द्विघात समीकरण \(ax^2+bx+c=0\) में मूलों की प्रकृति जानने के लिए किस राशि का उपयोग किया जाता है?
Which quantity is used to know the nature of roots of the quadratic equation \(ax^2+bx+c=0\)?
#quadratic equations
#nature of roots
#discriminant
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A विविक्तकर \(D=b^2-4ac\) / Discriminant \(D=b^2-4ac\)
B योग (a+b+c) / Sum (a+b+c)
C गुणनफल (abc) / Product (abc)
D अनुपात \(\frac{a}{b}\) / Ratio \(\frac{a}{b}\)
Explanation opens after your attempt
Correct Answer
A. विविक्तकर \(D=b^2-4ac\) / Discriminant \(D=b^2-4ac\)
Step 1
Concept
The nature of roots is decided by \(D=b^2-4ac\). In exams first identify (a), (b), and (c).
Step 2
Why this answer is correct
The correct answer is A. विविक्तकर \(D=b^2-4ac\) / Discriminant \(D=b^2-4ac\). The nature of roots is decided by \(D=b^2-4ac\). In exams first identify (a), (b), and (c).
Step 3
Exam Tip
मूलों की प्रकृति \(D=b^2-4ac\) से तय होती है। परीक्षा में पहले (a), (b), (c) पहचानें।
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यदि (D>0) हो तो \(ax^2+bx+c=0\) के मूल कैसे होंगे?
If (D>0), what will be the roots of \(ax^2+bx+c=0\)?
#quadratic equations
#D positive
#real roots
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A दो वास्तविक और भिन्न / Two real and distinct
B दो वास्तविक और समान / Two real and equal
C कोई वास्तविक मूल नहीं / No real roots
D एक ही मूल नहीं मिलेगा / No root will be found
Explanation opens after your attempt
Correct Answer
A. दो वास्तविक और भिन्न / Two real and distinct
Step 1
Concept
When (D>0), two different real roots are obtained. Watch the sign (>) carefully.
Step 2
Why this answer is correct
The correct answer is A. दो वास्तविक और भिन्न / Two real and distinct. When (D>0), two different real roots are obtained. Watch the sign (>) carefully.
Step 3
Exam Tip
जब (D>0) होता है तो दो अलग वास्तविक मूल मिलते हैं। संकेत (>) को ध्यान से देखें।
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यदि (D=0) हो तो द्विघात समीकरण के मूलों की प्रकृति क्या होगी?
If (D=0), what will be the nature of roots of the quadratic equation?
#quadratic equations
#D zero
#equal roots
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A दो वास्तविक और समान / Two real and equal
B दो वास्तविक और भिन्न / Two real and distinct
C कोई वास्तविक मूल नहीं / No real roots
D तीन वास्तविक मूल / Three real roots
Explanation opens after your attempt
Correct Answer
A. दो वास्तविक और समान / Two real and equal
Step 1
Concept
When (D=0), both roots are equal. In such questions choose the equal roots option.
Step 2
Why this answer is correct
The correct answer is A. दो वास्तविक और समान / Two real and equal. When (D=0), both roots are equal. In such questions choose the equal roots option.
Step 3
Exam Tip
(D=0) होने पर दोनों मूल बराबर होते हैं। ऐसे प्रश्नों में बराबर मूल वाला विकल्प चुनें।
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समीकरण \(x^2-5x+6=0\) के लिए (D) का मान क्या है?
What is the value of (D) for the equation \(x^2-5x+6=0\)?
#quadratic equations
#discriminant
#numerical
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A (1)
B (0)
C (-1)
D (49)
Explanation opens after your attempt
Step 1
Concept
Here (D=(-5)2 -4(1)(6)=1). So calculate \(b^2\) first.
Step 2
Why this answer is correct
The correct answer is A. (1). Here (D=(-5)2 -4(1)(6)=1). So calculate \(b^2\) first.
Step 3
Exam Tip
यहाँ (D=(-5)2 -4(1)(6)=1) है। इसलिए पहले \(b^2\) निकालें।
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समीकरण \(x^2-4x+4=0\) के मूलों की प्रकृति बताइए।
State the nature of roots of the equation \(x^2-4x+4=0\).
#quadratic equations
#equal roots
#D zero
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A वास्तविक और समान / Real and equal
B वास्तविक और भिन्न / Real and distinct
C वास्तविक नहीं / Not real
D अपरिमेय और भिन्न / Irrational and distinct
Explanation opens after your attempt
Correct Answer
A. वास्तविक और समान / Real and equal
Step 1
Concept
Here (D=(-4)2 -4(1)(4)=0). Hence both roots will be equal.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक और समान / Real and equal. Here (D=(-4)2 -4(1)(4)=0). Hence both roots will be equal.
Step 3
Exam Tip
यहाँ (D=(-4)2 -4(1)(4)=0) है। इसलिए दोनों मूल समान होंगे।
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समीकरण \(2x^2-3x+1=0\) के लिए (D) कितना होगा?
What will be (D) for the equation \(2x^2-3x+1=0\)?
#quadratic equations
#discriminant
#coefficient
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A (1)
B (17)
C (-1)
D (0)
Explanation opens after your attempt
Step 1
Concept
Here (D=(-3)2 -4(2)(1)=1). Do not forget to put (a=2) in the formula.
Step 2
Why this answer is correct
The correct answer is A. (1). Here (D=(-3)2 -4(2)(1)=1). Do not forget to put (a=2) in the formula.
