?
#roots
#parameter
#sum_of_roots
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A (11)
B (-11)
C (30)
D (-30)
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Explanation
Simple Explanation
मूलों का योग (-11) है और (-m=-11) होगा। इसलिए (m=11) है। / The sum of roots is (-11), and (-m=-11). Therefore (m=11).
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यदि द्विघात समीकरण के मूल (6) और \(\frac{1}{6}\) हैं तो उनके बारे में सही कथन कौन सा है?
If the roots of a quadratic equation are (6) and \(\frac{1}{6}\), which statement is correct about them?
#roots
#reciprocal_roots
#concept
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A वे एक दूसरे के व्युत्क्रम हैं / They are reciprocals of each other
B वे बराबर मूल हैं / They are equal roots
C उनका योग (1) है / Their sum is (1)
D उनका गुणनफल (0) है / Their product is (0)
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Correct Answer
A. वे एक दूसरे के व्युत्क्रम हैं / They are reciprocals of each other
Explanation
Simple Explanation
\(6\cdot\frac{1}{6}=1\) है इसलिए दोनों व्युत्क्रम मूल हैं। व्युत्क्रम मूलों का गुणनफल (1) होता है। / \(6\cdot\frac{1}{6}=1\), so the roots are reciprocals. Reciprocal roots have product (1).
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समीकरण \(x^2-7x-30=0\) का धनात्मक मूल कौन सा है?
What is the positive root of \(x^2-7x-30=0\)?
#roots
#positive_root
#factorisation
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A (10)
B (-3)
C (3)
D (-10)
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Explanation
Simple Explanation
(x-2 -7x-30=(x-10)(x+3)) है। इसलिए मूल (10) और (-3) हैं तथा धनात्मक मूल (10) है। / (x-2 -7x-30=(x-10)(x+3)). Therefore the roots are (10) and (-3), and the positive root is (10).
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यदि (D=0) और मूलों का योग (14) है तो प्रत्येक मूल क्या होगा?
If (D=0) and the sum of roots is (14), what will each root be?
#roots
#equal_roots
#sum
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A (7)
B (14)
C (0)
D (28)
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Explanation
Simple Explanation
(D=0) होने पर दोनों मूल बराबर होते हैं। इसलिए प्रत्येक मूल \(\frac{14}{2}=7\) होगा। / When (D=0), both roots are equal. Therefore each root is \(\frac{14}{2}=7\).
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समीकरण \(x^2-8x-9=0\) के मूलों का अंतर कितना है?
What is the difference between the roots of \(x^2-8x-9=0\)?
#roots
#difference_of_roots
#factorisation
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A (10)
B (8)
C (9)
D (1)
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Explanation
Simple Explanation
(x-2 -8x-9=(x-9)(x+1)) है। मूल (9) और (-1) हैं इसलिए अंतर (10) है। / (x-2 -8x-9=(x-9)(x+1)). The roots are (9) and (-1), so the difference is (10).
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यदि \(\alpha\) और \(\beta\) समीकरण \(4x^2-20x+7=0\) के मूल हैं तो \(\alpha+\beta\) क्या है?
If \(\alpha\) and \(\beta\) are roots of \(4x^2-20x+7=0\), what is \(\alpha+\beta\)?
#roots
#sum_of_roots
#calculation
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A (5)
B (-5)
C \(\frac{7}{4}\)
D -\(\frac{7}{4}\)
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Explanation
Simple Explanation
मूलों का योग \(-\frac{b}{a}=-\frac{-20}{4}=5\) है। सूत्र में (b) का चिन्ह सही रखें। / The sum of roots is \(-\frac{b}{a}=-\frac{-20}{4}=5\). Keep the sign of (b) correct in the formula.
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यदि \(\alpha\) और \(\beta\) समीकरण \(6x^2+5x-4=0\) के मूल हैं तो \(\alpha\beta\) क्या है?
If \(\alpha\) and \(\beta\) are roots of \(6x^2+5x-4=0\), what is \(\alpha\beta\)?
