बहुपद (p(x)=ax-3 +bx-2 +cx+d) में \(a\ne0\) है। यदि (p(x)) को रैखिक बहुपद कहा जाए, तो यह कथन क्यों गलत है?
In the polynomial (p(x)=ax-3 +bx-2 +cx+d), \(a\ne0\). Why is it wrong to call (p(x)) a linear polynomial?
#polynomials
#degree
#cubic
#hard
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A क्योंकि घात (1) है / Because degree is (1)
B क्योंकि घात (2) है / Because degree is (2)
C क्योंकि घात (3) है / Because degree is (3)
D क्योंकि स्थिर पद (d) है / Because constant term is (d)
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Correct Answer
C. क्योंकि घात (3) है / Because degree is (3)
Explanation
Simple Explanation
सबसे बड़ी घात (3) है इसलिए बहुपद घन है। परीक्षा में हमेशा सबसे बड़े शून्येतर घातांक को देखें। / The highest power is (3), so the polynomial is cubic. In exams, always check the highest non-zero exponent.
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यदि (p(x)=kx-2 -5x+6) का एक शून्यक (2) है, तो (k) का मान क्या होगा?
If (2) is a zero of (p(x)=kx-2 -5x+6), what is the value of (k)?
#polynomials
#zero
#value
#hard
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A (1)
B (2)
C (-1)
D (-2)
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Explanation
Simple Explanation
(p(2)=0) रखने पर (4k-10+6=0) से (k=1) मिलता है। शून्यक के लिए (p\(\alpha\)=0) लगाएं। / Putting (p(2)=0) gives (4k-10+6=0), so (k=1). For a zero, use (p\(\alpha\)=0).
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यदि (p(x)=x-2 -(m+3)x+12) के शून्यक (3) और (4) हैं, तो (m) का मान क्या है?
If the zeroes of (p(x)=x-2 -(m+3)x+12) are (3) and (4), what is (m)?
#polynomials
#zeroes
#sum
#hard
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A (3)
B (4)
C (5)
D (7)
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Explanation
Simple Explanation
शून्यकों का योग (7) है और यह (m+3) के बराबर है। इसलिए (m=4) मिलेगा। / The sum of zeroes is (7), which equals (m+3). Therefore, (m=4).
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द्विघात बहुपद \(2x^2-7x+3\) के शून्यकों का गुणनफल क्या है?
What is the product of the zeroes of the quadratic polynomial \(2x^2-7x+3\)?
#polynomials
#zeroes
#product
#hard
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A (3)
B \(\frac{3}{2}\)
C \(-\frac{7}{2}\)
D (2)
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Correct Answer
B. \(\frac{3}{2}\)
Explanation
Simple Explanation
द्विघात \(ax^2+bx+c\) में गुणनफल \(\frac{c}{a}\) होता है। यहां \(\frac{3}{2}\) सही है। / For \(ax^2+bx+c\), the product is \(\frac{c}{a}\). Here, \(\frac{3}{2}\) is correct.
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यदि \(x^2-9x+k\) के शून्यक बराबर हैं, तो (k) का मान क्या है?
If the zeroes of \(x^2-9x+k\) are equal, what is the value of (k)?
#polynomials
#equal_zeroes
#discriminant
#hard
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A (18)
B (81)
C \(\frac{81}{4}\)
D \(\frac{9}{4}\)
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Correct Answer
C. \(\frac{81}{4}\)
Explanation
Simple Explanation
बराबर शून्यकों के लिए विविक्तकर \(b^2-4ac=0\) होता है। इसलिए (81-4k=0) से \(k=\frac{81}{4}\) मिलता है। / For equal zeroes, the discriminant \(b^2-4ac=0\). Thus (81-4k=0), giving \(k=\frac{81}{4}\).
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बहुपद \(x^3-4x^2+x+6\) में (x=2) रखने पर क्या निष्कर्ष निकलेगा?
What conclusion follows when (x=2) is substituted in \(x^3-4x^2+x+6\)?
#polynomials
#zero
#substitution
#hard
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A (2) शून्यक है / (2) is a zero
B (2) शून्यक नहीं है / (2) is not a zero
C बहुपद रैखिक है / The polynomial is linear
D बहुपद स्थिर है / The polynomial is constant
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Correct Answer
A. (2) शून्यक है / (2) is a zero
Explanation
Simple Explanation
(p(2)=8-16+2+6=0) है इसलिए (2) शून्यक है। सीधे प्रतिस्थापन से जल्दी जांच होती है। / (p(2)=8-16+2+6=0), so (2) is a zero. Direct substitution is the quickest check.