Step 3
Exam Tip
यहाँ (D=(-3)2 -4(2)(1)=1) है। (a=2) को सूत्र में रखना न भूलें।
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समीकरण \(2x^2-3x+1=0\) के मूल कैसे होंगे?
What will be the roots of the equation \(2x^2-3x+1=0\)?
#quadratic equations
#D positive
#roots nature
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A वास्तविक और भिन्न / Real and distinct
B वास्तविक और समान / Real and equal
C वास्तविक नहीं / Not real
D केवल काल्पनिक / Only imaginary
Explanation opens after your attempt
Correct Answer
A. वास्तविक और भिन्न / Real and distinct
Step 1
Concept
Here (D=1>0), so the roots will be two real and distinct. Apply this rule when (D) is positive.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक और भिन्न / Real and distinct. Here (D=1>0), so the roots will be two real and distinct. Apply this rule when (D) is positive.
Step 3
Exam Tip
इसमें (D=1>0) है इसलिए दो वास्तविक और भिन्न मूल होंगे। (D) धनात्मक हो तो यही नियम लगाएँ।
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समीकरण \(3x^2+6x+3=0\) के लिए मूलों की प्रकृति क्या है?
What is the nature of roots for the equation \(3x^2+6x+3=0\)?
#quadratic equations
#D zero
#equal roots
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A वास्तविक और समान / Real and equal
B वास्तविक और भिन्न / Real and distinct
C वास्तविक नहीं / Not real
D दो धनात्मक और भिन्न / Two positive and distinct
Explanation opens after your attempt
Correct Answer
A. वास्तविक और समान / Real and equal
Step 1
Concept
Here (D=(6)2 -4(3)(3)=0). Hence the roots are equal.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक और समान / Real and equal. Here (D=(6)2 -4(3)(3)=0). Hence the roots are equal.
Step 3
Exam Tip
यहाँ (D=(6)2 -4(3)(3)=0) है। इसलिए मूल बराबर हैं।
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समीकरण \(x^2+x+1=0\) के लिए (D) का चिन्ह क्या है?
What is the sign of (D) for the equation \(x^2+x+1=0\)?
#quadratic equations
#sign of D
#no real roots
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A ऋणात्मक (D<0) / Negative (D<0)
B धनात्मक (D>0) / Positive (D>0)
C शून्य (D=0) / Zero (D=0)
D एक (D=1) / One (D=1)
Explanation opens after your attempt
Correct Answer
A. ऋणात्मक (D<0) / Negative (D<0)
Step 1
Concept
Here (D=(1)2 -4(1)(1)=-3). So (D) is negative.
Step 2
Why this answer is correct
The correct answer is A. ऋणात्मक (D<0) / Negative (D<0). Here (D=(1)2 -4(1)(1)=-3). So (D) is negative.
Step 3
Exam Tip
यहाँ (D=(1)2 -4(1)(1)=-3) है। इसलिए (D) ऋणात्मक है।
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यदि किसी द्विघात समीकरण में \(b^2-4ac=9\) है तो मूलों की प्रकृति क्या होगी?
If \(b^2-4ac=9\) in a quadratic equation, what will be the nature of roots?
#quadratic equations
#D value
#real distinct roots
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A वास्तविक और भिन्न / Real and distinct
B वास्तविक और समान / Real and equal
C वास्तविक नहीं / Not real
D एक मूल शून्य / One root zero
Explanation opens after your attempt
Correct Answer
A. वास्तविक और भिन्न / Real and distinct
Step 1
Concept
Since (9>0), we have (D>0). Therefore the roots are real and distinct.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक और भिन्न / Real and distinct. Since (9>0), we have (D>0). Therefore the roots are real and distinct.
Step 3
Exam Tip
क्योंकि (9>0) है इसलिए (D>0) होगा। अतः मूल वास्तविक और भिन्न होंगे।
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यदि \(b^2-4ac=-7\) हो तो सही निष्कर्ष कौन सा है?
If \(b^2-4ac=-7\), which conclusion is correct?
#quadratic equations
#D negative
#conclusion
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A कोई वास्तविक मूल नहीं / No real roots
B दो वास्तविक और भिन्न / Two real and distinct
C दो वास्तविक और समान / Two real and equal
D दो शून्य मूल / Two zero roots
Explanation opens after your attempt
Correct Answer
A. कोई वास्तविक मूल नहीं / No real roots
Step 1
Concept
Since (-7<0), we have (D<0). In this case real roots are not obtained.
Step 2
Why this answer is correct
The correct answer is A. कोई वास्तविक मूल नहीं / No real roots. Since (-7<0), we have (D<0). In this case real roots are not obtained.
Step 3
Exam Tip
(-7<0) है इसलिए (D<0) होगा। ऐसे में वास्तविक मूल नहीं मिलते।
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यदि \(b^2-4ac=0\) है तो द्विघात समीकरण में कितने अलग वास्तविक मूल होंगे?
If \(b^2-4ac=0\), how many different real roots will a quadratic equation have?
#quadratic equations
#D zero
#distinct root count
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A (1) अलग वास्तविक मूल / (1) different real root
B (2) अलग वास्तविक मूल / (2) different real roots
C (0) वास्तविक मूल / (0) real roots
D (3) वास्तविक मूल / (3) real roots
Explanation opens after your attempt
Correct Answer
A. (1) अलग वास्तविक मूल / (1) different real root
Step 1
Concept
When (D=0), both roots are equal, so there is (1) different real root. Remember the difference between equal and distinct.
Step 2
Why this answer is correct
The correct answer is A. (1) अलग वास्तविक मूल / (1) different real root. When (D=0), both roots are equal, so there is (1) different real root. Remember the difference between equal and distinct.
