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) \(x^2 + 5\sqrt{5}\,x - 70 = 0\)
(i) Real and distinct: \(x = -\dfrac{3}{4},\; \dfrac{1}{2}\).
(ii) Real and distinct: \(x = -\dfrac{1}{2},\; 2\).
(iii) Real and distinct: \(x = \dfrac{1 - \sqrt{51}}{5},\; \dfrac{1 + \sqrt{51}}{5}\).
(iv) Real and distinct: \(x = 4 \pm \dfrac{3\sqrt{2}}{2}\).
(v) Real and distinct: \(x = -7\sqrt{5},\; 2\sqrt{5}\).
Find a natural number whose square diminished by \(84\) is equal to thrice of \(8\) more than the given number.
12
A natural number, when increased by \(12\), equals \(160\) times its reciprocal. Find the number.
8
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.
45 km/h
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?
14 years
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.
Nisha: 5 years; Asha: 27 years
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.
Length = 34 m, Breadth = 24 m
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 \(t\).
\(t = 6\) minutes