l, m, n are the $$p^{th}$$, $$q^{th}$$ and $$r^{th}$$ term of a G.P all positive, then\,\left| {\matrix{ {\log \,l} & p & 1 \cr {\log \,m} & q & 1 \cr {\log \,n} & r & 1 \cr } } \right|\,equals

If 1, $${\log }_9(3^{1-x}+2),{\log }_3(4.3^x-1)$$ are in A.P. then x equals

Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is

Fifth term of a GP is 2, then the product of its 9 terms is

The value of $$2^{1/4}.4^{1/8}.8^{1/16}...\infty$$ is

The maximum distance from origin of a point on the curve x = a\sin t - b\sin \left( {{{at} \over b}} \right) y = a\cos t - b\cos \left( {{{at} \over b}} \right), both $$a,b>0$$ is

If $$2a+3b+6c=0,$$ $$\left(a,b,c\in R\right)$$ then the quadratic equation $$a{x}^2+bx+c=0$$ has

The area bounded by the curves $$y=\ln x,y=\ln \left|x\right|,y=\left|\ln x\right|$$ and $$y=\left|\ln \left|x\right|\right|$$ is :

Identify the pair whose dimensions are equal

If two mirrors are kept at $$60^{\circ }$$ to each other, then the number of images formed by them is

An astronomical telescope has a large aperture to

Which of the following is used in optical fibres?

Wavelength of light used in an optical instrument are $${\lambda }_1=4000\stackrel{\circ }{A}$$ and $${\lambda }_2=5000\stackrel{\circ }{A},$$ then ratio of their respective resolving powers (corresponding to $${\lambda }_1$$ and $${\lambda }_2$$ ) is :

If an ammeter is to be used in place of a voltmeter, then we must connect with the ammeter a

A wire when connected to $$220$$ $$V$$ mains supply has power dissipation $$P_1.$$ Now the wire is cut into two equal pieces which are connected in parallel to the same supply. Power dissipation in this case is $$P_2.$$ Then $$P_2:P_1$$ is

If $${\theta }_1,$$ is the inversion temperature, $${\theta }_n$$ is the neutral temperature, $${\theta }_c$$ is the temperature of the cold junction, then

The mass of product liberated on anode in an electrochemical cell depends on (where $$t$$ is the time period for which the current is passed).

If in the circuit, power dissipation is $$150W,$$ then $$R$$ is

Two identical particles move towards each other with velocity $$2v$$ and $$v$$ respectively. The velocity of center of mass is