An air filled cylindrical capacitor (capacitance $C_0$ ) of length $L$, with $a$ and $b$ as its inner and outer radii, respectively, consists of two coaxial conducting surfaces. Its crosssectional view is shown in Figure (i). In order to increase the capacitance, a dielectric material of relative permittivity $\varepsilon_r$ is inserted inside $50 \%$ of the annular region as shown in figure (ii). The value of $\varepsilon_r$ for which the capacitance of the capacitor in figure (ii), becomes $5 C_0$ is

The relationship between two variables $x$ and $y$ is given by $x+p y+q=0$ and is shown in the figure. Find the values of $p$ and $q$.

Note: The figure shown is representative.

Consider the set $S$ of points $(x, y) \in R^2$ which minimize the real valued function

$$ f(x, y)=(x+y-1)^2+(x+y)^2 $$

Which of the following statements is true about the set $S$ ?

Let $v_1$ and $v_2$ be the two eigen vectors corresponding to distinct eigen values of a $3 \times 3$ real symmetric matrix. Which one of the following statements is true?