#roots
#product_of_roots
#calculation
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A -\(\frac{2}{3}\)
B \(\frac{2}{3}\)
C -\(\frac{5}{6}\)
D \(\frac{5}{6}\)
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Correct Answer
A. -\(\frac{2}{3}\)
Explanation
Simple Explanation
मूलों का गुणनफल \(\frac{c}{a}=\frac{-4}{6}=-\frac{2}{3}\) होता है। (c) का चिन्ह न भूलें। / The product of roots is \(\frac{c}{a}=\frac{-4}{6}=-\frac{2}{3}\). Do not forget the sign of (c).
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समीकरण \(x^2+6x+k=0\) के वास्तविक मूल होने के लिए (k) के बारे में कौन सी शर्त सही है?
For \(x^2+6x+k=0\) to have real roots, which condition on (k) is correct?
#roots
#real_roots
#parameter_condition
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A \(k\le9\)
B (k>9)
C (k=12)
D (k<0) ही होगा / Only (k<0)
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Correct Answer
A. \(k\le9\)
Explanation
Simple Explanation
वास्तविक मूलों के लिए \(D\ge0\) चाहिए। \(36-4k\ge0\) से \(k\le9\) मिलता है। / For real roots \(D\ge0\) is required. From \(36-4k\ge0\), we get \(k\le9\).
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समीकरण \(x^2+13x+42=0\) के दोनों मूलों के चिन्ह कैसे होंगे?
What will be the signs of both roots of \(x^2+13x+42=0\)?
#roots
#sign_of_roots
#reasoning
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A दोनों ऋणात्मक / Both negative
B दोनों धनात्मक / Both positive
C एक धनात्मक और एक ऋणात्मक / One positive and one negative
D दोनों शून्य / Both zero
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Correct Answer
A. दोनों ऋणात्मक / Both negative
Explanation
Simple Explanation
मूल (-6) और (-7) हैं क्योंकि ((x+6)(x+7)=0)। योग ऋणात्मक और गुणनफल धनात्मक होने पर दोनों मूल ऋणात्मक होते हैं। / The roots are (-6) and (-7) because ((x+6)(x+7)=0). When the sum is negative and product is positive, both roots are negative.
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यदि \(x^2+ax+25=0\) के मूल (5) और (5) हैं तो (a) का मान क्या है?
If the roots of \(x^2+ax+25=0\) are (5) and (5), what is the value of (a)?
#roots
#parameter
#equal_roots
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A (-10)
B (10)
C (5)
D (-5)
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Explanation
Simple Explanation
मूलों का योग (10) है और (-a=10) होगा। इसलिए (a=-10) है। / The sum of roots is (10), and (-a=10). Therefore (a=-10).
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समीकरण \(10x^2-17x+3=0\) के मूल कौन से हैं?
What are the roots of \(10x^2-17x+3=0\)?
#roots
#factorisation
#fraction_roots
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A \(\frac{3}{2}\) और \(\frac{1}{5}\) / \(\frac{3}{2}\) and \(\frac{1}{5}\)
B -\(\frac{3}{2}\) और \(-\frac{1}{5}\) / \(-\frac{3}{2}\) and \(-\frac{1}{5}\)
C (2) और (3) / (2) and (3)
D \(\frac{5}{3}\) और \(\frac{1}{2}\) / \(\frac{5}{3}\) and \(\frac{1}{2}\)
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Correct Answer
A. \(\frac{3}{2}\) और \(\frac{1}{5}\) / \(\frac{3}{2}\) and \(\frac{1}{5}\)
Explanation
Simple Explanation
(10x-2 -17x+3=(2x-3)(5x-1)) है। इसलिए मूल \(\frac{3}{2}\) और \(\frac{1}{5}\) हैं। / (10x-2 -17x+3=(2x-3)(5x-1)). Therefore the roots are \(\frac{3}{2}\) and \(\frac{1}{5}\).
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यदि किसी द्विघात समीकरण के मूल (c) और (-c) हैं तो अचर पद और अग्र गुणांक के अनुपात \(\frac{d}{a}\) का मान क्या होगा?
If the roots of a quadratic equation are (c) and (-c), what is the value of the ratio \(\frac{d}{a}\) of constant term to leading coefficient?
