Write whether the

(i) Every quadratic equation has exactly one root: False

A quadratic equation has exactly two roots.

(ii) Every quadratic equation has at least one real root: False

A quadratic equation may have real or imaginary roots.

(iii) Every quadratic equation has at least two roots: False

A quadratic equation has exactly two roots, no more no less.

(iv) Every quadratic equation has at most two roots: True

A quadratic equation can’t have more than two roots.

(v) If the coefficient of x² and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots: True

The standard form of a quadratic equation is given by,

ax² + bx + c = 0, a ≠ 0

So, if coefficient of x²(a) and constant term(c) have opposite signs then, the roots will always be real i.e.,

ac < 0

⇒ b² - 4ac > 0

(vi) If the coefficient of x² and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots: True

The standard form of a quadratic equation is given by,

ax² + bx + c = 0, a ≠ 0

So, if coefficient of x²(a) and constant term(c) have the same sign and coefficient of x(b) is zero, then the roots will be imaginary.

ac > 0

⇒ b² - 4ac < 0