समीकरण \(x^2-5x+6=0\) के मूलों की प्रकृति क्या है?
What is the nature of roots of the equation \(x^2-5x+6=0\)?
#quadratic equations
#numerical
#distinct real roots
A दो वास्तविक और असमान / two real and distinct
B दो वास्तविक और समान / two real and equal
C वास्तविक मूल नहीं / no real roots
D केवल एक वास्तविक मूल / only one real root
Explanation opens after your attempt
Correct Answer
A. दो वास्तविक और असमान / two real and distinct
Step 1
Concept
Here (D=(-5)2 -4(1)(6)=1>0). So the roots are real and distinct.
Step 2
Why this answer is correct
The correct answer is A. दो वास्तविक और असमान / two real and distinct. Here (D=(-5)2 -4(1)(6)=1>0). So the roots are real and distinct.
Step 3
Exam Tip
यहां (D=(-5)2 -4(1)(6)=1>0) है। इसलिए मूल वास्तविक और असमान हैं।
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यदि \(4x^2-20x+9=0\) की जड़ें \(\alpha,\beta\) हैं, तो \(\alpha-\beta\) का सही धनात्मक मान क्या है?
If \(\alpha,\beta\) are roots of \(4x^2-20x+9=0\), what is the correct positive value of \(\alpha-\beta\)?
#quadratic-roots
#difference-of-roots
#numerical
A (4)
B \(\frac{7}{2}\)
C \(\frac{9}{2}\)
D (5)
Explanation opens after your attempt
Step 1
Concept
Here \(\alpha+\beta=5\) and \(\alpha\beta=\frac{9}{4}\). Since (\(\alpha-\beta\)2 =25-9=16), the positive difference is (4).
Step 2
Why this answer is correct
The correct answer is A. (4). Here \(\alpha+\beta=5\) and \(\alpha\beta=\frac{9}{4}\). Since (\(\alpha-\beta\)2 =25-9=16), the positive difference is (4).
Step 3
Exam Tip
\(\alpha+\beta=5\) और \(\alpha\beta=\frac{9}{4}\) है। (\(\alpha-\beta\)2 =25-9=16), इसलिए धनात्मक अंतर (4) है।
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यदि \(3x^2-13x+4=0\) की जड़ें \(\alpha,\beta\) हैं, तो \(\alpha-\beta\) का धनात्मक मान क्या है?
If \(\alpha,\beta\) are roots of \(3x^2-13x+4=0\), what is the positive value of \(\alpha-\beta\)?
#quadratic-roots
#difference-of-roots
#numerical
A \(\frac{7}{3}\)
B \(\frac{11}{3}\)
C \(\frac{13}{3}\)
D (3)
Explanation opens after your attempt
Correct Answer
B. \(\frac{11}{3}\)
Step 1
Concept
Use (\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta). With \(\alpha+\beta=\frac{13}{3}\) and \(\alpha\beta=\frac{4}{3}\), the positive difference is \(\frac{11}{3}\).
Step 2
Why this answer is correct
The correct answer is B. \(\frac{11}{3}\). Use (\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta). With \(\alpha+\beta=\frac{13}{3}\) and \(\alpha\beta=\frac{4}{3}\), the positive difference is \(\frac{11}{3}\).
Step 3
Exam Tip
(\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta) लगाएँ। \(\alpha+\beta=\frac{13}{3}\) और \(\alpha\beta=\frac{4}{3}\), इसलिए धनात्मक अंतर \(\frac{11}{3}\) है।
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यदि \(2x^2-7x+3=0\) की जड़ें \(\alpha,\beta\) हैं, तो \(\alpha-\beta\) का धनात्मक मान क्या है?
If \(\alpha,\beta\) are roots of \(2x^2-7x+3=0\), what is the positive value of \(\alpha-\beta\)?
#quadratic-roots
#difference-of-roots
#numerical
A \(\frac{5}{2}\)
B \(\frac{3}{2}\)
C \(\frac{7}{2}\)
D \(\frac{1}{2}\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{5}{2}\)
Step 1
Concept
Use (\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta). With \(\alpha+\beta=\frac{7}{2}\) and \(\alpha\beta=\frac{3}{2}\), the positive difference is \(\frac{5}{2}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{5}{2}\). Use (\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta). With \(\alpha+\beta=\frac{7}{2}\) and \(\alpha\beta=\frac{3}{2}\), the positive difference is \(\frac{5}{2}\).
Step 3
Exam Tip
(\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta) लगाएँ। \(\alpha+\beta=\frac{7}{2}\) और \(\alpha\beta=\frac{3}{2}\), इसलिए धनात्मक अंतर \(\frac{5}{2}\) है।
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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
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)
Explanation opens after your attempt
Correct Answer
A. (5) और (9) / (5) and (9)
Step 1
Concept
(x-2 -14x+45=(x-5)(x-9)). Therefore the roots are (5) and (9).