#roots
#opposite_roots
#product
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A \(-c^2\)
B \(c^2\)
C (0)
D (2c)
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Correct Answer
A. \(-c^2\)
Explanation
Simple Explanation
\(\frac{d}{a}\) मूलों का गुणनफल होता है। यहां (c(-c)=-c-2 ) है। / \(\frac{d}{a}\) is the product of roots. Here (c(-c)=-c-2 ).
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यदि \(\alpha+\beta=-7\) और \(\alpha\beta=-18\) है तो \(\alpha\) और \(\beta\) के लिए मोनिक समीकरण कौन सा है?
If \(\alpha+\beta=-7\) and \(\alpha\beta=-18\), which monic equation has roots \(\alpha\) and \(\beta\)?
#roots
#equation_from_roots
#sum_product
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A \(x^2+7x-18=0\)
B \(x^2-7x-18=0\)
C \(x^2+18x-7=0\)
D \(x^2-18x+7=0\)
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Correct Answer
A. \(x^2+7x-18=0\)
Explanation
Simple Explanation
मोनिक समीकरण (x-2 -\(\alpha+\beta\)x+\alpha\beta=0) होता है। इसलिए \(x^2+7x-18=0\) सही है। / The monic equation is (x-2 -\(\alpha+\beta\)x+\alpha\beta=0). Therefore \(x^2+7x-18=0\) is correct.
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समीकरण \(x^2-9x+14=0\) के मूलों के वर्गों का योग क्या है?
What is the sum of squares of the roots of \(x^2-9x+14=0\)?
#roots
#sum_of_squares
#identity
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A (53)
B (81)
C (28)
D (67)
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Explanation
Simple Explanation
यहां \(\alpha+\beta=9\) और \(\alpha\beta=14\) है। \(\alpha^2+\beta^2=81-28=53\) होगा। / Here \(\alpha+\beta=9\) and \(\alpha\beta=14\). Thus \(\alpha^2+\beta^2=81-28=53\).
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यदि (x=4) समीकरण \(ax^2-10x+8=0\) का मूल है तो (a) का मान क्या है?
If (x=4) is a root of \(ax^2-10x+8=0\), what is the value of (a)?
#roots
#parameter
#leading_coefficient
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A (2)
B (1)
C (-2)
D (-1)
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Explanation
Simple Explanation
(x=4) रखने पर (16a-40+8=0) इसलिए (16a=32) और (a=2)। अज्ञात गुणांक में प्रतिस्थापन करें। / Putting (x=4) gives (16a-40+8=0), so (16a=32) and (a=2). Substitute when a coefficient is unknown.
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समीकरण \(x^2-22x+121=0\) के मूलों का योग और गुणनफल क्रमशः क्या हैं?
For \(x^2-22x+121=0\), what are the sum and product of roots respectively?
#roots
#repeated_root
#sum_product
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A (22) और (121) / (22) and (121)
B -(22) और (121) / (-22) and (121)
C (11) और (11) / (11) and (11)
D (121) और (22) / (121) and (22)
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Correct Answer
A. (22) और (121) / (22) and (121)
Explanation
Simple Explanation
यह ((x-11)2 =0) है इसलिए दोनों मूल (11) हैं। योग (22) और गुणनफल (121) होगा। / It is ((x-11)2 =0), so both roots are (11). The sum is (22) and the product is (121).
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यदि \(\alpha\) और \(\beta\) समीकरण \(4x^2-5x-6=0\) के मूल हैं तो \(\alpha+\beta-2\alpha\beta\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(4x^2-5x-6=0\), what is the value of \(\alpha+\beta-2\alpha\beta\)?
#roots
#sum_product
#expression
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A \(\frac{17}{4}\)
B \(-\frac{17}{4}\)
C \(\frac{5}{4}\)
D \(-\frac{3}{2}\)
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Correct Answer
A. \(\frac{17}{4}\)
Explanation
Simple Explanation
यहां \(\alpha+\beta=\frac{5}{4}\) और \(\alpha\beta=-\frac{3}{2}\) है। इसलिए \(\alpha+\beta-2\alpha\beta=\frac{5}{4}+3=\frac{17}{4}\) होगा। / Here \(\alpha+\beta=\frac{5}{4}\) and \(\alpha\beta=-\frac{3}{2}\). Therefore \(\alpha+\beta-2\alpha\beta=\frac{5}{4}+3=\frac{17}{4}\).
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