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 real numbers \(p\) and \(q\), if \(q\ne0\), what is the nature of the roots of the equation \(x^2-2px+(p^2+q^2)=0\)?ExpertLevel 37Assertion: For the equation \(x^2-2(a+b)x+(a-b)^2=0\), if \(ab>0\), its roots are real and distinct. Reason: The discriminant of this equation is \(D=16ab\). Choose the correct option.ExpertLevel 37Assertion: The graph of x² + 2x + 5 = 0 does not cut the x-axis. Reason: Its discriminant D = −16. Choose the correct option.MediumLevel 37What is the correct discriminant \(D\) of the quadratic equation \(x^2-2(k+1)x+k^2=0\)?ExpertLevel 37Which of the following quadratic equations has equal roots?ExpertLevel 37Which of the following equations has real, irrational, and distinct roots?ExpertLevel 37Which quadratic equation will have two distinct real roots?ExpertLevel 37The discriminant of a quadratic equation is D = (s − 2)^2. What must be the value of s for the equation to have equal roots?ExpertLevel 37If the discriminant of a quadratic equation is \(D=(u+1)(u-5)\), which interval of \(u\) results in no real roots?ExpertLevel 37If the discriminant of a quadratic equation is \(D=8m-24\), what condition on \(m\) is necessary for the equation to have real roots?ExpertLevel 37If a quadratic has discriminant D = 25 - 4q², what are the values of q for equal roots?MediumLevel 37What is the nature of the roots of the equation \(x^2+2(1-\sqrt{3})x+4=0\)?ExpertLevel 37What is the nature of the roots of \\(x^2-2(3+\sqrt{2})x+(17+12\sqrt{2})=0\\)?ExpertLevel 37If x² - 2rx + (r² - 16) = 0, what is the nature of the roots for any real value of r?MediumLevel 37If \(r\) is any real number, what is the correct conclusion about the nature of the roots of the equation \(x^2-2rx+(r^2+9)=0\)?ExpertLevel 37For a real constant \(k\), how many points of intersection are there between the parabola \(y=x^2-2kx+k^2+1\) and the \(x\)-axis?ExpertLevel 37How will the parabola \(y=x^2-2kx+k^2\) intersect the x-axis?ExpertLevel 37If the equation of a parabola is \(y=x^2-2kx+(k^2-4)\), how will it intersect the \(x\)-axis?ExpertLevel 37An area situation gives l² - 2(a + 3)l + (a² + 10) = 0. What is the condition on a for real length values?HardLevel 37A number puzzle gives the equation \(n^2-2pn+(p^2-5p)=0\). What condition on \(p\) is necessary for the equation to have two real and distinct values of \(n\)?ExpertLevel 37