Step 2
Why this answer is correct
The correct answer is A. (5) और (9) / (5) and (9). (x-2 -14x+45=(x-5)(x-9)). Therefore the roots are (5) and (9).
Step 3
Exam Tip
(x-2 -14x+45=(x-5)(x-9)) है। इसलिए मूल (5) और (9) हैं।
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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
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)
Explanation opens after your attempt
Correct Answer
A. (5) और (7) / (5) and (7)
Step 1
Concept
(x-2 -12x+35=(x-5)(x-7)). Therefore the roots are (5) and (7).
Step 2
Why this answer is correct
The correct answer is A. (5) और (7) / (5) and (7). (x-2 -12x+35=(x-5)(x-7)). Therefore the roots are (5) and (7).
Step 3
Exam Tip
(x-2 -12x+35=(x-5)(x-7)) है। इसलिए मूल (5) और (7) हैं।
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समीकरण \(x^2-11x+28=0\) के मूल कौन से हैं?
What are the roots of \(x^2-11x+28=0\)?
#roots
#factorisation
#numerical
A (4) और (7) / (4) and (7)
B (2) और (14) / (2) and (14)
C (-4) और (-7) / (-4) and (-7)
D (1) और (28) / (1) and (28)
Explanation opens after your attempt
Correct Answer
A. (4) और (7) / (4) and (7)
Step 1
Concept
(x-2 -11x+28=(x-4)(x-7)). Therefore the roots are (4) and (7).
Step 2
Why this answer is correct
The correct answer is A. (4) और (7) / (4) and (7). (x-2 -11x+28=(x-4)(x-7)). Therefore the roots are (4) and (7).
Step 3
Exam Tip
(x-2 -11x+28=(x-4)(x-7)) है। इसलिए मूल (4) और (7) हैं।
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समीकरण \(7x^2+5x-14=0\) के मूलों का गुणनफल क्या है?
What is the product of roots of \(7x^2+5x-14=0\)?
#roots
#product_of_roots
#numerical
A (2)
B (-2)
C \(\frac{5}{7}\)
D \(-\frac{5}{7}\)
Explanation opens after your attempt
Step 1
Concept
The product of roots is \(\frac{c}{a}\). Here \(\frac{-14}{7}=-2\).
Step 2
Why this answer is correct
The correct answer is B. (-2). The product of roots is \(\frac{c}{a}\). Here \(\frac{-14}{7}=-2\).
Step 3
Exam Tip
मूलों का गुणनफल \(\frac{c}{a}\) होता है। यहां \(\frac{-14}{7}=-2\) है।
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समीकरण \(x^2-13x+42=0\) के मूलों का योग क्या है?
What is the sum of roots of \(x^2-13x+42=0\)?
#roots
#sum_of_roots
#numerical
A (13)
B (-13)
C (42)
D (-42)
Explanation opens after your attempt
Step 1
Concept
The sum of roots is \(-\frac{b}{a}\). Here (b=-13), so the sum is (13).
Step 2
Why this answer is correct
The correct answer is A. (13). The sum of roots is \(-\frac{b}{a}\). Here (b=-13), so the sum is (13).
Step 3
Exam Tip
मूलों का योग \(-\frac{b}{a}\) है। यहां (b=-13) इसलिए योग (13) है।
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यदि (5) समीकरण \(x^2+tx-30=0\) का मूल है तो (t) का मान क्या है?
If (5) is a root of \(x^2+tx-30=0\), what is the value of (t)?
#roots
#parameter
#numerical
A (1)
B (5)
C (-1)
D (-5)
Explanation opens after your attempt
Step 1
Concept
Putting (x=5) gives (25+5t-30=0), so (t=1). If the root is known, substitute directly.
Step 2
Why this answer is correct
The correct answer is A. (1). Putting (x=5) gives (25+5t-30=0), so (t=1). If the root is known, substitute directly.
Step 3
Exam Tip
(x=5) रखने पर (25+5t-30=0) इसलिए (t=1)। मूल ज्ञात हो तो सीधे प्रतिस्थापन करें।
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समीकरण \(x^2-8x+15=0\) के मूल कौन से हैं?
What are the roots of \(x^2-8x+15=0\)?
#roots
#factorisation
#numerical
A (3) और (5) / (3) and (5)
B (-3) और (-5) / (-3) and (-5)
C (1) और (15) / (1) and (15)
D (0) और (8) / (0) and (8)
Explanation opens after your attempt
Correct Answer
A. (3) और (5) / (3) and (5)
Step 1
Concept
(x-2 -8x+15=(x-3)(x-5)). Therefore the roots are (3) and (5).
Step 2
Why this answer is correct
The correct answer is A. (3) और (5) / (3) and (5). (x-2 -8x+15=(x-3)(x-5)). Therefore the roots are (3) and (5).