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यदि (p(x)=x-3 +px-2 +qx-6) के शून्यक (1), (2) और (3) हैं, तो (p+q) का मान क्या है?
If the zeroes of (p(x)=x-3 +px-2 +qx-6) are (1), (2), and (3), what is (p+q)?
#polynomials
#cubic
#coefficients
#hard
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A (-5)
B (5)
C (11)
D (0)
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Explanation
Simple Explanation
बहुपद ((x-1)(x-2)(x-3)=x-3 -6x-2 +11x-6) है। इसलिए (p=-6), (q=11) और (p+q=5) नहीं बल्कि (5) है। / The polynomial is ((x-1)(x-2)(x-3)=x-3 -6x-2 +11x-6). Hence (p=-6), (q=11), so (p+q=5).
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यदि (p(x)=x-3 +px-2 +qx-6) के शून्यक (1), (2) और (3) हैं, तो (p+q) का सही मान क्या है?
If the zeroes of (p(x)=x-3 +px-2 +qx-6) are (1), (2), and (3), what is the correct value of (p+q)?
#polynomials
#cubic
#coefficient_matching
#hard
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A (5)
B (-5)
C (17)
D (-17)
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Explanation
Simple Explanation
((x-1)(x-2)(x-3)=x-3 -6x-2 +11x-6) से (p=-6) और (q=11)। अतः (p+q=5) है। / From ((x-1)(x-2)(x-3)=x-3 -6x-2 +11x-6), (p=-6) and (q=11). Hence (p+q=5).
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द्विघात बहुपद जिसके शून्यक \(\frac{2}{3}\) और \(-\frac{5}{4}\) हैं, कौन सा हो सकता है?
Which can be a quadratic polynomial whose zeroes are \(\frac{2}{3}\) and \(-\frac{5}{4}\)?
#polynomials
#formation
#zeroes
#hard
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A \(12x^2+7x-10\)
B \(12x^2-7x-10\)
C \(12x^2+7x+10\)
D \(12x^2-7x+10\)
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Correct Answer
A. \(12x^2+7x-10\)
Explanation
Simple Explanation
शून्यकों का योग \(-\frac{7}{12}\) और गुणनफल \(-\frac{5}{6}\) है। इसलिए \(12x^2+7x-10\) सही है। / The sum is \(-\frac{7}{12}\) and the product is \(-\frac{5}{6}\). So \(12x^2+7x-10\) is correct.
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यदि \(x^2+ax+b\) के शून्यक (4) और (-7) हैं, तो (a-b) क्या होगा?
If the zeroes of \(x^2+ax+b\) are (4) and (-7), what is (a-b)?
#polynomials
#zeroes
#coefficient_relation
#hard
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A (25)
B (31)
C (-25)
D (-31)
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Explanation
Simple Explanation
योग (-3) है इसलिए (a=3) और गुणनफल (-28) है इसलिए (b=-28)। अतः (a-b=31) है। / The sum is (-3), so (a=3), and the product is (-28), so (b=-28). Hence (a-b=31).
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किस बहुपद की घात (4) है?
Which polynomial has degree (4)?
#polynomials
#degree
#zero_coefficient
#hard
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A \(7x^4+0x^5-3x+1\)
B \(0x^6+5x^3-2\)
C \(x^2+x+1\)
D (9)
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Correct Answer
A. \(7x^4+0x^5-3x+1\)
Explanation
Simple Explanation
\(0x^5\) पद घात नहीं बढ़ाता क्योंकि उसका गुणांक (0) है। सबसे बड़ा शून्येतर घात (4) है। / The term \(0x^5\) does not affect degree because its coefficient is (0). The highest non-zero power is (4).
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यदि (p(x)=x-2 -5x+6) है, तो (p(3)-p(2)) का मान क्या है?
If (p(x)=x-2 -5x+6), what is the value of (p(3)-p(2))?
#polynomials
#evaluation
#zeroes
#hard
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A (0)
B (1)
C (-1)
D (6)
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Explanation
Simple Explanation
(2) और (3) दोनों शून्यक हैं, इसलिए (p(3)=0) और (p(2)=0)। अतः अंतर (0) है। / Both (2) and (3) are zeroes, so (p(3)=0) and (p(2)=0). Therefore, the difference is (0).
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यदि (p(x)=2x-2 +kx+8) के शून्यकों का योग (3) है, तो (k) क्या होगा?
If the sum of zeroes of (p(x)=2x-2 +kx+8) is (3), what is (k)?
