Find the value$(s)$ of $k$ such that the distance between the two parallel lines represented by $(x-2y)^2 + k(x-2y) = 0$ is $3$ units.

The circumcentre of the triangle with vertices at $(-2, 3)$,$(1, -2)$ and $(2, 1)$ is

$A$ man wants to reach a certain destination. One-sixth of the total distance is muddy while half the distance is tar road. For the remaining distance,he takes a boat. His speed of travelling in mud,in water,and on tar road is in the ratio $3:4:5$. The ratio of the durations he requires to cross the patch of mud,stream,and tar road is

$A$ small metal ball is suspended in a uniform electric field with the help of an insulated thread. If a high-energy $X-$ray beam falls on the ball,then the ball

$A$ metal wire $PQ$ slides on parallel metallic rails having separation $0.25 \ m,$ each having negligible resistance. There is a $2 \ \Omega$ resistor and $10 \ V$ battery as shown in the figure. There is a uniform magnetic field directed into the plane of the paper of magnitude $0.5 \ T.$ $A$ force of $0.5 \ N$ to the left is required to keep the wire $PQ$ moving with constant speed to the right. With what speed is the wire $PQ$ moving? (Neglect self-inductance of the loop).

On the interval $[0, 1]$,the function $f(x) = x^{25}(1-x)^{75}$ takes its maximum value at the point