Let $a \neq 0$ and $p(x)$ be a polynomial of degree greater than $2$. If $p(x)$ leaves remainders $a$ and $-a$ when divided respectively by $x+a$ and $x-a$,then the remainder when $p(x)$ is divided by $x^2-a^2$ is:

$A$ whistle of frequency $540 \ Hz$ rotates in a horizontal circle of radius $2 \ m$ at an angular speed of $15 \ rad/s$. What is the highest frequency heard by a listener at rest with respect to the centre of the circle (in $Hz$)? (Velocity of sound in air $= 330 \ m/s$)

If $\alpha$ and $\beta$ are the roots of the equation $ax^2+bx+c=0$ and the equation having roots $\frac{1-\alpha}{\alpha}$ and $\frac{1-\beta}{\beta}$ is $px^2+qx+r=0$,then $r=$

In the following circuit,$PN$-junction diodes $D_1, D_2$ and $D_3$ are ideal. For the following potentials of $A$ and $B$,find the correct increasing order of equivalent resistance between $A$ and $B$:
$(i) V_A = -10\,V, V_B = -5\,V$
$(ii) V_A = -5\,V, V_B = -10\,V$
$(iii) V_A = -4\,V, V_B = -12\,V$

$A$ thin semi-circular ring of radius $r$ has a positive charge $q$ distributed uniformly over it. The net electric field $\vec{E}$ at the center $O$ is:

$A$ thin magnetic iron rod of length $30 \ cm$ is suspended in a uniform magnetic field. Its time period of oscillation is $4 \ s$. It is broken into three equal parts. The time period in seconds of oscillation of one part when suspended in the same magnetic field is