The radius of the larger circle lying in the first quadrant and touching the line $4x + 3y - 12 = 0$ and the coordinate axes is:

$A$ ray of light is incident along a line which meets another line,$7x - y + 1 = 0$,at the point $(0, 1)$. The ray is then reflected from this point along the line $y + 2x = 1$. Find the equation of the line of incidence of the ray of light.

$A$ meter scale is supported on a wedge at its center of gravity. $A$ body of weight '$w$' is suspended from the $20 \text{ cm}$ mark and another weight of $25 \text{ g}$ is suspended from the $74 \text{ cm}$ mark. The system is balanced and the meter scale remains perfectly horizontal. Neglecting the weight of the meter scale,the weight of the body '$w$' is: (in $\text{ g}$)

$A$ particle of mass $10\, g$ is kept on the surface of a uniform sphere of mass $100\, kg$ and a radius of $10\, cm$. Find the work to be done against the gravitational force between them to take the particle far away from the sphere. (Take $G = 6.67 \times 10^{-11} \, Nm^2/kg^2$)

Let $f: R \rightarrow R$ be defined by $f(x) = \begin{cases} x+2, & x \leq -1 \\ x^2, & -1 < x < 1 \\ 2-x, & x \geq 1 \end{cases}$. Then the value of $f(-1.75) + f(0.5) + f(1.5)$ is

$A$ body is executing simple harmonic motion. At a displacement $x$ its potential energy is $E_1$ and at a displacement $y$ its potential energy is $E_2$. The potential energy $(E)$ at displacement $(x+y)$ is