Mathematics
Nature of Roots
मूलों की प्रकृति
In Class 10 Mathematics, Nature of Roots explains how to determine the type and number of solutions of a quadratic equation. Students use the discriminant, b² − 4ac, to identify whether an equation has two distinct real roots, two equal real roots, or no real roots. The topic connects algebraic calculations with the graph of a quadratic function and helps learners interpret equations, compare cases, and solve related problems from the chapter Quadratic Equations.
Practice questions
For which value of \(k\) will the two roots of the quadratic equation \(2x^2-(k+4)x+2k=0\) be equal?ExpertLevel 37If (x^2+2(k-1)x+(k+5)=0) has no real roots, in which interval will (k) lie?ExpertLevel 37If \(k\) is a real number, what is the correct condition for the equation \(x^2+2(k-1)x+(k+5)=0\) to have no real roots?ExpertLevel 37If \(k\neq 0\) and \(kx^2-2(k+1)x+(k+3)=0\) is a quadratic equation, what must be the value of \(k\) for it to have equal roots?ExpertLevel 37In ((k-2)x^2+2kx+(k+3)=0), with (k\neq2), what is the correct condition for real roots?ExpertLevel 37Which condition is correct for real roots of ((k-2)x^2+2kx+(k+3)=0), if (k\neq2)?ExpertLevel 37For real numbers a and b, if the equation \(x^2-2(a+b)x+(a-b)^2=0\) has real and distinct roots, which condition on a and b is correct?ExpertLevel 37If \(a\) and \(b\) are real numbers, which of the following relations is necessary for the equation \(x^2-2(a+b)x+(a^2+b^2)=0\) to have real roots?ExpertLevel 37Which quadratic equation has real and distinct roots?ExpertLevel 37What is the nature of the roots of the quadratic equation \(3x^2-2\sqrt{6}x+2=0\)?ExpertLevel 37Which root nature is correct for 2x² − 5√2x + 12 = 0?HardLevel 37If (x^2-2\lambda x+\lambda=0) has two real and distinct roots, which condition on (\lambda) is correct?ExpertLevel 37For which values of \(\lambda\) does the equation \(x^2-2\lambda x+\lambda=0\) have equal roots?ExpertLevel 37Given that \\(\alpha\ne -1\\), what condition on \\(\alpha\\) is necessary for the equation \\((\alpha+1)x^2-2\alpha x+\alpha=0\\) to have real roots?ExpertLevel 37The equation (x^2-2(2t-1)x+(t^2+2)=0) has no real roots. Choose the correct interval for (t).ExpertLevel 37If \\(x^2+2(m-2)x+(m^2-3m+4)=0\\) has equal roots, what is the value of \\(m\\)?ExpertLevel 37If the equation \(x^2+2(m-2)x+(m^2-3m+4)=0\) has no real roots, what is the correct condition on \(m\)?ExpertLevel 37For the quadratic equation \(3x^2-2(2k+1)x+(k+1)^2=0\) to have equal roots, what are the values of \(k\)?ExpertLevel 37For the equation \(3x^2-2(2k+1)x+(k+1)^2=0\) to have two distinct real roots, which condition must \(k\) satisfy?ExpertLevel 37Let \(p\) and \(q\) be real numbers. If \(x^2-2px+(p^2-q^2)=0\) and \(q\neq 0\), what is the nature of the roots of this quadratic equation?ExpertLevel 37