Step 3
Exam Tip
(D=0) होने पर दोनों मूल समान होते हैं इसलिए अलग वास्तविक मूल (1) होता है। समान और भिन्न में अंतर याद रखें।
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समीकरण \(4x^2+4x+1=0\) में (D) का मान क्या है?
What is the value of (D) in the equation \(4x^2+4x+1=0\)?
#quadratic equations
#discriminant
#equal roots
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A (0)
B (8)
C (16)
D (-8)
Explanation opens after your attempt
Step 1
Concept
Here (D=(4)2 -4(4)(1)=0). Thus it is a case of equal roots.
Step 2
Why this answer is correct
The correct answer is A. (0). Here (D=(4)2 -4(4)(1)=0). Thus it is a case of equal roots.
Step 3
Exam Tip
यहाँ (D=(4)2 -4(4)(1)=0) है। अतः यह बराबर मूलों का मामला है।
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समीकरण \(4x^2+4x+1=0\) के मूल कैसे होंगे?
What will be the roots of the equation \(4x^2+4x+1=0\)?
#quadratic equations
#D zero
#root nature
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A वास्तविक और समान / Real and equal
B वास्तविक और भिन्न / Real and distinct
C वास्तविक नहीं / Not real
D दो ऋणात्मक और भिन्न / Two negative and distinct
Explanation opens after your attempt
Correct Answer
A. वास्तविक और समान / Real and equal
Step 1
Concept
Here (D=0), so both roots are equal. Never treat (D=0) as (D>0).
Step 2
Why this answer is correct
The correct answer is A. वास्तविक और समान / Real and equal. Here (D=0), so both roots are equal. Never treat (D=0) as (D>0).
Step 3
Exam Tip
इसमें (D=0) है इसलिए दोनों मूल समान हैं। (D=0) को कभी (D>0) न मानें।
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समीकरण \(x^2-2x-3=0\) के लिए (D) क्या है?
What is (D) for the equation \(x^2-2x-3=0\)?
#quadratic equations
#negative c
#discriminant
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A (16)
B (8)
C (0)
D (-16)
Explanation opens after your attempt
Step 1
Concept
Here (D=(-2)2 -4(1)(-3)=16). When (c) is negative, handle signs carefully.
Step 2
Why this answer is correct
The correct answer is A. (16). Here (D=(-2)2 -4(1)(-3)=16). When (c) is negative, handle signs carefully.
Step 3
Exam Tip
यहाँ (D=(-2)2 -4(1)(-3)=16) है। (c) ऋणात्मक हो तो चिन्ह ध्यान से लगाएँ।
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समीकरण \(x^2-2x-3=0\) के मूलों की प्रकृति क्या है?
What is the nature of roots of the equation \(x^2-2x-3=0\)?
#quadratic equations
#D positive
#real distinct roots
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A वास्तविक और भिन्न / Real and distinct
B वास्तविक और समान / Real and equal
C वास्तविक नहीं / Not real
D दो समान ऋणात्मक / Two equal negative
Explanation opens after your attempt
Correct Answer
A. वास्तविक और भिन्न / Real and distinct
Step 1
Concept
Since (D=16>0), the roots are real and distinct. Positive (D) always gives distinct real roots.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक और भिन्न / Real and distinct. Since (D=16>0), the roots are real and distinct. Positive (D) always gives distinct real roots.
Step 3
Exam Tip
(D=16>0) है इसलिए मूल वास्तविक और भिन्न हैं। धनात्मक (D) हमेशा भिन्न वास्तविक मूल देता है।
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समीकरण \(5x^2+2x+1=0\) के लिए (D) का मान ज्ञात कीजिए।
Find the value of (D) for the equation \(5x^2+2x+1=0\).
#quadratic equations
#D negative
#numerical
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A (-16)
B (16)
C (0)
D (24)
Explanation opens after your attempt
Step 1
Concept
Here (D=(2)2 -4(5)(1)=-16). Hence the roots will not be real.
Step 2
Why this answer is correct
The correct answer is A. (-16). Here (D=(2)2 -4(5)(1)=-16). Hence the roots will not be real.
Step 3
Exam Tip
यहाँ (D=(2)2 -4(5)(1)=-16) है। इसलिए मूल वास्तविक नहीं होंगे।
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समीकरण \(5x^2+2x+1=0\) के वास्तविक मूलों की संख्या कितनी है?
How many real roots does the equation \(5x^2+2x+1=0\) have?
#quadratic equations
#no real roots
#root count
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A (0)
B (1)
C (2)
D (5)
Explanation opens after your attempt
Step 1
Concept
Because (D=-16<0), the number of real roots is (0). In exams write this conclusion as soon as (D<0) appears.
Step 2
Why this answer is correct
The correct answer is A. (0). Because (D=-16<0), the number of real roots is (0). In exams write this conclusion as soon as (D<0) appears.
Step 3
Exam Tip
क्योंकि (D=-16<0) है इसलिए वास्तविक मूल (0) हैं। परीक्षा में (D<0) दिखते ही यही निष्कर्ष लिखें।
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समीकरण \(9x^2-6x+1=0\) में मूलों की प्रकृति कौन सी है?
Which is the nature of roots in the equation \(9x^2-6x+1=0\)?
#quadratic equations
#equal roots
#perfect square
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A वास्तविक और समान / Real and equal
B वास्तविक और भिन्न / Real and distinct
C वास्तविक नहीं / Not real
D दो धनात्मक भिन्न / Two positive distinct
Explanation opens after your attempt
Correct Answer
A. वास्तविक और समान / Real and equal
Step 1
Concept
Here (D=(-6)2 -4(9)(1)=0). Hence the roots are equal.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक और समान / Real and equal. Here (D=(-6)2 -4(9)(1)=0). Hence the roots are equal.
