$x$ = length $\therefore \,[X]\, = \,[L]$ and $ [dx]\, = \,[L]$ $\left[ {\frac{x}{a}} \right]\, =$ dimensionless $\therefore \,[a] = [x]\, =

$x$ = length $\therefore \,[X]\, = \,[L]$ and $ [dx]\, = \,[L]$ $\left[ {\frac{x}{a}} \right]\, =$ dimensionless $\therefore \,[a] = [x]\, = \,[L]$ $\frac{{[L]}}{{{{[{L^2} - {L^2}]}^{1/2}}}} = [{L^n}]$ $\therefore \,\,n\, = \,0$

1.Units, Dimensions and MeasurementDifficult

$\int_{}^{} {\frac{{dx}}{{{{(2ax - {x^2})}^{1/2}}}} = {a^n}{{\sin }^{ - 1}}\left( {\frac{x}{a} - 1} \right)} $ in this formula $n =$ _____

A
$1$
B
$-1$
C
$0$
D
$\frac{1}{2}$

Std 11
Physics

Choose the correct match

List I

List II

$(i)$ Curie

$(A)$ $ML{T^{ - 2}}$

$(ii)$ Light year

$(B)$ $M$

$(iii)$ Dielectric strength

$(C)$ Dimensionless

$(iv)$ Atomic weight

$(D)$ $T$

$(v)$ Decibel

$(E)$ $M{L^2}{T^{ - 2}}$

$(F)$ $M{T^{ - 3}}$

$(G)$ ${T^{ - 1}}$

$(H)$ $L$

$(I)$ $ML{T^{ - 3}}{I^{ - 1}}$

$(J)$ $L{T^{ - 1}}$

List I	List II
$(i)$ Curie	$(A)$ $ML{T^{ - 2}}$
$(ii)$ Light year	$(B)$ $M$
$(iii)$ Dielectric strength	$(C)$ Dimensionless
$(iv)$ Atomic weight	$(D)$ $T$
$(v)$ Decibel	$(E)$ $M{L^2}{T^{ - 2}}$
	$(F)$ $M{T^{ - 3}}$
	$(G)$ ${T^{ - 1}}$
	$(H)$ $L$
	$(I)$ $ML{T^{ - 3}}{I^{ - 1}}$
	$(J)$ $L{T^{ - 1}}$

Medium

[IIT 1992]

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Difficult

[AIPMT 2015]

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$\int_{}^{} {\frac{{dx}}{{{{(2ax - {x^2})}^{1/2}}}} = {a^n}{{\sin }^{ - 1}}\left( {\frac{x}{a} - 1} \right)} $ in this formula $n =$ _____

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If dimensions of critical velocity $v_c$ of a liquid flowing through a tube are expressed as$ [\eta ^x \rho ^yr^z]$ where $\eta ,\rho $ and $r $ are the coefficient of viscosity of liquid, density of liquid and radius of the tube respectively, then the values of $x, y$ and $z$ are given by

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