#polynomials
#sum_zeroes
#hard
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A (6)
B (-6)
C (3)
D (-3)
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Explanation
Simple Explanation
योग \(-\frac{k}{2}\) होता है और यह (3) है। इसलिए (k=-6) है। / The sum is \(-\frac{k}{2}\), and it equals (3). Hence (k=-6).
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यदि (p(x)=3x-2 -10x+m) के शून्यकों का गुणनफल \(\frac{4}{3}\) है, तो (m) का मान क्या है?
If the product of zeroes of (p(x)=3x-2 -10x+m) is \(\frac{4}{3}\), what is (m)?
#polynomials
#product_zeroes
#hard
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A (4)
B (3)
C \(\frac{4}{9}\)
D (12)
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Explanation
Simple Explanation
गुणनफल \(\frac{m}{3}\) है। \(\frac{m}{3}=\frac{4}{3}\) से (m=4) मिलता है। / The product is \(\frac{m}{3}\). From \(\frac{m}{3}=\frac{4}{3}\), we get (m=4).
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बहुपद \(x^2-6x+9\) के शून्यकों के बारे में सही कथन कौन सा है?
Which statement about the zeroes of \(x^2-6x+9\) is correct?
#polynomials
#equal_zeroes
#factorisation
#hard
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A दोनों शून्यक अलग हैं / Both zeroes are distinct
B दोनों शून्यक बराबर हैं / Both zeroes are equal
C कोई वास्तविक शून्यक नहीं है / There is no real zero
D शून्यकों का गुणनफल (6) है / Product of zeroes is (6)
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Correct Answer
B. दोनों शून्यक बराबर हैं / Both zeroes are equal
Explanation
Simple Explanation
(x-2 -6x+9=(x-3)2 ) है इसलिए दोनों शून्यक (3) हैं। वर्ग रूप को पहचानना परीक्षा में तेज तरीका है। / (x-2 -6x+9=(x-3)2 ), so both zeroes are (3). Recognizing square form is a fast exam method.
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यदि (x-a), (x-b) और (x-c) किसी घन बहुपद के गुणनखंड हैं, तो उसका एक संभावित बहुपद कौन सा है?
If (x-a), (x-b), and (x-c) are factors of a cubic polynomial, which is a possible polynomial?
#polynomials
#cubic
#factors
#hard
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A (x-3 -(a+b+c)x-2 +(ab+bc+ca)x-abc)
B (x-3 +(a+b+c)x-2 +(ab+bc+ca)x+abc)
C (x-3 -(ab+bc+ca)x-2 +(a+b+c)x-abc)
D (x-3 +abcx-2 -(a+b+c)x+ab+bc+ca)
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Correct Answer
A. (x-3 -(a+b+c)x-2 +(ab+bc+ca)x-abc)
Explanation
Simple Explanation
((x-a)(x-b)(x-c)) फैलाने पर पहला रूप मिलता है। शून्यकों और गुणनखंडों का संबंध याद रखें। / Expanding ((x-a)(x-b)(x-c)) gives the first form. Remember the link between zeroes and factors.
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बहुपद \(x^3-6x^2+11x-6\) के शून्यक कौन से हैं?
What are the zeroes of \(x^3-6x^2+11x-6\)?
#polynomials
#cubic_zeroes
#hard
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A (1,2,3)
B (-1,-2,-3)
C (1,1,6)
D (2,2,2)
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Correct Answer
A. (1,2,3)
Explanation
Simple Explanation
यह ((x-1)(x-2)(x-3)) के बराबर है। अतः शून्यक (1), (2) और (3) हैं। / It equals ((x-1)(x-2)(x-3)). Therefore, the zeroes are (1), (2), and (3).
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यदि (p(x)=x-3 -3x-2 -4x+12) है, तो कौन सा (p(x)) का गुणनखंड है?
If (p(x)=x-3 -3x-2 -4x+12), which is a factor of (p(x))?
#polynomials
#factor_theorem
#hard
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A (x-2)
B (x+2)
C (x-3)
D (x+3)
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Explanation
Simple Explanation
(p(2)=8-12-8+12=0) है इसलिए (x-2) गुणनखंड है। गुणनखंड प्रमेय का प्रयोग करें। / Since (p(2)=8-12-8+12=0), (x-2) is a factor. Use the factor theorem.
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यदि (x+1) बहुपद \(2x^3+kx^2-5x+2\) का गुणनखंड है, तो (k) का मान क्या है?