Step 3
Exam Tip
(x-2 -8x+15=(x-3)(x-5)) है। इसलिए मूल (3) और (5) हैं।
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समीकरण \(5x^2-2x-8=0\) के मूलों का गुणनफल क्या है?
What is the product of roots of \(5x^2-2x-8=0\)?
#roots
#product_of_roots
#numerical
A (8)
B (-8)
C \(-\frac{8}{5}\)
D \(\frac{2}{5}\)
Explanation opens after your attempt
Correct Answer
C. \(-\frac{8}{5}\)
Step 1
Concept
The product of roots is \(\frac{c}{a}\). Here \(\frac{-8}{5}\) is correct.
Step 2
Why this answer is correct
The correct answer is C. \(-\frac{8}{5}\). The product of roots is \(\frac{c}{a}\). Here \(\frac{-8}{5}\) is correct.
Step 3
Exam Tip
मूलों का गुणनफल \(\frac{c}{a}\) होता है। यहां \(\frac{-8}{5}\) सही है।
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समीकरण \(x^2+11x+30=0\) के मूलों का योग क्या है?
What is the sum of roots of \(x^2+11x+30=0\)?
#roots
#sum_of_roots
#numerical
A (11)
B (-11)
C (30)
D (-30)
Explanation opens after your attempt
Step 1
Concept
The sum of roots is \(-\frac{b}{a}\). Here the sum is (-11).
Step 2
Why this answer is correct
The correct answer is B. (-11). The sum of roots is \(-\frac{b}{a}\). Here the sum is (-11).
Step 3
Exam Tip
मूलों का योग \(-\frac{b}{a}\) है। यहां योग (-11) होगा।
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यदि (4) समीकरण \(x^2+mx-20=0\) का मूल है तो (m) का मान क्या है?
If (4) is a root of \(x^2+mx-20=0\), what is the value of (m)?
#roots
#parameter
#numerical
A (1)
B (-1)
C (4)
D (-4)
Explanation opens after your attempt
Step 1
Concept
Putting (x=4) gives (16+4m-20=0), so (m=1). For a parameter, substitute the root directly.
Step 2
Why this answer is correct
The correct answer is A. (1). Putting (x=4) gives (16+4m-20=0), so (m=1). For a parameter, substitute the root directly.
Step 3
Exam Tip
(x=4) रखने पर (16+4m-20=0) इसलिए (m=1)। पैरामीटर के लिए मूल को सीधे रखें।
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समीकरण \(x^2-7x+12=0\) के मूल कौन से हैं?
What are the roots of \(x^2-7x+12=0\)?
#roots
#factorisation
#numerical
A (3) और (4) / (3) and (4)
B (-3) और (-4) / (-3) and (-4)
C (2) और (6) / (2) and (6)
D (1) और (12) / (1) and (12)
Explanation opens after your attempt
Correct Answer
A. (3) और (4) / (3) and (4)
Step 1
Concept
(x-2 -7x+12=(x-3)(x-4)). Therefore the roots are (3) and (4).
Step 2
Why this answer is correct
The correct answer is A. (3) और (4) / (3) and (4). (x-2 -7x+12=(x-3)(x-4)). Therefore the roots are (3) and (4).
Step 3
Exam Tip
(x-2 -7x+12=(x-3)(x-4)) है। इसलिए मूल (3) और (4) हैं।
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समीकरण \(2x^2-3x-2=0\) के मूलों का गुणनफल क्या है?
What is the product of roots of \(2x^2-3x-2=0\)?
#roots
#product_of_roots
#numerical
A (-1)
B (1)
C (-2)
D (2)
Explanation opens after your attempt
Step 1
Concept
The product of roots is \(\frac{c}{a}\) so \(\frac{-2}{2}=-1\). Identify (a) and (c) first.
Step 2
Why this answer is correct
The correct answer is A. (-1). The product of roots is \(\frac{c}{a}\) so \(\frac{-2}{2}=-1\). Identify (a) and (c) first.
Step 3
Exam Tip
मूलों का गुणनफल \(\frac{c}{a}\) है इसलिए \(\frac{-2}{2}=-1\)। पहले (a) और (c) पहचानें।
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समीकरण \(x^2+7x+10=0\) के मूलों का योग क्या है?
What is the sum of roots of \(x^2+7x+10=0\)?
#roots
#sum_of_roots
#numerical
A (7)
B (10)
C (-7)
D (-10)
Explanation opens after your attempt
Step 1
Concept
The sum of roots is \(-\frac{b}{a}\) so here it is (-7). This can be found quickly without factorising.
Step 2
Why this answer is correct
The correct answer is C. (-7). The sum of roots is \(-\frac{b}{a}\) so here it is (-7). This can be found quickly without factorising.