Step 3
Exam Tip
यहाँ (D=(-6)2 -4(9)(1)=0) है। इसलिए मूल समान हैं।
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समीकरण \(2x^2+5x+2=0\) के लिए (D) क्या होगा?
What will be (D) for the equation \(2x^2+5x+2=0\)?
#quadratic equations
#discriminant
#D positive
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A (9)
B (25)
C (0)
D (-9)
Explanation opens after your attempt
Step 1
Concept
Here (D=(5)2 -4(2)(2)=9). Since (9>0), the roots will be distinct.
Step 2
Why this answer is correct
The correct answer is A. (9). Here (D=(5)2 -4(2)(2)=9). Since (9>0), the roots will be distinct.
Step 3
Exam Tip
यहाँ (D=(5)2 -4(2)(2)=9) है। (9>0) इसलिए मूल भिन्न होंगे।
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समीकरण \(2x^2+5x+2=0\) के मूलों की प्रकृति बताइए।
State the nature of roots of the equation \(2x^2+5x+2=0\).
#quadratic equations
#root nature
#D positive
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A वास्तविक और भिन्न / Real and distinct
B वास्तविक और समान / Real and equal
C वास्तविक नहीं / Not real
D दो समान धनात्मक / Two equal positive
Explanation opens after your attempt
Correct Answer
A. वास्तविक और भिन्न / Real and distinct
Step 1
Concept
Here (D=9>0), so there will be two real and distinct roots. Finding (D) first is the safest method.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक और भिन्न / Real and distinct. Here (D=9>0), so there will be two real and distinct roots. Finding (D) first is the safest method.
Step 3
Exam Tip
इसमें (D=9>0) है इसलिए दो वास्तविक और भिन्न मूल होंगे। पहले (D) निकालना सबसे सुरक्षित तरीका है।
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समीकरण \(3x^2+2x+1=0\) में (D) का मान कौन सा है?
Which is the value of (D) in the equation \(3x^2+2x+1=0\)?
#quadratic equations
#D negative
#nature
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A (-8)
B (8)
C (0)
D (12)
Explanation opens after your attempt
Step 1
Concept
Here (D=(2)2 -4(3)(1)=-8). A negative (D) does not give real roots.
Step 2
Why this answer is correct
The correct answer is A. (-8). Here (D=(2)2 -4(3)(1)=-8). A negative (D) does not give real roots.
Step 3
Exam Tip
यहाँ (D=(2)2 -4(3)(1)=-8) है। ऋणात्मक (D) वास्तविक मूल नहीं देता।
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समीकरण \(3x^2+2x+1=0\) के मूल कैसे होंगे?
What will be the roots of the equation \(3x^2+2x+1=0\)?
#quadratic equations
#no real roots
#D sign
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A वास्तविक नहीं / Not real
B वास्तविक और भिन्न / Real and distinct
C वास्तविक और समान / Real and equal
D दो शून्य मूल / Two zero roots
Explanation opens after your attempt
Correct Answer
A. वास्तविक नहीं / Not real
Step 1
Concept
Here (D=-8<0), so there are no real roots. The sign of (D) decides the answer.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक नहीं / Not real. Here (D=-8<0), so there are no real roots. The sign of (D) decides the answer.
Step 3
Exam Tip
यहाँ (D=-8<0) है इसलिए वास्तविक मूल नहीं हैं। (D) का चिन्ह उत्तर तय करता है।
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समीकरण \(x^2+6x+9=0\) के लिए सही कथन कौन सा है?
Which statement is correct for the equation \(x^2+6x+9=0\)?
#quadratic equations
#statement based
#D zero
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A (D=0) और मूल समान हैं / (D=0) and roots are equal
B (D>0) और मूल भिन्न हैं / (D>0) and roots are distinct
C (D<0) और मूल नहीं हैं / (D<0) and roots do not exist
D (D=1) और मूल भिन्न हैं / (D=1) and roots are distinct
Explanation opens after your attempt
Correct Answer
A. (D=0) और मूल समान हैं / (D=0) and roots are equal
Step 1
Concept
Here (D=(6)2 -4(1)(9)=0). Therefore the statement with equal roots is correct.
Step 2
Why this answer is correct
The correct answer is A. (D=0) और मूल समान हैं / (D=0) and roots are equal. Here (D=(6)2 -4(1)(9)=0). Therefore the statement with equal roots is correct.
Step 3
Exam Tip
यहाँ (D=(6)2 -4(1)(9)=0) है। इसलिए बराबर मूल वाला कथन सही है।
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समीकरण \(x^2+7x+10=0\) में (D) किसके बराबर है?
What is (D) equal to in the equation \(x^2+7x+10=0\)?
#quadratic equations
#discriminant
#positive D
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A (9)
B (49)
C (0)
D (-9)
Explanation opens after your attempt
Step 1
Concept
Here (D=(7)2 -4(1)(10)=9). (D) is positive, so the roots are distinct.
Step 2
Why this answer is correct
The correct answer is A. (9). Here (D=(7)2 -4(1)(10)=9). (D) is positive, so the roots are distinct.
Step 3
Exam Tip
यहाँ (D=(7)2 -4(1)(10)=9) है। (D) धनात्मक है इसलिए मूल भिन्न हैं।
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समीकरण \(x^2+7x+10=0\) की प्रकृति क्या होगी?