If (x+1) is a factor of \(2x^3+kx^2-5x+2\), what is (k)?
#polynomials
#factor_theorem
#parameter
#hard
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A (5)
B (-5)
C (1)
D (-1)
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Explanation
Simple Explanation
(x+1) गुणनखंड होने पर (p(-1)=0) होगा। (-2+k+5+2=0) से (k=-5) नहीं बल्कि (k=-5) मिलता है। / If (x+1) is a factor, then (p(-1)=0). From (-2+k+5+2=0), we get (k=-5).
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यदि (x+1) बहुपद \(2x^3+kx^2-5x+2\) का गुणनखंड है, तो (k) का सही मान क्या है?
If (x+1) is a factor of \(2x^3+kx^2-5x+2\), what is the correct value of (k)?
#polynomials
#factor_theorem
#parameter
#hard
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A (-5)
B (5)
C (-1)
D (1)
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Explanation
Simple Explanation
(p(-1)=0) रखने पर (-2+k+5+2=0) मिलता है। इसलिए (k=-5) सही है। / Putting (p(-1)=0) gives (-2+k+5+2=0). Therefore, (k=-5) is correct.
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\(x^3-5x^2+8x-4\) को (x-1) से भाग देने पर शेषफल क्या होगा?
What is the remainder when \(x^3-5x^2+8x-4\) is divided by (x-1)?
#polynomials
#remainder_theorem
#hard
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A (0)
B (1)
C (-1)
D (4)
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Explanation
Simple Explanation
शेषफल प्रमेय से शेषफल (p(1)) है। (1-5+8-4=0) मिलता है। / By the remainder theorem, the remainder is (p(1)). We get (1-5+8-4=0).
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\(3x^3-2x^2+x+7\) को (x+2) से भाग देने पर शेषफल क्या है?
What is the remainder when \(3x^3-2x^2+x+7\) is divided by (x+2)?
#polynomials
#remainder
#substitution
#hard
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A (-27)
B (-25)
C (25)
D (27)
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Explanation
Simple Explanation
शेषफल (p(-2)) होगा। (3(-8)-2(4)-2+7=-27) है। / The remainder is (p(-2)). (3(-8)-2(4)-2+7=-27).
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यदि (p(x)) को (x-4) से भाग देने पर शेषफल (7) है, तो (p(4)) क्या होगा?
If the remainder when (p(x)) is divided by (x-4) is (7), what is (p(4))?
#polynomials
#remainder_theorem
#concept
#hard
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A (4)
B (7)
C (-7)
D (0)
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Explanation
Simple Explanation
शेषफल प्रमेय के अनुसार (x-4) से भाग देने पर शेषफल (p(4)) होता है। इसलिए (p(4)=7)। / By the remainder theorem, the remainder on division by (x-4) is (p(4)). Therefore, (p(4)=7).
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\(x^4-1\) के रैखिक गुणनखंडों में से कौन सा गलत है?
Which one is not a linear factor of \(x^4-1\)?
#polynomials
#factors
#degree
#hard
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A (x-1)
B (x+1)
C \(x^2+1\)
D (x-i)
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Correct Answer
C. \(x^2+1\)
Explanation
Simple Explanation
\(x^2+1\) रैखिक नहीं है क्योंकि उसकी घात (2) है। वास्तविक संख्याओं में इसे रैखिक गुणनखंड नहीं माना जाता। / \(x^2+1\) is not linear because its degree is (2). Over real numbers, it is not a linear factor.
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यदि (p(x)=x-2 -2x+5), तो (p(1)) का मान क्या है और क्या (1) शून्यक है?
If (p(x)=x-2 -2x+5), what is (p(1)), and is (1) a zero?
#polynomials
#evaluation
#zero_check
#hard
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A (4), नहीं / (4), no
B (0), हां / (0), yes
C (2), नहीं / (2), no
D (5), नहीं / (5), no
Explanation opens after your attempt
Correct Answer
A. (4), नहीं / (4), no
Explanation
Simple Explanation
(p(1)=1-2+5=4) है इसलिए (1) शून्यक नहीं है। शून्यक तभी होगा जब मान (0) हो। / (p(1)=1-2+5=4), so (1) is not a zero. A zero must make the value (0).
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यदि द्विघात बहुपद के शून्यक \(\alpha\) और \(\beta\) हैं तथा \(\alpha+\beta=5\), \(\alpha\beta=6\), तो मोनिक बहुपद क्या है?
If a quadratic polynomial has zeroes \(\alpha\) and \(\beta\), with \(\alpha+\beta=5\) and \(\alpha\beta=6\), what is the monic polynomial?