Step 3
Exam Tip
मूलों का योग \(-\frac{b}{a}\) है इसलिए यहां (-7) होगा। गुणनखंड न बनाकर भी यह जल्दी निकलता है।
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क्या (x=2) समीकरण \(x^2-5x+6=0\) का मूल है?
Is (x=2) a root of \(x^2-5x+6=0\)?
#roots
#checking
#numerical
A हाँ / Yes
B नहीं / No
C केवल (x=0) पर / Only at (x=0)
D निर्धारित नहीं / Cannot be determined
Explanation opens after your attempt
Correct Answer
A. हाँ / Yes
Step 1
Concept
Putting (x=2) gives (4-10+6=0) so it is a root. In exams always check the final sum after substitution.
Step 2
Why this answer is correct
The correct answer is A. हाँ / Yes. Putting (x=2) gives (4-10+6=0) so it is a root. In exams always check the final sum after substitution.
Step 3
Exam Tip
(x=2) रखने पर (4-10+6=0) मिलता है इसलिए यह मूल है। परीक्षा में मान रखने के बाद अंतिम योग जरूर देखें।
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यदि (p(x)=2x-2 -8) है तो ग्राफ (x)-अक्ष को किन (x)-मानों पर काटेगा?
If (p(x)=2x-2 -8), at which (x)-values will the graph cut the (x)-axis?
#quadratic
#symmetric-zeroes
#numerical
A (-2) और (2) / (-2) and (2)
B (-4) और (4) / (-4) and (4)
C (0) और (2) / (0) and (2)
D केवल (4) / Only (4)
Explanation opens after your attempt
Correct Answer
A. (-2) और (2) / (-2) and (2)
Step 1
Concept
From \(2x^2-8=0\), \(x^2=4\). Hence the zeroes are (-2) and (2).
Step 2
Why this answer is correct
The correct answer is A. (-2) और (2) / (-2) and (2). From \(2x^2-8=0\), \(x^2=4\). Hence the zeroes are (-2) and (2).
Step 3
Exam Tip
\(2x^2-8=0\) से \(x^2=4\) मिलता है। इसलिए शून्यक (-2) और (2) हैं।
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यदि (p(x)=x-2 -2x-8) है तो ग्राफ (x)-अक्ष को किन बिंदुओं पर मिलेगा?
If (p(x)=x-2 -2x-8), at which points will the graph meet the (x)-axis?
#quadratic
#numerical
#x-intercepts
A ((-2,0)) और ((4,0)) / ((-2,0)) and ((4,0))
B ((2,0)) और ((-4,0)) / ((2,0)) and ((-4,0))
C ((0,-2)) और ((0,4)) / ((0,-2)) and ((0,4))
D ((-8,0)) और ((1,0)) / ((-8,0)) and ((1,0))
Explanation opens after your attempt
Correct Answer
A. ((-2,0)) और ((4,0)) / ((-2,0)) and ((4,0))
Step 1
Concept
(x-2 -2x-8=(x-4)(x+2)). So the graph meets the (x)-axis at (x=4) and (x=-2).
Step 2
Why this answer is correct
The correct answer is A. ((-2,0)) और ((4,0)) / ((-2,0)) and ((4,0)). (x-2 -2x-8=(x-4)(x+2)). So the graph meets the (x)-axis at (x=4) and (x=-2).
Step 3
Exam Tip
(x-2 -2x-8=(x-4)(x+2)) है। अतः (x=4) और (x=-2) पर ग्राफ (x)-अक्ष से मिलेगा।
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यदि (p(x)=x-2 -9) है तो इसका ग्राफ (x)-अक्ष को किन बिंदुओं पर काटेगा?
If (p(x)=x-2 -9), at which points will its graph cut the (x)-axis?
#polynomials
#x-intercepts
#numerical
A ((-3,0)) और ((3,0)) / ((-3,0)) and ((3,0))
B ((0,-3)) और ((0,3)) / ((0,-3)) and ((0,3))
C ((-9,0)) और ((9,0)) / ((-9,0)) and ((9,0))
D ((0,-9)) और ((0,9)) / ((0,-9)) and ((0,9))
Explanation opens after your attempt
Correct Answer
A. ((-3,0)) और ((3,0)) / ((-3,0)) and ((3,0))
Step 1
Concept
Solving \(x^2-9=0\) gives \(x=\pm3\). Zeroes appear on the (x)-axis as ((x,0)).
Step 2
Why this answer is correct
The correct answer is A. ((-3,0)) और ((3,0)) / ((-3,0)) and ((3,0)). Solving \(x^2-9=0\) gives \(x=\pm3\). Zeroes appear on the (x)-axis as ((x,0)).
Step 3
Exam Tip
\(x^2-9=0\) से \(x=\pm3\) मिलता है। शून्यक हमेशा (x)-अक्ष पर ((x,0)) रूप में दिखते हैं।
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