What will be the nature of the equation \(x^2+7x+10=0\)?
#quadratic equations
#real distinct roots
#D positive
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A दो वास्तविक और भिन्न मूल / Two real and distinct roots
B दो वास्तविक और समान मूल / Two real and equal roots
C कोई वास्तविक मूल नहीं / No real roots
D केवल एक काल्पनिक मूल / Only one imaginary root
Explanation opens after your attempt
Correct Answer
A. दो वास्तविक और भिन्न मूल / Two real and distinct roots
Step 1
Concept
Since (D=9>0), two real and distinct roots are obtained. (D>0) means two different real roots.
Step 2
Why this answer is correct
The correct answer is A. दो वास्तविक और भिन्न मूल / Two real and distinct roots. Since (D=9>0), two real and distinct roots are obtained. (D>0) means two different real roots.
Step 3
Exam Tip
(D=9>0) होने से दो वास्तविक और भिन्न मूल मिलेंगे। (D>0) का अर्थ दो अलग वास्तविक मूल है।
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समीकरण \(x^2+4x+8=0\) में (D) का मान ज्ञात करें।
Find the value of (D) in the equation \(x^2+4x+8=0\).
#quadratic equations
#discriminant
#negative D
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A (-16)
B (16)
C (0)
D (48)
Explanation opens after your attempt
Step 1
Concept
Here (D=(4)2 -4(1)(8)=-16). Thus its (D) is negative.
Step 2
Why this answer is correct
The correct answer is A. (-16). Here (D=(4)2 -4(1)(8)=-16). Thus its (D) is negative.
Step 3
Exam Tip
यहाँ (D=(4)2 -4(1)(8)=-16) है। अतः इसका (D) ऋणात्मक है।
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समीकरण \(x^2+4x+8=0\) के लिए कौन सा निष्कर्ष सही है?
Which conclusion is correct for the equation \(x^2+4x+8=0\)?
#quadratic equations
#no real roots
#conclusion
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A वास्तविक मूल नहीं हैं / There are no real roots
B मूल वास्तविक और समान हैं / Roots are real and equal
C मूल वास्तविक और भिन्न हैं / Roots are real and distinct
D दोनों मूल (0) हैं / Both roots are (0)
Explanation opens after your attempt
Correct Answer
A. वास्तविक मूल नहीं हैं / There are no real roots
Step 1
Concept
Because (D=-16<0), there are no real roots. This is the most direct conclusion.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक मूल नहीं हैं / There are no real roots. Because (D=-16<0), there are no real roots. This is the most direct conclusion.
Step 3
Exam Tip
क्योंकि (D=-16<0) है इसलिए वास्तविक मूल नहीं हैं। यह सबसे सीधा निष्कर्ष है।
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यदि \(x^2+kx+9=0\) के मूल समान हैं तो (k) के लिए सही शर्त क्या है?
If the roots of \(x^2+kx+9=0\) are equal, what is the correct condition for (k)?
#quadratic equations
#equal roots
#parameter
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A \(k^2=36\)
B \(k^2=9\)
C \(k^2=0\)
D \(k^2=18\)
Explanation opens after your attempt
Correct Answer
A. \(k^2=36\)
Step 1
Concept
For equal roots (D=0), so \(k^2-36=0\). Hence \(k^2=36\) is correct.
Step 2
Why this answer is correct
The correct answer is A. \(k^2=36\). For equal roots (D=0), so \(k^2-36=0\). Hence \(k^2=36\) is correct.
Step 3
Exam Tip
समान मूलों के लिए (D=0) होता है इसलिए \(k^2-36=0\)। अतः \(k^2=36\) सही है।
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यदि \(x^2+kx+9=0\) के मूल समान हों तो (k) का संभावित मान कौन सा है?
If the roots of \(x^2+kx+9=0\) are equal, which is a possible value of (k)?
#quadratic equations
#parameter
#equal roots
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A (6)
B (3)
C (0)
D (9)
Explanation opens after your attempt
Step 1
Concept
From \(k^2=36\), we get (k=6) or (k=-6). Among the given options (6) is correct.
Step 2
Why this answer is correct
The correct answer is A. (6). From \(k^2=36\), we get (k=6) or (k=-6). Among the given options (6) is correct.
Step 3
Exam Tip
\(k^2=36\) से (k=6) या (k=-6) मिलता है। दिए गए विकल्पों में (6) सही है।
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यदि \(x^2-2kx+16=0\) के मूल समान हैं तो \(k^2\) कितना होगा?
If the roots of \(x^2-2kx+16=0\) are equal, what will be \(k^2\)?
#quadratic equations
#parameter
#D zero
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A (16)
B (8)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
For equal roots (D=0), so ((-2k)2 -4(1)(16)=0). This gives \(k^2=16\).
Step 2
Why this answer is correct
The correct answer is A. (16). For equal roots (D=0), so ((-2k)2 -4(1)(16)=0). This gives \(k^2=16\).
Step 3
Exam Tip
समान मूलों के लिए (D=0) है इसलिए ((-2k)2 -4(1)(16)=0)। इससे \(k^2=16\) मिलता है।
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समीकरण \(x^2+2kx+25=0\) के समान मूलों के लिए \(k^2\) का मान क्या होगा?
For equal roots of \(x^2+2kx+25=0\), what will be the value of \(k^2\)?
#quadratic equations
#equal roots
#parameter k
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A (25)
B (5)
C (50)
D (100)
Explanation opens after your attempt
Step 1
Concept
Putting (D=0) gives ((2k)2 -4(1)(25)=0). Therefore \(k^2=25\).
