If the roots of the equation $x^2 - 2ax + a^2 + a - 3 = 0$ are real and less than $3$,then:

An engine has an efficiency of $1/6$. When the temperature of the sink is reduced by $62^{\circ}C$,its efficiency is doubled. The temperature of the source is $.........^{\circ}C$.

Limestone is present in the blast furnace production of iron in order to:
$(I)$ provide a source of $CaO$
$(II)$ remove some impurities
$(III)$ supply $CO_2$

The plates of a parallel plate capacitor are charged up to $100\, V$. $A$ $2\, mm$ thick plate is inserted between the plates. To maintain the same potential difference,the distance between the capacitor plates is increased by $1.6\, mm$. The dielectric constant of the plate is:

Three charges $2q, -q$ and $-q$ are located at the vertices of an equilateral triangle. At the centre of the triangle,

The wavelengths of light used in an optical instrument are ${\lambda _1 = 4000 \ \mathring{A}}$ and ${\lambda _2 = 5000 \ \mathring{A}}$. The ratio of their respective resolving powers (corresponding to ${\lambda _1}$ and ${\lambda _2}$) is: