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
Assertion: In the equation \(x^2-2(a-3b)x+(a+3b)^2=0\), if \(ab<0\), then its roots are real and distinct. Reason: The discriminant of this equation is \(D=-48ab\). Choose the correct option.ExpertLevel 39Assertion: The graph of the parabola \(y=3x^2-6x+11\) does not intersect the \(x\)-axis. Reason: The discriminant of the corresponding quadratic equation is \(D=-96\). Choose the correct option.ExpertLevel 39What is the discriminant \(D\) of the quadratic equation \(x^2-2(k-3)x+k^2=0\)?ExpertLevel 39If the two possible values of the discriminant of a quadratic equation are \\(D_1=(y+5)^2\\) and \\(D_2=-(y+5)^2\\), with \\(y\neq -5\\), which case gives two distinct real roots?ExpertLevel 39If the discriminant of a quadratic equation is \(D=(m-8)^2\), what must be the value of \(m\) for the equation to have equal roots?ExpertLevel 39If the discriminant of a quadratic equation is D = (z − 9)(z + 1), which interval of z is required for the equation to have no real roots?ExpertLevel 39If the discriminant of a quadratic equation is \(D=20n-80\), what condition on \(n\) is required for the equation to have two real and distinct roots?ExpertLevel 39If a quadratic equation has discriminant D = 49 − 16h², what are the values of h for equal roots?MediumLevel 39What is the nature of the roots of \\(x^2+2(3-\sqrt{10})x+16=0\\)?ExpertLevel 39What is the nature of the roots of the equation \(x^2-2(3+\sqrt{5})x+(14+6\sqrt{5})=0\)?ExpertLevel 39If \(x^2-2rx+(r^2-81)=0\) is a quadratic equation and \(r\) is any real number, what will be the nature of its roots?ExpertLevel 39If \(r\) is any real number in the equation \(x^2-2rx+(r^2+49)=0\), what is the correct conclusion about its roots?ExpertLevel 39How many points of intersection are there between the parabola \(y=x^2-2kx+k^2+9\) and the \(x\)-axis?ExpertLevel 39How does the parabola \(y=x^2-2(k+2)x+(k+2)^2\) meet the \(x\)-axis?ExpertLevel 39For the parabola \(y=x^2-2kx+(k^2-16)\), at how many distinct points will it intersect the \(x\)-axis?ExpertLevel 39For the quadratic equation \(ax^2+bx+c=0\) with real coefficients, where \(a\ne0\), if \(\frac{c}{a}<0\), which statement about its roots is correct?ExpertLevel 39Suppose the quadratic equation \(ax^2+bx+c=0\) has two distinct real roots. Which condition definitely indicates that the roots have opposite signs?ExpertLevel 39A number problem leads to the equation \(n^2-2pn+(p^2-11p)=0\). What condition on \(p\) is necessary for the equation to have two real and distinct values of \(n\)?ExpertLevel 39If \(x^2-(2a+5)x+(a+2)(a+3)=0\), where \(a\) is a real constant, what will be the nature of its roots?ExpertLevel 39If x² − (a + b + 1)x + ab + a + b = 0, on what does the nature of the roots depend for a and b?HardLevel 39