#polynomials
#monic
#formation
#hard
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A \(x^2-5x+6\)
B \(x^2+5x+6\)
C \(x^2-6x+5\)
D \(x^2+6x+5\)
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Correct Answer
A. \(x^2-5x+6\)
Explanation
Simple Explanation
मोनिक बहुपद (x-2 -\(\alpha+\beta\)x+\alpha\beta) होता है। इसलिए \(x^2-5x+6\) सही है। / The monic polynomial is (x-2 -\(\alpha+\beta\)x+\alpha\beta). Hence \(x^2-5x+6\) is correct.
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यदि \(2x^2-3x-5\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो \(\frac{1}{\alpha}+\frac{1}{\beta}\) क्या होगा?
If \(\alpha\) and \(\beta\) are zeroes of \(2x^2-3x-5\), what is \(\frac{1}{\alpha}+\frac{1}{\beta}\)?
#polynomials
#zeroes
#reciprocal
#hard
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A \(-\frac{3}{5}\)
B \(\frac{3}{5}\)
C \(-\frac{5}{3}\)
D \(\frac{5}{3}\)
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Correct Answer
A. \(-\frac{3}{5}\)
Explanation
Simple Explanation
\(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}\) है। यहां \(\frac{3}{2}\div-\frac{5}{2}=-\frac{3}{5}\)। / \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}\). Here \(\frac{3}{2}\div-\frac{5}{2}=-\frac{3}{5}\).
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यदि \(x^2-7x+10\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो \(\alpha^2+\beta^2\) क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(x^2-7x+10\), what is \(\alpha^2+\beta^2\)?
#polynomials
#zeroes
#identity
#hard
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A (29)
B (49)
C (20)
D (39)
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Explanation
Simple Explanation
(\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta) होता है। (72 -2(10)=29) है। / (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta). Thus (72 -2(10)=29).
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यदि \(3x^2+2x-1\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो (\(\alpha+\beta\)2 ) का मान क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(3x^2+2x-1\), what is (\(\alpha+\beta\)2 )?
#polynomials
#sum_zeroes
#square
#hard
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A \(\frac{4}{9}\)
B \(-\frac{4}{9}\)
C \(\frac{1}{9}\)
D \(\frac{2}{3}\)
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Correct Answer
A. \(\frac{4}{9}\)
Explanation
Simple Explanation
\(\alpha+\beta=-\frac{2}{3}\) है। इसलिए (\(\alpha+\beta\)2 =\frac{4}{9}) होगा। / \(\alpha+\beta=-\frac{2}{3}\). Therefore, (\(\alpha+\beta\)2 =\frac{4}{9}).
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यदि \(x^2+px+16\) के शून्यक परस्पर बराबर और ऋणात्मक हैं, तो (p) का मान क्या है?
If the zeroes of \(x^2+px+16\) are equal and negative, what is (p)?
#polynomials
#equal_negative_zeroes
#hard
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A (8)
B (-8)
C (4)
D (-4)
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Explanation
Simple Explanation
बराबर ऋणात्मक शून्यक (-4) और (-4) होंगे क्योंकि गुणनफल (16) है। योग (-8) है इसलिए (p=8)। / The equal negative zeroes are (-4) and (-4) because the product is (16). The sum is (-8), so (p=8).
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यदि (p(x)=x-2 -4x+k) और (p(1)=p(3)), तो (k) के बारे में क्या कहा जा सकता है?
If (p(x)=x-2 -4x+k) and (p(1)=p(3)), what can be said about (k)?
#polynomials
#evaluation
#parameter
#hard
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A (k=1)
B (k=3)
C कोई भी वास्तविक मान / Any real value
D कोई मान संभव नहीं / No value possible
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Correct Answer
C. कोई भी वास्तविक मान / Any real value
Explanation
Simple Explanation
(p(1)=k-3) और (p(3)=k-3) दोनों बराबर हैं। इसलिए (k) कोई भी वास्तविक मान हो सकता है। / (p(1)=k-3) and (p(3)=k-3), so they are equal. Hence (k) can be any real value.
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बहुपद \(2x^3-9x^2+13x-6\) का एक गुणनखंड कौन सा है?
Which is a factor of \(2x^3-9x^2+13x-6\)?
