NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 4: Quadatric Equation

Exercise 4.2

State whether the following quadratic equations have two distinct real roots. Justify your answer.

(i) \(x^2 - 3x + 4 = 0\)

(ii) \(2x^2 + x - 1 = 0\)

(iii) \(2x^2 - 6x + \dfrac{9}{2} = 0\)

(iv) \(3x^2 - 4x + 1 = 0\)

(v) \((x+4)^2 - 8x = 0\)

(vi) \((x - \sqrt{2})^2 - 2(x+1) = 0\)

(vii) \(\sqrt{2}x^2 - \dfrac{3}{\sqrt{2}}x + \dfrac{1}{\sqrt{2}} = 0\)

(viii) \(x(1-x) - 2 = 0\)

(ix) \((x-1)(x+2)+2=0\)

(x) \((x+1)(x-2)+x=0\)

Write whether the following statements are true or false. Justify.

(i) Every quadratic has exactly one root.

(ii) Every quadratic has at least one real root.

(iii) Every quadratic has at least two roots.

(iv) Every quadratic has at most two roots.

(v) If coefficient of \(x^2\) and constant term have opposite signs, the quadratic has real roots.

(vi) If coefficient of \(x^2\) and constant have same sign and coefficient of x term is 0, then quadratic has no real roots.

A quadratic equation with integral coefficients has integral roots. Justify.

Does there exist a quadratic equation with rational coefficients but irrational roots? Justify.

Does there exist a quadratic equation with irrational coefficients but rational roots? Why?

Is 0.2 a root of \(x^2-0.4=0\)? Justify.

If b=0 and c<0, are the roots of \(x^2+bx+c=0\) numerically equal and opposite in sign? Justify.