Step 2
Why this answer is correct
The correct answer is A. (25). Putting (D=0) gives ((2k)2 -4(1)(25)=0). Therefore \(k^2=25\).
Step 3
Exam Tip
(D=0) रखने पर ((2k)2 -4(1)(25)=0) मिलता है। इसलिए \(k^2=25\) है।
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यदि \(kx^2+4x+1=0\) के मूल समान हैं तो (k) का मान क्या होगा?
If the roots of \(kx^2+4x+1=0\) are equal, what will be the value of (k)?
#quadratic equations
#parameter k
#equal roots
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A (4)
B (2)
C (1)
D (8)
Explanation opens after your attempt
Step 1
Concept
For equal roots (D=0), so (42 -4(k)(1)=0). This gives (k=4).
Step 2
Why this answer is correct
The correct answer is A. (4). For equal roots (D=0), so (42 -4(k)(1)=0). This gives (k=4).
Step 3
Exam Tip
समान मूलों के लिए (D=0) है इसलिए (42 -4(k)(1)=0)। इससे (k=4) मिलता है।
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समीकरण \(2x^2+kx+2=0\) के समान मूलों के लिए \(k^2\) क्या होगा?
For equal roots of \(2x^2+kx+2=0\), what will \(k^2\) be?
#quadratic equations
#equal roots
#discriminant
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A (16)
B (8)
C (4)
D (2)
Explanation opens after your attempt
Step 1
Concept
From (D=0), we get (k-2 -4(2)(2)=0). Therefore \(k^2=16\).
Step 2
Why this answer is correct
The correct answer is A. (16). From (D=0), we get (k-2 -4(2)(2)=0). Therefore \(k^2=16\).
Step 3
Exam Tip
(D=0) से (k-2 -4(2)(2)=0) मिलेगा। इसलिए \(k^2=16\) है।
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कौन सा समीकरण वास्तविक और भिन्न मूल रखता है?
Which equation has real and distinct roots?
#quadratic equations
#choose equation
#D positive
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A \(x^2-3x+2=0\)
B \(x^2+2x+1=0\)
C \(x^2+x+1=0\)
D \(4x^2+4x+1=0\)
Explanation opens after your attempt
Correct Answer
A. \(x^2-3x+2=0\)
Step 1
Concept
In the first equation (D=(-3)2 -4(1)(2)=1>0). Hence its roots are real and distinct.
Step 2
Why this answer is correct
The correct answer is A. \(x^2-3x+2=0\). In the first equation (D=(-3)2 -4(1)(2)=1>0). Hence its roots are real and distinct.
Step 3
Exam Tip
पहले समीकरण में (D=(-3)2 -4(1)(2)=1>0) है। इसलिए उसके मूल वास्तविक और भिन्न हैं।
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कौन सा समीकरण वास्तविक और समान मूल रखता है?
Which equation has real and equal roots?
#quadratic equations
#choose equation
#equal roots
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A \(x^2-8x+16=0\)
B \(x^2-8x+15=0\)
C \(x^2-8x+20=0\)
D \(x^2+8x+12=0\)
Explanation opens after your attempt
Correct Answer
A. \(x^2-8x+16=0\)
Step 1
Concept
In the first equation (D=(-8)2 -4(1)(16)=0). So its roots are real and equal.
Step 2
Why this answer is correct
The correct answer is A. \(x^2-8x+16=0\). In the first equation (D=(-8)2 -4(1)(16)=0). So its roots are real and equal.
Step 3
Exam Tip
पहले समीकरण में (D=(-8)2 -4(1)(16)=0) है। इसलिए इसके मूल वास्तविक और समान हैं।
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कौन सा समीकरण वास्तविक मूल नहीं रखता है?
Which equation has no real roots?
#quadratic equations
#choose equation
#no real roots
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A \(x^2+2x+10=0\)
B \(x^2-2x+1=0\)
C \(x^2-2x-10=0\)
D \(x^2+10x+2=0\)
Explanation opens after your attempt
Correct Answer
A. \(x^2+2x+10=0\)
Step 1
Concept
In the first equation (D=(2)2 -4(1)(10)=-36<0). Therefore there is no real root.
Step 2
Why this answer is correct
The correct answer is A. \(x^2+2x+10=0\). In the first equation (D=(2)2 -4(1)(10)=-36<0). Therefore there is no real root.
Step 3
Exam Tip
पहले समीकरण में (D=(2)2 -4(1)(10)=-36<0) है। इसलिए कोई वास्तविक मूल नहीं है।
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समीकरण \(x^2-10x+25=0\) के लिए सही विकल्प चुनिए।
Choose the correct option for the equation \(x^2-10x+25=0\).
#quadratic equations
#D zero
#correct option
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A (D=0) और मूल समान / (D=0) and roots equal
B (D>0) और मूल भिन्न / (D>0) and roots distinct
C (D<0) और मूल वास्तविक नहीं / (D<0) and roots not real
D (D=25) और मूल भिन्न / (D=25) and roots distinct
Explanation opens after your attempt
Correct Answer
A. (D=0) और मूल समान / (D=0) and roots equal
Step 1
Concept
Here (D=(-10)2 -4(1)(25)=0). So the option with equal roots is correct.
Step 2
Why this answer is correct
The correct answer is A. (D=0) और मूल समान / (D=0) and roots equal. Here (D=(-10)2 -4(1)(25)=0). So the option with equal roots is correct.
