NCERT Exemplar Solutions
Class 10 - Mathematics

CHAPTER 4: Quadatric Equation

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.

Find the roots of the quadratic equations by using the quadratic formula in each of the following:

(i) \(2x^2 - 3x - 5 = 0\)

(ii) \(5x^2 + 13x + 8 = 0\)

(iii) \(-3x^2 + 5x + 12 = 0\)

(iv) \(-x^2 + 7x - 10 = 0\)

(v) \(x^2 + 2\sqrt{2}\,x - 6 = 0\)

(vi) \(x^2 - 3\sqrt{5}\,x + 10 = 0\)

(vii) \(\dfrac{1}{2}x^2 - \sqrt{11}\,x + 1 = 0\)

Find the roots of the following quadratic equations by the factorisation method:

(i) \(2x^2 + \dfrac{5}{3}x - 2 = 0\)

(ii) \(\dfrac{2}{5}x^2 - x - \dfrac{3}{5} = 0\)

(iii) \(3\sqrt{2}\,x^2 - 5x - \sqrt{2} = 0\)

(iv) \(3x^2 + 5\sqrt{5}\,x - 10 = 0\)

(v) \(21x^2 - 2x + \dfrac{1}{21} = 0\)

Find whether the following equations have real roots. If real roots exist, find them.

(i) \(8x^2 + 2x - 3 = 0\)

(ii) \(-2x^2 + 3x + 2 = 0\)

(iii) \(5x^2 - 2x - 10 = 0\)

(iv) \(\dfrac{1}{2x-3} + \dfrac{1}{x-5} = 1\),   \(x \neq \dfrac{3}{2},\; 5\)

(v) \(x^2 + 5\sqrt{5}\,x - 70 = 0\)

Find a natural number whose square diminished by \(84\) is equal to thrice of \(8\) more than the given number.

A natural number, when increased by \(12\), equals \(160\) times its reciprocal. Find the number.

A train, travelling at a uniform speed for \(360\,\text{km}\), would have taken \(48\) minutes less to travel the same distance if its speed were \(5\,\text{km/h}\) more. Find the original speed of the train.

If Zeba were younger by \(5\) years than what she really is, then the square of her age (in years) would have been \(11\) more than five times her actual age. What is her age now?

At present Asha’s age (in years) is \(2\) more than the square of her daughter Nisha’s age. When Nisha grows to her mother’s present age, Asha’s age would be one year less than \(10\) times the present age of Nisha. Find their present ages.

In the centre of a rectangular lawn of dimensions \(50\,\text{m} \times 40\,\text{m}\), a rectangular pond is to be constructed so that the area of grass surrounding the pond is \(1184\,\text{m}^2\) (see Fig. 4.1). Find the length and breadth of the pond.

At \(t\) minutes past 2 pm, the time needed by the minute hand of a clock to show 3 pm was found to be \(3\) minutes less than \(\dfrac{t^2}{4}\) minutes. Find \(t\).