#polynomials
#factor_check
#cubic
#hard
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A (x-1)
B (x+1)
C (x-4)
D (x+3)
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Explanation
Simple Explanation
(p(1)=2-9+13-6=0) है इसलिए (x-1) गुणनखंड है। विकल्प जांच में छोटे मान पहले लगाएं। / (p(1)=2-9+13-6=0), so (x-1) is a factor. In option checking, try small values first.
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यदि (p(x)=x-3 -2x-2 -5x+6), तो निम्न में से कौन सा शून्यक नहीं है?
If (p(x)=x-3 -2x-2 -5x+6), which of the following is not a zero?
#polynomials
#not_zero
#hard
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A (1)
B (2)
C (3)
D (-2)
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Explanation
Simple Explanation
(p(2)=8-8-10+6=-4) है इसलिए (2) शून्यक नहीं है। बाकी विकल्पों को भी प्रतिस्थापन से जांच सकते हैं। / (p(2)=8-8-10+6=-4), so (2) is not a zero. The other options can be checked by substitution.
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यदि \(x^2+5x+6\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो नया बहुपद जिसके शून्यक \(\alpha+1\) और \(\beta+1\) हैं, क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(x^2+5x+6\), what is the new polynomial whose zeroes are \(\alpha+1\) and \(\beta+1\)?
#polynomials
#transformed_zeroes
#hard
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A \(x^2+3x+2\)
B \(x^2+5x+6\)
C \(x^2+7x+12\)
D \(x^2-3x+2\)
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Correct Answer
A. \(x^2+3x+2\)
Explanation
Simple Explanation
मूल शून्यक (-2) और (-3) हैं, इसलिए नए शून्यक (-1) और (-2) हैं। नया बहुपद \(x^2+3x+2\) है। / The original zeroes are (-2) and (-3), so the new zeroes are (-1) and (-2). The new polynomial is \(x^2+3x+2\).
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यदि \(x^2-4x+1\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो \(\alpha^3+\beta^3\) का मान क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(x^2-4x+1\), what is \(\alpha^3+\beta^3\)?
#polynomials
#zeroes
#cube_identity
#hard
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A (52)
B (64)
C (40)
D (28)
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Explanation
Simple Explanation
\(\alpha+\beta=4\) और \(\alpha\beta=1\) है। (\alpha-3 +\beta-3 =\(\alpha+\beta\)3 -3\alpha\beta\(\alpha+\beta\)=64-12=52)। / \(\alpha+\beta=4\) and \(\alpha\beta=1\). (\alpha-3 +\beta-3 =\(\alpha+\beta\)3 -3\alpha\beta\(\alpha+\beta\)=64-12=52).
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किस (k) के लिए \(x^2+kx+9\) का एक शून्यक दूसरे का तीन गुना है और दोनों धनात्मक हैं?
For which (k) does \(x^2+kx+9\) have one zero three times the other and both positive?
#polynomials
#zeroes
#ratio
#hard
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A \(-4\sqrt{3}\)
B \(4\sqrt{3}\)
C (-6)
D (6)
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Correct Answer
A. \(-4\sqrt{3}\)
Explanation
Simple Explanation
शून्यक (t) और (3t) मानें, तो \(3t^2=9\) से \(t=\sqrt{3}\) है। योग \(4\sqrt{3}\) है इसलिए \(k=-4\sqrt{3}\)। / Let the zeroes be (t) and (3t), so \(3t^2=9\) gives \(t=\sqrt{3}\). The sum is \(4\sqrt{3}\), hence \(k=-4\sqrt{3}\).
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यदि \(x^2-8x+k\) के शून्यक \(\alpha\) और \(\beta\) हैं तथा \(\alpha-\beta=2\), तो (k) क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(x^2-8x+k\) and \(\alpha-\beta=2\), what is (k)?
#polynomials
#zeroes
#difference
#hard
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A (15)
B (16)
C (12)
D (10)
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Explanation
Simple Explanation
योग (8) और अंतर (2) से शून्यक (5) और (3) हैं। गुणनफल (15) है इसलिए (k=15)। / From sum (8) and difference (2), the zeroes are (5) and (3). Their product is (15), so (k=15).
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द्विघात बहुपद \(kx^2+6x+4\) के शून्यकों का योग (-3) है। (k) का मान क्या है?
The sum of zeroes of the quadratic polynomial \(kx^2+6x+4\) is (-3). What is (k)?
#polynomials
#parameter
#sum_zeroes
#hard
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A (2)
B (-2)
C (3)
D (-3)
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Explanation
Simple Explanation
योग \(-\frac{6}{k}\) है और यह (-3) है। इसलिए (k=2) होगा। / The sum is \(-\frac{6}{k}\), and it equals (-3). Therefore, (k=2).