Step 3
Exam Tip
यहाँ (D=(-10)2 -4(1)(25)=0) है। इसलिए समान मूल वाला विकल्प सही है।
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समीकरण \(2x^2-7x+3=0\) में मूलों की प्रकृति क्या है?
What is the nature of roots in the equation \(2x^2-7x+3=0\)?
#quadratic equations
#D positive
#nature
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A वास्तविक और भिन्न / Real and distinct
B वास्तविक और समान / Real and equal
C वास्तविक नहीं / Not real
D दो समान शून्य / Two equal zeros
Explanation opens after your attempt
Correct Answer
A. वास्तविक और भिन्न / Real and distinct
Step 1
Concept
Here (D=(-7)2 -4(2)(3)=25>0). Hence the roots will be real and distinct.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक और भिन्न / Real and distinct. Here (D=(-7)2 -4(2)(3)=25>0). Hence the roots will be real and distinct.
Step 3
Exam Tip
यहाँ (D=(-7)2 -4(2)(3)=25>0) है। अतः मूल वास्तविक और भिन्न होंगे।
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समीकरण \(6x^2+x+1=0\) के वास्तविक मूलों के बारे में क्या सही है?
What is correct about the real roots of the equation \(6x^2+x+1=0\)?
#quadratic equations
#D negative
#real roots
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A वास्तविक मूल नहीं हैं / There are no real roots
B दो वास्तविक और भिन्न मूल हैं / There are two real and distinct roots
C दो वास्तविक और समान मूल हैं / There are two real and equal roots
D एक वास्तविक और एक शून्य मूल है / There is one real and one zero root
Explanation opens after your attempt
Correct Answer
A. वास्तविक मूल नहीं हैं / There are no real roots
Step 1
Concept
Here (D=(1)2 -4(6)(1)=-23<0). So there are no real roots.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक मूल नहीं हैं / There are no real roots. Here (D=(1)2 -4(6)(1)=-23<0). So there are no real roots.
Step 3
Exam Tip
यहाँ (D=(1)2 -4(6)(1)=-23<0) है। इसलिए वास्तविक मूल नहीं हैं।
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समीकरण \(7x^2-10x+3=0\) के लिए (D) कितना है?
What is (D) for the equation \(7x^2-10x+3=0\)?
#quadratic equations
#discriminant
#D positive
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A (16)
B (100)
C (0)
D (-16)
Explanation opens after your attempt
Step 1
Concept
Here (D=(-10)2 -4(7)(3)=16). Since (16>0), the roots will be distinct.
Step 2
Why this answer is correct
The correct answer is A. (16). Here (D=(-10)2 -4(7)(3)=16). Since (16>0), the roots will be distinct.
Step 3
Exam Tip
यहाँ (D=(-10)2 -4(7)(3)=16) है। (16>0) होने से मूल भिन्न होंगे।
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समीकरण \(7x^2-10x+3=0\) के मूलों की प्रकृति क्या है?
What is the nature of roots of the equation \(7x^2-10x+3=0\)?
#quadratic equations
#root nature
#D positive
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A वास्तविक और भिन्न / Real and distinct
B वास्तविक और समान / Real and equal
C वास्तविक नहीं / Not real
D दो समान ऋणात्मक / Two equal negative
Explanation opens after your attempt
Correct Answer
A. वास्तविक और भिन्न / Real and distinct
Step 1
Concept
Here (D=16>0), so the roots are real and distinct. When (D) is positive, distinct roots are obtained.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक और भिन्न / Real and distinct. Here (D=16>0), so the roots are real and distinct. When (D) is positive, distinct roots are obtained.
Step 3
Exam Tip
इसमें (D=16>0) है इसलिए मूल वास्तविक और भिन्न हैं। (D) धनात्मक हो तो अलग मूल मिलते हैं।
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समीकरण \(8x^2+8x+2=0\) में (D) का मान क्या है?
What is the value of (D) in the equation \(8x^2+8x+2=0\)?
#quadratic equations
#D zero
#numerical
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A (0)
B (64)
C (-64)
D (16)
Explanation opens after your attempt
Step 1
Concept
Here (D=(8)2 -4(8)(2)=0). Therefore the roots will be equal.
Step 2
Why this answer is correct
The correct answer is A. (0). Here (D=(8)2 -4(8)(2)=0). Therefore the roots will be equal.
Step 3
Exam Tip
यहाँ (D=(8)2 -4(8)(2)=0) है। इसलिए मूल समान होंगे।
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यदि किसी समीकरण में (D=4) है तो कौन सा कथन सही है?
If an equation has (D=4), which statement is correct?
#quadratic equations
#D value
#concept
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A मूल वास्तविक और भिन्न हैं / Roots are real and distinct
B मूल वास्तविक और समान हैं / Roots are real and equal
C मूल वास्तविक नहीं हैं / Roots are not real
D मूलों की प्रकृति नहीं बता सकते / Nature of roots cannot be told
Explanation opens after your attempt
Correct Answer
A. मूल वास्तविक और भिन्न हैं / Roots are real and distinct
Step 1
Concept
Since (4>0), we have (D>0). Therefore the roots are real and distinct.
Step 2
Why this answer is correct
The correct answer is A. मूल वास्तविक और भिन्न हैं / Roots are real and distinct. Since (4>0), we have (D>0). Therefore the roots are real and distinct.
Step 3
Exam Tip
(4>0) है इसलिए (D>0) होगा। अतः मूल वास्तविक और भिन्न होंगे।
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यदि किसी समीकरण में (D=-1) है तो सही विकल्प कौन सा है?
If an equation has (D=-1), which option is correct?
