The abscissae of $A$ and $B$ are the roots of the equation $x^2+2ax-b^2=0$ and their ordinates are the roots of the equation $y^2+2py-q^2=0$. Find the equation of the circle with $AB$ as diameter.

If the function $f: R \rightarrow R$ defined by $f(x) = \begin{cases} \frac{\sin(a+1)x + \sin x}{x}, & x < 0 \\ b, & x = 0 \\ \frac{\sqrt{x+x^2} - \sqrt{x}}{x^{3/2}}, & x > 0 \end{cases}$ is continuous on $R$,then $a+b =$

Assertion : Free electrons always keep on moving in a conductor even then no magnetic force acts on them in a magnetic field unless a current is passed through it.
Reason : The average velocity of free electrons is zero.

In a $p-n$ junction diode,the thickness of the depletion layer is $2 \times 10^{-6} ~m$ and the barrier potential is $0.3 ~V$. The intensity of the electric field at the junction is:

Match the following columns:

Column-$I$	Column-$II$
$A$. Halophiles	$i$. Gram-negative bacteria
$B$. Thermoacidophiles	$ii$. Inhabit extremely salty areas
$C$. Spirochetes	$iii$. Responsible for methane production
$D$. Firmicutes	$iv$. Survive in acidic and high-temperature conditions
	$v$. Gram-positive bacteria

$A$ clock pendulum made of invar has a period of $0.5 \, s$ at $20^{\circ} C$. If the clock is used in a climate where the temperature averages to $30^{\circ} C$, how much time does the clock lose in each oscillation? (For invar, $\alpha = 9 \times 10^{-7} /{ }^{\circ} C$, $g = \text{constant}$)