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यदि \(x^2-10x+q\) के शून्यक (2r) और (3r) हैं, तो (q) का मान क्या है?
If the zeroes of \(x^2-10x+q\) are (2r) and (3r), what is (q)?
#polynomials
#ratio_zeroes
#product
#hard
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A (24)
B (20)
C (12)
D (30)
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Explanation
Simple Explanation
योग (5r=10) से (r=2) है। गुणनफल \(2r\cdot3r=6r^2=24\) है। / From the sum (5r=10), (r=2). The product is \(2r\cdot3r=6r^2=24\).
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यदि (p(x)=x-3 +ax-2 +bx+8) के शून्यक (-1), (-2) और (-4) हैं, तो (a+b) क्या है?
If the zeroes of (p(x)=x-3 +ax-2 +bx+8) are (-1), (-2), and (-4), what is (a+b)?
#polynomials
#cubic_zeroes
#coefficients
#hard
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A (21)
B (7)
C (14)
D (-7)
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Explanation
Simple Explanation
बहुपद ((x+1)(x+2)(x+4)=x-3 +7x-2 +14x+8) है। इसलिए (a+b=21)। / The polynomial is ((x+1)(x+2)(x+4)=x-3 +7x-2 +14x+8). Hence (a+b=21).
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यदि \(x^2-3x-10\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो \(\alpha^2\beta+\alpha\beta^2\) क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(x^2-3x-10\), what is \(\alpha^2\beta+\alpha\beta^2\)?
#polynomials
#zeroes
#symmetric_expression
#hard
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A (-30)
B (30)
C (-13)
D (13)
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Explanation
Simple Explanation
(\alpha-2 \beta+\alpha\beta-2 =\alpha\beta\(\alpha+\beta\)) होता है। यहां \(\alpha\beta=-10\) और \(\alpha+\beta=3\), इसलिए मान (-30) है। / (\alpha-2 \beta+\alpha\beta-2 =\alpha\beta\(\alpha+\beta\)). Here \(\alpha\beta=-10\) and \(\alpha+\beta=3\), so the value is (-30).
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यदि \(2x^2+5x-3\) को (\(x-\alpha\)\(x-\beta\)) के रूप में लिखा जाए, तो \(\alpha+\beta\) क्या होगा?
If \(2x^2+5x-3\) is considered with factors involving (\(x-\alpha\)\(x-\beta\)), what is \(\alpha+\beta\)?
#polynomials
#sum_zeroes
#hard
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A \(-\frac{5}{2}\)
B \(\frac{5}{2}\)
C (-3)
D (3)
Explanation opens after your attempt
Correct Answer
A. \(-\frac{5}{2}\)
Explanation
Simple Explanation
द्विघात \(ax^2+bx+c\) के लिए शून्यकों का योग \(-\frac{b}{a}\) होता है। इसलिए \(\alpha+\beta=-\frac{5}{2}\)। / For \(ax^2+bx+c\), the sum of zeroes is \(-\frac{b}{a}\). Hence \(\alpha+\beta=-\frac{5}{2}\).
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किस बहुपद के शून्यक (0), (2) और (-3) हैं?
Which polynomial has zeroes (0), (2), and (-3)?
#polynomials
#formation
#cubic
#hard
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A \(x^3+x^2-6x\)
B \(x^3-x^2-6x\)
C \(x^3+x^2+6x\)
D \(x^3-5x^2+6x\)
Explanation opens after your attempt
Correct Answer
A. \(x^3+x^2-6x\)
Explanation
Simple Explanation
शून्यकों से बहुपद (x(x-2)(x+3)) बनेगा। इसे फैलाने पर \(x^3+x^2-6x\) मिलता है। / From the zeroes, the polynomial is (x(x-2)(x+3)). Expanding gives \(x^3+x^2-6x\).
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यदि (p(x)=x-4 -5x-2 +4), तो कौन सा मान (p(x)) का शून्यक नहीं है?
If (p(x)=x-4 -5x-2 +4), which value is not a zero of (p(x))?
#polynomials
#quartic
#zero_check
#hard
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A (1)
B (-1)
C (2)
D (3)
Explanation opens after your attempt
Explanation
Simple Explanation
(p(3)=81-45+4=40) है, इसलिए (3) शून्यक नहीं है। शेष (1), (-1) और (2) पर मान (0) मिलता है। / (p(3)=81-45+4=40), so (3) is not a zero. The values (1), (-1), and (2) make the polynomial (0).