#quadratic equations
#D negative
#concept
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A कोई वास्तविक मूल नहीं / No real roots
B दो वास्तविक और भिन्न मूल / Two real and distinct roots
C दो वास्तविक और समान मूल / Two real and equal roots
D एक वास्तविक मूल और एक काल्पनिक मूल / One real root and one imaginary root
Explanation opens after your attempt
Correct Answer
A. कोई वास्तविक मूल नहीं / No real roots
Step 1
Concept
Since (-1<0), we have (D<0). In such a case there are no real roots.
Step 2
Why this answer is correct
The correct answer is A. कोई वास्तविक मूल नहीं / No real roots. Since (-1<0), we have (D<0). In such a case there are no real roots.
Step 3
Exam Tip
(-1<0) है इसलिए (D<0) होगा। ऐसे में वास्तविक मूल नहीं होते।
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यदि किसी समीकरण में (D=0) है तो दोनों मूलों के बारे में क्या कहा जाएगा?
If an equation has (D=0), what will be said about both roots?
#quadratic equations
#D zero
#concept
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A दोनों मूल बराबर हैं / Both roots are equal
B दोनों मूल भिन्न हैं / Both roots are different
C दोनों मूल वास्तविक नहीं हैं / Both roots are not real
D दोनों मूल अपरिभाषित हैं / Both roots are undefined
Explanation opens after your attempt
Correct Answer
A. दोनों मूल बराबर हैं / Both roots are equal
Step 1
Concept
When (D=0), both roots are equal. This is a main rule to remember for nature of roots.
Step 2
Why this answer is correct
The correct answer is A. दोनों मूल बराबर हैं / Both roots are equal. When (D=0), both roots are equal. This is a main rule to remember for nature of roots.
Step 3
Exam Tip
(D=0) होने पर दोनों मूल बराबर होते हैं। यह प्रकृति याद करने का मुख्य नियम है।
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समीकरण \(x^2-12x+36=0\) के मूलों की प्रकृति बताइए।
State the nature of roots of the equation \(x^2-12x+36=0\).
#quadratic equations
#equal roots
#D zero
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A वास्तविक और समान / Real and equal
B वास्तविक और भिन्न / Real and distinct
C वास्तविक नहीं / Not real
D दो ऋणात्मक और भिन्न / Two negative and distinct
Explanation opens after your attempt
Correct Answer
A. वास्तविक और समान / Real and equal
Step 1
Concept
Here (D=(-12)2 -4(1)(36)=0). Therefore the roots are real and equal.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक और समान / Real and equal. Here (D=(-12)2 -4(1)(36)=0). Therefore the roots are real and equal.
Step 3
Exam Tip
यहाँ (D=(-12)2 -4(1)(36)=0) है। इसलिए मूल वास्तविक और समान हैं।
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समीकरण \(x^2-6x+13=0\) के वास्तविक मूलों के बारे में क्या सही है?
What is correct about the real roots of the equation \(x^2-6x+13=0\)?
#quadratic equations
#nature of roots
#no real roots
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A कोई वास्तविक मूल नहीं / No real roots
B दो वास्तविक और भिन्न मूल / Two real and distinct roots
C दो वास्तविक और समान मूल / Two real and equal roots
D एक वास्तविक मूल / One real root
Explanation opens after your attempt
Correct Answer
A. कोई वास्तविक मूल नहीं / No real roots
Step 1
Concept
Here (D=(-6)2 -4(1)(13)=-16<0). Therefore there is no real root.
Step 2
Why this answer is correct
The correct answer is A. कोई वास्तविक मूल नहीं / No real roots. Here (D=(-6)2 -4(1)(13)=-16<0). Therefore there is no real root.
Step 3
Exam Tip
यहाँ (D=(-6)2 -4(1)(13)=-16<0) है। इसलिए कोई वास्तविक मूल नहीं है।
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यदि \(x^2+px+4=0\) के मूल वास्तविक और भिन्न हैं तो (p) के लिए सही शर्त क्या है?
If the roots of \(x^2+px+4=0\) are real and distinct, what is the correct condition for (p)?
#quadratic equations
#parameter
#real distinct roots
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A \(p^2>16\)
B \(p^2=16\)
C \(p^2<16\)
D \(p^2=4\)
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Correct Answer
A. \(p^2>16\)
Step 1
Concept
For real and distinct roots (D>0), so \(p^2-16>0\). Therefore \(p^2>16\) is correct.
Step 2
Why this answer is correct
The correct answer is A. \(p^2>16\). For real and distinct roots (D>0), so \(p^2-16>0\). Therefore \(p^2>16\) is correct.
Step 3
Exam Tip
वास्तविक और भिन्न मूलों के लिए (D>0) होता है इसलिए \(p^2-16>0\)। अतः \(p^2>16\) सही है।
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यदि \(3x^2+mx+12=0\) के मूल वास्तविक और समान हैं तो \(m^2\) का मान क्या होगा?
If the roots of \(3x^2+mx+12=0\) are real and equal, what will be the value of \(m^2\)?
#quadratic equations
#equal roots
#parameter
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A (144)
B (36)
C (72)
D (12)
Explanation opens after your attempt
Step 1
Concept
For equal roots (D=0), so (m-2 -4(3)(12)=0). Therefore \(m^2=144\).
Step 2
Why this answer is correct
The correct answer is A. (144). For equal roots (D=0), so (m-2 -4(3)(12)=0). Therefore \(m^2=144\).
Step 3
Exam Tip
समान मूलों के लिए (D=0) होता है इसलिए (m-2 -4(3)(12)=0)। अतः \(m^2=144\) है।
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