$A$ man has some identical resistors,each with a resistance $R = 10 \, \Omega$ and a maximum current rating of $1 \, \text{A}$. He needs to create a circuit with an equivalent resistance of $5 \, \Omega$ that can handle a total current of $4 \, \text{A}$. What is the minimum number of such resistors required?

Three identical resistances $A$,$B$,and $C$ are connected as shown in the given figure. The heat produced will be maximum in:

Obtain the expression for the equivalent resistance for $3$ resistors connected in parallel and also write the expression of equivalent resistance for connection of $n$ resistors.

$A$ wire of resistance $R$ is bent into a triangular pyramid as shown in the figure,with each segment having the same length. The resistance between points $A$ and $B$ is $R / n$. The value of $n$ is

$(a)$ Three resistors $2 \; \Omega$,$4 \; \Omega$,and $5 \; \Omega$ are combined in parallel. What is the total resistance of the combination?
$(b)$ If the combination is connected to a battery of $emf \; 20 \; V$ and negligible internal resistance,determine the current through each resistor,and the total current drawn from the battery.

$A$ ring is made of a wire having a resistance $R_0 = 12 \,\Omega$. Find the points $A$ and $B$,as shown in the figure,at which a current-carrying conductor should be connected so that the resistance $R$ of the sub-circuit between these points is equal to $\frac{8}{3} \,\Omega$.