Practice focused topic-wise MCQs with answers and explanations for quick revision and exam preparation.
Difficulty select karke sirf is topic ke focused MCQs practice karein. Score, timer aur explanations student-friendly flow me milenge.
Showing 421-430 of 600 questions.
If (x-2+(k+2)x+9=0) has equal roots, what are the values of (k)?
A. (k=4) या (k=-8)(k=4) or (k=-8)
(D=0) से ((k+2)2-36=0), इसलिए \(k+2=\pm6\)। अतः (k=4) या (k=-8)। / From (D=0), ((k+2)2-36=0), so \(k+2=\pm6\). Hence (k=4) or (k=-8).
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For which (t) will \(x^2-10x+t=0\) have real and equal roots?
C. (t=25)
बराबर मूलों के लिए (D=0), अतः (100-4t=0) से (t=25)। परीक्षा में \(b^2=4ac\) बराबर मूल की तेज पहचान है। / For equal roots, (D=0), so (100-4t=0) gives (t=25). In exams, \(b^2=4ac\) is a quick sign of equal roots.
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What is the nature of roots of \(2x^2-7x+6=0\)?
A. दो वास्तविक और भिन्न मूलTwo real and distinct roots
यहां (D=(-7)2-4\cdot2\cdot6=1), इसलिए दो भिन्न वास्तविक मूल हैं। परीक्षा में (D=1) भी धनात्मक पूर्ण वर्ग है। / Here (D=(-7)2-4\cdot2\cdot6=1), so the roots are two distinct real roots. In exams, (D=1) is also a positive perfect square.
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If \(4x^2+4kx+k=0\) has equal roots, what will be the values of (k)?
A. (k=0) या (k=1)(k=0) or (k=1)
(D=0) से ((4k)2-4\cdot4\cdot k=0), यानी (16k(k-1)=0)। इसलिए (k=0) या (k=1), और \(a=4\neq0\) है। / From (D=0), ((4k)2-4\cdot4\cdot k=0), that is (16k(k-1)=0). So (k=0) or (k=1), and \(a=4\neq0\).
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What is the nature of roots of \(x^2+2\sqrt{3}x+3=0\)?
A. दो बराबर वास्तविक मूलTwo equal real roots
यहां (D=\(2\sqrt{3}\)2-4\cdot1\cdot3=0), इसलिए मूल बराबर वास्तविक हैं। परीक्षा में surd coefficient को square करते समय सावधानी रखें। / Here (D=\(2\sqrt{3}\)2-4\cdot1\cdot3=0), so the roots are equal and real. In exams, be careful while squaring a surd coefficient.
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What will be the nature of roots of \(x^2+2\sqrt{2}x+1=0\)?
A. दो भिन्न वास्तविक मूलTwo distinct real roots
यहां (D=\(2\sqrt{2}\)2-4\cdot1\cdot1=4), इसलिए (D>0)। परीक्षा में (D=4) होने पर दो भिन्न वास्तविक मूल होते हैं। / Here (D=\(2\sqrt{2}\)2-4\cdot1\cdot1=4), so (D>0). In exams, (D=4) gives two distinct real roots.
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If (x-2-2(m+1)x+16=0) has equal roots, what are the values of (m)?
A. (m=3) या (m=-5)(m=3) or (m=-5)
(D=0) से ([-2(m+1)]2-64=0), इसलिए ((m+1)2=16)। अतः (m=3) या (m=-5)। / From (D=0), ([-2(m+1)]2-64=0), so ((m+1)2=16). Hence (m=3) or (m=-5).
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For which (r) will \(rx^2-5x+r=0\) have two distinct real roots?
A. \(r^2<\frac{25}{4}\) और \(r\neq0\)\(r^2<\frac{25}{4}\) and \(r\neq0\)
दो भिन्न वास्तविक मूलों के लिए (D>0), इसलिए \(25-4r^2>0\)। साथ में \(r\neq0\) चाहिए क्योंकि समीकरण द्विघात होना चाहिए। / For two distinct real roots, (D>0), so \(25-4r^2>0\). Also \(r\neq0\) is needed because the equation must remain quadratic.
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If \(x^2+2x+\lambda=0\) has no real roots, what is the correct condition for \(\lambda\)?
A. \(\lambda>1\)
वास्तविक मूल न होने के लिए (D<0), अतः \(4-4\lambda<0\) से \(\lambda>1\)। परीक्षा में strict inequality को बराबरी से अलग रखें। / For no real roots, (D<0), so \(4-4\lambda<0\) gives \(\lambda>1\). In exams, keep strict inequality separate from equality.
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State the nature of roots of \(5x^2+20x+20=0\).
A. दो बराबर वास्तविक मूलTwo equal real roots
यहां \(D=20^2-4\cdot5\cdot20=0\), इसलिए मूल बराबर वास्तविक हैं। परीक्षा में सामान्य गुणनखंड हटाने पर भी प्रकृति नहीं बदलती। / Here \(D=20^2-4\cdot5\cdot20=0\), so the roots are equal and real. In exams, removing a common factor does not change the nature.
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