Practice focused topic-wise MCQs with answers and explanations for quick revision and exam preparation.
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If \(\alpha,\beta\) are roots of \(x^2+x-2=0\), what is \(\alpha^5+\beta^5\)?
B. (-31)
समीकरण की जड़ें (1) और (-2) हैं। इसलिए (\alpha-5+\beta-5=15+(-2)5=-31)। / The roots of the equation are (1) and (-2). Therefore (\alpha-5+\beta-5=15+(-2)5=-31).
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If (x-2-(u+v+2)x+(u+1)(v+1)=0) has equal roots, which statement is correct?
C. (u=v)
इस समीकरण की जड़ें (u+1) और (v+1) हैं। समान जड़ों के लिए (u+1=v+1), इसलिए (u=v)। / The roots of this equation are (u+1) and (v+1). For equal roots, (u+1=v+1), so (u=v).
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A. \(x^2-11x+30=0\)
\(\alpha+\beta=5\) और \(\alpha\beta=6\) हैं। नई जड़ें (5) और (6) हैं, इसलिए समीकरण \(x^2-11x+30=0\) है। / Here \(\alpha+\beta=5\) and \(\alpha\beta=6\). The new roots are (5) and (6), so the equation is \(x^2-11x+30=0\).
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What is the positive difference between the roots of \(x^2-2kx+k^2-25=0\)?
B. (10)
समीकरण ((x-k)2-25=0) है। जड़ें (k+5) और (k-5) हैं, इसलिए धनात्मक अंतर (10) है। / The equation is ((x-k)2-25=0). The roots are (k+5) and (k-5), so the positive difference is (10).
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If (x-2-2mx+\(m^2-m\)=0) has real roots, what is the correct condition on (m)?
C. \(m\ge0\)
वास्तविक जड़ों के लिए \(D\ge0\) चाहिए। यहाँ (D=4m), इसलिए \(m\ge0\)। / For real roots, \(D\ge0\) is required. Here (D=4m), so \(m\ge0\).
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If the sum of roots of a quadratic equation is (7) and the product is (10), which is the equation?
C. \(x^2-7x+10=0\)
\(जड़ों का समीकरण (x^2-(\)योग)x+गुणनफल=0) होता है। इसलिए \(x^2-7x+10=0\) सही है। \(/ The equation is (x^2-(\)sum)x+product\(=0). Hence (x^2-7x+10=0) is correct.\)
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A. \(2x^2-5x+2=0\)
व्युत्क्रम जड़ों का गुणनफल (1) होता है। समीकरण \(x^2-\frac{5}{2}x+1=0\) को (2) से गुणा करने पर \(2x^2-5x+2=0\) मिलता है। / Reciprocal roots have product (1). Multiplying \(x^2-\frac{5}{2}x+1=0\) by (2) gives \(2x^2-5x+2=0\).
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If ((m-1)x-2+2(m+1)x+(m-1)=0) has real reciprocal roots, what is the correct condition on (m)?
A. \(m\ge0\) और \(m\neq1\)\(m\ge0\) and \(m\neq1\)
जड़ों का गुणनफल \(\frac{m-1}{m-1}=1\) है, इसलिए \(m\neq1\) चाहिए। वास्तविक जड़ों के लिए \(D=16m\ge0\), अतः \(m\ge0\) और \(m\neq1\)। / The product of roots is \(\frac{m-1}{m-1}=1\), so \(m\neq1\) is needed. For real roots, \(D=16m\ge0\), hence \(m\ge0\) and \(m\neq1\).
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A. (4)
यहाँ समीकरण ((x-(a+1))(x-(a+5))=0) बनता है। इसलिए जड़ें (a+1) और (a+5) हैं, अतः धनात्मक अंतर (4) है। / The equation becomes ((x-(a+1))(x-(a+5))=0). So the roots are (a+1) and (a+5), hence the positive difference is (4).
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If one root of (2x-2-(3p+2)x+p(p+2)=0) is (p), what will be the other root?
A. \(\frac{p+2}{2}\)
जड़ों का गुणनफल (\frac{p(p+2)}{2}) है। एक जड़ (p) होने पर दूसरी जड़ \(\frac{p+2}{2}\) होगी। / The product of roots is (\frac{p(p+2)}{2}). If one root is (p), the other root is \(\frac{p+2}{2}\).
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