In a triangle $$ABC$$, medians $$AD$$ and $$BE$$ are drawn. If $$AD=4$$, \angle DAB = {\pi \over 6} and \angle ABE = {\pi \over 3}, then the area of the $$\angle \Delta ABC$$ is :

If in a $$\Delta ABC$$ a\,{\cos ^2}\left( {{C \over 2}} \right) + c\,{\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2}, then the sides $$a,b$$ and $$c$$ :

The shortest distance from the plane $$12x+4y+3z=327$$ to the sphere $$x^2+y^2+z^2+4x-2y-6z=155$$ is

The two lines $$x=ay+b,z=cy+d$$ and $$x=a^{\prime }y+b^{\prime },z=c^{\prime }y+d^{\prime }$$ will be perpendicular, if and only if :

The lines {{x - 2} \over 1} = {{y - 3} \over 1} = {{z - 4} \over { - k}} and {{x - 1} \over k} = {{y - 4} \over 2} = {{z - 5} \over 1} are coplanar if :

The radius of the circle in which the sphere $$x^2+y^2+z^2+2x-2y-4z-19=0$$ is cut by the plane $$x+2y+2z+7=0$$ is

Two systems of rectangular axes have the same origin. If a plane cuts then at distances $$a,b,c$$ and $$a^{\prime },b^{\prime },c^{\prime }$$ from the origin then

In an experiment with 15 observations on $$x$$, then following results were available: $$\sum x^2=2830$$, $$\sum x=170$$ One observation that was 20 was found to be wrong and was replaced by the correct value 30. Then the corrected variance is :

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set :

The sum of the serier {1 \over {1.2}} - {1 \over {2.3}} + {1 \over {3.4}}.............. up to $$\infty$$ is equal to

The real number $$x$$ when added to its inverse gives the minimum sum at $$x$$ equal :

If the function $$f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1,$$ where $$a>0,$$ attains its maximum and minimum at $$p$$ and $$q$$ respectively such that $$p^2=q$$ , then $$a$$ equals

The area of the region bounded by the curves $$y=\left|x-1\right|$$ and $$y=3-\left|x\right|$$ is :

Dimensions of {1 \over {{\mu _0}{\varepsilon _0}}}, where symbols have their usual meaning, are

The physical quantities not having same dimensions are

The image formed by an objective of a compound microscope is

To get three images of a single object, one should have two plane mirrors at an angle of

The length of a given cylindrical wire is increased by $$100\%$$. Due to the consequent decrease in diameter the change in the resistance of the wire will be

A $$220$$ volt, $$1000$$ watt bulb is connected across a $$110$$ $$volt$$ mains supply. The power consumed will be

A $$3$$ volt battery with negligible internal resistance is connected in a circuit as shown in the figure. The current $$\mathrm{I}$$, in the circuit will be