For any integer $n \geq 2$,let $I_n = \int \tan^n x \, dx$. If $I_n = \frac{1}{a} \tan^{n-1} x - b I_{n-2}$ for $n \geq 2$,then the ordered pair $(a, b)$ equals to

Let $P = \{ (x, y) | x^2 + y^2 = 1, x, y \in R \}$. Then $P$ is

$A$ convex lens of focal length $0.15 \ m$ is made of a material of refractive index $\frac{3}{2}$. When it is placed in a liquid,its focal length increases by $0.225 \ m$. The refractive index of the liquid is

If $u=\sin ^{-1}\left(\frac{x}{y}\right)+\tan ^{-1}\left(\frac{y}{x}\right)$,then the value of $x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}$ is

$A$ transistor is used as a common emitter amplifier with a load resistance of $2 \ k\Omega$. The input resistance is $150 \ \Omega$. Base current is changed by $20 \ \mu A$,which results in a change in collector current by $1.5 \ mA$. The voltage gain of the amplifier is

Which of the following pairs is correctly matched?
$(i)$. Fungi - Regeneration
$(ii)$. Mosses - Fragmentation
$(iii)$. Planaria - Budding