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यदि \(x^2+mx+n\) का एक शून्यक (0) है और दूसरा शून्यक (5) है, तो (m+n) क्या होगा?
If one zero of \(x^2+mx+n\) is (0) and the other zero is (5), what is (m+n)?
#polynomials
#zeroes
#constant_term
#hard
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A (-5)
B (5)
C (0)
D (10)
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Explanation
Simple Explanation
योग (5) है इसलिए (m=-5), और गुणनफल (0) है इसलिए (n=0)। अतः (m+n=-5)। / The sum is (5), so (m=-5), and the product is (0), so (n=0). Hence (m+n=-5).
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यदि \(x^2-11x+30\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो (\(\alpha-\beta\)2 ) क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(x^2-11x+30\), what is (\(\alpha-\beta\)2 )?
#polynomials
#zeroes
#difference_square
#hard
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A (1)
B (121)
C (60)
D (49)
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Explanation
Simple Explanation
(\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta) है। (121-120=1) मिलता है। / (\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta). We get (121-120=1).
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यदि (p(x)=4x-2 -12x+9), तो इसके शून्यकों के बारे में कौन सा कथन सही है?
If (p(x)=4x-2 -12x+9), which statement about its zeroes is correct?
#polynomials
#perfect_square
#zeroes
#hard
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A दोनों शून्यक \(\frac{3}{2}\) हैं / Both zeroes are \(\frac{3}{2}\)
B दोनों शून्यक \(-\frac{3}{2}\) हैं / Both zeroes are \(-\frac{3}{2}\)
C शून्यक \(\frac{2}{3}\) और \(\frac{3}{2}\) हैं / Zeroes are \(\frac{2}{3}\) and \(\frac{3}{2}\)
D कोई शून्यक नहीं है / There is no zero
Explanation opens after your attempt
Correct Answer
A. दोनों शून्यक \(\frac{3}{2}\) हैं / Both zeroes are \(\frac{3}{2}\)
Explanation
Simple Explanation
(4x-2 -12x+9=(2x-3)2 ) है। इसलिए दोनों शून्यक \(\frac{3}{2}\) हैं। / (4x-2 -12x+9=(2x-3)2 ). Therefore, both zeroes are \(\frac{3}{2}\).
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\(x^3-7x+6\) के लिए कौन सा कथन सही है?
Which statement is correct for \(x^3-7x+6\)?
#polynomials
#factor_theorem
#two_factors
#hard
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A (x-1) और (x-2) दोनों गुणनखंड हैं / Both (x-1) and (x-2) are factors
B केवल (x-1) गुणनखंड है / Only (x-1) is a factor
C केवल (x-2) गुणनखंड है / Only (x-2) is a factor
D दोनों गुणनखंड नहीं हैं / Neither is a factor
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Correct Answer
A. (x-1) और (x-2) दोनों गुणनखंड हैं / Both (x-1) and (x-2) are factors
Explanation
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(p(1)=0) और (p(2)=0) दोनों हैं। इसलिए (x-1) और (x-2) दोनों गुणनखंड हैं। / Both (p(1)=0) and (p(2)=0). Hence (x-1) and (x-2) are both factors.
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यदि (p(x)=x-3 +mx-2 -4x-4) में (x+2) एक गुणनखंड है, तो (m) का मान क्या होगा?
If (x+2) is a factor of (p(x)=x-3 +mx-2 -4x-4), what is (m)?
#polynomials
#factor_parameter
#hard
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A (1)
B (-1)
C (2)
D (-2)
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Explanation
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(p(-2)=0) रखने पर (-8+4m+8-4=0) मिलता है। इससे (4m-4=0) और (m=1) है। / Putting (p(-2)=0) gives (-8+4m+8-4=0). Thus (4m-4=0), so (m=1).
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यदि द्विघात बहुपद \(x^2-6x+k\) के शून्यक \(\alpha\) और \(\beta\) हैं तथा \(\alpha^2+\beta^2=20\) है, तो (k) का मान क्या होगा?
If \(\alpha\) and \(\beta\) are zeroes of the quadratic polynomial \(x^2-6x+k\) and \(\alpha^2+\beta^2=20\), what is the value of (k)?
#polynomials
#zeroes
#identity
#hard
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A (8)
B (6)
C (10)
D (16)
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Explanation
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\(\alpha+\beta=6\) और (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta) होता है। इसलिए (20=36-2k) से (k=8) मिलता है। / Here \(\alpha+\beta=6\) and (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta). So (20=36-2k), giving (k=8).
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