If ${\left( \frac{2}{3} \right)^{x + 2}} = {\left( \frac{3}{2} \right)^{2 - 2x}}$,then $x =$

If $\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x$,then $x \in$

In the following pairs,the one in which both transition metal ions are colourless is

$A$ galvanometer,having a resistance of $50 \Omega$,gives a full-scale deflection for a current of $0.05 \text{ A}$. The length in metres of a resistance wire of cross-sectional area $2.97 \times 10^{-2} \text{ cm}^2$ that can be used to convert the galvanometer into an ammeter which can read a maximum of $5 \text{ A}$ current is: (Specific resistance of the wire $= 5 \times 10^{-7} \Omega\text{-m}$)

Let $E$ be the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ and $C$ be the circle $x^2 + y^2 = 9$. Let $P$ and $Q$ be the points $(1, 2)$ and $(2, 1)$ respectively. Then:

Four smooth steel balls of equal mass at rest are free to move along a straight line without friction. The first ball is given a velocity of $0.4 \, m/s$. It collides head-on with the second elastically,the second one similarly with the third,and so on. The velocity of the last ball is .......... $m/s$.